Elimination Geometry

arXiv:2608.17646v1 [cs.LG] 18 Aug 2026

Elimination Geometry From Local Optima to Global Realizability Structural Realizability, Certification, and Repair in Statistical Learning and AI

Mian Huang

Xueqin Wang

Research monograph

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Elimination Geometry: From Local Optima to Global Realizability Structural Realizability, Certification, and Repair in Statistical Learning and AI Copyright © 2026 Mian Huang and Xueqin Wang Draft for research circulation. All theorem statements, proofs, examples, and computational protocols remain subject to author review before formal publication. Typeset in LATEX. Computational outputs are reproduced only where their status and validation scope are explicitly stated in the text.

Citation note. Until a publisher version is available, cite this work by title, authors, version number, and year.

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Preface Modern statistical learning and artificial intelligence increasingly rely on a common mechanism being reused across many local problems. Amortized inference replaces repeated optimization by one inference network; multitask systems share representations or parameters; sequential decision systems compress growing histories into finite states; scientific learning pipelines seek common labels or comparable representations across samples, batches, or conditions. Reuse creates efficiency, but it also creates a structural question that is distinct from ordinary finite-sample error and numerical optimization. Suppose each input 𝑥 has a locally optimal object 𝑎★𝑥. Pointwise solvability asks only ∀𝑥 ∃𝑎★𝑥 . Deployment asks for one admissible rule 𝐴(𝑥) ≈ 𝑎★𝑥,

∃𝐴 ∈ 𝔄 ∀𝑥,

where 𝔄 may enforce shared parameters, continuity, finite memory, a fixed output type, communication limits, or a computation budget. The first statement does not imply the second. When the implication fails, more data, longer training, or greater width inside the same deployment contract need not remove the resulting loss; the required intervention may instead be a different representation, sharing pattern, memory state, output semantics, or resource allocation. The phenomenon is not new. Approximation theory, amortized inference, multitask learning, representation learning, control, and topology all study nearby failures. The book therefore makes a narrower claim. Once a local statistical or optimization problem has fixed its oracle objects, can the loss caused specifically by a shared deployment contract be separated from local model approximation, finite-sample generalization, and implementation error? Can that loss be expressed in the scale generated by the original objective rather than in a convenient post hoc distance? Three levels must be kept distinct. Local solvability asks whether the pointwise oracle is well defined and what it costs to deviate from it. Global realizability asks whether one rule satisfying the declared deployment contract can realize the collection of local optima simultaneously. Finite-sample certifiability asks whether the available data are sufficient to distinguish realizability, nonrealizability, and an unresolved case with stated error control. None of these levels subsumes the others: a population obstruction need not be identifiable from finite data, and statistical unresolvedness does not imply that the population structure is absent. The technical starting point is elimination. If an auxiliary object 𝑎 is optimized out of a local objective, 𝐽 (𝑥) = inf 𝐻 𝑥 (𝑎), 𝑎

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vi then D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) ≥ 0

is the native defect: the exact excess objective incurred by deploying 𝑎 rather than a local optimum. This quantity is inherited from the likelihood, divergence, proper score, Bellman objective, free energy, or other declared criterion; it is not an arbitrary distance chosen after the fact. For a deployment class 𝔄 and population law 𝑃, inf E 𝑃 𝐷 𝑋 { 𝐴(𝑋)} 𝐴∈𝔄

is the architecture obstruction: the smallest native population risk left by the deployment contract after the local oracle has been fixed. The book assembles an audit workflow rather than a new universal approximation or representation theory. Its mathematical ingredients come from several established traditions, including convex duality, approximation and representation theory, information theory, control, topology, partial identification, and statistical learning. The proposed contribution is to type their assumptions and conclusions on a common native-loss interface and to expose the additional assumptions needed when moving from one interface to the next. The ordering is methodological, not a theorem saying that a native defect implies an architecture obstruction, or that an obstruction implies a successful repair. The resulting workflow is deliberately falsifiable. It derives a native loss, audits a declared deployment contract, constructs an appropriate population or statistical certificate, and proposes a mechanism-matched intervention. A stronger architecture-choice claim additionally requires evidence of saturation within the fixed contract, independent validation, and parameter-, compute-, and selectionmatched negative controls. The empirical studies in this edition illustrate only parts of that workflow; none completes a general certificate-saturation-repair validation chain. This Version closes one architecture-specific mathematical chain for full-column-rank nonnegative sparse inference: strict active-set neighborhoods yield a computable one-pass native-loss floor, and the declared proximal repair has an explicit sufficient crossing depth. The sealed diabetes case also closes a separate two-step, patient-average, equal-information and equal-compute decision. These two advances are deliberately not conflated: the theorem-alignment calculation is post-confirmation, its priority remains independently unresolved, and no general architecture-choice theory or external replication is claimed. This scope imposes important limits. Architecture obstruction is not a new name for generic approximation error. A positive worst-case topological cost does not automatically imply a positive average risk. A deterministic population obstruction is not automatically identifiable from finite data. A native defect need not be visible to the final scientific endpoint. And a structural repair does not by itself guarantee held-out improvement. Each transition requires its own theorem or empirical validation. The recurring examples—variational inference and EM, graph-indexed distribution learning, shared policies and finite memory, roots and eigenvectors, resource-limited representations, and multi-sample population alignment—serve one purpose: to test whether the same sequence of questions can be answered across different mathematical objects. What is locally optimal? What must be reused globally? Which deployment constraint creates the obstruction? What is the minimal structural change that removes it? Can finite data support that diagnosis? The intended contribution is therefore a typed, native-loss, audit-oriented synthesis complementary to existing learning theory. It organizes results about when local optima exist, when a declared shared contract can realize them, how particular lower bounds are expressed in objective units, and what

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additional statistical and experimental evidence would be needed before an architecture intervention is justified. It does not claim that these questions constitute a wholly new approximation or representation theory. Mian Huang and Xueqin Wang August 2026

Notation and Standing Conventions Symbol

Meaning

Core elimination and architecture X instance, covariate, history, or problem-index space 𝑃 population law on X or on observed biological/statistical units A𝑥 auxiliary fiber at instance 𝑥 𝐻 𝑥 (𝑎) lifted objective before elimination 𝐽 (𝑥) eliminated objective inf 𝑎∈ A 𝑥 𝐻 𝑥 (𝑎) O (𝑥) oracle set arg min𝑎∈ A 𝑥 𝐻 𝑥 (𝑎) D 𝑥 (𝑎) native defect 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) □ typed infimal (min-plus) composition of compatible stage costs 𝔄 declared class of deployable fields or architectures 𝐴 one deployed field 𝑥 ↦→ 𝐴(𝑥) 𝔒 𝑃 (𝔄) population architecture obstruction inf 𝐴∈𝔄 E 𝑃 D𝑋 { 𝐴(𝑋)} 𝔒∞ (𝔄) uniform obstruction inf 𝐴∈𝔄 sup 𝑥 D 𝑥 { 𝐴(𝑥)} Δimpl ( 𝐴; 𝔄) excess defect of 𝐴 above the architecture infimum 𝑍 carrier or representation 𝑔 decoder or readout † conditional dual oracle signature E(𝑆 | 𝑈) and its canonical Bregman decoder 𝑆𝑈 , 𝑎𝑈 ∇Φ★ (𝑆𝑈 ) CΦ (𝑈), vcΦ (𝑈) carrier information loss and visible signature cardinality 𝔉𝐽 , K (𝑚, 𝑟, 𝛿) quotient-reduced lift frontier at cone, coherence, and fidelity budgets 𝔊𝑚,𝑟 , Ψ𝑚,𝑟 (𝑍) resource grammar and decoder nonsaturation for carrier 𝑍 Dnat (𝑚, 𝑟) optimal native architecture distortion under the resource grammar 𝑊𝑖 , 𝜌 𝑖 rectangular Bellman value and its local coordination residual at stage 𝑖 𝔗 declared downstream task contract 𝐾 min-plus or operational kernel C typed finite-data certificate Δmodel local model or variational approximation component Δarch global architecture obstruction component Δgen generalization component Δopt optimization or implementation component Certified learning and intervention W, 𝐶𝑛 , (𝐶𝑡 )𝑡 ≥0 Population-world universe, fixed-record confidence world, and anytime confidence-world sequence. Their declared coverage events contain the true population world at the protected records or times.

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Symbol

Meaning

𝑞 𝐽𝐶 (𝑞), 𝛿𝐶 , 𝑚 𝐶 (𝑞)

Identified image, standard typed certificate, and certificate margin for query 𝑞 over confidence world 𝐶. Declared certificate truth or architecture-action color; Alt𝑞 (𝜃) = {𝜆 : 𝑞(𝜆) ≠ 𝑞(𝜃)}. Raw-data max–min KL rate and its inverse characteristic time. Retained-evidence counterparts; data processing gives I𝑍𝑞,∗ ≤ I𝑋𝑞,∗ and 𝑇𝑍𝑞,∗ ≥ 𝑇𝑋𝑞,∗ . Deployment carrier and action decoder; exactness requires 𝑎★ = 𝑑dep ◦ 𝐻dep . World-independent channel from raw observations or transcripts to retained evidence 𝑍. Recursive carrier preserving the declared color and every observation-labeled successor law. Joint deployment, certificate, and contextual-equivalence color. Class counts of the ℎ-step, stable, and joint-color stable recursive quotients. Worldwise feasible set and adapted common witness; certification requires Ñ 𝜂 𝐴 𝜏 ∈ 𝜆∈𝐶𝜏 𝔉𝑅 (𝜆).

𝑞:Θ→Q I𝑋𝑞,∗ (𝜃), 𝑇𝑋𝑞,∗ (𝜃) I𝑍𝑞,∗ (𝜃), 𝑇𝑍𝑞,∗ (𝜃) 𝐻dep , 𝑑dep 𝑇ev 𝜙rec 𝜁 = (𝑐 dep , 𝑐 cert , 𝑐 ctx ) 𝑁 ℎ , 𝑁∞ , 𝑁joint 𝜂 𝔉𝑅 (𝜆), 𝐴 𝜏

Orientation of asymmetric divergences Throughout the book, 𝐷 Φ (𝑎∥𝑏) = Φ(𝑎) − Φ(𝑏) − ⟨∇Φ(𝑏), 𝑎 − 𝑏⟩, so the first argument is the evaluation point and the second is the tangent or reference point. Likewise,   ∫ d𝑄 KL(𝑄∥𝑃) = log d𝑄 d𝑃 when 𝑄 ≪ 𝑃, and KL(𝑄∥𝑃) = +∞ otherwise. The first argument is the integration law and the second the reference; conditioning and pushforward preserve this order. Native defects therefore place the trial or deployed object first and the oracle second.

Standing distinctions The following distinctions are maintained throughout. • A zero infimum does not imply that an exact minimizer or global section is attained. • Uniform, average, and task-weighted risks are different contracts and are never interchanged without an explicit theorem. • Local oracle error, architecture obstruction, generalization error, and implementation error are recorded separately unless an exact identity combines them. • Point-valued, set-valued, quotient-valued, projector-valued, and distribution-valued outputs are different semantic contracts. • Deterministic obstruction, statistical impossibility, posterior credibility, and computational intractability are different claims.

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• The independent statistical unit is the declared sampling unit, not automatically an individual cell, token, edge, or repeated measurement.

Infima and measurability Unless stated otherwise, all infima are over nonempty declared classes. Measurability, lower semicontinuity, compactness, coercivity, or closure assumptions needed for attainment are stated locally rather than imposed globally. When only approximation is required, the book works with infima and 𝜀-optimal selections.

Contents Preface

v

Notation and Standing Conventions

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I

The Structural Realizability Problem

1

1

Structural Realizability under Shared Deployment 1.1 Reuse changes the learning problem . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Three levels: solvability, realizability, and certifiability . . . . . . . . . . . . . . . 1.3 Native risk and architecture obstruction . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mission and falsifiable consequences . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3 4 5

2

Certified Elimination Systems 2.1 The primitive interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A complete learning contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Pointwise, population, and uniform risk functionals . . . . . . . . . . . . . . . . . 2.4 Regular and singular oracle geometry . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Defect fibrations and fine-to-coarse maps . . . . . . . . . . . . . . . . . . . . . . 2.6 Common loss scales, attainment, and approximation . . . . . . . . . . . . . . . .

6 6 7 7 8 8 9

3 A Four-Component Decomposition of Population Risk 3.1 Reference risks and the four components . . . . . . . . . . . . . . . . . . . . . . . 3.2 A high-probability learning bound . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Structural meaning and mechanism coupling . . . . . . . . . . . . . . . . . . . . 3.4 What an empirical study should separate . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 3: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 10 12 13 14 15

4

17 17 17 18 19 20 22

Four Running Laboratories 4.1 Four recurring structural problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Laboratory I: oracle-preserving coarse-graining . . . . . . . . . . . . . . . . . . . 4.3 Laboratory II: graph-indexed distribution learning . . . . . . . . . . . . . . . . . . 4.4 Laboratory III: shared conditional decisions . . . . . . . . . . . . . . . . . . . . . 4.5 Laboratory IV: one-pass prediction and local refinement . . . . . . . . . . . . . . 4.6 A common table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

CONTENTS

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II

Native Defects and Elimination Calculus

24

5

Conjugate Lifts and Native Loss Scales 5.1 Legendre conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Examples of native Bregman defects . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Further certified eliminations and boundaries . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 5: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 26 27 30

6

Elimination Towers and the P/G/X/V/C Calculus 6.1 Five distinct operations on an elimination system . . . . . . . . . . . . . . . . . . 6.2 A compact classification test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Horizontal operations and multistage towers . . . . . . . . . . . . . . . . . . . . . 6.4 Worked decompositions and coupling boundaries . . . . . . . . . . . . . . . . . . Appendix to Chapter 6: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 33 35 37

7

Exactification and Optimization Certificates 7.1 Jet exactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The converse normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Approximation, transmission, and certification error . . . . . . . . . . . . . . . . 7.4 Practical algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 7: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 38 40 41 42 43

8

Probability Validity, Graph Smoothing, and CRPS 8.1 The CDF set and projection repair . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 CRPS risk and its observable decomposition . . . . . . . . . . . . . . . . . . . . 8.3 The probability-validity barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Sharp path rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 8: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 45 46 47 48 49

III 9

Validity of Defects and Representations

51

Integrability, Hodge Repair, and Global Defect Consistency 9.1 When local defect reports may be inconsistent . . . . . . . . . . . . . . . . . . . . 9.2 Square curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Approximate flatness and Hodge repair . . . . . . . . . . . . . . . . . . . . . . . 9.4 Repair, falsification, and audit boundaries . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 9: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 52 52 54 55 57

10 Lift Admissibility, Slack Factorization, and Complexity 10.1 Why exact lifts can be vacuous . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Epigraph geometry and intrinsic lift structure . . . . . . . . . . . . . . . . . . . . 10.3 Gauge reduction and lift frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Target-visible reduction and extraction . . . . . . . . . . . . . . . . . . . . . . . . 10.5 From lift size to native defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 10: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

58 58 59 60 61 63 65

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IV

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Structural Obstructions under Shared Deployment

11 Architecture Obstruction as a Second Elimination 11.1 From a local defect to a class-level floor . . . . . . . . . . . . . . . . . . . . . . . 11.2 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Architecture comparison and witnesses . . . . . . . . . . . . . . . . . . . . . . . 11.4 Architecture obstruction versus approximation and optimization . . . . . . . . . . 11.5 Output, risk, and theorem contracts . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 70 70 71 71

12 Regular Obstruction Transfer and Flow Duality 12.1 The regular regime and quarter transport bound . . . . . . . . . . . . . . . . . . . 12.2 Finite calibration, dual flow, and population thickening . . . . . . . . . . . . . . . 12.3 Set-valued oracles and certificate boundaries . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 12: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 74 75 77

13 Rectangularity, Coordination Tax, and Memory 13.1 Conditional architectures and their rectangular hull . . . . . . . . . . . . . . . . . 13.2 The coordination decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Memory compression as base change . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 13: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

78 78 79 82 82 84

14 Singular Fibers, Monodromy, and Catastrophe Taxes 14.1 The radical normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Risk level, general loops, and degree . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Beyond radicals: roots, eigenlines, and partial monodromy . . . . . . . . . . . . . 14.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 14: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 87 87 88 89

15 Atlases, Quotients, and Randomized Repair 15.1 From obstruction diagnosis to a repair contract . . . . . . . . . . . . . . . . . . . 15.2 Local and symmetry-respecting repair . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Repairing deployment mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Selection, validation, and repair frontiers . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 15: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

90 90 91 92 93 94

V

96

Resources, Operational Semantics, and Composition

16 Resource-Constrained Architecture Rate–Distortion 16.1 The architecture as carrier and decoder . . . . . . . . . . . . . . . . . . . . . . . 16.2 Canonical Bregman carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Discrete and continuous carrier extraction . . . . . . . . . . . . . . . . . . . . . . 16.4 Finite frontiers and phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 An exact data-selection frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Resource limits and operational boundaries . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 16: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 98 99 100 100 102 103

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17 Operational Semantics and Contextual Observability 17.1 From internal difference to task-observable kernels . . . . . . . . . . . . . . . . . 17.2 Task envelopes and contextual completion . . . . . . . . . . . . . . . . . . . . . . 17.3 From native defect to operational separation . . . . . . . . . . . . . . . . . . . . . 17.4 Ordering curvature and contextual witnesses . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 17: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

104 104 105 107 107 109

18 Composition, Base Change, and Dequantization 18.1 Conditional chain rules and architecture composition . . . . . . . . . . . . . . . . 18.2 Base change and data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 When conditional elimination destroys encoder semantics . . . . . . . . . . . . . . 18.4 Hard limits and composition boundaries . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 18: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 113 114 116 118

VI

120

Statistical Certification

19 Confidence Worlds and Statistical Elimination Geometry 19.1 Confidence worlds and finite-data permission . . . . . . . . . . . . . . . . . . . . 19.2 Identified images, margins, and honest certificates . . . . . . . . . . . . . . . . . . 19.3 Defect envelopes and architecture grammars . . . . . . . . . . . . . . . . . . . . . 19.4 Persistent atlases and operational transport . . . . . . . . . . . . . . . . . . . . . . 19.5 Pointwise-oracle sparse model selection . . . . . . . . . . . . . . . . . . . . . . . 19.6 Observational overlap and deployment conflict . . . . . . . . . . . . . . . . . . . 19.7 Active-set conflict and repair depth . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8 Identification, hidden winding, and data-dependent architecture . . . . . . . . . . Appendix to Chapter 19: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

121 122 123 125 126 130 133 135 139 140

20 Certificate Statistics and Resolution Complexity 20.1 The certificate record and four nonexchangeable objects . . . . . . . . . . . . . . . 20.2 Resolution complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Evidence carriers and task-relative sufficiency . . . . . . . . . . . . . . . . . . . . 20.4 Recursive certificate posterior quotient . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 20: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 148 150 152 157

VII

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Obstruction-Aware Learning and Structural Repair

21 Toward a Certified Structural Learning Theory 21.1 Three nonexchangeable carrier contracts . . . . . . . . . . . . . . . . . . . . . . . 21.2 Typed architecture realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Finite-library comparison and data-dependent selection . . . . . . . . . . . . . . . 21.4 Confidence, stopping, and the learning-theory frontier . . . . . . . . . . . . . . . Appendix to Chapter 21: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 166 169 171 173

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22 Obstruction-Aware Learning and Inference 22.1 From diagnosis to controlled intervention . . . . . . . . . . . . . . . . . . . . . . 22.2 The repair must match the diagnosed mechanism . . . . . . . . . . . . . . . . . . 22.3 A falsifiable saturation–repair–validation chain . . . . . . . . . . . . . . . . . . . 22.4 When a certificate implies population improvement . . . . . . . . . . . . . . . . . Appendix to Chapter 22: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

176 176 177 178 178 180

23 Partial Transport, Cycle Holonomy, and Recovery 23.1 Local branch sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Partial matching objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Noisy recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Path transport, holonomy, and global sections . . . . . . . . . . . . . . . . . . . . 23.5 Multi-cycle witness, atlas construction, and uncertainty . . . . . . . . . . . . . . . 23.6 Deterministic code-path audit of the radical repair . . . . . . . . . . . . . . . . . . 23.7 Closing the radical loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix to Chapter 23: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 182 182 183 185 186 188 190

24 Four Applied Laboratories 24.1 How the laboratories close the loop . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Laboratory I: one-pass inference audits . . . . . . . . . . . . . . . . . . . . . . . 24.3 Laboratory II: certified Monte Carlo reuse . . . . . . . . . . . . . . . . . . . . . . 24.4 Laboratory III: multi-sample cytometry . . . . . . . . . . . . . . . . . . . . . . . 24.5 Laboratory IV: policies, world models, and memory . . . . . . . . . . . . . . . . .

193 193 194 200 202 204

VIII

207

Stress Tests, Boundaries, and Research Program

25 Quantum Elimination Geometry 25.1 Universal Gibbs base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Commuting sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Replacement, measurement, and operational meaning . . . . . . . . . . . . . . . . Appendix to Chapter 25: complete proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

208 208 210 210 212

26 Boundaries of Structural Explanation 26.1 Failures that invalidate the structural object . . . . . . . . . . . . . . . . . . . . . 26.2 Interfaces that require additional theorems . . . . . . . . . . . . . . . . . . . . . . 26.3 Alternative explanations and falsification . . . . . . . . . . . . . . . . . . . . . . 26.4 Conditions that must remain visible . . . . . . . . . . . . . . . . . . . . . . . . .

216 216 217 217 218

27 A Falsifiable Research Program for Structural Learning 27.1 Data-dependent and computation-dependent deployment . . . . . . . . . . . . . . 27.2 Population effects of singularity and composition . . . . . . . . . . . . . . . . . . 27.3 Scientific AI and foundation-model contracts . . . . . . . . . . . . . . . . . . . . 27.4 Success criteria and stopping rules . . . . . . . . . . . . . . . . . . . . . . . . . .

219 219 220 220 221

xvi

CONTENTS

Appendices: Technical Background, Terminology, and Crosswalks

225

A Convex Analysis and Bregman Geometry A.1 Convex conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Bregman divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Conditional Bregman projection . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Strong convexity and smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Metric projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Primal–dual gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 225 226 226 226 226

B Graph Operators, Min-Plus Algebra, and Partial Maps B.1 Weighted graph Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Spectral filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Min-plus algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Cycle rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Partial bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 227 227 228 228

C Covering Spaces, Degree, and Monodromy C.1 Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Winding number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Topological degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Schwarz genus and atlas number . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229 229 230 230 230

D Statistical Tools for Honest Certification D.1 Uniform concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Sample splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 Confidence sets and three-way decisions . . . . . . . . . . . . . . . . . . . . . . . D.4 Confidence sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 Testing lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.6 Clustered data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 231 231 231 232 232 232

E Validation and Dependency Map E.1 Constructive spine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Validity spine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.3 Architecture spine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4 Resources, semantics, and composition . . . . . . . . . . . . . . . . . . . . . . . E.5 Statistical and intervention spine . . . . . . . . . . . . . . . . . . . . . . . . . . . E.6 No circularity in the graph-CRPS result . . . . . . . . . . . . . . . . . . . . . . . E.7 No circularity in exactification . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233 233 234 234 234 234 235 235

F Controlled Glossary of Structural Terms

236

G Source Manuscript Crosswalk 243 G.1 Principal-result audit ledger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 G.2 Editorial principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

CONTENTS

xvii

Thematic Index

263

List of Figures Three distinct questions. Local optimization, global deployment, and finite-data certification require different assumptions and different remedies. . . . . . . . . . . .

3

Reference risks in the four-component decomposition. The first two gaps are populationlevel structural terms; generalization and implementation control the distance from the class optimum to the learned rule. . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

An elementary elimination square. Nonzero circulation records order dependence. On a general complex, zero face curvature must still be supplemented by a period audit. .

53

10.1 The admissibility chain. Exact elimination alone is insufficient: a lift must be tied to target slack, stripped of slack-invisible gauge, and then connected to a deployment state whose distinctions are measured by native defect. . . . . . . . . . . . . . . . .

58

13.1 Rectangularization preserves the locally available kernels but removes the cross-history constraint that forces them to share one parameterization or memory state. . . . . . .

79

14.1 Monodromy depends on the output contract. A point-valued representative can fail to return to itself after a loop, while a quotient-valued object such as a projector or unordered root set can remain globally well defined. . . . . . . . . . . . . . . . . . .

85

1.1

3.1

9.1

17.1 Contextual observability. Two internal kernels are operationally equivalent only if no legal precontext, postcontext, and terminal task can distinguish them. . . . . . . . . . 104 19.1 Finite-data certification transports a covered set of population worlds through a declared query before producing a terminal label. . . . . . . . . . . . . . . . . . . . . . . . . 121 21.1 The three architecture chains after finite-information closure. Each has its own domain, exactness obligation, and reporting scale; they meet only at a common certified action. 164 22.1 The proposed OALI workflow. Arrows denote audit stages rather than theorem implications. A certificate motivates a repair candidate but does not establish saturation, mechanism exclusivity, or held-out improvement. . . . . . . . . . . . . . . . . . . . 176 23.1 Synthetic multi-cycle graph illustrating partial transports, cycle-space obstruction, and atlas coverage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

xviii

List of Tables 3.1

Evidence associated with the four components of risk. . . . . . . . . . . . . . . . . .

14

4.1 4.2

Exact one-pass and local-refinement ledger for the three-point quadratic laboratory. . The four laboratories under one structural vocabulary. . . . . . . . . . . . . . . . . .

21 22

6.1

The five operational contracts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

19.1 The confidence-world construction contract. . . . . . . . . . . . . . . . . . . . . . . 122 19.2 Four confidence-world deployment states under attainment. . . . . . . . . . . . . . . 129 20.1 Three statistical objects that must not be conflated. . . . . . . . . . . . . . . . . . . 152 20.2 Department admission-rate differences and 95% simultaneous exact intervals, in percentage points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 22.1 Structural mechanisms and corresponding repair families. . . . . . . . . . . . . . . . 177 23.1 Radical-loop common-witness deterministic code-path audit. . . . . . . . . . . . . . 187 23.2 The radical loop as a seven-step certification chain. . . . . . . . . . . . . . . . . . . 188 24.1 Prospectively frozen MNIST audit. The first three rows report mean test amortization gaps in nats across five independently trained decoders. The final two rows are the seed-level primary endpoints; parentheses contain the seed standard deviation and confidence bounds are one-sided 95% Student 𝑡 bounds. . . . . . . . . . . . . . . . . 24.2 Digits R2 equal-information, equal-compute control upgrade. Bounds are one-sided 95% paired bootstrap bounds over 360 confirmatory images. The table groups all 12 frozen gates; every gate passed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 Frozen patient-level diabetes confirmation. One-sided 95% paired bootstrap bounds treat the patient as the independent unit. All 12 frozen gates passed. . . . . . . . . . 24.4 Three diagnostic scales for posterior-family Monte Carlo reuse. The final column reports only the fixed analytic pilots described above. . . . . . . . . . . . . . . . . . 24.5 Synthetic cytometry smoke test with 95% intervals. Sample-level endpoints use 5,000 bootstrap resamples of the ten samples; discrete edge and cycle endpoints use Wilson intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1

195

197 198 201

203

Controlled glossary of book-level terms. . . . . . . . . . . . . . . . . . . . . . . . . 236

xix

xx

LIST OF TABLES

Part I

The Structural Realizability Problem Modern learning systems reuse representations, parameterizations, memories, and output interfaces across many local problems. The first question is therefore not only whether local optima exist, but whether one declared deployment system can realize them simultaneously.

1

Chapter 1

Structural Realizability under Shared Deployment 1.1

Reuse changes the learning problem

Many modern learning systems solve a family of local problems and then reuse a single mechanism across them. Let 𝑎★𝑥 ∈ arg min 𝐻 𝑥 (𝑎) 𝑎∈ A 𝑥

denote a local posterior, conditional policy, spectral object, population identity, or inner optimization state. A pointwise procedure asks only that each local problem be solvable. Deployment asks for one field Ä 𝐴 : X −→ A𝑥 , 𝐴(𝑥) ∈ A 𝑥 , 𝑥 ∈X

subject to a contract such as continuity, Lipschitz regularity, finite memory, shared parameters, a fixed output type, bounded communication, or a prescribed computation graph. The quantifiers are different: ∀𝑥 ∃𝑎★𝑥

⇏

∃𝐴 ∈ 𝔄 ∀𝑥, 𝐴(𝑥) = 𝑎★𝑥 .

The right-hand statement is a global realizability problem. Its failure is structural when it persists after the local oracles have been fixed and after optimization within the declared deployment class has been completed. A two-point example isolates the issue. Let 𝑋 ∈ {0, 1} be equiprobable and let the native loss be D 𝑥 (𝑎) = (𝑎 − 𝑥) 2 . Each local optimum is exact: 𝑎★ (𝑥) = 𝑥. If deployment is restricted to constant rules 𝐴(𝑥) ≡ 𝑐, then 1 1 inf E(𝑐 − 𝑋) 2 = inf {𝑐2 + (𝑐 − 1) 2 } = . 𝑐 𝑐 2 4 No reparameterization of the same constant-output contract removes this 1/4. The obstruction disappears only when the deployment rule is allowed to depend on 𝑥. The same pattern appears in less trivial forms. Amortized inference asks one encoder to reproduce many instancewise variational optima. Sequential decision systems reuse parameters or a compressed memory state across many histories. Spectral procedures may need one continuous representative of 2

1.2. THREE LEVELS: SOLVABILITY, REALIZABILITY, AND CERTIFIABILITY

3

an eigenspace whose oriented eigenvector changes sign around a loop. The mechanisms differ, but the quantifier mismatch is the same: local solvability does not determine global realizability under reuse.

1.2

Three levels: solvability, realizability, and certifiability

The book separates three questions that are often mixed together. 1. Local solvability. For each input, is the local oracle well defined, and what excess objective is paid by deviating from it? 2. Global realizability. Does one rule satisfying the declared representation, sharing, memory, regularity, and resource constraints realize the local oracle family simultaneously? 3. Finite-sample certifiability. When the local oracles and population structure must themselves be estimated, do the available data support a statistically valid conclusion of realizability, nonrealizability, or unresolvedness? These levels do not substitute for one another. A genuine population obstruction need not be identifiable at a realistic sample size. Conversely, an unresolved statistical certificate does not imply that the obstruction is absent. If the local model itself is misspecified, then a perfectly realizable deployment can still be scientifically inadequate. sampling experiment

deployment contract

Local solvability ∀𝑥 ∃𝑎★𝑥

Global realizability ∃𝐴 ∈ 𝔄

Finite-sample certifiability resolved / unresolved

Figure 1.1: Three distinct questions. Local optimization, global deployment, and finite-data certification require different assumptions and different remedies.

This separation determines the intervention. Model misspecification calls for a richer local model; finite-sample ambiguity calls for more informative data or a different experiment; implementation error calls for a better solver. Structural nonrealizability calls instead for a change in the deployment contract—for example a quotient representation, an atlas, additional memory, a different sharing pattern, or a larger resource budget. Position relative to neighboring theory. The book does not claim priority for the observation that sharing can create suboptimality. The amortization gap has been separated from variational-family approximation since early work on VAEs [Cremer et al., 2018], and recent results characterize when amortized variational inference can attain its instancewise optimum [Margossian and Blei, 2024]. Multitask representation learning studies benefits and limits of shared features [Li et al., 2025], while finite-memory RL quantifies errors caused by compressed histories [Eberhard et al., 2025]. Elimination geometry makes a more specific claim: once the local oracle and its native excess objective are fixed, it seeks objective-denominated lower bounds, finite-data certificates, and mechanism-specific repairs for the remaining deployment constraint. The architecture-specific candidate in Chapter 19 is correspondingly narrower. O’Neill et al. [2025] already establish a global amortisation gap for a one-layer linear–nonlinear sparse autoencoder and study inference-time optimization. This book does not reclaim that conclusion. It asks whether

CHAPTER 1. STRUCTURAL REALIZABILITY UNDER SHARED DEPLOYMENT

4

two strict KKT neighborhoods in one fitted nonnegative sparse problem expose incompatible oracle Jacobian rows, converts that conflict into a quantitative native-loss floor for the declared one-pass class, and computes a sufficient proximal repair depth. Its exact priority remains unresolved and is recorded as such in Appendix G.

1.3

Native risk and architecture obstruction

A structural lower bound is useful only if it is measured in a loss scale connected to the original problem. If an auxiliary object 𝑎 is eliminated from 𝐽 (𝑥) = inf 𝐻 𝑥 (𝑎), 𝑎∈ A 𝑥

then the native defect is D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) ≥ 0.

It is an exact excess objective, not a post hoc parameter distance. The population risk of a deployment 𝐴 is L 𝑃 ( 𝐴) = E 𝑃 D𝑋 { 𝐴(𝑋)}, and the architecture obstruction is 𝔒 𝑃 (𝔄) = inf L 𝑃 ( 𝐴). 𝐴∈𝔄

When 𝔒 𝑃 (𝔄) > 0, the declared deployment class has a strictly positive population floor relative to the already fixed local oracle family. This is the book’s organizing interface with generic expressivity and approximation theory. In settings with a unique oracle and a norm-equivalent defect, the resulting obstruction may reduce to an ordinary approximation error. The useful additional discipline is then not a new mathematical mechanism, but the requirement that the local target, loss scale, deployment contract, and population quantifier remain explicit. Boundary of the claim Architecture obstruction is not a synonym for generic approximation error. It has an independent interpretation only after the local oracle, native defect, representation semantics, deployment class, statistical unit, and implementation error have been declared separately.

1.4. MISSION AND FALSIFIABLE CONSEQUENCES

1.4

5

Mission and falsifiable consequences

The purpose of this book is to provide a common audit language for identifying, quantifying, and testing claims of structural nonrealizability under shared deployment. The chapters that follow ask five questions in a recommended workflow order: 1. Which native loss scale is generated by the original elimination? 2. Are the local defect descriptions and lifted representations structurally valid? 3. What regular, coordination, singular, or resource obstruction is forced by the deployment contract? 4. Can finite data certify the relevant structural conclusion, and can the declared downstream task observe it? 5. Which minimal structural intervention is predicted to remove the obstruction, and does that intervention improve independent held-out risk? A structural explanation should be falsifiable. A performance plateau alone is not evidence of architecture obstruction: misspecification, optimization failure, data scarcity, regularization, and metric ceilings can produce the same curve. A stronger empirical signature has several parts. First, a structural certificate predicts a nonzero floor under a fixed deployment contract. Second, capacity or compute increases that preserve that contract do not remove the diagnosed gap. Third, a mechanism-matched change of representation, sharing, memory, or output semantics reduces the native defect. Finally, the gain survives independent validation and matched controls. Failure of any step weakens the structural interpretation. The book therefore does not propose a replacement for approximation, generalization, optimization, information theory, or representation learning. It develops a complementary structural question: once local optima are defined, what can a shared learning system coherently realize, what native risk is unavoidable when it cannot, and what evidence justifies changing the deployment contract?

Exercises Exercise 1.1. Construct a finite instance space on which every local optimum exists but a constant deployment class has positive architecture obstruction. Then enlarge the deployment class minimally so that the obstruction vanishes. Exercise 1.2. For

 𝑣(𝜃) =

 cos(𝜃/2) , sin(𝜃/2)

𝑃(𝜃) = 𝑣(𝜃)𝑣(𝜃) ⊤ ,

show that 𝑃(0) = 𝑃(2𝜋) while 𝑣(2𝜋) = −𝑣(0). Explain why the projector representation changes the realizability problem without discarding the eigenspace. Exercise 1.3. Choose a modern learning system and write three separate questions: local solvability, global realizability, and finite-sample certifiability. Propose one negative control that could falsify a structural explanation.

Chapter 2

Certified Elimination Systems 2.1

The primitive interface

The theory begins with a family of inner optimization problems. Let X be an instance space. Let 𝜋 : E → X be a fibered auxiliary space with fiber A 𝑥 = 𝜋 −1 (𝑥). A lifted objective is a function 𝐻 : E → R ∪ {+∞},

𝐻 𝑥 (𝑎) = 𝐻 (𝑥, 𝑎).

Definition 2.1 (Certified elimination). A certified elimination is a triple (𝐻, 𝐽, D) satisfying 𝐽 (𝑥) = inf 𝐻 𝑥 (𝑎) ∈ R, 𝑎∈ A 𝑥

D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) ∈ [0, +∞].

Thus every auxiliary fiber is nonempty and its lifted objective is proper and bounded below by a finite value; attainment is not required. These conditions make the residual well defined even when a trial state has infinite lifted cost. The oracle fiber is O (𝑥) = {𝑎 ∈ A 𝑥 : D 𝑥 (𝑎) = 0}. The adjective certified refers to the exact objective identity. A distance from 𝑎 to a chosen optimizer may still be useful, but it is not a certificate until a proved exchange inequality links that distance to 𝐻 𝑥 (𝑎) − 𝐽 (𝑥). Example 2.2 (Quadratic profiling). Let 𝐻 𝑥 (𝑎) = 12 ∥𝑎 − 𝑚(𝑥) ∥ 2 + 𝑐(𝑥). Then 𝐽 (𝑥) = 𝑐(𝑥), O (𝑥) = {𝑚(𝑥)}, and 1 D 𝑥 (𝑎) = ∥𝑎 − 𝑚(𝑥) ∥ 2 . 2 Here Euclidean distance and native defect coincide up to scale. Example 2.3 (Nonidentifiable mixture). Suppose 𝐻 𝑥 (𝑎) is invariant under a permutation group 𝐺. The oracle fiber is an orbit rather than a point. Parameter distance between two representatives may be positive while the native defect is zero. The correct geometry lives on a quotient or an unordered output space.

6

2.2. A COMPLETE LEARNING CONTRACT

2.2

7

A complete learning contract

A learning system is not determined by 𝐻 alone. We use the following package. Definition 2.4 (Certified learning system). A certified learning system is 𝔈 = ( X, 𝑃, E, 𝐻, 𝐽, D, 𝔄, R, 𝔗, P), where: • ( X, 𝑃) is the population probability space; • E → X is the auxiliary fibration; • (𝐻, 𝐽, D) is a certified elimination; • 𝔄 is a class of 𝑃-measurable sections 𝐴 of E → X for which 𝑥 ↦→ D 𝑥 { 𝐴(𝑥)} is measurable; • R is the representation and resource grammar; • 𝔗 is the legal downstream task/context contract; • P is the statistical experiment generating observed data. The components play different roles: the objective determines the defect; deployment determines the architecture class; the task contract determines operational visibility; and the statistical experiment determines what can be certified. Boundary of the claim Changing the architecture or representation while keeping the objective fixed changes the obstruction but not the native defect. Changing the lifted objective can change the statistical target itself. These are distinct interventions.

2.3

Pointwise, population, and uniform risk functionals

For a deployed field 𝐴 ∈ 𝔄, define pointwise and population defects D 𝐴 (𝑥) = D 𝑥 { 𝐴(𝑥)},

L 𝑃 ( 𝐴) = E 𝑃 D 𝐴 (𝑋).

The expectation is allowed to be extended-valued. For the finite population decomposition below, assume that 𝔄 is nonempty and contains at least one field of finite population defect, and take 𝐴 itself to have finite population defect. Then the architecture obstruction is finite: 𝔒 𝑃 (𝔄) = inf L 𝑃 ( 𝐴). 𝐴∈𝔄

Every deployed field satisfies the exact architecture decomposition impl

L 𝑃 ( 𝐴) = 𝔒 𝑃 (𝔄) + Δ 𝑃 ( 𝐴; 𝔄), where impl

Δ 𝑃 ( 𝐴; 𝔄) = L 𝑃 ( 𝐴) − 𝔒 𝑃 (𝔄) ≥ 0.

CHAPTER 2. CERTIFIED ELIMINATION SYSTEMS

8

The identity is simple, but conceptually decisive. It separates an irreducible class-level floor from failure to reach the class optimum. Uniform deployment uses L∞ ( 𝐴) = sup D 𝐴 (𝑥),

𝔒∞ (𝔄) = inf L∞ ( 𝐴). 𝐴∈𝔄

𝑥 ∈X

Uniform and average risks can differ sharply. A continuous field may be forced to incur a fixed worst-case tax at a topological seam while concentrating that seam on a set of arbitrarily small 𝑃-mass.

2.4

Regular and singular oracle geometry

The oracle incidence set O = {(𝑥, 𝑎) : 𝑎 ∈ O (𝑥)} ⊆ E can have several geometries. Unique regular oracle. Each fiber contains one oracle 𝑎★ (𝑥) and the map 𝑥 ↦→ 𝑎★ (𝑥) is regular. Obstruction then comes from limited variation, resources, or decoder capacity. Separated finite cover. Each fiber contains finitely many separated branches. Local branches form a covering space; a global labeled section may fail because of monodromy. Singular oracle family. Branches collide, disappear, or change multiplicity. The incidence map is stratified rather than a covering. Strong convexity and uniform branch separation fail, and the defect itself must price the singularity. Set- or distribution-valued oracle. The natural oracle object may be an orbit, a projector, an unordered set, or a law. A point-valued representation can introduce an artificial obstruction.

2.5

Defect fibrations and fine-to-coarse maps

Suppose 𝜌 : E → B maps a fine auxiliary object to a coarse one. A useful elimination hierarchy has an exact decomposition D 𝑥E (𝑒) = D 𝑥B {𝜌(𝑒)} + 𝛿 𝑥 (𝑒),

𝛿 𝑥 (𝑒) ≥ 0.

The coarse defect measures the price of choosing the wrong coarse state; the vertical term measures the price of realizing that coarse state in the fine fiber. For a restricted fine architecture 𝔄 E , define the fiber realization tax Ψ𝑥 (𝑏) = inf{𝛿 𝑥 (𝑒) : 𝑒 ∈ 𝔄 E (𝑥), 𝜌(𝑒) = 𝑏}. Then the fine obstruction takes the infimal form  𝔒 E = inf D B (𝑏) + Ψ(𝑏) . 𝑏

2.6. COMMON LOSS SCALES, ATTAINMENT, AND APPROXIMATION

9

This is the basic obstruction tower. It is a Bellman or min-plus recursion over representation levels. Certified decomposition The tower separates two questions: which coarse state should be selected, and how much must the architecture pay to realize its canonical fine fiber? An objective-faithful coarse representation need not be obstruction-faithful.

2.6

Common loss scales, attainment, and approximation

A central discipline of elimination geometry is that different nonnegative quantities are not added merely because they appear in the same problem. Structural principle Two terms may be summed only if they are parts of an exact identity, an infimal decomposition, or a theorem that converts them to a common declared loss scale. For example, KL defect, topological degree, parameter distance, runtime, and task regret have different units and semantics. A theorem may exchange one for another under strong convexity, data processing, an exposure gate, or a complexity model. Without that theorem, the quantities should be reported separately. Attainment and approximation.

A zero architecture obstruction has several interpretations.

• If the infimum is attained and equals zero, the architecture contains an exact oracle section. • If the infimum is zero but unattained, the architecture approximates the oracle arbitrarily well in the declared risk. • If the average obstruction is zero but the uniform obstruction is positive, the architecture can squeeze failure into a vanishing seam but cannot remove it everywhere. Compactness, lower semicontinuity, coercivity, or finite-dimensional closure are needed for attainment. The notation min must not replace inf without such a theorem.

Exercises Exercise 2.5. For a finite instance space and finite auxiliary fibers, prove that the population architecture obstruction is attained for every nonempty architecture class. Exercise 2.6. Construct an example with 𝔒 𝑃 (𝔄) = 0 but 𝔒∞ (𝔄) > 0. Hint: use a continuous selector on a circle with a seam of shrinking measure. Exercise 2.7. Let 𝜌 forget a label permutation in a mixture model. Describe a coarse oracle object for which the coarse defect is zero while a labeled fine architecture pays a positive realization tax.

Chapter 3

A Four-Component Decomposition of Population Risk The same excess risk can have different causes, and those causes call for different interventions. The local statistical model may be inadequate; a shared deployment contract may fail to realize the local oracle family; finite data may not identify a good deployment; or the numerical procedure may not have reached the best member of the declared class. The decomposition below separates these mechanisms by their reference objects before any probability bound is applied.

3.1

Reference risks and the four components

Let 𝑅★ be the scientifically relevant population optimum. Let 𝑅oracle be the population risk attained by the local oracle family generated by the declared local model or variational approximation. Let 𝔄 be b be the learned deployment. Define the deployment class, and let 𝐴 Δmodel = 𝑅oracle − 𝑅★,

Δarch = inf 𝑅( 𝐴) − 𝑅oracle , 𝐴∈𝔄

and opt b − inf 𝑅 b𝑛 ( 𝐴) b𝑛 ( 𝐴). Δ𝑛 = 𝑅 𝐴∈𝔄

For a reference deployment 𝐴† ∈ 𝔄, define b𝑛 ( 𝐴† ) − inf 𝑅 b𝑛 ( 𝐴) b 𝜌 𝑛 ( 𝐴† ) = 𝑅 𝐴∈𝔄

and 𝜌( 𝐴† ) = 𝑅( 𝐴† ) − inf 𝑅( 𝐴). 𝐴∈𝔄

The four principal components have different meanings. • Δmodel is the model approximation component: it compares the best local oracle family with the scientific target. • Δarch is the architecture obstruction component: after the local oracle has been fixed, it measures the smallest population loss forced by the deployment contract. 10

3.1. REFERENCE RISKS AND THE FOUR COMPONENTS

11

b and a comparator form the generalization component. • The population–empirical deviations at 𝐴 • Δ𝑛 is the implementation component: it measures empirical suboptimality within the declared deployment class. opt

All components that are added quantitatively must refer to a common risk scale, or be connected by an explicit exchange theorem. A KL defect, a parameter distance, a topological degree, a runtime, and a scientific utility difference are not automatically commensurable. Δmodel 𝑅★ scientific optimum

data + implementation

Δarch inf 𝐴∈𝔄 𝑅 ( 𝐴) best deployment

𝑅oracle local oracle

b 𝑅 ( 𝐴) learned rule

Figure 3.1: Reference risks in the four-component decomposition. The first two gaps are population-level structural terms; generalization and implementation control the distance from the class optimum to the learned rule.

How to read the assumptions The population and empirical class infima and every risk evaluation appearing in the identity are finite, and 𝐴† belongs to 𝔄; hence no undefined extended-real subtraction occurs. No stochastic assumption or minimizer attainment is needed for the exact identity below. The generalization bracket and the comparator correction may have either sign; only the architecture and model components are structural population floors under the stated nesting of targets. Proposition 3.1 (Exact four-component risk identity). For any reference 𝐴† ∈ 𝔄, b − 𝑅★ = Δmodel + Δarch 𝑅( 𝐴) h i b −𝑅 b +𝑅 b𝑛 ( 𝐴) b𝑛 ( 𝐴† ) − 𝑅( 𝐴† ) + 𝑅( 𝐴) + Δ𝑛 + 𝜌( 𝐴† ) − b 𝜌 𝑛 ( 𝐴† ). opt

If 𝐴† is a population minimizer, then 𝜌( 𝐴† ) = 0; if it is an empirical minimizer, then b 𝜌 𝑛 ( 𝐴† ) = 0. Interpretation and boundary The proposition is an algebraic identity, not a probabilistic bound. Its role is to fix the reference objects. In particular, the architecture obstruction is measured relative to an already declared local oracle family, so it cannot be silently merged with local model misspecification. Proof roadmap b b𝑛 ( 𝐴† ), the empirical class infimum, and 𝑅 b𝑛 ( 𝐴). Insert, in order, 𝑅oracle , inf 𝐴∈𝔄 𝑅( 𝐴), 𝑅( 𝐴† ), 𝑅 The chapter appendix gives the full telescoping calculation. When the population risk has the native form 𝑅( 𝐴) = 𝑅oracle + ED𝑋 { 𝐴(𝑋)},

CHAPTER 3. A FOUR-COMPONENT DECOMPOSITION OF POPULATION RISK

12

the second component becomes exactly Δarch = inf ED𝑋 { 𝐴(𝑋)} = 𝔒 𝑃 (𝔄). 𝐴∈𝔄

This is the principal interface between elimination geometry and statistical learning: generic approximation relative to 𝑅★ is refined into local model inadequacy and structural nonrealizability under shared deployment. In amortized variational inference, the distinction is especially simple. Fix the decoder and local variational family and take 𝐻 𝑥 = − ELBO 𝑥 . Then D 𝑥 is the per-datum amortization gap of Cremer et al. [2018], while Δarch is the smallest population-averaged amortization gap attainable by the declared encoder class—not the gap of a particular fitted encoder. For a deployed encoder 𝐴, the population-averaged amortization gap is therefore the sum of this class-level floor and a within-class impl residual, Δarch + Δ 𝑃 ( 𝐴; 𝔄). Margossian and Blei [2024] characterize when the first term can vanish under the sharing rule they study: one deterministic inference function must produce all local variational parameters.

3.2

A high-probability learning bound

Assume that, with probability at least 1 − 𝛿, b𝑛 ( 𝐴)| ≤ 𝜀 𝑛 (𝛿), sup |𝑅( 𝐴) − 𝑅 𝐴∈𝔄

and that the training algorithm satisfies b ≤ inf 𝑅 b𝑛 ( 𝐴) b𝑛 ( 𝐴) + 𝜂 𝑛 . 𝑅 𝐴∈𝔄

How to read the assumptions The deployment class is fixed on the event being analyzed, the population–empirical deviation is uniform over that class, and the training procedure is an 𝜂 𝑛 -approximate empirical minimizer. In addition, 𝑅★ ≤ 𝑅oracle ≤ inf 𝐴∈𝔄 𝑅( 𝐴) so that the model and architecture components are nonnegative. Theorem 3.2 (Four-component certified learning bound). On the uniform generalization event, b − 𝑅★ ≤ Δmodel + Δarch + 2𝜀 𝑛 (𝛿) + 𝜂 𝑛 . Δmodel + Δarch ≤ 𝑅( 𝐴) Interpretation and boundary The theorem places the learned excess risk around a structural population floor. More data can reduce 𝜀 𝑛 (𝛿) and better optimization can reduce 𝜂 𝑛 ; neither operation changes a positive Δmodel + Δarch unless the local model or deployment contract itself is changed. If the architecture is selected from the same data, the fixed-class uniform event must be replaced by sample splitting, a finite-library bound, stability, or a data-dependent random-set argument; PAC–Bayesian bounds for random hypothesis sets give one such route [Dupuis et al., 2024].

3.3. STRUCTURAL MEANING AND MECHANISM COUPLING

13

Proof roadmap b ∈ 𝔄. For the upper bound, compare 𝐴 b with an empirical class The lower bound uses only 𝐴 minimizer or a minimizing sequence and apply the uniform deviation twice. The appendix gives the argument without assuming attainment.

3.3

Structural meaning and mechanism coupling

The decomposition is exact, but its mechanisms are not independent. Enlarging a local variational family can reduce Δmodel while producing a multibranch oracle family that increases Δarch . A data-driven atlas makes the deployment class itself random. Test-time computation can reduce implementation error while also enlarging the effective deployment class. A nonconvex optimizer may reach only a subset of a nominal architecture class. The correct use of the decomposition is therefore diagnostic, not merely algebraic. The order of analysis is: local oracle −→ architecture floor −→ finite-sample control −→ implementation residual. The intervention then follows the dominant mechanism. Component

Typical cause

Model approximation

local family omits relevant structure Architecture obstruc- representation, sharing, memtion ory, regularity, or resource restriction Generalization limited independent data or data-dependent selection Implementation incomplete training or numerical solution

First intervention to examine change the local model or scientific target change the deployment contract

collect information or control statistical complexity improve the solver or computation budget

A positive architecture obstruction can itself admit more refined exact or infimal decompositions. Carrier information loss and decoder nonsaturation are separated in Chapter 16; rectangularization and coordination are separated in Chapter 13; fine-to-coarse representations generate fiber realization costs in Chapter 2. These decompositions are used only under their declared assumptions. They are not combined into a universal multi-term formula without an exact identity or a proved exchange bound. Boundary of the claim Different loss scales cannot be added by analogy. Topological degree, native KL defect, parameter error, communication, runtime, and downstream utility remain distinct quantities until a theorem converts one into another.

CHAPTER 3. A FOUR-COMPONENT DECOMPOSITION OF POPULATION RISK

14

3.4

What an empirical study should separate

A convincing structural study should report evidence for the four components rather than infer mechanism from one performance gap. Table 3.1: Evidence associated with the four components of risk.

Component

Diagnostic evidence

Model approximation

stronger local oracle family, oracle refit, simulation truth, or local-family comparison Architecture obstruc- transport or monodromy certificate, coordination decomposition, resource tion lower bound, or zero-obstruction witness independent-unit split, confidence sequence, stability, or data-dependent Generalization complexity bound Implementation primal–dual gap, KKT residual, multiple restarts, or certified inner-solver tolerance A performance plateau is therefore only a symptom. It supports a structural interpretation only after local model inadequacy, finite-sample uncertainty, and implementation error have been separately controlled. The prospectively frozen MNIST audit in Laboratory 24.2.1 applies this separation to a finite one-pass inference grammar: it holds the local variational contract fixed, uses held-out selection and evaluation, audits the imagewise reference, and tests data, width, and training-effort alternatives before changing the deployment computation.

Exercises Exercise 3.3. Verify the exact four-component identity term by term. Simplify it when 𝐴† is a population minimizer and when it is an empirical minimizer. Exercise 3.4. Suppose 𝔄1 ⊆ 𝔄2 . Prove that the architecture obstruction is monotone nonincreasing. Explain why the generalization component need not be. Exercise 3.5. Construct a problem in which Δmodel = 0, Δarch > 0, and both the generalization and implementation terms vanish asymptotically.

Appendix to Chapter 3: complete proofs

15

Appendix to Chapter 3: complete proofs Proof of the exact four-component risk identity Dependencies. Only the definitions of the four components and finiteness of every displayed population/empirical evaluation and class infimum are used. No stochastic assumption and no attainment of the architecture infimum are needed. Proof. Write b − 𝑅★ = {𝑅oracle − 𝑅★ } + { inf 𝑅( 𝐴) − 𝑅oracle } + {𝑅( 𝐴) b − inf 𝑅( 𝐴)}. 𝑅( 𝐴) 𝐴∈𝔄

𝐴∈𝔄

For an arbitrary reference 𝐴† ∈ 𝔄, decompose the last bracket by adding and subtracting 𝑅( 𝐴† ), b b𝑛 ( 𝐴† ), the empirical class infimum, and 𝑅 b𝑛 ( 𝐴): 𝑅 b − inf 𝑅( 𝐴) = 𝑅( 𝐴) b −𝑅 b b𝑛 ( 𝐴) 𝑅( 𝐴) 𝐴∈𝔄   b b b + 𝑅𝑛 ( 𝐴) − inf 𝑅𝑛 ( 𝐴) 𝐴∈𝔄   † b𝑛 ( 𝐴 ) − inf 𝑅 b𝑛 ( 𝐴) − 𝑅 𝐴∈𝔄

†

b𝑛 ( 𝐴 ) − 𝑅( 𝐴† ) +𝑅 + 𝑅( 𝐴† ) − inf 𝑅( 𝐴). 𝐴∈𝔄

The first two population brackets are respectively Δmodel and Δarch ; the two empirical-infimum brackets opt are Δ𝑛 and b 𝜌 𝑛 ( 𝐴† ); and the final population term is 𝜌( 𝐴† ). Substitution proves the identity. The two residuals vanish under population and empirical attainment, respectively. □

Proof of the four-component certified learning bound b ∈ 𝔄. The upper bound uses the uniform Dependencies. The lower bound uses only membership 𝐴 generalization event and the empirical optimization residual. Population attainment is not required. b ∈ 𝔄, Proof. Since 𝐴 b ≥ inf 𝑅( 𝐴) = 𝑅★ + Δmodel + Δarch , 𝑅( 𝐴) 𝐴∈𝔄

which proves the lower bound. For the upper bound, let ( 𝐴 𝑘 ) 𝑘 ≥1 ⊆ 𝔄 be a minimizing sequence such that 𝑅( 𝐴 𝑘 ) ↓ inf 𝑅( 𝐴). 𝐴∈𝔄

b𝑛 ( 𝐴)| ≤ 𝜀 𝑛 (𝛿), we have, for every 𝑘, On the event sup 𝐴∈𝔄 |𝑅( 𝐴) − 𝑅 b ≤𝑅 b + 𝜀 𝑛 (𝛿) b𝑛 ( 𝐴) 𝑅( 𝐴) b𝑛 ( 𝐴) + 𝜂 𝑛 + 𝜀 𝑛 (𝛿) ≤ inf 𝑅 𝐴∈𝔄

b𝑛 ( 𝐴 𝑘 ) + 𝜂 𝑛 + 𝜀 𝑛 (𝛿) ≤𝑅 ≤ 𝑅( 𝐴 𝑘 ) + 2𝜀 𝑛 (𝛿) + 𝜂 𝑛 .

16

CHAPTER 3. A FOUR-COMPONENT DECOMPOSITION OF POPULATION RISK

Letting 𝑘 → ∞ gives b ≤ inf 𝑅( 𝐴) + 2𝜀 𝑛 (𝛿) + 𝜂 𝑛 . 𝑅( 𝐴) 𝐴∈𝔄

Subtracting 𝑅★ and using inf

𝐴∈𝔄 𝑅( 𝐴) − 𝑅

★ = Δmodel + Δarch proves the upper bound.

□

Chapter 4

Four Running Laboratories 4.1

Four recurring structural problems

Four recurring laboratories keep the abstract machinery tied to concrete failure modes. Each isolates one obstruction and its matched repair; Chapter 24 later recombines them inside more complex scientific interfaces. 1. Oracle-preserving coarse-graining illustrates carrier information loss, safe quotienting, backmapping, and representation refinement. 2. Graph-indexed distribution learning illustrates signed high-order smoothing, probability validity, nonlinear projection, and sharp statistical rates. 3. Sequential conditional decision making illustrates rectangularity, shared-parameter coordination, memory compression, and operational context. 4. One-pass prediction illustrates architecture obstruction, local refinement, and an exact computation–defect frontier indexed by deployment budget. The same terminology will mean the same thing in all four settings.

4.2

Laboratory I: oracle-preserving coarse-graining

e have oracle output 𝑎★ , and let 𝑍 = 𝑞( 𝑋) e What may a representation safely forget? Let a fine state 𝑋 e 𝑋 be the carrier exposed to a decoder. Under squared native loss, the best unrestricted decoder returns E(𝑎★e | 𝑍) and pays the carrier-information component 𝑋

2 1 C(𝑍) = E 𝑎★𝑋e − E(𝑎★𝑋e | 𝑍) . 2

If 𝑊 = ℎ(𝑍) is a further coarsening, the tower property gives the exact tax 2 1 C(𝑊) − C(𝑍) = E E(𝑎★𝑋e | 𝑍) − E(𝑎★𝑋e | 𝑊) . 2

Thus forgetting is free exactly when 𝑊 preserves the conditional oracle signature; raw state count and hidden dimension are not the relevant capacity. 17

CHAPTER 4. FOUR RUNNING LABORATORIES

18

Molecular coarse-graining supplies a concrete fiber model. A reaction coordinate 𝑧 = 𝑞(e 𝑥) descends a fine potential 𝑈 to the potential of mean force   ∫ 1 −𝛽𝑈 (e 𝑥) W 𝛽 (𝑧) = − log 𝑣(𝑧) 𝑒 𝜈 𝑧 (𝑑e 𝑥) . 𝛽 𝑞 −1 (𝑧) Here 𝑣(𝑧)𝜈 𝑧 records the declared disintegration of the fine reference measure. e 𝛽 is the fine Gibbs law, Π W its coarse marginal, and 𝜂 𝛽,𝑧 its oracle fiber kernel, then any trial If Π 𝛽 fine law 𝑄, with coarse marginal 𝑃 and conditional back-mapping 𝑄 𝑧 , obeys the exact conditional-KL decomposition 1 e 𝛽 ) = 1 KL(𝑃∥Π𝛽W ) + 1 E 𝑃 KL(𝑄 𝑧 ∥𝜂 𝛽,𝑧 ). KL(𝑄∥ Π 𝛽 𝛽 𝛽 When 𝑃 is induced by a learned coarse potential, enriching that potential can reduce only the first component. The second is controlled by the conditional back-map; if the scientific oracle requires distinctions not measurable from 𝑧, no coarse predictor can restore them. The matched repair is to refine the reaction coordinate or retain a faithful conditional back-mapping, according to which component is dominant. Not every quotient is destructive. Replacing a subspace frame 𝑈 by the projector 𝑈𝑈 ⊤ removes only the 𝑈 ∼ 𝑈𝑄 basis gauge and preserves the subspace oracle. The criterion is always oracle survival, not whether the representation became smaller. Structural prediction. Scaling a predictor behind a fixed coarse carrier saturates at C(𝑍) when distinct conditional oracle means have been merged. Witness-directed carrier refinement can reduce that floor. For fine-law reconstruction, a faithful stochastic back-mapping instead removes the separate conditional-fiber term; dummy states and irrelevant gauge coordinates reduce neither.

4.3

Laboratory II: graph-indexed distribution learning

Let 𝑡 = 1, . . . , 𝑇 index vertices of a graph with Laplacian 𝐿. At vertex 𝑡, observations 𝑌𝑡1 , . . . , 𝑌𝑡 𝑛𝑡 b𝑡 . A graph filter 𝑆 produces a raw field define an empirical CDF 𝐹 e = 𝑆 𝐹. b 𝐹 e𝑡 need not be a For higher-order qualification, the filter may have signed off-diagonal weights. Then 𝐹 CDF. Fix a compact outcome interval 𝐼 = [𝑎, 𝑏] and work in the product space 𝐿 2 (𝐼) 𝑇 . The valid set C = {𝐹 : 𝐹𝑡 is a CDF for every 𝑡} is closed and convex in that space. Projection e 𝐹 = Π C ( 𝐹) restores validity. Because the truth 𝐹 ★ belongs to C, e − 𝐹★ ∥ 2 . ∥𝐹 − 𝐹 ★ ∥ 2 ≤ ∥ 𝐹

4.4. LABORATORY III: SHARED CONDITIONAL DECISIONS

19

The projection can also contract graph roughness under the declared product geometry. On an unbounded outcome line, CDFs themselves need not lie in 𝐿 2 ; the corresponding statement must instead be formulated for an affine difference class (for example, functions whose difference from a fixed reference CDF is square integrable). The laboratory separates three objects that are often conflated: 1. the raw ambient variational solution; 2. the nonlinear validity-repaired field; 3. the constrained variational optimizer over the CDF set. The first and second are not generally the same as the third. Structural prediction. For smoothness beyond the graph-universal Markov range, linear high-order validity fails. Nonlinear repair can retain the statistical rate while restoring the semantic contract.

4.4

Laboratory III: shared conditional decisions

Consider a finite history tree. At history ℎ, a local oracle kernel is 𝑞★ℎ (· | ℎ). The formal rectangular-hull construction is given in Chapter 13; here we use only its motivating contrast. A fully rectangular architecture can choose every conditional kernel independently, whereas a shared architecture may require 𝑞 ℎ (· | ℎ) = 𝑞 𝜃 (· | 𝜙(ℎ)) for one parameter 𝜃 and perhaps one compressed memory state 𝜙(ℎ). The rectangular hull Rect(𝔄), defined formally in Chapter 13, frees the locally available kernels and allows independent pasting. The rectangular value obeys an exact Bellman recursion. The gap Tcoord (𝔄) = 𝔒(𝔄) − 𝔒{Rect(𝔄)}

prices the cross-history coupling induced by sharing. A coarse memory representation can be objective-faithful for one task while failing to realize the canonical fine conditional fibers. The resulting fiber tax depends on occupancy and on the declared oracle law. Structural prediction. Increasing network width without refining memory may not remove the coordination tax. Rectangularization, memory refinement, or a task-specific quotient can.

CHAPTER 4. FOUR RUNNING LABORATORIES

20

4.5

Laboratory IV: one-pass prediction and local refinement

When does test-time computation change the deployed class rather than merely improve training? At instance 𝑥, consider the scalar local problem 𝐻 𝑥 (𝑎) =

1 {𝑎 − 𝑚(𝑥)}2 , 2

𝐽 (𝑥) = inf 𝐻 𝑥 (𝑎) = 0,

D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − 𝐽 (𝑥).

𝑎

The local oracle is 𝑎★𝑥 = 𝑚(𝑥). Let 𝑃 be the deployment law and let 𝔄0 be a declared class of one-pass predictors. Its architecture obstruction is 𝔒 𝑃 (𝔄0 ) = inf E 𝑃 D𝑋 { 𝐴(𝑋)}. 𝐴∈𝔄0

This infimum is architecture-relative by definition. A measured positive residual estimates it only after reachability error inside 𝔄0 has been controlled. Starting from 𝑎 0 = 𝐴(𝑥), now permit exact local-gradient steps with a fixed 0 < 𝜂 < 1: 𝑎 𝑗+1 = 𝑎 𝑗 − 𝜂{𝑎 𝑗 − 𝑚(𝑥)}. The recursion is explicit, 𝑎 𝑗 = 𝑚(𝑥) + (1 − 𝜂) 𝑗 { 𝐴(𝑥) − 𝑚(𝑥)},

D 𝑥 (𝑎 𝑗 ) = (1 − 𝜂) 2 𝑗 D 𝑥 { 𝐴(𝑥)}.

Write 𝑅 𝑗 𝐴 for the resulting predictor and define the budgeted deployment class by 𝔄 ≤ 𝑘 = {𝑅 𝑗 𝐴 : 𝐴 ∈ 𝔄0 , 0 ≤ 𝑗 ≤ 𝑘 }, where 𝑗 is integer. These classes are nested, and the pointwise identity gives the exact class-level resource–defect frontier 𝔒 𝑃 (𝔄 ≤ 𝑘 ) = (1 − 𝜂) 2𝑘 𝔒 𝑃 (𝔄0 ). (4.1) An exact three-point ledger. Let 𝑃 be uniform on {−1, 0, 1}, set 𝑚(𝑥) = 𝑥 2 , and take the one-pass grammar 𝔄0 = 𝔄aff = {𝑥 ↦→ 𝛼 + 𝛽𝑥 : 𝛼, 𝛽 ∈ R}. Its exact population risk is E 𝑃 D𝑋 {𝛼 + 𝛽𝑋 } =

 1 (𝛼 − 𝛽 − 1) 2 + 𝛼2 + (𝛼 + 𝛽 − 1) 2 . 6

The unique minimizer is (𝛼★, 𝛽★) = (2/3, 0), so 𝔒 𝑃 (𝔄aff ) =

1 ≈ 0.111111. 9

With 𝜂 = 1/2, the class-level identity gives the complete numerical ledger in Table 4.1. These are exact population values under the declared three-point law, not Monte Carlo estimates.

4.5. LABORATORY IV: ONE-PASS REFINEMENT

21

Table 4.1: Exact one-pass and local-refinement ledger for the three-point quadratic laboratory. Deployment contract Affine one pass Affine plus local refinement Affine plus local refinement Affine plus local refinement Quadratic-feature one pass

Step budget 𝑘

Residual factor

Exact minimum (decimal)

0 1 2 4 0

1 1/4 1/16 1/256 —

1/9 (0.111111) 1/36 (0.027778) 1/144 (0.006944) 1/2304 (0.000434) 0 (0.000000)

The last row enlarges the one-pass grammar to {𝛼 + 𝛽𝑥 + 𝛾𝑥 2 } and realizes the oracle with (𝛼, 𝛽, 𝛾) = (0, 0, 1). It therefore removes the same baseline obstruction through a representation change rather than test-time compute. Operationally the refined system runs a local optimizer; structurally it has moved from 𝔄0 to the computation-indexed contract 𝔄 ≤ 𝑘 , not merely trained the same class more accurately. Interface accounting. Changing predictor width or features changes 𝔄0 ; adding training data changes estimation; improving the one-pass optimizer changes reachability inside 𝔄0 ; and increasing 𝑘 changes the deployment computation budget. Calling all four changes “more model capacity” loses the intervention map. Structural prediction and boundary. If a well-optimized one-pass class has 𝔒 𝑃 (𝔄0 ) > 0, local refinement traces the monotone frontier in (4.1). The quadratic calculation transfers an existing baseline floor; it does not prove that the floor is positive. For fixed-variance Gaussian variational inference, the negative-ELBO defect is this same laboratory up to a constant scale factor. Nonquadratic problems replace the equality by a curvature-controlled bound or an empirical audit; Chapter 24 later applies that audit logic to learned one-pass inference on MNIST.

CHAPTER 4. FOUR RUNNING LABORATORIES

22

4.6

A common table

The four laboratories can now be compared row by row. The table aligns their oracle, native defect, deployment contract, obstruction, repair, and statistical evidence before the final cross-laboratory lesson. Table 4.2: The four laboratories under one structural vocabulary. Laboratory

Oracle and native defect

Contract and obstruction

Repair and evidence

Coarse-graining

conditional oracle/fiber law; carrier loss/fiber KL

Graph distributions Sequential decisions

vertexwise CDF; integrated squared-CDF loss historywise kernel; conditional KL or Bellman defect instancewise local optimum; excess local objective

carrier plus decoder/back-map; lost oracle distinctions graph filter plus validity interface; validity barrier shared network/memory; coordination/memory tax

refine carrier/faithful back-map; marginal and transfer risk nonlinear projection; CRPS/minimax rate rectangular hull/memory refinement; held-out policy value representation/local refinement; exact frontier and held-out audit

One-pass refinement

one-pass class plus step budget; architecture floor

Why one example is not enough. A theory based only on coarse-graining could be dismissed as information compression. A theory based only on graph CDFs could be dismissed as shape-constrained smoothing. A theory based only on sequential kernels could be dismissed as robust control or memory approximation. A theory based only on one-pass refinement could be dismissed as amortized inference or local optimization. The common lesson is deliberately limited: the same decomposition discipline applies in all four laboratories, while the obstruction mechanism and the repair remain model specific.

Exercises Exercise 4.1. Under squared native loss, prove the displayed coarsening identity for 𝑊 = ℎ(𝑍). Give one coarsening with zero tax and one that merges distinct conditional oracle means and therefore has positive tax. Exercise 4.2. Show that the coordinatewise projection of an arbitrary function onto the set of CDFs is nonexpansive in 𝐿 2 . Which step uses convexity of the CDF set? Exercise 4.3. Construct a two-history shared-kernel example in which both local oracle kernels belong to the architecture locally, but no single shared parameter realizes them simultaneously. Exercise 4.4. For the three-point one-pass laboratory, derive the optimizer (2/3, 0) and the obstruction 1/9. Prove the class-level frontier for general 0 < 𝜂 < 1, verify the 𝑘 = 1, 2, 4 rows of Table 4.1, and explain why adding the quadratic feature changes 𝔄0 whereas increasing 𝑘 changes the deployment computation budget.

Part I Historical and Conceptual Notes

23

Part I Historical and Conceptual Notes Where the four accounts sit in learning theory Classical statistical learning separates approximation and estimation, and computational analyses add optimization error [Bartlett and Mendelson, 2002, Bousquet and Elisseeff, 2002, Bottou and Bousquet, 2008]. Amortized inference then makes a further distinction between the best member of a local variational family and the output delivered by a shared inference mechanism [Cremer et al., 2018, Amos, 2023, Margossian and Blei, 2024]. Part I does not claim priority for these decompositions or for the telescoping identity in Proposition 3.1. The narrower book contribution is to insert a contract-relative reference between local approximation and learned deployment: b 𝑅★ −→ 𝑅oracle −→ inf 𝑅( 𝐴) −→ 𝑅( 𝐴). 𝐴∈𝔄

The first arrow measures whether the declared local family is scientifically adequate. The second asks whether one declared global architecture can realize the already-fixed local oracle family. The last arrow is governed by data and implementation. This split can coincide with an amortization gap in a variational-inference example, but it is not defined by amortization: it also applies to shared policies, finite memories, continuous selectors, compressed carriers, and other deployment contracts. Account

Historical neighbor

Local model gap

approximation or variational- compares the local oracle with the scientific family gap target before global sharing is imposed amortization, approximation under fixes one deployment contract and prices a restricted function class failure to realize the whole oracle field in population-risk units estimation and uniform conver- controls population versus empirical behavgence ior for the declared, possibly selected, class optimization or reachability error compares the fitted rule with the empirical class optimum and is not folded into structural nonrealizability

Architecture gap

Generalization Implementation

Part I distinction

Ownership of the Part I results The exact identity is a book-level accounting synthesis. The certified bound is the standard two-deviation empirical-risk argument specialized to that accounting chain. Their value is diagnostic separation: changing the local family, the shared architecture, the data, or the optimizer changes a different reference comparison. The detailed row-by-row provenance is in Appendix G.1.

From structural realizability to native loss scales Part I separated local solvability, global realizability, and finite-sample certifiability, and introduced a four-component risk decomposition. Part II now derives the loss scale in which structural nonrealizability should be measured: the excess objective generated by the original elimination itself.

Part II

Native Defects and Elimination Calculus Elimination does more than simplify an objective. It generates an objective-native notion of excess loss and determines which transformations preserve the statistical target.

24

Chapter 5

Conjugate Lifts and Native Loss Scales Suppose an inner state is introduced because it makes an objective easier to optimize. After elimination, a trial inner state should be evaluated in the same units as the objective. A Euclidean penalty may be algorithmically convenient, but it is not automatically the insertion price. The canonical residual is D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − inf 𝐻 𝑥 (𝑏). 𝑏

In conjugate-complete lifts this residual is a directed Bregman divergence. This observation fixes the orientation of KL and other asymmetric divergences.

5.1

Legendre conjugacy

Let Φ : A → R ∪ {+∞} be a proper, closed, strictly convex Legendre function. Its conjugate is Φ★ (𝑢) = sup{⟨𝑢, 𝑎⟩ − Φ(𝑎)}. 𝑎

For 𝑢 in the interior of the dual domain, 𝑎★ (𝑢) = ∇Φ★ (𝑢) solves inf {Φ(𝑎) − ⟨𝑢, 𝑎⟩} = −Φ★ (𝑢). 𝑎

The Bregman divergence generated by Φ is [Bregman, 1967] 𝐷 Φ (𝑎∥𝑏) = Φ(𝑎) − Φ(𝑏) − ⟨∇Φ(𝑏), 𝑎 − 𝑏⟩. How to read the assumptions The generator Φ is differentiable and strictly convex on the relevant domain, and the lift is conjugate-complete: every auxiliary occurrence is paired with the full conjugate term Φ∗ . The pointwise oracle 𝑎★ lies in the common Legendre domain. These assumptions fix both the nonnegativity and the orientation of the Bregman divergence. 25

CHAPTER 5. CONJUGATE LIFTS AND NATIVE LOSS SCALES

26

Theorem 5.1 (Conjugate defect identity). Let 𝐻𝑢 (𝑎) = Φ(𝑎) − ⟨𝑢, 𝑎⟩ + 𝑐(𝑢). Then the eliminated objective is 𝐽 (𝑢) = 𝑐(𝑢) − Φ★ (𝑢), and 𝐻𝑢 (𝑎) − 𝐽 (𝑢) = 𝐷 Φ {𝑎∥𝑎★ (𝑢)}. Interpretation and boundary The theorem says that the insertion price is not a chosen regularizer: it is exactly the directed Bregman divergence generated by the eliminated objective. Reversing the arguments generally changes the loss. In the entropy case this distinction fixes the correct direction of KL used by the algorithm. Proof roadmap Use the Fenchel–Young equality at the oracle and subtract the lifted objective evaluated at a trial auxiliary state. Rearranging the conjugate terms produces the three-point definition of 𝐷 Φ (𝑎∥𝑎★). The chapter appendix supplies the full calculation. The order of the arguments is not cosmetic. The trial state appears in the first argument and the oracle in the second. Reversing the divergence generally changes both geometry and performance guarantees.

5.2

Examples of native Bregman defects

5.2.1

Relative entropy

For probability vectors 𝑞 and 𝑝 on the same finite set, take ∑︁ Φ(𝑞) = 𝑞 𝑧 log 𝑞 𝑧 . 𝑧

Then 𝐷 Φ (𝑞∥ 𝑝) = KL(𝑞∥ 𝑝). Classical 𝐼-divergence geometry supplies the projection background for this probability-simplex example [Csiszár, 1975]. A variational lift based on entropy nevertheless generates one particular KL direction. The reverse direction is not certified unless it arises from a different elimination.

5.2.2

LogDet divergence

𝑚 , take For positive-definite matrices 𝑋, 𝑌 ∈ S++

Φ(𝑋) = − log det 𝑋.

5.3. FURTHER CERTIFIED ELIMINATIONS AND BOUNDARIES

27

The corresponding Bregman divergence is 𝐷 Φ (𝑋 ∥𝑌 ) = tr(𝑌 −1 𝑋) − log det(𝑌 −1 𝑋) − 𝑚. This is the Burg or LogDet divergence [Kulis et al., 2009]. Its natural centroid and consensus endpoint differ from Euclidean matrix averaging.

5.2.3

Quantum relative entropy

On the faithful density-matrix domain, the entropy functional Φ(𝜌) = tr(𝜌 log 𝜌) induces Umegaki relative entropy [Umegaki, 1962] 𝐷 (𝜌∥𝜎) = tr 𝜌(log 𝜌 − log 𝜎). For general density matrices, this formula is understood through its lower-semicontinuous extension: 𝐷 (𝜌∥𝜎) < ∞ only when supp(𝜌) ⊆ supp(𝜎), and it is +∞ otherwise. On the faithful domain both logarithms and the Bregman gradient are ordinary finite-dimensional operators. The noncommutative case preserves the defect orientation but changes the validity of base change—the fine-to-coarse passage formalized in Chapter 18—and measurement.

5.3

Further certified eliminations and boundaries

EM as a certified elimination. Let 𝑝 𝜃 (𝑥, 𝑧) be a latent-variable model. For a trial conditional law 𝑞(𝑧) define 𝐻 𝑥 (𝜃, 𝑞) = −E𝑞 log 𝑝 𝜃 (𝑥, 𝑍) − H (𝑞), where H (𝑞) = −E𝑞 log 𝑞(𝑍). The Gibbs variational identity gives − log 𝑝 𝜃 (𝑥) = inf 𝐻 𝑥 (𝜃, 𝑞). 𝑞

Moreover, 𝐻 𝑥 (𝜃, 𝑞) + log 𝑝 𝜃 (𝑥) = KL{𝑞∥ 𝑝 𝜃 (· | 𝑥)}. Thus inserting a shared or approximated posterior pays the forward variational KL from the trial law to the exact posterior. This is the variational identity underlying classical EM and its free-energy interpretation [Dempster et al., 1977, Neal and Hinton, 1998]. Certified decomposition In EM and variational inference, the native insertion certificate is not “posterior distance” in the abstract. It is the KL orientation generated by the Gibbs–Fenchel lift.

CHAPTER 5. CONJUGATE LIFTS AND NATIVE LOSS SCALES

28

Left and right Bregman pooling. For interior points 𝑎 1 , . . . , 𝑎𝑇 with weights 𝑤 𝑡 ≥ 0 satisfying Í 𝑡 𝑤 𝑡 = 1, two pooling problems differ: ∑︁ ∑︁ min 𝑤 𝑡 𝐷 Φ (𝑎 𝑡 ∥𝑎), min 𝑤 𝑡 𝐷 Φ (𝑎∥𝑎 𝑡 ). 𝑎

𝑎

𝑡

𝑡

When the displayed means remain in the relevant Legendre domain, the first produces the primal arithmetic average ∑︁ 𝑎= 𝑤𝑡 𝑎𝑡 , 𝑡

whereas the second produces the dual-coordinate average ∑︁ ∇Φ(𝑎) = 𝑤 𝑡 ∇Φ(𝑎 𝑡 ). 𝑡

For KL on probability vectors, these are respectively arithmetic mixture pooling and normalized geometric pooling; the distinction is standard in Bregman centroid geometry [Banerjee et al., 2005b]. This distinction matters when a shared auxiliary field is designed by averaging. The correct pooling direction depends on the certified defect orientation. Saddle elimination.

Certified residuals also arise from convex–concave elimination. Suppose 𝐽 (𝑥) = inf sup 𝐿 𝑥 (𝑎, 𝑏) 𝑎

𝑏

under strong duality. Let (𝑎★𝑥, 𝑏★𝑥) be a saddle point. A trial pair can be audited by primal and dual insertion gaps, 𝐿 𝑥 (𝑎, 𝑏★𝑥) − 𝐽 (𝑥) ≥ 0, 𝐽 (𝑥) − 𝐿 𝑥 (𝑎★𝑥, 𝑏) ≥ 0. Their sum is a primal–dual certificate. The same orientation discipline applies: the two signs are fixed by the min–max order. Exchange inequalities. The native defect can be exchanged for a more geometric error when the objective has curvature. If 𝜇 D 𝑥 (𝑎) ≥ 𝑑 2 {𝑎, O (𝑥)}, 2 then √︁ 𝑑{𝑎, O (𝑥)} ≤ 2D 𝑥 (𝑎)/𝜇. Conversely, work in Euclidean or Hilbert geometry, with 𝑑 (𝑎, 𝑏) = ∥𝑎 − 𝑏∥. Suppose a nearest oracle 𝑏 ∈ arg min ∥𝑎 − 𝑐∥ 𝑐∈ O ( 𝑥 )

exists, D 𝑥 is differentiable on a neighborhood containing the whole segment [𝑏, 𝑎], ∇D 𝑥 (𝑏) = 0, and ∇D 𝑥 is 𝐿-Lipschitz on that neighborhood. The descent lemma and D 𝑥 (𝑏) = 0 then give D 𝑥 (𝑎) ≤ D 𝑥 (𝑏) + ⟨∇D 𝑥 (𝑏), 𝑎 − 𝑏⟩ +

𝐿 𝐿 ∥𝑎 − 𝑏∥ 2 = 𝑑 2 {𝑎, O (𝑥)}. 2 2

These exchanges are local analytic theorems, not definitions of the defect.

5.3. FURTHER CERTIFIED ELIMINATIONS AND BOUNDARIES When the defect is set-valued.

29

If O (𝑥) has multiple elements, the residual still satisfies D 𝑥 (𝑎) = 0 ⇐⇒ 𝑎 ∈ O (𝑥).

A distance-to-oracle bound may use 𝑑{𝑎, O (𝑥)} =

inf 𝑑 (𝑎, 𝑏),

𝑏∈ O ( 𝑥 )

but a particular labeled representative should not be selected unless the output contract requires it. This is the first place where representation semantics enters.

Exercises Exercise 5.2. Derive the Bregman defect for the Poisson log-partition function and identify its statistical interpretation. Exercise 5.3. For KL divergence, compute the left and right centroids of two Bernoulli distributions. Compare their limiting behavior near the boundary of the simplex. Exercise 5.4. Show that strong convexity gives a defect-to-distance exchange, but not a distance-totask-regret exchange without an additional task theorem.

CHAPTER 5. CONJUGATE LIFTS AND NATIVE LOSS SCALES

30

Appendix to Chapter 5: complete proofs Proof of the conjugate defect identity Dependencies. Fenchel conjugacy and the Legendre identity ∇Φ(𝑎★ (𝑢)) = 𝑢 are the only ingredients. Strict convexity is used for uniqueness of the oracle, not for the algebraic equality itself. Proof. By definition of the convex conjugate, Φ★ (𝑢) = ⟨𝑢, 𝑎★ (𝑢)⟩ − Φ{𝑎★ (𝑢)}, where 𝑎★ (𝑢) = ∇Φ★ (𝑢) and 𝑢 = ∇Φ{𝑎★ (𝑢)}. Therefore 𝐽 (𝑢) = inf {Φ(𝑎) − ⟨𝑢, 𝑎⟩ + 𝑐(𝑢)} 𝑎

= 𝑐(𝑢) − Φ★ (𝑢). For an arbitrary trial state 𝑎, 𝐻𝑢 (𝑎) − 𝐽 (𝑢) = Φ(𝑎) − ⟨𝑢, 𝑎⟩ + Φ★ (𝑢) = Φ(𝑎) − Φ(𝑎★) − ⟨𝑢, 𝑎 − 𝑎★⟩ = Φ(𝑎) − Φ(𝑎★) − ∇Φ(𝑎★), 𝑎 − 𝑎★ = 𝐷 Φ (𝑎∥𝑎★). Nonnegativity and equality only at 𝑎 = 𝑎★ follow from strict convexity of Φ.

□

Chapter 6

Elimination Towers and the P/G/X/V/C Calculus The P/G/X/V/C calculus and an elimination tower describe orthogonal directions. The five letters classify what happens within one elimination level after its local oracle has been exposed, whereas a tower records how states and defects pass between levels. The mode says what was changed; the tower says where its cost propagates.

6.1

Five distinct operations on an elimination system

The calculus begins from one noncommutation: Eliminate ◦ Couple ≠ Couple ◦ Eliminate . The decisive question is not which numerical smoother is used, but where the coupling enters the declared contract. Suppose 𝐽 (Θ) = inf 𝐴 𝐻0 (Θ, 𝐴) and 𝐴★ (Θ) is the pointwise oracle. Across an index set T of locations, tasks, or replicas, the oracles form a field 𝐴★ = ( 𝐴★ 𝑡 ) 𝑡 ∈ T . A practitioner may wish to smooth, pool, share, or couple that field. The common contract fixes the original lift 𝐻0 , its eliminated target 𝐽, and the oracle field 𝐴★. It may also declare an external field operator 𝑆, a coherence penalty 𝑅, or marginal laws (𝜇𝑡 ) whose couplings form Γ(𝜇𝑡 : 𝑡 ∈ T ). A P/G/X/V/C label records which of these objects is acted on and what remains invariant. It does not name a numerical algorithm: the same smoother can implement different operations under different contracts.

6.1.1

P: pointwise elimination

The baseline is 𝐴𝑡 = 𝐴★ 𝑡 (Θ). No cross-index coherence is imposed. The target remains exactly 𝐽 and every local defect is zero. P is the reference contract against which the other four operations are measured. 31

CHAPTER 6. ELIMINATION TOWERS AND THE P/G/X/V/C CALCULUS

32

6.1.2

G: plug-in globalization

An external operator 𝑆 transforms the oracle field: e = 𝑆 𝐴★ . 𝐴 The field is inserted back into the original lift. The target value becomes e = 𝐽 (Θ) + DΘ ( 𝐴). e 𝐻0 (Θ, 𝐴) The insertion defect is exact. G does not optimize a new joint objective; it deploys an externally constructed field. Thus G changes the deployed field while holding 𝐻0 and 𝐽 fixed.

6.1.3

X: exactification

If a frozen G step at anchor Θ̄ gives 𝑄 𝐺 (Θ) = 𝐽 (Θ) + 𝑑 (Θ), X forms 𝑄 𝑋(𝑘 ) (Θ | Θ̄) = 𝑄 𝐺 (Θ) − 𝑇𝑘 𝑑 (Θ; Θ̄),

𝑗Θ̄𝑘 𝑄 𝑋(𝑘 ) = 𝑗Θ̄𝑘 𝐽.

X changes the frozen surrogate, not the field or the joint law. Chapter 7 develops the construction and its converse.

6.1.4

V: variational coupling

V places the field and a coherence penalty inside a new objective: 𝐽𝜆 (Θ) = inf {𝐻0 (Θ, 𝐴) + 𝜆𝑅( 𝐴)}. 𝐴

The statistical or optimization target changes. Block monotonicity can be exact, but the estimator need not target the original 𝐽. The returned estimand is 𝐽𝜆 , not merely 𝐽 evaluated at an external field.

6.1.5

C: fixed-marginal coupling

C keeps every coordinate procedure and marginal law fixed, choosing only 𝛾 ∈ Γ(𝜇𝑡 : 𝑡 ∈ T ). It is relevant to common random numbers, antithetic couplings, ensemble dependence, and variance reduction. Additive coordinate expectations cannot improve under a fixed-marginal coupling; only genuinely joint functionals can. Table 6.1: The five operational contracts. Mode P G X V C

Acts on local oracle choice deployed field frozen surrogate joint objective joint law

Held fixed

Returned object

Certified consequence

original lift and target

𝐴★

zero local defect

𝐻0 and 𝐽 original target 𝐽 declared feasible class every marginal 𝜇𝑡

𝑆 𝐴★ 𝑄 𝐺 − 𝑇𝑘 𝑑 optimizer of 𝐻0 + 𝜆𝑅 𝛾 ∈ Γ(𝜇𝑡 )

exact insertion defect target jet restored target generally changes additive means fixed

6.2. A COMPACT CLASSIFICATION TEST

6.2

33

A compact classification test

Classify a procedure from the provenance of its returned object, not from the last numerical array it produces. 1. If it stops after independent pointwise elimination, label it P. 2. If it transforms the oracle field outside the original objective and then evaluates that field in 𝐻0 , label it G. 3. If it subtracts a defect jet to restore target contact at an anchor, label that correction X. 4. If it moves coherence inside an objective and reoptimizes jointly, label it V. 5. If it fixes all marginals and changes only their joint coupling, label it C. A workflow may answer more than one question affirmatively. Record the sequence—for example, G → X—because the last step does not erase the contract of the earlier one. Boundary of the claim Numerical equality is weaker than operational equality. G and V can return the same field in a special quadratic problem while targeting different objects; X repairs a target jet without turning G into V; and C cannot remove an insertion defect or alter an additive coordinate expectation. A change of architecture, output semantics, or resource grammar is a separate contract change and must not be hidden inside one of the five letters.

6.3

Horizontal operations and multistage towers

If a stage uses one or more horizontal modes, its certified defect contribution enters the vertical decomposition; the stage costs then compose by infimal convolution. A multistage elimination tower. Let a fine state 𝑒 map to an intermediate state 𝑏 and then to a coarse state 𝑐: 𝜋 𝜎 E− → B −→ C. Suppose the defect splits at each stage, D E (𝑒) = D B (𝜋𝑒) + 𝛿 𝜋 (𝑒), D B (𝑏) = D C (𝜎𝑏) + 𝛿 𝜎 (𝑏).

Then D E (𝑒) = D C (𝜎𝜋𝑒) + 𝛿 𝜎 (𝜋𝑒) + 𝛿 𝜋 (𝑒).

After minimizing over architecture-constrained fibers, the realization costs compose by infimal convolution: Ψ 𝜎 𝜋 (𝑐) = inf {𝛿 𝜎 (𝑏) + Ψ 𝜋 (𝑏)}. 𝑏:𝜎𝑏=𝑐

This is the usual Bellman/min-plus composition law [Bellman, 1957, Baccelli et al., 1992].

CHAPTER 6. ELIMINATION TOWERS AND THE P/G/X/V/C CALCULUS

34

How to read the assumptions The successive coarse and fine states must be compatible, and infeasible transitions are encoded by the value +∞. All intermediate infima are taken over the same declared fibers. No minimizer need be unique; the statement concerns values of infimal composition. Proposition 6.1 (Tower associativity). For compatible consecutive state spaces, define the typed stage-cost composition by (𝑐 1 □ 𝑐 2 ) (𝑥 2 , 𝑥0 ) :=

inf

𝑥1 compatible

{𝑐 1 (𝑥1 , 𝑥0 ) + 𝑐 2 (𝑥2 , 𝑥1 )}.

Whenever the relevant feasible sets are nonempty, the following identity holds pointwise for every compatible endpoint pair (𝑥 3 , 𝑥0 ): {(𝑐 1 □ 𝑐 2 ) □ 𝑐 3 }(𝑥 3 , 𝑥0 ) = {𝑐 1 □ (𝑐 2 □ 𝑐 3 )}(𝑥 3 , 𝑥0 ). Interpretation and boundary A tower can be bracketed in any order because elimination is min-plus composition. This justifies dynamic-programming and multistage obstruction decompositions. It does not imply that two numerical algorithms have the same runtime, conditioning, or finite-precision behavior. Proof roadmap Write both bracketings as a nested infimum over all intermediate states, then exchange the order of the finite or extended-value infima. The chapter appendix gives the explicit Bellman/min-plus calculation. Why the tower matters. The tower converts a vague statement such as “the representation loses information” into a decomposition with identifiable stages. • A coarse representation can choose the wrong coarse object. • Even the correct coarse object may be expensive to realize in the fine architecture. • A deployed algorithm may then incur additional implementation error inside the chosen fine class. The same structure appears in COT under conditional-law base change, in resource-constrained architectures under carrier–decoder decomposition, and in certificate statistics under evidence compression.

6.4. WORKED DECOMPOSITIONS AND COUPLING BOUNDARIES

6.4

35

Worked decompositions and coupling boundaries Suppose C ⊆ A is convex and the Bregman projection in the

Bregman Pythagorean identities. first argument exists:

𝑎 C ∈ arg min 𝐷 Φ (𝑐∥𝑎★). 𝑐∈ C

Thus the projected point varies in the first argument while 𝑎★ remains the second argument. Under the standard differentiability and convex projection conditions [Bauschke and Borwein, 1997], every comparison point 𝑎 ∈ C satisfies 𝐷 Φ (𝑎∥𝑎★) ≥ 𝐷 Φ (𝑎∥𝑎 C ) + 𝐷 Φ (𝑎 C ∥𝑎★). This gives a three-level interpretation: 1. irreducible constraint defect; 2. within-constraint implementation defect; 3. possible nonorthogonality remainder if exact Pythagoras fails. A worked Gaussian field.

Let 𝑎★𝑡 be local Gaussian means and let 𝐻𝑡 (𝑎 𝑡 ) =

1 (𝑎 𝑡 − 𝑎★𝑡) 2 . 2

On a graph with Laplacian 𝐿, V solves 1 𝜆 min ∥𝑎 − 𝑎★ ∥ 2 + 𝑎 ⊤ 𝐿𝑎. 𝑎 2 2 The solution is

𝑎 V = (𝐼 + 𝜆𝐿) −1 𝑎★ .

If G uses the same filter externally, the deployed field coincides numerically with 𝑎 V in this quadratic unconstrained example. Nevertheless the interpretation differs: G evaluates the original lift at an external field; V defines the field as the optimizer of a changed objective. Under constraints or nonlinear repair, the numerical equality generally disappears. C coupling and its boundary. Let 𝑌1 , . . . , 𝑌𝑇 have fixed marginals. Whenever every ℓ𝑡 (𝑌𝑡 ) is integrable, any additive target satisfies ∑︁ ∑︁ E ℓ𝑡 (𝑌𝑡 ) = Eℓ𝑡 (𝑌𝑡 ), 𝑡

𝑡

changing the joint coupling cannot change the expectation. A benefit requires a joint functional, such as variance of an average, maximum loss, simultaneous coverage, or a path-dependent objective. Boundary of the claim C is not a hidden way to improve every coordinatewise estimator. Fixed marginals protect all additive coordinate expectations.

36

CHAPTER 6. ELIMINATION TOWERS AND THE P/G/X/V/C CALCULUS

Exercises Exercise 6.2. For the quadratic graph example, compute the exact G insertion defect and V objective improvement. Explain why they are numerically related but conceptually distinct. Exercise 6.3. Prove tower associativity for finite state spaces using min-plus matrix multiplication. Exercise 6.4. Give a joint functional for which fixed-marginal antithetic coupling improves performance, and an additive functional for which it cannot.

Appendix to Chapter 6: complete proofs

37

Appendix to Chapter 6: complete proofs Proof of tower associativity Dependencies. Only the compatibility of intermediate states and the convention that the infimum of an empty feasible set is +∞ are used. Proof. Let 𝑥 0 , 𝑥1 , 𝑥2 , 𝑥3 denote compatible states at four consecutive levels, and let the three stage costs be 𝑐 1 (𝑥1 , 𝑥0 ), 𝑐 2 (𝑥2 , 𝑥1 ), and 𝑐 3 (𝑥3 , 𝑥2 ). For fixed endpoints (𝑥3 , 𝑥0 ),   {(𝑐 1 □ 𝑐 2 ) □ 𝑐 3 }(𝑥 3 , 𝑥0 ) = inf inf {𝑐 1 (𝑥1 , 𝑥0 ) + 𝑐 2 (𝑥2 , 𝑥1 )} + 𝑐 3 (𝑥3 , 𝑥2 ) 𝑥2

𝑥1

= inf {𝑐 1 (𝑥1 , 𝑥0 ) + 𝑐 2 (𝑥2 , 𝑥1 ) + 𝑐 3 (𝑥3 , 𝑥2 )}. 𝑥1 , 𝑥2

Similarly, 



{𝑐 1 □ (𝑐 2 □ 𝑐 3 )}(𝑥 3 , 𝑥0 ) = inf 𝑐 1 (𝑥1 , 𝑥0 ) + inf {𝑐 2 (𝑥2 , 𝑥1 ) + 𝑐 3 (𝑥3 , 𝑥2 )} 𝑥1

𝑥2

= inf {𝑐 1 (𝑥1 , 𝑥0 ) + 𝑐 2 (𝑥2 , 𝑥1 ) + 𝑐 3 (𝑥3 , 𝑥2 )}. 𝑥1 , 𝑥2

The two expressions are identical. The same argument applies to extended-valued realization costs and proves the stated tower law. □

Chapter 7

Exactification and Optimization Certificates e The field may have been Fix an anchor Θ̄ and freeze a feasible, possibly nonlocal, auxiliary field 𝐴. constructed by a globalization step at the anchor, but it is held fixed as the outer parameter varies. Define e = 𝐽 (Θ) + 𝑑 (Θ), 𝑄 𝐺 (Θ) = 𝐻0 (Θ, 𝐴) where e = 𝐻0 (Θ, 𝐴) e − 𝐽 (Θ) ≥ 0. 𝑑 (Θ) = DΘ ( 𝐴) Even if the frozen lift is smooth in Θ, 𝑄 𝐺 need not touch 𝐽 at the current iterate. Minimizing 𝑄 𝐺 can therefore optimize the defect rather than the target. Exactification repairs this problem by subtracting the local defect jet. In the P/G/X/V/C calculus of Chapter 6, the frozen construction is G and the jet correction is X; no V objective or C coupling is introduced. In this chapter, an optimization certificate combines target-jet contact with explicit inner-oracle, defect-score, and step-acceptance error budgets; exactification supplies the target-contact component.

7.1

Jet exactification

Let 𝑇𝑘 𝑑 (Θ; Θ̄) be the order-𝑘 Taylor polynomial of the frozen defect 𝑑 at Θ̄. Define 𝑄 𝑋(𝑘 ) (Θ | Θ̄) = 𝐽 (Θ) + 𝑑 (Θ) − 𝑇𝑘 𝑑 (Θ; Θ̄). Then the 𝑘-jet of 𝑄 𝑋(𝑘 ) agrees with that of 𝐽 at the anchor. How to read the assumptions The order 𝑘 is a nonnegative integer, and 𝐽 and the frozen defect 𝑑 are 𝐶 𝑘 on a neighborhood of Θ̄ in one common local coordinate chart. The polynomial 𝑇𝑘 𝑑 is their ordinary order-𝑘 Taylor e fixed; no global convexity or descent is claimed. polynomial. Taylor expansion is in Θ with 𝐴 A function is called 𝑘-flat at Θ̄ when all of its derivatives through order 𝑘 vanish there. 38

7.1. JET EXACTIFICATION

39

Theorem 7.1 (Jet exactification). If both 𝐽 and 𝑑 are 𝐶 𝑘 on a neighborhood of Θ̄, then 𝑗Θ̄𝑘 𝑄 𝑋(𝑘 ) = 𝑗Θ̄𝑘 𝐽. In particular, for 𝑘 = 1, 𝑄 𝑋 ( Θ̄ | Θ̄) = 𝐽 ( Θ̄),

∇𝑄 𝑋 ( Θ̄ | Θ̄) = ∇𝐽 ( Θ̄).

Interpretation and boundary Subtracting the low-order defect jet makes the surrogate agree with the original target in value and derivatives through order 𝑘, while retaining higher-order auxiliary geometry. Exactification therefore repairs target tangency; it does not by itself guarantee a useful step size or global convergence. Proof roadmap Express the frozen surrogate as 𝐽 + 𝑑, subtract the order-𝑘 Taylor polynomial of 𝑑, and compare derivatives at the anchor. Every derivative of 𝑑 − 𝑇𝑘 𝑑 through order 𝑘 vanishes. The chapter appendix records the complete jet calculation. The result is elementary once 𝑑 is known. The substantive point is that 𝑑 is the exact native insertion defect, not an arbitrary correction. The construction and converse below are the exactification normal form of the companion framework; their relation to first-order surrogate functions is discussed there as well [Huang and Wang, 2026g, Mairal, 2013]. Candidate-wise descent identity.

For first-order exactification,

𝑄 𝑋 (Θ | Θ̄) = 𝐽 (Θ) + 𝑑 (Θ) − 𝑑 ( Θ̄) − ⟨∇𝑑 ( Θ̄), Θ − Θ̄⟩. If a candidate Θ+ satisfies 𝑄 𝑋 (Θ+ | Θ̄) ≤ 𝑄 𝑋 ( Θ̄ | Θ̄) − 𝛾∥Θ+ − Θ̄∥ 2 , then 𝐽 (Θ+ ) − 𝐽 ( Θ̄) = 𝑄 𝑋 (Θ+ | Θ̄) − 𝑄 𝑋 ( Θ̄ | Θ̄) h i − 𝑑 (Θ+ ) − 𝑑 ( Θ̄) − ⟨∇𝑑 ( Θ̄), Θ+ − Θ̄⟩ . If the defect gradient is locally 𝐿 𝑑 -Lipschitz, the bracket is bounded below by −𝐿 𝑑 ∥Θ+ − Θ̄∥ 2 /2. Hence 𝐽 (Θ+ ) − 𝐽 ( Θ̄) ≤ −(𝛾 − 𝐿 𝑑 /2) ∥Θ+ − Θ̄∥ 2 . A trust region or backtracking line search enforces 𝛾 > 𝐿 𝑑 /2.

CHAPTER 7. EXACTIFICATION AND OPTIMIZATION CERTIFICATES

40

7.2

The converse normal form

The forward construction subtracts the Taylor jet of the frozen defect. The converse asks whether any scalar correction can preserve the target 𝑘-jet without doing the same. The hypothesis box fixes the local comparison under which the theorem gives the answer. How to read the assumptions A candidate scalar correction 𝑐 − 𝑆 is compared with the actual frozen defect 𝑑 in a neighborhood of the anchor, and all objects possess the required 𝑘-jets. The conclusion is modulo a 𝑘-flat remainder; it does not identify a unique global surrogate. Theorem 7.2 (Converse exactification normal form). Suppose 𝐽, 𝑑, 𝑐, 𝑆 are 𝐶 𝑘 on a neighborhood of Θ̄ and a surrogate can be written as 𝑄(Θ) = 𝐽 (Θ) + 𝑑 (Θ) − 𝑐(Θ) + 𝑆(Θ). Then 𝑗Θ̄𝑘 𝑄 = 𝑗Θ̄𝑘 𝐽 if and only if 𝑗Θ̄𝑘 {𝑐 − 𝑆} = 𝑗Θ̄𝑘 𝑑. Thus any target-𝑘-jet-preserving correction subtracts the defect jet, modulo a 𝑘-flat term. Equivalently, the canonical representative is (𝑘 ) e − 𝑇𝑘 𝑑 (Θ; Θ̄), 𝑄 𝑋,can (Θ | Θ̄) = 𝐻0 (Θ, 𝐴) (𝑘 ) and every other target-𝑘-jet-preserving surrogate of the displayed form is 𝑄 𝑋,can + 𝑅, where 𝑅 is 𝑘-flat at Θ̄. Conversely, every such 𝑘-flat perturbation preserves the target 𝑘-jet.

Interpretation and boundary The defect-jet correction is not merely one convenient construction. Any scalar surrogate that preserves the target jet through order 𝑘 must subtract the same defect jet, up to a term whose derivatives through order 𝑘 vanish. This is a local normal form, not a statement about global optimizer equivalence. Proof roadmap Write the candidate surrogate as 𝑄 = 𝐽 + 𝑑 − 𝑐 + 𝑆 and take its 𝑘-jet at the anchor. Equality with the target jet is equivalent to equality of the 𝑘-jets of 𝑐 − 𝑆 and 𝑑. The chapter appendix makes both implications explicit. The converse gives exactification a structural status: first-order target preservation forces a defect-score correction, not just any tangent surrogate.

7.3. APPROXIMATION, TRANSMISSION, AND CERTIFICATION ERROR

7.3

41

Approximation, transmission, and certification error

Approximate inner oracles. In practice, the auxiliary oracle and its defect gradient are computed approximately. Suppose 𝐻0 (Θ, ·) is differentiable and 𝜇-strongly convex, its minimizer 𝑎★ is interior and satisfies ∇𝑎 𝐻0 (Θ, 𝑎★) = 0, and the outer score ∇Θ 𝐻0 (Θ, 𝑎) is 𝐿-Lipschitz in 𝑎. Then an inner gradient residual ∥∇𝑎 𝐻0 (Θ, e 𝑎) ∥ ≤ 𝜖 implies ∥e 𝑎 − 𝑎★ ∥ ≤ 𝜖/𝜇, 0 ≤ 𝐻0 (Θ, e 𝑎 ) − 𝐽 (Θ) ≤ 𝜖 2 /(2𝜇), and, under the usual envelope/Danskin regularity giving ∇𝐽 (Θ) = ∇Θ 𝐻0 (Θ, 𝑎★), ∥∇Θ 𝐻0 (Θ, e 𝑎 ) − ∇𝐽 (Θ) ∥ ≤ 𝐿𝜖/𝜇. These bounds turn numerical stopping tolerances into objective and gradient certificates. Summable certification error. Let 𝑄 𝑘 be an approximate exactified surrogate at anchor Θ 𝑘 and write 𝑟 𝑘 = 𝑄 𝑘 − 𝐽. Suppose an accepted candidate satisfies 𝑄 𝑘 (Θ 𝑘+1 ) ≤ 𝑄 𝑘 (Θ 𝑘 ) − Δ 𝑘 ,

𝑟 𝑘 (Θ 𝑘+1 ) − 𝑟 𝑘 (Θ 𝑘 ) ≥ −𝜂 𝑘 Δ 𝑘 − 𝜀 𝑘 ,

where Δ 𝑘 ≥ 0, 0 ≤ 𝜂 𝑘 < 1, and 𝜀 𝑘 ≥ 0. If 𝐽 is bounded below, sup 𝑘 𝜂 𝑘 < 1, Δ 𝑘 ≥ 𝑐∥Θ 𝑘+1 − Θ 𝑘 ∥ 2 then

Í

Í

𝑘 𝜀 𝑘 < ∞, and

for some 𝑐 > 0,

𝑘 Δ 𝑘 < ∞ and

∑︁

∥Θ 𝑘+1 − Θ 𝑘 ∥ 2 < ∞.

𝑘

For the stationary-point conclusion, additionally assume ∥∇𝑟 𝑘 (Θ 𝑘 ) ∥ ≤ 𝛿 𝑘 → 0,

∥∇𝑄 𝑘 (Θ 𝑘+1 ) ∥ ≤ 𝜁 𝑘 → 0,

and that ∇𝑄 𝑘 is 𝐾-Lipschitz on the segment from Θ 𝑘 to Θ 𝑘+1 , uniformly in 𝑘. Then ∥∇𝐽 (Θ 𝑘 ) ∥ ≤ 𝐾 ∥Θ 𝑘+1 − Θ 𝑘 ∥ + 𝜁 𝑘 + 𝛿 𝑘 −→ 0. If ∇𝐽 is continuous, every cluster point is stationary for 𝐽. Compactness or level boundedness is needed only to guarantee that cluster points exist. Boundary of the claim Stationarity is a target-level conclusion only because exactification restores the target score. A generic majorization or smoothed surrogate may converge to a stationary point of a different objective.

CHAPTER 7. EXACTIFICATION AND OPTIMIZATION CERTIFICATES

42

Conjugate transmission.

For the conjugate lift 𝐻0 (Θ, 𝐴) = 𝑐(Θ) + Φ( 𝐴) − ⟨𝐴, 𝜂(Θ)⟩,

(𝑘 ) (𝑘 ) let 𝑄 𝑋, 𝐴 and 𝑄 𝑋,𝐵 be the canonical exactifications obtained by freezing 𝐴 and 𝐵. Direct cancellation gives the transmission identity (𝑘 ) (𝑘 ) 𝑄 𝑋, 𝐴 − 𝑄 𝑋,𝐵 = −⟨𝐴 − 𝐵, 𝜂 − 𝑇𝑘 𝜂⟩.

Thus, for 𝑘 = 1, their Hessians at the anchor differ by −⟨𝐴 − 𝐵, 𝐷 2 𝜂( Θ̄)⟩. If the natural-parameter map 𝜂 is affine, the Taylor tail vanishes and no nonlinear curvature is transmitted. This is the conjugate-transmission calculation in the companion framework [Huang and Wang, 2026g].

7.4

Practical algorithm

The preceding error bounds become an implementable outer loop only when the acceptance test keeps the inner-oracle, defect-score, and outer-solver budgets separate. The algorithm records that interface. Algorithm 1 First-order exactification with certified backtracking 1: Input anchor Θ 𝑘 and predeclared inner-oracle, defect-score, and outer-solver tolerances.

e𝑘 ; certify the value and score ( 𝑑e𝑘 , e 2: Compute and freeze an approximate globalized field 𝐴 𝑔 𝑘 ) of

e𝑘 ) − 𝐽 (Θ) at Θ 𝑘 . 𝑑 𝑘 (Θ) = 𝐻0 (Θ, 𝐴 e𝑘 (Θ) = 𝐻0 (Θ, 𝐴 e𝑘 ) − 𝑑e𝑘 − ⟨e 3: Form 𝑄 𝑔 𝑘 , Θ − Θ 𝑘 ⟩. e𝑘 in a trust region. 4: Propose Θ 𝑘+1 by approximately minimizing 𝑄 5: Accept when the certified surrogate decrease exceeds the defect-remainder and inner-solver budgets. 6: Otherwise shrink the trust region and repeat. The algorithm separates three tolerances: inner-oracle error, defect-score error, and outer minimization error. Collapsing them into one generic tolerance obscures the proof.

Exercises ¯ Exercise 7.3. For 𝑑 (𝜃) = 𝜃 4 , write the first- and second-order exactified surrogates at an anchor 𝜃. Verify jet matching directly. Exercise 7.4. Under 𝜇-strong convexity, prove the value bound 𝐻 (e 𝑎 ) − 𝐻 (𝑎★) ≤ ∥∇𝐻 (e 𝑎 ) ∥ 2 /(2𝜇). Exercise 7.5. Construct a surrogate that is value-touching but not score-touching. Show that its stationary points need not be stationary for the target.

Appendix to Chapter 7: complete proofs

43

Appendix to Chapter 7: complete proofs Proof of jet exactification Dependencies. The frozen defect 𝑑 and target 𝐽 are 𝐶 𝑘 on a neighborhood of the anchor. The proof then uses only the defining derivative identities of the order-𝑘 Taylor polynomial; no convexity, majorization, or optimization assumption is required. Proof. Let 𝑅 𝑘 (Θ) = 𝑑 (Θ) − 𝑇𝑘 𝑑 (Θ; Θ̄). By the defining property of the Taylor polynomial, for every multi-index 𝛼 with |𝛼| ≤ 𝑘, 𝜕 𝛼 𝑅 𝑘 ( Θ̄) = 0. Since

𝑄 𝑋(𝑘 ) (Θ | Θ̄) = 𝐽 (Θ) + 𝑅 𝑘 (Θ),

all derivatives of 𝑄 𝑋(𝑘 ) − 𝐽 through order 𝑘 vanish at Θ̄. Hence 𝑗Θ̄𝑘 𝑄 𝑋(𝑘 ) = 𝑗 Θ̄𝑘 𝐽. For 𝑘 = 1, the zerothand first-order identities are precisely 𝑄 𝑋 ( Θ̄ | Θ̄) = 𝐽 ( Θ̄),

∇𝑄 𝑋 ( Θ̄ | Θ̄) = ∇𝐽 ( Θ̄). □

Proof of the converse exactification normal form Dependencies. All functions are 𝐶 𝑘 on a neighborhood of the anchor. The conclusion is local: it identifies the correction jet and a canonical representative, uniquely only modulo a 𝑘-flat remainder. Proof. From 𝑄−𝐽 = 𝑑−𝑐+𝑆 we obtain 𝑗Θ̄𝑘 𝑄 = 𝑗Θ̄𝑘 𝐽

⇐⇒

𝑗Θ̄𝑘 (𝑑 − 𝑐 + 𝑆) = 0.

Linearity of the jet operator gives 𝑗Θ̄𝑘 (𝑑 − 𝑐 + 𝑆) = 0

⇐⇒

𝑗Θ̄𝑘 (𝑐 − 𝑆) = 𝑗 Θ̄𝑘 𝑑.

This proves both directions. Because 𝑇𝑘 𝑑 has the same 𝑘-jet as 𝑑, the difference e − 𝑇𝑘 𝑑 (Θ; Θ̄)} = −𝑐 + 𝑆 + 𝑇𝑘 𝑑 𝑄 − {𝐻0 (Θ, 𝐴) is 𝑘-flat whenever the target-jet condition holds. Conversely, adding any 𝑘-flat remainder to the canonical representative leaves the target 𝑘-jet unchanged. This proves the claimed local uniqueness modulo 𝑘-flat terms. □

Chapter 8

Probability Validity, Graph Smoothing, and CRPS The native statistical loss is the continuous ranked probability score (CRPS), whose expected excess equals squared 𝐿 2 (𝐼) distance between the forecast and true CDFs in the compact-support setup declared below. Let 𝐺 = (𝑉, 𝐸) be a graph with 𝑇 = |𝑉 | and Laplacian 𝐿. Vertex 𝑡 has distribution function 𝐹𝑡★ and observations 𝑌𝑡1 , . . . , 𝑌𝑡 𝑛𝑡 . Throughout this chapter’s projection and risk statements, outcomes are supported on a declared compact interval 𝐼 = [𝑎, 𝑏]. Put H = 𝐿 2 (𝐼),

Ccdf = {𝐹 ∈ H : 0 ≤ 𝐹 ≤ 1 and 𝐹 is nondecreasing a.e.}.

Choose right-continuous representatives with 𝐹 (𝑏) = 1, extended by zero to the left of 𝑎 and by one to the right of 𝑏. Then Ccdf is the nonempty closed convex set of CDF restrictions in H. The empirical CDF restriction is 𝑛𝑡 ∑︁ b𝑡 (𝑦) = 1 𝐹 1{𝑌𝑡𝑖 ≤ 𝑦}. 𝑛𝑡 𝑖=1 b(𝑦) ∈ R𝑇 . Stack the CDFs as a vector-valued function 𝐹 For a self-adjoint graph filter 𝑆, define the raw estimator e(𝑦) = 𝑆 𝐹 b(𝑦). 𝐹 A higher-order Richardson filter can cancel low-order bias but may use signed weights. The raw coordinates may then fail monotonicity or leave [0, 1].

44

8.1. THE CDF SET AND PROJECTION REPAIR

8.1

45

The CDF set and projection repair

𝑇 ⊂ H𝑇 , and let 𝚷 Let C = Ccdf cdf be its coordinatewise metric projection. Define

e = Π C ( 𝐹). e 𝐹 = 𝚷 cdf ( 𝐹) How to read the assumptions The metric projection is taken in H𝑇 = 𝐿 2 (𝐼) 𝑇 , the population target belongs to C, and the graph-roughness conclusion uses the same nonexpansive coordinate projection Πcdf at every vertex. On an unbounded outcome space, one must instead formulate the argument in an affine 𝐿 2 -difference class under suitable moment conditions; no 𝐿 2 (R) membership of each individual CDF is asserted here. Proposition 8.1 (Validity repair). For every true CDF field 𝐹 ★ ∈ C, e − 𝐹★ ∥ 2 . ∥𝐹 − 𝐹 ★ ∥ 2 ≤ ∥ 𝐹 For the coordinatewise product projection above, graph roughness also satisfies e 𝐿 𝐹⟩. e ⟨𝐹, 𝐿𝐹⟩ ≤ ⟨𝐹, Interpretation and boundary Projection can repair an invalid signed high-order proposal without increasing squared CDF error, hence without increasing CRPS risk. Under the graph product geometry it also cannot increase edge roughness. The projected G estimator is not automatically the constrained V optimizer in a different metric. Proof roadmap Apply the Hilbert projection variational inequality with the true CDF field as a feasible comparison. The Pythagorean inequality yields risk contraction; coordinatewise nonexpansiveness yields edgewise roughness contraction. Full details appear in the chapter appendix. In the P/G/X/V/C calculus of Chapter 6, the repaired field is a G construction: an ambient proposal followed by a validity interface. It is not generally equal to the V solution obtained by minimizing a penalized objective directly over C.

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46

8.2

CRPS risk and its observable decomposition

CRPS as squared-CDF risk. For a forecast CDF 𝐹 and observation 𝑌 , the usual real-line CRPS integral reduces, under the support convention above, to ∫ CRPS(𝐹, 𝑌 ) = {𝐹 (𝑦) − 1(𝑌 ≤ 𝑦)}2 d𝑦. 𝐼

Its expected excess risk is ∫ {𝐹 (𝑦) − 𝐹 ★ (𝑦)}2 d𝑦.

E CRPS(𝐹, 𝑌 ) − E CRPS(𝐹 ★, 𝑌 ) = 𝐼

The continuous-distribution scoring-rule lineage and the strict-propriety framework are classical [Matheson and Winkler, 1976, Gneiting and Raftery, 2007]. What is specific here is the probabilityvalid graph repair and its matching rate analysis. The displayed identity implies that the Hilbert projection repair cannot increase excess CRPS. Exact bias–variance decomposition. Assume that observations are i.i.d. within each vertex and that the samples are independent across vertices. Let 𝐵 = (𝑆 − 𝐼)𝐹 ★ be the smoothing bias. Then e − 𝐹 ★ = 𝐵 + 𝑆( 𝐹 b − 𝐹 ★). 𝐹 The cross term vanishes after expectation, giving e − 𝐹 ★ ∥ 2 = ∥𝐵∥ 2 + E∥ 𝐹

𝑇 ∑︁

(𝑆 2 )𝑡𝑡

𝜅𝑡 , 𝑛𝑡

𝑡=1

where

∫ 𝐹𝑡★ (𝑦){1 − 𝐹𝑡★ (𝑦)} d𝑦.

𝜅𝑡 = 𝐼

The variance complexity is therefore controlled by tr(𝑆 2 ), not merely by tr(𝑆). Certified decomposition The graph effective dimension for squared-CDF risk is tr(𝑆 2 ) because the observation noise is filtered twice in the second moment. An observable raw-risk estimate. For a fixed data-independent filter, pairwise empirical-CDF identities yield an unbiased estimate of the raw risk after correcting the diagonal self-influence. The correction involves 2𝑆𝑡𝑡 − 1. The nonlinear projection then provides the pathwise Pythagorean bound e − 𝐹 ∥2 ≤ ∥ 𝐹 e − 𝐹★ ∥ 2 . ∥𝐹 − 𝐹 ★ ∥ 2 + ∥ 𝐹 e − 𝐹 ∥ 2 is an observable guaranteed deduction from the raw squared error. The raw estimator Thus ∥ 𝐹 can therefore be tuned using an exact risk decomposition while the final projection restores validity.

8.3. THE PROBABILITY-VALIDITY BARRIER

8.3

47

The probability-validity barrier

Projection repairs a raw forecast after it has left the CDF set. A sharper question is when the ambient linear filter preserves probability validity without repair. The source box fixes the provenance and scope of that classification and of the path-rate result used in the next section. Source result; proof not reproduced here The graph-universal Markov-validity classification is established in Elimination Geometry II, whereas the sharp path minimax theorem is established in Elimination Geometry I and audited in Elimination Geometry II [Huang and Wang, 2026g,f]. The book records their statements and operative scope but does not count them among its principal formal results. The chapter appendix proves only Proposition 8.1. Within that scope, suppose the ambient V filter is 𝑆 = (𝐼 + 𝜆𝜓(𝐿)) −1 . Here 𝜓 : [0, ∞) → [0, ∞) and 𝜓(0) = 0; the latter condition is the spectral mass-preservation gate. A positive-semidefinite contraction need not be entrywise nonnegative or mass preserving. Graph-universal Markov validity requires the generator 𝜓(𝐿) to be a graph Laplacian for every weighted graph. Every Bernstein function with 𝜓(0) = 0 satisfies this Laplacian gate, possibly producing a dense graph; this uses the classical link among Bernstein functions, matrix functions, and graph generators [Schilling et al., 2012, Micchelli and Willoughby, 1979, Gregorio and Mugnolo, 2020]. If exact kernel preservation on every connected graph is also required, assume in addition that 𝜓(𝑥) > 0 for 𝑥 > 0. In particular, (𝐼 + 𝜆𝐿 𝜈 ) −1 is Markov for every finite weighted graph and every 𝜆 > 0 exactly when 0 < 𝜈 ≤ 1. For 𝜈 > 1, a weighted three-node path produces a negative entry for a range of 𝜆. Likewise, among real polynomials 𝜓 with 𝜓(0) = 0, the only graph-universally valid spectral penalties are nonnegative multiples of 𝐿. This barrier applies to linear probability-preserving mechanisms. It does not rule out nonlinear projection repair.

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8.4

Sharp path rates

On a growing path, suppose the graph-source smoothness is 𝑠 > 0 and a Richardson filter has integer qualification 𝑚 ≥ 𝑠. With calibrated bandwidth, the repaired forecast attains average excess CRPS 𝑁 −1 + 𝑅 2/(4𝑠+1) 𝑁 −4𝑠/(4𝑠+1) ,

𝑁 = 𝑛𝑇,

in the stated regime. An embedded Bernoulli experiment gives a matching lower bound. The rate has two pieces. The 𝑁 −1 term reflects the graph nullspace or global average. The nonparametric term balances spectral bias and effective dimension. Boundary of the claim The source path-rate theorem assumes the declared growing-path source class, compact outcome support, independent vertex samples, and calibrated qualification. It is not a universal rate for arbitrary graphs, dependence, or unbounded outcomes.

The broader design principle. The example suggests a general workflow for constrained statistical objects: high-order ambient proposal −→ diagnose semantic invalidity −→ nonlinear validity repair −→ risk and rate certificate. The workflow is relevant to covariance matrices, densities, quantiles, stochastic matrices, projectors, and other shape-constrained outputs, but each case requires its own projection and loss geometry.

Exercises Exercise 8.2. Derive the expected excess CRPS identity from the definition of CRPS. Exercise 8.3. For a two-vertex graph, compute tr(𝑆 2 ) for 𝑆 = (𝐼 + 𝜆𝐿) −1 . Interpret the limiting values as 𝜆 ↓ 0 and 𝜆 ↑ ∞. Exercise 8.4. Construct a signed linear combination of two valid CDFs that is not a CDF. Show that projection repairs the violation without increasing squared-CDF error to any valid target.

Appendix to Chapter 8: complete proofs

49

Appendix to Chapter 8: complete proofs Proof of validity repair Dependencies. The first assertion uses the Hilbert-space projection theorem. The graph-roughness assertion additionally uses coordinatewise nonexpansiveness of the projection and the edge representation of the graph Dirichlet form. Proof. Let 𝑃 = Π C be metric projection onto the nonempty closed convex set C. The projection variational inequality states that ⟨𝑢 − 𝑃𝑢, 𝑣 − 𝑃𝑢⟩ ≤ 0

(𝑣 ∈ C).

e and 𝑣 = 𝐹 ★ and expanding the square gives Taking 𝑢 = 𝐹 e − 𝐹★ ∥ 2 = ∥ 𝐹 e − 𝐹 ∥ 2 + ∥𝐹 − 𝐹 ★ ∥ 2 + 2⟨𝐹 e − 𝐹, 𝐹 − 𝐹 ★⟩ ∥𝐹 ≥ ∥𝐹 − 𝐹 ★ ∥ 2 , which proves the risk contraction. For the graph claim, write the weighted Dirichlet form as ⟨𝐺, 𝐿𝐺⟩ =

1 ∑︁ 𝑤 𝑢𝑣 ∥𝐺 𝑢 − 𝐺 𝑣 ∥ 2H . 2 𝑢,𝑣

The metric projection 𝑃0 = Π Ccdf is nonexpansive, so for every edge e𝑢 ) − 𝑃0 ( 𝐹 e𝑣 ) ∥ H ≤ ∥ 𝐹 e𝑢 − 𝐹 e𝑣 ∥ H . ∥𝑃0 ( 𝐹 Squaring, multiplying by 𝑤 𝑢𝑣 /2, and summing over edges yields e 𝐿 𝐹⟩. e ⟨𝐹, 𝐿𝐹⟩ ≤ ⟨𝐹, □

Part II Historical and Conceptual Notes Classical engines Part II uses four mature mathematical mechanisms. Fenchel–Young equality produces Bregman divergences [Bregman, 1967, Rockafellar, 1970]; infimal composition is Bellman/min-plus algebra [Bellman, 1957, Baccelli et al., 1992]; Taylor subtraction produces a touching jet and is adjacent to first-order surrogate methodology [Mairal, 2013]; and metric projection onto a convex CDF set gives the validity-repair contraction. CRPS as squared-CDF risk and its propriety are likewise established [Matheson and Winkler, 1976, Gneiting and Raftery, 2007].

CHAPTER 8. PROBABILITY VALIDITY, GRAPH SMOOTHING, AND CRPS

50 Book object

Classical core

Conjugate defect

Fenchel–Bregman identity

Obstruction tower Exactification

CDF repair

Elimination-calculus use

the eliminated objective fixes the loss units and the direction of an asymmetric divergence infimal convolution and dynamic intermediate states are typed and every stage programming cost is read as an objective-generated tax Taylor touching and surrogate cor- the subtracted jet is the actual frozen inserrection tion defect; the converse is stated modulo a flat remainder within that class Hilbert projection and nonexpan- validity is restored without increasing siveness squared-CDF/CRPS risk or the declared graph roughness

What is and is not claimed The conjugate identity, tower associativity, jet-matching calculation, and projection inequalities are not claimed as new classical mathematics. The program-specific layer is the insistence that a defect be generated by the same elimination as the target, that transformations preserve its units and orientation, and that target-jet preservation be audited against that exact defect. The converse exactification normal form is recorded as a source- program claimed increment, not as an exhaustive priority assertion.

From native loss to valid structural objects A defect can enter an architecture theorem only after two questions have been answered. Do locally reported increments integrate to one global objective? If an auxiliary lift is not directly native, does it arise from an intrinsic factorization rather than an arbitrary target-calling coordinate? Part III closes these validity gates before obstruction is quantified.

Part III

Validity of Defects and Representations An architecture obstruction is meaningful only when its local defect system is integrable and its carrier is intrinsic to the declared target.

51

Chapter 9

Integrability, Hodge Repair, and Global Defect Consistency 9.1

When local defect reports may be inconsistent

A complex learning pipeline may eliminate several coordinates in different orders. Local modules can report nonnegative, touching residuals while failing to arise from one global objective. The first structural audit is therefore integrability. Consider a finite elimination complex whose vertices represent partial eliminations and whose edges represent eliminating one additional coordinate. Let 𝑒 assign a reported increment to every oriented edge. If 𝑒 is the gradient of a potential 𝑉, 𝑒 = 𝑑0𝑉, then every path sum between two vertices is the same. Conversely, on a simply connected complex, zero circulation around every elementary face implies path independence. This discrete potential/cycle condition is the same integrability mechanism that underlies exact potential games [Monderer and Shapley, 1996]; here the edge quantities are elimination defects rather than unilateral payoffs.

9.2

Square curvature

For the elementary square shown below, 𝑒 1 , 𝑒 4 form the lower-then-right path and 𝑒 2 , 𝑒 3 form the left-then-upper path. Define Ω = 𝑒1 + 𝑒4 − 𝑒2 − 𝑒3 . A nonzero Ω measures elimination-order dependence. 52

9.2. SQUARE CURVATURE

53 𝑒3

𝑒2

𝑒4

Ω = 𝑒1 + 𝑒4 − 𝑒2 − 𝑒3 local curvature

𝑒1

Figure 9.1: An elementary elimination square. Nonzero circulation records order dependence. On a general complex, zero face curvature must still be supplemented by a period audit.

How to read the assumptions The elimination index is a finite product of nonempty local choice sets. For every 𝑆 ⊆ [𝐾], 𝑗 ∉ 𝑆, and fixed partial state 𝑎 𝑆 , the edge report satisfies 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ) ≥ 0 for every 𝑎 𝑗 and inf 𝑎 𝑗 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ) = 0. If a touching witness is declared, one must be supplied in each fiber and attain zero there. Every elementary square has zero circulation. These are compatibility conditions on a complete local defect system, not regularity assumptions on a statistical estimator. Theorem 9.1 (Flat elimination criterion). On a product-of-stars elimination complex, let a real-valued local increment 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ) be given for every 𝑆 ⊆ [𝐾] and 𝑗 ∉ 𝑆. There are potentials 𝑉𝑆 satisfying 𝑒 𝑗 |𝑆 = 𝑉𝑆∪{ 𝑗 } − 𝑉𝑆 ,

𝑉𝑆 (𝑎 𝑆 ) = inf 𝑉𝑆∪{ 𝑗 } (𝑎 𝑆 , 𝑎 𝑗 ), 𝑎𝑗

if and only if, for every 𝑆, 𝑗, and fixed 𝑎 𝑆 , the edge reports are nonnegative with inf 𝑎 𝑗 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ) = 0, and every elementary square has zero circulation. Interpretation and boundary Local nonnegative gaps come from one global elimination tower exactly when their discrete curvature vanishes and fiberwise touching identifies every partial minimum. Nonzero square curvature is therefore a falsification certificate. On a general complex, harmonic periods must be added to the audit. Proof roadmap Integrate the edge reports along a path from the base vertex. Square-flatness makes the path sum invariant under adjacent swaps, hence well defined. Fiberwise touching identifies the resulting vertex potential with the required partial minima. The converse follows by telescoping a genuine potential around every square. The chapter appendix gives both directions. The nonnegativity and fiberwise touching conditions make the potential a certified defect potential rather than an arbitrary scalar potential.

CHAPTER 9. INTEGRABILITY AND DEFECT CONSISTENCY

54

9.3

Approximate flatness and Hodge repair

Approximate flatness. Two elimination orders differ by adjacent swaps. Every swap crosses one elementary square. Hence the discrepancy between path sums is bounded by the sum of absolute square curvatures along a swap sequence. A Kendall-distance bound follows when a uniform curvature bound is available. This provides a quantitative diagnostic: small local curvature implies limited order ambiguity, but does not guarantee exact integrability. Hodge decomposition. Source result; proof not reproduced here The full weighted Hodge decomposition, nearest-flat and nearest-exact projections, period tax, and nearest-certified repair summarized below are established in Elimination Geometry II. The chapter appendix proves the flat elimination criterion, not this broader repair theory [Huang and Wang, 2026g]. Let 𝐶 0 , 𝐶 1 , 𝐶 2 be weighted cochain spaces with coboundary maps 𝑑0 : 𝐶 0 → 𝐶 1 ,

𝑑1 : 𝐶 1 → 𝐶 2 .

An observed edge-defect field 𝑒 ∈ 𝐶 1 decomposes orthogonally as 𝑒 = 𝑑 0𝑉 + 𝑑 ★ 1 𝜓 + ℎ, where • 𝑑0𝑉 is exact; • 𝑑★ 1 𝜓 carries local curvature; • ℎ is harmonic and carries global periods. Combinatorial Hodge decompositions have been used to separate gradient, cyclic, and harmonic components of graph data, notably in statistical ranking [Jiang et al., 2011]. The present use is narrower and auditable: the cochain is an elimination defect field, and exactness is the condition for a global potential with the declared units and orientation. The exact and flat projections must be distinguished. The nearest exact defect field is 𝑒 ex = 𝑃im 𝑑0 𝑒 = 𝑑0𝑉, whereas the nearest flat defect field is 𝑒 fl = 𝑃ker 𝑑1 𝑒 = 𝑑0𝑉 + ℎ. On a contractible elimination complex the harmonic component vanishes and the two projections coincide. On a general complex, local face tests cannot detect ℎ. Certified decomposition A complete repair may pay three taxes: curvature removal, global period removal, and projection onto the cone of certified nonnegative/touching defect fields.

9.4. REPAIR, FALSIFICATION, AND AUDIT BOUNDARIES

9.4

55

Repair, falsification, and audit boundaries

Nearest certified repair. Let 𝐶cert ⊆ im 𝑑0 be a declared nonempty closed convex set of exact defect fields satisfying witness constraints. Starting from the exact projection 𝑒 ex , the certified repair is 𝑒 cert = 𝑃𝐶cert 𝑒 ex . Orthogonality gives 2 2 ex 2 ∥𝑒 − 𝑒 cert ∥ 2𝑊 = ∥𝑑★ 1 𝜓∥ 𝑊 + ∥ℎ∥ 𝑊 + ∥𝑒 − 𝑒 cert ∥ 𝑊 .

The three terms are curvature, period, and certification taxes. Local falsification. If a square has circulation Ω, then at least one of its four edge reports must be wrong by at least |Ω|/4 under the sup norm. This turns local curvature into a falsification certificate for the collection of local claims. When integrability must be audited. Architecture obstruction assumes that the local defect field belongs to one declared objective. This contract is automatic when the defect is derived directly as 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) from a single global objective. It is not automatic when local modules report increments that are later assembled into a defect field. In the latter case, optimizing before validation can produce a precise answer to an incoherent question. This chapter closes the integrability gate before the architecture obstruction results of Part IV are invoked. The logical validation order is global derivation =⇒ interpret obstruction, assembled defect field =⇒ audit/repair =⇒ interpret obstruction. A reader may postpone the full Hodge repair construction, but any theorem that combines separately reported local increments must either invoke this audit or derive its defect directly from one global objective.

9.4.1

Local generative vector fields under density and transport contracts

Score-based generation supplies a useful contract-sensitive example. If a reported field 𝑠 : X → R𝑑 is declared to be the score of one positive density, 𝑠(𝑥) = ∇ log 𝑝(𝑥), then, on a simply connected smooth domain, a continuously differentiable field must have symmetric Jacobian: 𝜕𝑖 𝑠 𝑗 = 𝜕 𝑗 𝑠𝑖 . On a domain with holes, vanishing local curl must additionally be supplemented by zero periods around noncontractible loops. An antisymmetric Jacobian entry or a nonzero loop integral is therefore a finite falsification certificate for the declared global density or energy interpretation. Projecting the field onto an exact component, or parameterizing it directly as a scalar-energy gradient, repairs the gradient-consistency part of that contract; the removed coexact and harmonic components are

CHAPTER 9. INTEGRABILITY AND DEFECT CONSISTENCY

56

the corresponding integrability taxes. A density claim still requires the exponential potential to be integrable and normalized. This viewpoint is consistent with empirical audits of conservativeness in learned score fields [Chao et al., 2023]. The conclusion changes when the output contract changes. A vector field used only to define a transport ODE or sampler need not be conservative. Nonzero curl alone then does not certify poor samples, invalid transport, or excess native generative loss. The audit must instead use the declared terminal law or transport objective. Thus the same numerical field can fail a global-score contract while remaining admissible under a sampling contract; integrability is a typed obligation, not a model-name diagnosis. Boundary of the claim The gradient criterion, period obstruction, and Hodge projection are classical Poincaré/Hodge facts. This example introduces no new theorem about diffusion or score-based models; it instantiates the chapter’s audit sequence and marks the precise boundary at which a density certificate ceases to be a sampling certificate.

Operational ordering curvature. Algebraic elimination curvature asks whether two orders produce the same internal defect field. Operational curvature, developed in Chapter 17, asks whether a legal task and context can observe the difference. An internal discrepancy may be operationally invisible; conversely, a context can expose a difference hidden at the base input.

Exercises Exercise 9.2. For a square elimination complex, solve the least-squares projection of arbitrary edge values onto exact increments. Exercise 9.3. Construct a cochain on a cycle graph with zero local curvature but nonzero period. Explain why local face tests cannot detect it. Exercise 9.4. Prove the |Ω|/4 local falsification bound and show that the constant is sharp.

Appendix to Chapter 9: complete proofs

57

Appendix to Chapter 9: complete proofs Proof of the flat certified-elimination criterion Dependencies. The state space is a finite product of nonempty coordinate sets. Nonnegativity and the condition inf 𝑎 𝑗 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ) = 0 in every fixed partial-state fiber certify conditional minimization; square flatness certifies path independence. Proof. Assume first that potentials 𝑉𝑆 exist with 𝑒 𝑗 |𝑆 = 𝑉𝑆∪{ 𝑗 } − 𝑉𝑆 ,

𝑉𝑆 (𝑎 𝑆 ) = inf 𝑉𝑆∪{ 𝑗 } (𝑎 𝑆 , 𝑎 𝑗 ). 𝑎𝑗

For every fixed 𝑎 𝑆 , the second identity immediately yields 𝑒 𝑗 |𝑆 ≥ 0,

inf 𝑒 𝑗 |𝑆 = 0. 𝑎𝑗

For distinct 𝑖, 𝑗 ∉ 𝑆, both path sums around the elementary square telescope to 𝑉𝑆∪{𝑖, 𝑗 } − 𝑉𝑆 , proving square flatness. Conversely, fix the additive constant 𝑉∅ . For a nonempty subset 𝑆, choose an ordering 𝜋 = (𝜋1 , . . . , 𝜋 |𝑆 | ), put 𝑆 𝑘𝜋 = {𝜋1 , . . . , 𝜋 𝑘 }, and define 𝑉𝑆 (𝑎 𝑆 ) = 𝑉∅ +

|𝑆 | ∑︁

𝜋 (𝑎 𝑆 𝜋 ). 𝑒 𝜋𝑘 |𝑆𝑘−1 𝑘

𝑘=1

Any two orderings of 𝑆 are connected by adjacent transpositions. An adjacent transposition replaces one two-edge path around an elementary square by the other; square flatness says the two sums agree. Hence 𝑉𝑆 is independent of the chosen ordering. Append 𝑗 to an ordering of 𝑆. The definition then gives 𝑉𝑆∪{ 𝑗 } (𝑎 𝑆 , 𝑎 𝑗 ) − 𝑉𝑆 (𝑎 𝑆 ) = 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ). For the fixed partial state 𝑎 𝑆 , nonnegativity and fiberwise touching give inf 𝑉𝑆∪{ 𝑗 } (𝑎 𝑆 , 𝑎 𝑗 ) = 𝑉𝑆 (𝑎 𝑆 ) + inf 𝑒 𝑗 |𝑆 (𝑎 𝑆 , 𝑎 𝑗 ) = 𝑉𝑆 (𝑎 𝑆 ). 𝑎𝑗

𝑎𝑗

Iterating this identity over the complement of 𝑆 yields 𝑉𝑆 (𝑎 𝑆 ) = inf 𝑉[𝐾 ] (𝑎). 𝑎𝑆 𝑐

Finally, once 𝑉∅ is fixed, every 𝑉𝑆 is the path integral of the edge field and is therefore unique.

□

Chapter 10

Lift Admissibility, Slack Factorization, and Complexity In this chapter, lift admissibility means target-faithfulness of a nonnative lifted representation after slack, minimal-face, quotient, and gauge reduction. This is a representation-level gate. A lift used to support a deployment claim must also pass a second interface: quotient-faithful extraction into a state whose oracle-relevant distinctions are measured in the native defect. The audit therefore proceeds from target slack, through minimal-face and gauge reduction, to deployment extraction; finite auxiliary dimension alone certifies none of these steps. target epigraph and slack

minimal-face and gauge quotient

admissible cone factorization

target-visible deployment carrier

Figure 10.1: The admissibility chain. Exact elimination alone is insufficient: a lift must be tied to target slack, stripped of slack-invisible gauge, and then connected to a deployment state whose distinctions are measured by native defect.

10.1

Why exact lifts can be vacuous

Given any target objective 𝐽 (𝑥) and any chosen field 𝑔(𝑥), 1 𝐻𝑔 (𝑥, 𝑎) = 𝐽 (𝑥) + ∥𝑎 − 𝑔(𝑥) ∥ 2 2 is an exact lift. Its oracle is 𝑎★ (𝑥) = 𝑔(𝑥). By choosing 𝑔 we can manufacture arbitrary coherence, local metrics, or apparent geometric structure that was not present in the target. This first observation is analytical: it does not assert that an arbitrary 𝑔 has a finite representation in a declared lift grammar. The finite-size statement below therefore restricts 𝑔 to an affine field, or more generally to a field whose graph and penalty composition are explicitly representable by that grammar. A constant 𝑔 makes the oracle globally coherent. An affine 𝑔(𝑥) = 𝑀𝑥 + 𝑏 manufactures frozenoracle metric 𝑀 ⊤ 𝑀. With suitable primitives, arbitrary positive-semidefinite Hessians can be appended at fixed dimension-dependent overhead. 58

10.2. EPIGRAPH GEOMETRY AND INTRINSIC LIFT STRUCTURE

59

How to read the assumptions An exact lift 𝐿 of size 𝑚 is already available, with objective 𝐻 𝐿 (𝑥, 𝑧). The appended field is affine, 𝑔(𝑥) = 𝑀𝑥 + 𝑏; the same construction applies to a nonaffine field only when its graph and the composed penalty are explicitly representable in the grammar. A closed convex penalty 𝜓 : R𝑞 → [0, ∞] vanishes only at zero, its epigraph has fixed representation cost 𝑐 𝜓 (𝑞), and the cone dictionary is stable under the required product. Theorem 10.1 (Finite-complexity target-calling no-go). Under the preceding conditions, 𝐿 has an exact augmentation 𝐻 𝐿 [𝑔, 𝜓] (𝑥, 𝑧, 𝑎, 𝑠) = 𝐻 𝐿 (𝑥, 𝑧) + 𝑠 + 𝜄 {𝑠≥ 𝜓 (𝑎−𝑔 ( 𝑥 ) ) } of size at most 𝑚 + 𝑐 𝜓 (𝑞), whose appended exact oracle is 𝑎★ (𝑥) = 𝑔(𝑥). The overhead is independent of the coefficients 𝑀, 𝑏. Consequently a constant 𝑔 appends a dummy coherent coordinate; a polyhedral ℓ1 primitive appends any affine field at fixed dimension-dependent overhead; and a quadratic primitive produces frozen-oracle Hessian 𝑀 ⊤ 𝑀, hence any prescribed positive-semidefinite local metric. Interpretation and boundary A finite extension-size claim is meaningless if the grammar permits target-calling or dummy coordinates: one can manufacture an exact, apparently coherent lift without explaining the target. The theorem is a no-go for an insufficient notion of complexity, not for all finite lifts. Proof roadmap Append the representable epigraph of 𝜓(𝑎 − 𝑔(𝑥)), eliminate (𝑎, 𝑠), and verify that its unique minimum is (𝑔(𝑥), 0). Product stability gives the fixed overhead, while the affine and quadratic specializations yield the dummy and prescribed-metric conclusions. The chapter appendix gives the complete construction.

10.2

Epigraph geometry and intrinsic lift structure

Thus finite auxiliary dimension or extension size alone does not define lift admissibility. The targetcalling no-go and the quotient-faithful lift interface in this chapter come from the Lift Complexity companion [Huang and Wang, 2026i]. For a convex objective on a compact domain, consider a truncated epigraph body 𝐾 𝐽 = {(𝑥, 𝑡) : 𝑥 ∈ 𝑋, 𝐽 (𝑥) ≤ 𝑡 ≤ 𝑀 }. A proper conic lift of the objective corresponds to an extended formulation of 𝐾 𝐽 . The intrinsic target object is the slack operator 𝑆 𝐽 (𝑢, 𝑣) = 1 − ⟨𝑢, 𝑣⟩ after the appropriate normalization of extreme points and supporting functionals. Classical extension-complexity theory relates lift size to factorization of the slack operator: nonnegative rank governs polyhedral lifts, and cone factorizations extend the principle to semidefinite and other convex lifts [Yannakakis, 1991, Gouveia et al., 2013, Fawzi et al., 2015]. Semidefinite

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60

hierarchies and extension-complexity lower bounds provide complementary parts of this literature [Lasserre, 2001, Parrilo, 2003, Fiorini et al., 2015, 2012]. How to read the assumptions The target epigraph is a normalized compact convex body, the cone lift is proper and finite dimensional, and redundant directions invisible to the slack operator are quotiented out. Minimal-face reduction is used to remove artificial cone dimensions. Theorem 10.2 (Conic lift–factorization gate). Every proper K-lift of the truncated epigraph induces a K-factorization of its slack operator, and every K-factorization induces a K-lift. A nonproper lift is handled after restriction to the minimal face containing its affine slice. Polyhedral and semidefinite lifts correspond to nonnegative and positive-semidefinite factorizations. This is the lift–factorization theorem of Gouveia et al. [2013], extending the polyhedral theorem of Yannakakis [1991]. Interpretation and boundary Admissible conic lifts are exactly factorizations of the target slack operator, up to the declared quotient. This makes lift geometry target-generated and prevents dummy directions from creating fake coherence. The theorem classifies a representation interface; it does not equate extension size with unrestricted runtime. Proof roadmap From a proper conic lift, pair primal cone representatives with dual exposing functionals to factor the slack. Conversely, use a cone factorization to reconstruct a lifted feasible set and projection. Minimal-face reduction and the slack-kernel quotient establish target faithfulness. The chapter appendix gives both constructions. The theorem does not by itself define a useful auxiliary coherence field. It supplies the gate through which lift design must pass.

10.3

Gauge reduction and the fidelity–coherence–complexity frontier

Factorizations are nonunique. Cone automorphisms, duplicated factors, and slack-invisible directions can change raw coordinates without changing the represented objective. A reduced slack carrier removes directions not seen by the target slack operator. A coherence rule is admissible only if it is invariant under cone gauge and faithful under quotient refinement. Example 10.3 (Factor splitting artifact). Duplicate each nonnegative factor 𝑘 times and divide its weight among copies. A naive Euclidean factor-coherence penalty can decrease by a factor 1/𝑘 while the represented convex set is unchanged. On the quotient-reduced carrier the apparent improvement disappears.

10.4. TARGET-VISIBLE REDUCTION AND EXTRACTION

61

Fidelity–coherence–complexity frontier. For cone family K, size budget 𝑚, coherence budget 𝑟, and one-sided objective-fidelity error 𝛿, define 𝔉𝐽 , K (𝑚, 𝑟, 𝛿) as the least true objective defect achievable by a quotient-reduced K-lift satisfying the budgets. The exact-lift frontier is the slice 𝛿 = 0. Approximate lifts introduce a model or formulation bias that must be separated from coupling obstruction. A deployed lift admits the decomposition total defect = complexity-limited obstruction + formulation-selection tax + within-formulation implementation tax, under the declared decomposition convention.

10.4

Target-visible reduction and extraction

Slack reduction answers which lift directions belong to the target. It does not yet answer which distinctions in a retained deployment state matter to native defect. In the common-fiber Legendre– Bregman regime there is a second, canonical reduction. Let 𝑎★ be the unique oracle, put 𝑆 = ∇Φ(𝑎★),

𝑆𝑈 = E(𝑆 | 𝑈),

† 𝑎𝑈 = ∇Φ★ (𝑆𝑈 )

for a standard-Borel deployment statistic 𝑈, and define † CΦ (𝑈) = E𝐷 Φ (𝑎𝑈 ∥𝑎★) = E𝐷 Φ★ (𝑆∥𝑆𝑈 ).

For every factorized deployment b 𝑎 = 𝑞(𝑈), conditional Bregman projection gives the exact defect field † E𝐷 Φ (b 𝑎 ∥𝑎★) = CΦ (𝑈) + E𝐷 Φ (b 𝑎 ∥𝑎𝑈 ).

(10.1)

Thus CΦ (𝑈) is the native obstruction caused by the carrier, while the second term is a decoder tax. The identity is the classical conditional Bregman predictor decomposition [Banerjee et al., 2005a, Pfau, 2025, Adlam et al., 2022]; its use as a canonical quotient of an admissible lift is the additional step here. Count only distinct conditional dual signatures: vcΦ (𝑈) = essran 𝑆𝑈 ,

CΦ (𝑘) =

inf

vcΦ (𝑈) ≤ 𝑘

CΦ (𝑈).

Raw labels inducing the same 𝑆𝑈 therefore consume one visible state, not several. For finitely many separated oracle types, this is the ordinary Bregman quantization frontier and is positive below the number of distinct types. How to read the assumptions Φ is Legendre on a common open convex fiber; 𝑆 is integrable; all conditional means lie in ri(dom Φ★). The coarsening identity below is asserted only when CΦ (𝑈) and CΦ (𝑉) are both finite, so that its left-hand difference is defined. For the transfer clause below, every exact deployment counted by the lift frontier has a measurable, quotient-faithful extraction b 𝑎 𝐿,𝑧 = 𝑞 𝐿,𝑧 (𝑈 𝐿,𝑧 ), a uniform native exchange constant 𝑐 > 0, and the stated finite visiblecapacity bound.

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Theorem 10.4 (Visible quotient and lift-to-carrier transfer). Let 𝑅𝑈 = 𝑆𝑈 . Then 𝑅𝑈 is a quotient of 𝑈, E(𝑆 | 𝑅𝑈 ) = 𝑅𝑈 ,

CΦ (𝑅𝑈 ) = CΦ (𝑈).

If 𝑉 is a coarsening of 𝑈 and both CΦ (𝑈), CΦ (𝑉) < ∞, then CΦ (𝑉) − CΦ (𝑈) = E𝐷 Φ★ (𝑆𝑈 ∥𝑆𝑉 ) † = E𝐷 Φ (𝑎 †𝑉 ∥𝑎𝑈 ) ≥ 0,

(10.2)

with equality exactly when 𝑆𝑈 = 𝑆𝑉 almost surely. Hence 𝜎(𝑅𝑈 ) is, modulo null sets, the coarsest quotient of 𝑈 preserving optimal native defect. Suppose, in addition, that every quotient-reduced exact lift of size at most 𝑚 and every deployment of coherence at most 𝑟 admits the declared extraction and satisfies true defect ≥ 𝑐 E𝐷 Φ (b 𝑎 𝐿,𝑧 ∥𝑎★),

vcΦ (𝑈 𝐿,𝑧 ) ≤ 𝜅 K (𝑚, 𝑟).

Then the exact lift frontier obeys 𝔉𝐽 , K (𝑚, 𝑟, 0) ≥ 𝑐 CΦ (𝜅 K (𝑚, 𝑟)) .

(10.3)

Interpretation and boundary There are two different quotients. The reduced slack carrier removes representation gauge; 𝑅𝑈 then removes deployment labels invisible to the native defect. A quotient-faithful extraction theorem is the bridge between them. Cone size 𝑚 and visible capacity 𝑘 remain different units: there is no universal 𝜅 K determined by cone order alone. Proof roadmap Conditional expectation makes 𝑅𝑈 sufficient for its own dual mean. The Bregman predictor decomposition applied to a coarsening gives the exact tax (10.2). Applying (10.1) to each extracted deployment, then using the native exchange and taking the frontier infimum, gives (10.3). The chapter appendix supplies the complete argument. The upgraded lift chain can therefore be read as quotient-faithful extraction

visible Bregman frontier

(𝑚, 𝑟) −−−−−−−−−−−−−−−−−−−→ 𝜅 K (𝑚, 𝑟) −−−−−−−−−−−−−−−−−→ 𝑐 CΦ (𝜅 K (𝑚, 𝑟)) . Chapter 16 starts after such a deployment carrier has been declared or extracted and separates its information loss from decoder nonsaturation.

10.5. FROM LIFT SIZE TO NATIVE DEFECT

10.5

63

From lift size to native defect

Scaling obstruction. Positive rescaling of the objective preserves extension size while rescaling the native defect. Therefore extension size cannot be a universal defect scale. Complexity and objective loss live in different units until an exchange theorem is supplied. Barrier-to-defect exchange. For a fixed encoded, well-conditioned conic lift with a self-concordant barrier, a primal–dual gap provides an objective defect certificate. Iteration complexity depends on: • barrier parameter; • coefficient encoding length; • conditioning; • cone-oracle cost; • per-iteration linear algebra; • requested numerical accuracy. Boundary of the claim An extension-complexity lower bound counts the size of an exact extended formulation. It is not a bit-complexity or runtime lower bound: coefficient encoding length and numerical conditioning may grow even when extension size does not. A computational conclusion requires a declared oracle model, barrier, conditioning regime, and linear-algebra cost.

Relationship to architecture obstruction. When an obstruction argument uses a nonnative lift, lift complexity answers a logically earlier question than EOT or COT: do the auxiliary states arise from an intrinsic factorization of the target rather than an arbitrary appended coordinate? Native auxiliary states require no separate lift audit, and a lift whose target-faithfulness follows by construction has already discharged the gate. This chapter closes the target-faithfulness gate before nonnative lifts are used in the architectureobstruction results of Part IV. The logical order is target epigraph and slack −→ reduced admissible carrier −→ coherence grammar −→ architecture obstruction.

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Exercises Exercise 10.5. Verify that the target-calling lift is exact for every field 𝑔. Explain why this makes unrestricted lift geometry nonfalsifiable. Exercise 10.6. Construct the duplicated-factor artifact for a simplex slack matrix and compute the naive coherence before and after duplication. Exercise 10.7. List the additional assumptions needed to convert a semidefinite extension into a numerical runtime statement.

Appendix to Chapter 10: complete proofs

65

Appendix to Chapter 10: complete proofs Proof of the finite-complexity target-calling no-go Dependencies. The cone dictionary must represent the epigraph of the declared penalty primitive with fixed overhead and must be stable under products. The oracle map is affine so that its graph can be imposed by affine formulation constraints. A nonaffine field is covered only if the declared grammar separately represents its graph and the composed epigraph with the claimed overhead. Proof. Let the original exact lift be represented by auxiliary variable 𝑧 and objective 𝐻 𝐿 (𝑥, 𝑧), so that inf 𝐻 𝐿 (𝑥, 𝑧) = 𝐽 (𝑥). 𝑧

Append variables 𝑎 ∈ R𝑞 and 𝑠 ∈ R, together with the representable epigraph constraint 𝑠 ≥ 𝜓{𝑎 − 𝑔(𝑥)}. The enlarged lifted objective is 𝐻 𝐿 [𝑔, 𝜓] (𝑥, 𝑧, 𝑎, 𝑠) = 𝐻 𝐿 (𝑥, 𝑧) + 𝑠 + 𝜄 {𝑠≥ 𝜓 (𝑎−𝑔 ( 𝑥 ) ) } . Because 𝜓 ≥ 0 and vanishes only at zero, inf {𝑠 : 𝑠 ≥ 𝜓(𝑎 − 𝑔(𝑥))} = 0, 𝑎,𝑠

and the infimum is attained uniquely at 𝑎 = 𝑔(𝑥) and 𝑠 = 0. Eliminating first (𝑎, 𝑠) and then 𝑧 therefore returns 𝐽 (𝑥), so the enlarged formulation is exact. Product stability of the cone dictionary adds at most the fixed representation cost 𝑐 𝜓 (𝑞), proving the size bound. If 𝑔 is constant, the appended exact oracle is constant. If 𝜓 = ∥ · ∥ 1 , any affine field is appended with polyhedral overhead depending only on 𝑞. If 𝜓(𝑣) = 12 ∥𝑣∥ 22 and the auxiliary state is frozen at 𝑎 = 𝑔(𝑥 0 ), then 1 1 1 ∥𝑔(𝑥 0 ) − 𝑔(𝑥) ∥ 2 = ∥ 𝑀 (𝑥 − 𝑥0 ) ∥ 2 = (𝑥 − 𝑥0 ) ⊤ 𝑀 ⊤ 𝑀 (𝑥 − 𝑥0 ). 2 2 2 Its Hessian is 𝑀 ⊤ 𝑀. Every positive semidefinite matrix has such a factorization, establishing the final claim. □

Proof of the conic lift–factorization gate Dependencies. The body is full dimensional, compact, and affinely normalized with the origin in its interior. Properness ensures conic dual certificates on the ambient cone; a nonproper lift is first restricted to its minimal face. Proof. We give the standard lift–factorization construction. Suppose first that 𝐶 = 𝜋(𝐾 ∩ 𝐿)

66

CHAPTER 10. LIFT ADMISSIBILITY, SLACK FACTORIZATION, AND COMPLEXITY

is a proper 𝐾-lift, where 𝐿 is an affine subspace meeting int 𝐾. For every extreme point 𝑥 ∈ 𝐶, choose a lifted representative 𝐴(𝑥) ∈ 𝐾 ∩ 𝐿 with 𝜋 𝐴(𝑥) = 𝑥. For every 𝑢 ∈ ext(𝐶 ◦ ), the inequality ⟨𝑢, 𝑥⟩ ≤ 1 is valid on the projection. Conic strong duality for the proper slice supplies a dual vector 𝐵(𝑢) ∈ 𝐾 ∗ whose pairing with any lifted feasible point equals the slack of that inequality. In particular, 1 − ⟨𝑢, 𝑥⟩ = ⟨𝐴(𝑥), 𝐵(𝑢)⟩. Thus the intrinsic slack operator has a 𝐾-factorization. Conversely, suppose maps 𝐴 : ext(𝐶) → 𝐾,

𝐵 : ext(𝐶 ◦ ) → 𝐾 ∗

satisfy 1 − ⟨𝑢, 𝑥⟩ = ⟨𝐴(𝑥), 𝐵(𝑢)⟩. Form the affine set of pairs (𝑥, 𝑧) satisfying ⟨𝑧, 𝐵(𝑢)⟩ = 1 − ⟨𝑢, 𝑥⟩

for every 𝑢 ∈ ext(𝐶 ◦ ),

𝑧 ∈ 𝐾.

Every extreme point 𝑥 of 𝐶 has feasible witness 𝑧 = 𝐴(𝑥), and convexity supplies witnesses for all of 𝐶. Conversely, if (𝑥, 𝑧) satisfies the displayed system, then 1 − ⟨𝑢, 𝑥⟩ = ⟨𝑧, 𝐵(𝑢)⟩ ≥ 0 for every extreme point of 𝐶 ◦ , hence for every 𝑢 ∈ 𝐶 ◦ . By the bipolar theorem, 𝑥 ∈ 𝐶. Therefore the projection of this affine slice of 𝐾 is exactly 𝐶. If the original lift is not proper, intersect 𝐾 with the minimal face containing the affine slice. The slice is proper relative to that face, and the same argument applies there. □

Proof of the visible quotient and lift-to-carrier transfer Dependencies. The conditional Bregman decomposition uses the common-fiber Legendre assumptions stated in the chapter. Doob–Dynkin is applied to the standard-Borel statistic 𝑈. The frontier conclusion additionally uses the declared quotient-faithful extraction, native exchange, and capacity bound; none is inferred from cone order. Proof. The random variable 𝑅𝑈 = 𝑆𝑈 is a measurable function of 𝑈. Since it is also 𝑅𝑈 -measurable, the tower property gives E(𝑆 | 𝑅𝑈 ) = E{E(𝑆 | 𝑈) | 𝑅𝑈 } = 𝑅𝑈 . Substitution in the dual expression for CΦ shows that CΦ (𝑅𝑈 ) = CΦ (𝑈). If 𝜎(𝑉) ⊆ 𝜎(𝑈), then 𝑆𝑉 = E(𝑆𝑈 | 𝑉). The conditional Bregman predictor identity, now in dual coordinates, yields E𝐷 Φ★ (𝑆∥𝑆𝑉 ) = E𝐷 Φ★ (𝑆∥𝑆𝑈 ) + E𝐷 Φ★ (𝑆𝑈 ∥𝑆𝑉 ). † Legendre duality converts the last term to E𝐷 Φ (𝑎 †𝑉 ∥𝑎𝑈 ). Strict convexity makes it zero exactly when 𝑆𝑈 = 𝑆𝑉 almost surely. Any defect-preserving coarsening must therefore retain 𝑆𝑈 , proving the coarsest-quotient claim modulo null sets.

Part III Historical and Conceptual Notes

67

Finally fix a deployment counted by 𝔉𝐽 , K (𝑚, 𝑟, 0). Its factorized extraction and (10.1) imply true defect ≥ 𝑐 E𝐷 Φ (b 𝑎 𝐿,𝑧 ∥𝑎★) ≥ 𝑐 CΦ (𝑈 𝐿,𝑧 ) ≥ 𝑐 CΦ (𝜅 K (𝑚, 𝑟)) . The last inequality is the definition of the visible-capacity frontier. Taking the infimum over all exact deployments proves (10.3). □

Part III Historical and Conceptual Notes Two different validity questions The Part separates an integrability question from a representation question. Cycle-flat edge data are gradients of a potential under familiar discrete integrability conditions, as in exact potential games and combinatorial Hodge theory [Monderer and Shapley, 1996, Jiang et al., 2011]. Separately, a convex lift is intrinsic only when it factors the target slack operator: the polyhedral equivalence is due to Yannakakis [1991], and the general closed-cone equivalence to Gouveia et al. [2013]. These are not interchangeable audits. A perfectly integrable local defect field can be attached to a targetcalling auxiliary coordinate; a valid slack factorization can still carry duplicated or gauge directions whose raw coherence has no target meaning. Gate

Established antecedent

Flatness

potential/cycle tests

Lift admissibility

Target-calling no-go Visible quotient

Part III increment or specialization

adds nonnegativity and fiberwise touching so the potential is a partial-minimum elimination tower slack-factorization/extension complex- imports the classical iff theorem as a gate ity and removes minimal-face and gauge artifacts exact augmentation by representable elementary diagnostic specialization showpenalties ing why size alone is not admissibility conditional Bregman prediction retains only conditional dual signatures [Banerjee et al., 2005a] and then requires an explicit resource-tocarrier extraction theorem

The conic lift–factorization theorem is therefore explicitly classical. The finite target-calling no-go and the quotient/extraction bridge are the Part’s program-specific claims, with the qualified priority statuses stated in Appendix G.1.

From valid defects to structural nonrealizability Parts II–III have fixed the native loss scale and certified the defect system and carrier through which it is represented. Part IV changes the quantifiers: can one shared, compressed, smooth, or coordinated deployment realize the entire oracle family? Regular variation, parameter sharing, and branch inconsistency around loops (monodromy) produce different lower bounds and call for different repairs.

Part IV

Structural Obstructions under Shared Deployment A shared deployment contract turns local optimization into a global compatibility problem. Regularity, coordination, and singularity impose different structural limits and call for mechanism-matched repairs.

68

Chapter 11

Architecture Obstruction as a Second Elimination 11.1

From a local defect to a class-level floor

Once the native defect is fixed, architecture design becomes another optimization problem. Throughout Part IV, that defect is assumed either to be derived directly from one declared global objective or to come with an integrable local defect system. Any nonnative lifted carrier is assumed to satisfy the target-faithfulness gate. Chapters 9 and 10 audit these conditions retrospectively when they are not guaranteed by construction. For a population law 𝑃 and architecture class 𝔄, 𝔒 𝑃 (𝔄) = inf E 𝑃 D𝑋 { 𝐴(𝑋)}. 𝐴∈𝔄

This is a second elimination: the first eliminates the local auxiliary state at each 𝑥; the second eliminates over globally deployable fields. Every deployment 𝐴 ∈ 𝔄 has the exact defect field impl

E 𝑃 D𝑋 { 𝐴(𝑋)} = 𝔒 𝑃 (𝔄) + Δ 𝑃 ( 𝐴; 𝔄). The architecture obstruction is a property of the triple ( D, 𝑃, 𝔄), not of the architecture alone. Changing the loss, population weighting, or output contract can change the tax.

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70

11.2

Saturation

The first class-level question is whether the second elimination leaves any irreducible defect. The definition distinguishes a zero infimum from its attainment by an exact oracle section. Definition 11.1 (Saturation). An architecture class 𝔄 is saturated for an oracle family if its obstruction is zero under the declared contract. It is exactly saturated if it contains an exact oracle section. Saturation is the correct zero-tax condition. Universal approximation in a generic norm is neither necessary nor sufficient without an exchange theorem connecting that norm to the native defect. Example 11.2 (Quotient saturation). A vector-valued architecture may be nonsaturated for an eigenline oracle because of sign ambiguity. A projector-valued architecture can be exactly saturated while using no more intrinsic information.

11.3

Architecture comparison and witnesses

If 𝔄1 ⊆ 𝔄2 , then 𝔒 𝑃 (𝔄2 ) ≤ 𝔒 𝑃 (𝔄1 ). But a larger class can have worse generalization or optimization behavior. The structural comparison isolates only the irreducible floor. For a resource-indexed family {𝔄𝑟 : 𝑟 ≥ 0} with 𝔄𝑟 ⊆ 𝔄𝑟 ′ for 𝑟 ≤ 𝑟 ′ , the frontier 𝑟 ↦−→ 𝔒 𝑃 (𝔄𝑟 ) is nonincreasing. Plateaus identify ranges in which additional resources do not change realizability. Lower and upper architecture witnesses. The same native defect supports both directions of the theory: a concrete deployment gives a constructive upper witness, while a flow, topological, resource, or information argument gives a converse lower certificate; together, the two should bracket the obstruction. Lower witness. A dual flow, topological argument, resource counting theorem, or information contraction gives 𝔒 ≤ 𝔒 𝑃 (𝔄). Upper witness.

An explicit deployment 𝐴wit ∈ 𝔄 gives 𝔒 𝑃 (𝔄) ≤ E 𝑃 D𝑋 { 𝐴wit (𝑋)}.

A sharp theory seeks matching witnesses. Even when they do not match, the gap is itself an auditable unresolved region.

11.4. ARCHITECTURE OBSTRUCTION VERSUS APPROXIMATION AND OPTIMIZATION

11.4

71

Architecture obstruction versus approximation and optimization

Classical approximation theory studies inf ∥ 𝑓 − 𝑓 ★ ∥.

𝑓 ∈F

Architecture obstruction differs in four ways. 1. The target may be a fiber, orbit, or set rather than one function. 2. The error loss scale is generated by an eliminated objective. 3. The architecture contract may include sharing, memory, topology, and output semantics. 4. The obstruction may disappear under a quotient or atlas without enlarging ordinary function capacity. The two theories are complementary. When the oracle is a unique function and the defect is equivalent to a norm, architecture obstruction reduces to a familiar approximation problem. b A large Architecture obstruction versus optimization. Suppose a training algorithm returns 𝐴. achieved defect may arise because impl b Δ ( 𝐴; 𝔄) > 0 𝑃

or because 𝔒 𝑃 (𝔄) > 0. Training loss alone does not separate the two. One needs a class-level lower certificate and an upper witness. A particularly strong empirical design compares: 1. more optimization within the baseline class; 2. more parameters within the same contract; 3. a targeted contract change predicted by the obstruction; 4. random or capacity-matched repairs.

11.5

Output, risk, and theorem contracts

The same oracle incidence can have different taxes. Output contract. A point, unordered set, orbit, projector, probability law, or charted section are different outputs. Aggregation contract. Uniform, average, tail, or task-weighted risk can see different parts of the obstruction.

CHAPTER 11. ARCHITECTURE OBSTRUCTION AS A SECOND ELIMINATION

72

Regularity contract. Continuous, measurable, Lipschitz, finite-memory, and neural-parameterized deployments form different classes. Structural principle An architecture tax is never stated without naming the output object, regularity class, population aggregation, and native defect.

A minimal theorem template.

A complete architecture theorem should contain four clauses.

1. Oracle clause: identify the local oracle set. 2. Obstruction clause: prove a lower bound over the declared architecture. 3. Witness clause: exhibit an architecture or deployment attaining or approaching the bound. 4. Repair clause: identify a contract change that changes the bound, with its own proof. The regular, coordination, singular, and resource theories in the next chapters instantiate this template.

Exercises Exercise 11.3. Prove monotonicity of the obstruction under architecture inclusion and under weakening of the regularity contract. Exercise 11.4. Construct two output contracts on the same oracle family for which one has zero obstruction and the other has positive obstruction. Exercise 11.5. Explain why a class-level lower bound is needed to distinguish architecture saturation from an optimizer that simply failed.

Chapter 12

Regular Obstruction Transfer and Flow Duality 12.1

The regular regime and quarter transport bound

Assume a unique oracle 𝑎★ (𝑥) and a defect with local growth D 𝑥 (𝑎) ≥

𝜇 2 𝑑 {𝑎, 𝑎★ (𝑥)}. 2

Let the architecture class consist of 𝐿-Lipschitz maps 𝐴 : X → A. If the oracle field varies faster than 𝐿, the architecture must deviate somewhere. For two points 𝑥, 𝑦, 𝑑{ 𝐴(𝑥), 𝐴(𝑦)} ≤ 𝐿𝑑X (𝑥, 𝑦). By the triangle inequality, 𝑑{𝑎★ (𝑥), 𝑎★ (𝑦)} ≤ 𝑒 𝑥 + 𝐿𝑑X (𝑥, 𝑦) + 𝑒 𝑦 , where 𝑒 𝑥 = 𝑑{ 𝐴(𝑥), 𝑎★ (𝑥)}. Hence   𝑒 𝑥 + 𝑒 𝑦 ≥ 𝑑{𝑎★ (𝑥), 𝑎★ (𝑦)} − 𝐿𝑑X (𝑥, 𝑦) + . Using 𝑢 2 + 𝑣 2 ≥ (𝑢 + 𝑣) 2 /2, strong growth gives D 𝑥 { 𝐴(𝑥)} + D 𝑦 {𝐴(𝑦)} ≥

2 𝜇 𝑑{𝑎★ (𝑥), 𝑎★ (𝑦)} − 𝐿𝑑X (𝑥, 𝑦) + . 4

Averaging over selected pairs yields an architecture lower bound. How to read the assumptions The input law 𝑃 and oracle map 𝑎★ are measurable, the oracle is single-valued in a metric space, and every deployed architecture is 𝐿-Lipschitz. The displayed expectations must be finite (or interpreted as extended nonnegative values). The native-defect conclusion additionally requires the pointwise exchange inequality D 𝑥 (𝑎) ≥ (𝜇/2)𝑑 2 {𝑎, 𝑎★ (𝑥)} with 𝜇 > 0. 73

CHAPTER 12. REGULAR OBSTRUCTION TRANSFER AND FLOW DUALITY

74

Theorem 12.1 (Oracle-variation transport bound). Let Π(𝑃, 𝑃) denote the set of couplings of 𝑃 with itself. Then 𝔒 𝑃 (𝔄 𝐿 ) ≥

 2 𝜇 sup E (𝑋,𝑌 )∼ 𝜋 𝑑{𝑎★ (𝑋), 𝑎★ (𝑌 )} − 𝐿𝑑X (𝑋, 𝑌 ) + . 8 𝜋 ∈Π ( 𝑃, 𝑃)

Equivalently, every fixed self-coupling 𝜋 supplies a valid, possibly weaker, lower certificate by omitting the supremum. Interpretation and boundary The theorem converts variation of the oracle field that exceeds what an 𝐿-Lipschitz deployment can follow into a quantitative architecture lower bound. It may be zero when the oracle variation lies within the architecture budget; it is a certificate, not a universal claim that every smooth architecture fails. Proof roadmap For each coupled pair, use the triangle inequality to lower-bound the sum of the two pointwise deployment errors by the excess oracle displacement. Square, average, and sum over edges; then use the exchange inequality to translate metric error into native objective defect. The complete constants are verified in the chapter appendix. The coefficient 𝜇/8 is fixed by the convention that 𝜋 is a probability coupling with both marginals equal to 𝑃. Alternative unnormalized edge sums require their own normalization; this does not alter the population theorem.

12.2

Finite calibration, dual flow, and population thickening

Finite calibration graphs. In data analysis the instance space is represented by a graph 𝐺 = (𝑉, 𝐸). At vertex 𝑖, an estimated local oracle is e 𝑎 𝑖 with uncertainty radius 𝜖𝑖 . Define robust edge demand   𝑏 𝑖 𝑗 = 𝑑 (e 𝑎𝑖 , e 𝑎 𝑗 ) − 𝜖 𝑖 − 𝜖 𝑗 − 𝐿𝑑𝑖 𝑗 + . Every admissible field has vertex errors 𝑒 𝑖 ≥ 0 satisfying 𝑒𝑖 + 𝑒 𝑗 ≥ 𝑏𝑖 𝑗

((𝑖, 𝑗) ∈ 𝐸).

A quadratic lower certificate is obtained by minimizing ∑︁ min 𝑤 𝑖 𝑒 2𝑖 subject to 𝑒 𝑖 + 𝑒 𝑗 ≥ 𝑏 𝑖 𝑗 . 𝑒𝑖 ≥0

𝑖

Strong defect growth converts the solution to native objective units. Dual flow. The convex program has a flow-like dual. Introduce nonnegative edge multipliers 𝜆𝑖 𝑗 . The Lagrangian is ∑︁ ∑︁ 𝑤 𝑖 𝑒 2𝑖 + 𝜆𝑖 𝑗 (𝑏 𝑖 𝑗 − 𝑒 𝑖 − 𝑒 𝑗 ). 𝑖

(𝑖, 𝑗 )

12.3. SET-VALUED ORACLES AND CERTIFICATE BOUNDARIES

75

Minimization over 𝑒 𝑖 gives 𝑒𝑖 =

1 2𝑤 𝑖

∑︁

𝜆𝑖 𝑗

𝑗:(𝑖, 𝑗 ) ∈𝐸

when the nonnegativity constraint is inactive; the complete formula uses the positive part. The dual objective becomes !2 ∑︁ ∑︁ 1 ∑︁ 𝜆𝑖 𝑗 . 𝜆𝑖 𝑗 𝑏𝑖 𝑗 − 4𝑤 𝑖 𝑗 𝑖 (𝑖, 𝑗 )

Maximizing over 𝜆 ≥ 0 yields the exact convex lower certificate in the quadratic graph model. This calculation is a direct Lagrange dual of the displayed finite program, not a claim to originate network-flow duality or order-restricted regression. Its closest classical lineage includes the leastsquares isotonic formulation and its cumulative-sum/dual characterizations [Barlow and Brunk, 1972, Rockafellar, 1984]. The contribution here is the oracle-mismatch interpretation and conversion of the optimum into native objective units. Certified decomposition The dual variables select incompatible oracle differences and charge their total demand against congestion at the vertices. The certificate is objective-denominated after multiplication by the defect-growth constant.

Population thickening. A pointwise mismatch becomes a positive population tax only if the population places mass near the mismatch. Suppose 𝐴 and 𝑎★ are regular and at 𝑥 0 the distance is at least 𝛿. If both fields are Lipschitz, the distance remains at least 𝛿/2 on a ball of radius proportional to 𝛿. A lower mass condition 𝑃{𝐵(𝑥, 𝑟)} ≥ 𝑐𝑟 𝑑 then gives a population lower bound of order 𝜇𝑐 𝛿 𝑑+2 . This step is essential. A deterministic point obstruction on a nonatomic population can otherwise have zero average cost.

12.3

Set-valued oracles and certificate boundaries

For a finite oracle cover, replace point distance by 𝑑{ 𝐴(𝑥), O (𝑥)}. Local branch transport is needed to compare oracle elements across nearby points. If the cover is separated and transport-consistent, a regular lower bound can be applied branchwise. When branches collide, the assumptions fail and the singular theory of Chapter 14 is required.

CHAPTER 12. REGULAR OBSTRUCTION TRANSFER AND FLOW DUALITY

76

What the certificate does not prove. A positive graph certificate proves that the declared baseline architecture cannot follow the estimated oracle field within its uncertainty radii. It does not by itself prove: • that the local oracle model is scientifically correct; • that an atlas will generalize; • that the defect is visible to the final task; • that the graph faithfully represents the population geometry. Each claim requires a later gate.

Exercises Exercise 12.2. Derive the two-point quarter bound carefully and track constants under equal vertex weights. Exercise 12.3. Compute the primal and dual certificates for a three-vertex path with demands 𝑏 12 = 𝑏 23 = 1 and equal weights. Exercise 12.4. Show how a lower mass condition converts a pointwise defect lower bound into a population bound. Identify where regularity of the deployed field is used.

Appendix to Chapter 12: complete proofs

77

Appendix to Chapter 12: complete proofs Proof of the oracle-variation transport bound Dependencies. The metric part uses only the triangle inequality and the 𝐿-Lipschitz architecture contract. The native-risk conclusion additionally uses the quadratic exchange inequality D 𝑥 (𝑎) ≥ (𝜇/2)𝑑A2 (𝑎, 𝑎★𝑥). Proof. Fix 𝐴 ∈ 𝔄 𝐿 and write 𝑒 𝐴 (𝑥) = 𝑑A { 𝐴(𝑥), 𝑎★𝑥 }. For every pair (𝑥, 𝑦), the triangle inequality and the Lipschitz contract imply 𝑑A (𝑎★𝑥, 𝑎★𝑦) ≤ 𝑑A (𝑎★𝑥, 𝐴(𝑥)) + 𝑑A ( 𝐴(𝑥), 𝐴(𝑦)) + 𝑑A ( 𝐴(𝑦), 𝑎★𝑦) ≤ 𝑒 𝐴 (𝑥) + 𝐿𝑑X (𝑥, 𝑦) + 𝑒 𝐴 (𝑦). Therefore 

 𝑑A (𝑎★𝑥, 𝑎★𝑦) − 𝐿𝑑X (𝑥, 𝑦) + ≤ 𝑒 𝐴 (𝑥) + 𝑒 𝐴 (𝑦).

After squaring and using (𝑢 + 𝑣) 2 ≤ 2𝑢 2 + 2𝑣 2 , for every self-coupling 𝜋 of 𝑃 𝑋 ,  2 E 𝜋 𝑑A (𝑎★𝑋 , 𝑎𝑌★ ) − 𝐿𝑑X (𝑋, 𝑌 ) + ≤ 2E 𝜋 𝑒 2𝐴 (𝑋) + 2E 𝜋 𝑒 2𝐴 (𝑌 ) = 4E𝑒 2𝐴 (𝑋), because both marginals of 𝜋 equal 𝑃 𝑋 . Taking the supremum over 𝜋 and then the infimum over 𝐴 ∈ 𝔄 𝐿 proves 1 inf E𝑒 2𝐴 (𝑋) ≥ sup E 𝜋 [· · · ] 2+ . 4 𝜋 ∈Π ( 𝑃𝑋 , 𝑃𝑋 ) 𝐴∈𝔄 𝐿 Finally, the exchange inequality gives, for every 𝐴, ED𝑋 { 𝐴(𝑋)} ≥

𝜇 2 E𝑒 (𝑋). 2 𝐴

Taking the infimum over 𝐴 and combining the two bounds yields the coefficient 𝜇/8.

□

Chapter 13

Rectangularity, Coordination Tax, and Memory 13.1

Conditional architectures and their rectangular hull

Consider a finite history tree. A deployed law 𝑄 is determined by conditional kernels 𝑞 ℎ (· | ℎ) at each history ℎ. Let 𝑃★ be an oracle law with kernels 𝑝★ℎ. At inverse temperature 𝛽, the native defect is 1 𝐷 𝛽 (𝑄; 𝑃★) = KL(𝑄∥𝑃★). 𝛽 The architecture obstruction is 𝑂 𝛽 (𝔄; 𝑃★) = inf 𝐷 𝛽 (𝑄; 𝑃★). 𝑄∈𝔄

Every deployed law satisfies impl

𝐷 𝛽 (𝑄; 𝑃★) = 𝑂 𝛽 (𝔄; 𝑃★) + Δ𝛽,𝔄 (𝑄; 𝑃★). Rectangular hull. The local kernel sets available inside 𝔄 can be freed and pasted independently across histories. Definition 13.1 (Rectangular hull). Rect(𝔄) is the class obtained by allowing every conditional kernel that occurs locally in some member of 𝔄 to be chosen independently at every history where it is admissible. Write

𝐶𝑖𝔄 (ℎ) = {𝑞 𝑖 (· | ℎ) : 𝑄 𝑞 ∈ 𝔄}

for the locally available kernels at stage 𝑖 and history ℎ. The rectangular hull removes cross-history coordination constraints while preserving the local options. Its obstruction obeys a Bellman recursion because each history can then be optimized independently, conditional on occupancy. Rectangular uncertainty sets and robust Bellman recursions 78

13.2. THE COORDINATION DECOMPOSITION

79

ℎ1 𝐶 (ℎ1 )

shared parameter or memory

ℎ2 𝐶 (ℎ2 )

Rectangular hull independent choices from each 𝐶 (ℎ𝑖 )

Original architecture: local choices are coupled

ℎ3 𝐶 (ℎ3 )

Figure 13.1: Rectangularization preserves the locally available kernels but removes the cross-history constraint that forces them to share one parameterization or memory state.

are classical in robust dynamic programming and recursive multiple-priors models [Iyengar, 2005, Nilim and El Ghaoui, 2005, Epstein and Schneider, 2003], and tractable departures from rectangularity have also been studied [Goyal and Grand-Clément, 2023]. Here the object being rectangularized is not an uncertainty set but a deployment architecture. The loss created by forced cross-history sharing is retained explicitly as a coordination tax.

13.2

The coordination decomposition

Since 𝔄 ⊆ Rect(𝔄), 𝑂 𝛽 {Rect(𝔄); 𝑃★ } ≤ 𝑂 𝛽 (𝔄; 𝑃★). Define Tcoord (𝔄; 𝑃★) = 𝑂 𝛽 (𝔄; 𝑃★) − 𝑂 𝛽 {Rect(𝔄); 𝑃★ }. 𝛽

Source result; proof not reproduced here Under a uniform positivity floor and compactness, COT proves the universal converse: the original obstruction equals the rectangular Bellman value for every full-support oracle if and only if 𝔄 is rectangular. The book uses this classification as context but proves below only the Bellman and coordination-residual decomposition needed by later chapters [Huang and Wang, 2026b]. How to read the assumptions The history tree is finite, every reference conditional kernel has full support, and 𝔄 is nonempty and compact, so its local kernel projections are compact and the displayed minima are attained. The rectangular hull allows each history to choose its local rule independently while preserving exactly the sets 𝐶𝑖𝔄 (ℎ). Theorem 13.2 (Rectangular Bellman and coordination-residual decomposition). Set 𝑊𝑚 = 0 and

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define backward values by ( 𝑊𝑖−1 (ℎ) =

min

𝑞 ∈𝐶𝑖𝔄 (ℎ)

) ∑︁ 1 KL{𝑞∥ 𝑝★𝑖 (· | ℎ)} + 𝑞(𝑦)𝑊𝑖 (ℎ, 𝑦) . 𝛽 𝑦

For a locally available kernel define 𝜌𝑖 (𝑞; ℎ) =

∑︁ 1 KL{𝑞∥ 𝑝★𝑖 (· | ℎ)} + 𝑞(𝑦)𝑊𝑖 (ℎ, 𝑦) − 𝑊𝑖−1 (ℎ). 𝛽 𝑦

Then 𝑊0 = 𝑂 𝛽 {Rect(𝔄); 𝑃★ },

𝜌𝑖 (𝑞; ℎ) ≥ 0.

Every 𝑄 𝑞 ∈ Rect(𝔄) satisfies the exact chain rule 𝐷 𝛽 (𝑄 𝑞 ; 𝑃★) = 𝑊0 +

𝑚 ∑︁

E𝑄𝑞 𝜌𝑖 {𝑞 𝑖 (· | 𝐻𝑖−1 ); 𝐻𝑖−1 }.

𝑖=1

Moreover, Tcoord (𝔄; 𝑃★) = 𝛽

min

𝑚 ∑︁

𝑄 𝑞 ∈𝔄

E𝑄𝑞 𝜌𝑖 {𝑞 𝑖 (· | 𝐻𝑖−1 ); 𝐻𝑖−1 }.

𝑖=1

Consequently every 𝑄 ∈ 𝔄 obeys impl

𝐷 𝛽 (𝑄; 𝑃★) = 𝑂 𝛽 {Rect(𝔄); 𝑃★ } + Tcoord (𝔄; 𝑃★) + Δ𝛽,𝔄 (𝑄; 𝑃★). 𝛽 Interpretation and boundary The exact defect field separates local conditional inadequacy from the extra cost of forcing one shared parameter, memory state, or network to coordinate all histories. The coordination tax is measured in the native forward-KL/Bellman scale. Rectangularization is an analytical relaxation; it is not itself the deployed architecture. Proof roadmap Use the conditional KL chain rule to express every architecture loss as an expected sum of local Bellman residuals. Minimize first over the rectangular hull to obtain the local obstruction, then add and subtract that value inside the original architecture infimum. The chapter appendix gives the induction and exact decomposition. The theorem assigns a native cost to parameter sharing. A shared network couples decisions across histories; it is not merely a smaller list of local functions. Recent work studies when shared representations help in multitask learning [Li et al., 2025], how learned representations affect offline transfer RL [Bose et al., 2025], and how finite memory creates return error in partially observed RL [Eberhard et al., 2025]. Those results motivate the same deployment pressure. The theorem here isolates a narrower quantity: the exact native cost of cross-history coupling after the locally available kernels have been fixed. Rectangularization is also distinct from operation C in the P/G/X/V/C calculus of Chapter 6. It frees admissible historywise choices; C keeps the declared marginals fixed and changes only their joint coupling.

13.2. THE COORDINATION DECOMPOSITION

81

Local Bellman residuals. The rectangular dynamic program generates a nonnegative residual at each history. Under the occupancy induced by 𝑄, the expected sum of local residuals equals Tcoord + Δimpl .

This produces a diagnostic map: the global tax can be localized to histories where sharing is most expensive. A shared-kernel example. Suppose the same kernel 𝑞 is used at several histories with strictly positive oracle kernels 𝑝★ℎ on a common finite output set. Let the fixed occupancy weights satisfy 𝑤 ℎ ≥ 0 and ∑︁ 𝑊= 𝑤 ℎ > 0. ℎ

Assume also that each 𝑝★ℎ is locally admissible when histories are freed, so the corresponding rectangular obstruction is zero. The shared optimization problem is 1 ∑︁ 𝑤 ℎ KL(𝑞∥ 𝑝★ℎ). 𝑞 ∈Δ( Y ) 𝛽 ℎ inf

Define 𝑍𝑊 =

∑︁ Ö 𝑧

𝑝★ℎ (𝑧) 𝑤ℎ /𝑊 ,

ℎ

−1 𝑞★ (𝑧) = 𝑍𝑊

Ö

𝑝★ℎ (𝑧) 𝑤ℎ /𝑊 .

ℎ

Direct expansion gives the exact identity 𝑊 𝑊 1 ∑︁ 𝑤 ℎ KL(𝑞∥ 𝑝★ℎ) = KL(𝑞∥𝑞★) − log 𝑍𝑊 . 𝛽 ℎ 𝛽 𝛽 Hence the minimum, and therefore the exact fixed-occupancy shared-kernel coordination tax under the declared weights, is 𝑊 − log 𝑍𝑊 . 𝛽 This is not the full endogenous coordination tax unless a separate occupancy or fixed-point theorem shows that the weights 𝑤 ℎ are the occupancies induced by the optimizing shared law. When occupancies vary with 𝑞, they must remain inside the architecture-level optimization. Generalized Hölder gives 0 < 𝑍𝑊 ≤ 1, so this tax is nonnegative; it vanishes exactly when all oracle kernels carrying positive weight coincide. When the weights are normalized, 𝑊 = 1, this reduces to the usual normalized geometric-pooling formula [Genest and Zidek, 1986].

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82

13.3

Memory compression as base change

Rectangularization removes cross-history coupling; memory compression changes which histories are distinguishable before that coupling is assessed. The next identity treats this compression as a base change, and the source box states which results are imported. Source result; proof not reproduced here The memory base-change identity, canonical-fiber saturation criterion, approximate-saturation bound, and dequantization refinements summarized in the remainder of this chapter are established in COT. They are not counted among the principal book results; the chapter appendix proves the rectangular Bellman and coordination-residual theorem above [Huang and Wang, 2026b]. Within that imported scope, let a fine history ℎ map to a compressed memory state 𝑚 = 𝜌(ℎ). The KL chain rule decomposes a fine law into a coarse memory law and conditional fiber laws. At architecture level, 𝑂 fine = inf {𝑂 coarse (𝑄 𝑀 ) + Ψ(𝑄 𝑀 )} , 𝑄𝑀

where Ψ(𝑄 𝑀 ) is the minimum fine conditional realization tax compatible with the coarse law. A memory representation can preserve coarse objective values while failing to realize the canonical fine conditional kernels. This is objective faithfulness without obstruction faithfulness. Saturation and exact base change. Universal equality between fine and coarse architecture obstructions holds when the fine architecture is saturated by the canonical oracle fiber kernels. Approximate saturation yields an additive upper bound. This is stronger than saying that the coarse statistic is sufficient for one fixed oracle; it is an architecture-level statement over the declared class.

13.4

Zero temperature

As 𝛽 → ∞, soft KL-regularized functionals may converge to hard costs. For fixed architecture classes, uniform law-level dequantization passes to architecture minima. For varying classes, Γ-convergence and inner/outer limit conditions are needed. The zero-temperature limit can remain randomized. Temperature removal alone does not force a deterministic architecture. The admissible class and its closure determine the hard limit. Boundary of the claim Rectangularity is a property of conditional-kernel pasting. It is not equivalent to network width, and it is not automatically restored by more training data.

13.4. ZERO TEMPERATURE

83

Exercises Exercise 13.3. Derive the geometric-pooling solution for general weights and verify its normalizedweight specialization. Exercise 13.4. Give a nonrectangular architecture with two histories and compute its coordination tax explicitly. Exercise 13.5. Construct a coarse memory map that preserves one-step action distributions but loses a future-relevant conditional distinction.

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CHAPTER 13. RECTANGULARITY, COORDINATION TAX, AND MEMORY

Appendix to Chapter 13: complete proofs Proof of the rectangular Bellman and coordination-residual decomposition Dependencies. The finite history tree and strict positivity of the reference kernels make every local forward-KL term well defined. Compactness gives existence of the local Bellman minima and of the architecture minimum. The exact algebra uses the forward-KL chain rule. Proof. Let 𝑄 𝑞 be the path law generated by a conditional-kernel array 𝑞, and let 𝑃★ be the strictly positive oracle path law. The chain rule for forward relative entropy gives   𝑚 ∑︁ 1 1 𝑞 ★ ★ KL(𝑄 ∥𝑃 ) = E𝑄 𝑞 KL{𝑞 𝑖 (· | 𝐻𝑖−1 ) ∥ 𝑝 𝑖 (· | 𝐻𝑖−1 )} . 𝛽 𝛽 𝑖=1 In Rect(𝔄) the locally available kernel at every history can be selected independently. Backward induction therefore defines values 𝑊𝑚 = 0 and ( ) ∑︁ 1 ★ 𝑊𝑖−1 (ℎ) = min KL{𝑞∥ 𝑝 𝑖 (· | ℎ)} + 𝑞(𝑦)𝑊𝑖 (ℎ, 𝑦) , 𝑞 ∈𝐶𝑖𝔄 (ℎ) 𝛽 𝑦 and yields 𝑊0 = 𝑂 𝛽 {Rect(𝔄); 𝑃★ }. Define the coordination tax by Tcoord (𝔄; 𝑃★) = 𝑂 𝛽 (𝔄; 𝑃★) − 𝑂 𝛽 {Rect(𝔄); 𝑃★ }. 𝛽

It is nonnegative because 𝔄 ⊆ Rect(𝔄). For any 𝑄 ∈ 𝔄, add and subtract 𝑂 𝛽 (𝔄; 𝑃★): 𝐷 𝛽 (𝑄; 𝑃★) = 𝑂 𝛽 (𝔄; 𝑃★) + {𝐷 𝛽 (𝑄; 𝑃★) − 𝑂 𝛽 (𝔄; 𝑃★)} impl

= 𝑂 𝛽 {Rect(𝔄); 𝑃★ } + Tcoord (𝔄; 𝑃★) + Δ𝛽,𝔄 (𝑄; 𝑃★). 𝛽 This is the claimed decomposition. Define the local Bellman residual 𝜌𝑖 (𝑞; ℎ) =

∑︁ 1 KL{𝑞∥ 𝑝★𝑖 (· | ℎ)} + 𝑞(𝑦)𝑊𝑖 (ℎ, 𝑦) − 𝑊𝑖−1 (ℎ). 𝛽 𝑦

It is nonnegative for every locally available kernel by Bellman optimality. For every 𝑄 𝑞 ∈ Rect(𝔄), expanding and telescoping the value terms gives 𝐷 𝛽 (𝑄; 𝑃★) = 𝑊0 +

𝑚 ∑︁

E𝑄 𝜌𝑖 {𝑞 𝑖 (· | 𝐻𝑖−1 ); 𝐻𝑖−1 }.

𝑖=1

Subtracting 𝑊0 and minimizing over 𝑄 ∈ 𝔄 identifies the minimum expected residual sum with the coordination tax. Adding and subtracting the architecture minimum for an arbitrary 𝑄 ∈ 𝔄 leaves impl exactly Δ𝛽,𝔄 (𝑄; 𝑃★), proving the final three-term decomposition. □

Chapter 14

Singular Fibers, Monodromy, and Catastrophe Taxes The regular theory assumes a unique oracle or a finite separated cover. At a discriminant, branches collide, the vertical Hessian loses rank, and the oracle incidence map becomes stratified. Distance-tobranch arguments can degenerate precisely where the topology changes. Here monodromy is the permutation of locally tracked oracle branches after continuation around a loop; when the declared output contract cannot absorb that permutation, the resulting native loss is a catastrophe tax. The singular theory begins from a normal form whose native residual remains meaningful even when curvature vanishes. 𝑣(2𝜋) = −𝑣(0) or a branch permutation 𝑣𝑣 ⊤ or an unordered set returns to itself one circuit in input space

Figure 14.1: Monodromy depends on the output contract. A point-valued representative can fail to return to itself after a loop, while a quotient-valued object such as a projector or unordered root set can remain globally well defined.

85

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86

14.1

The radical normal form

Let 𝐻 𝜃 (𝑎) = |𝑎 𝑘 − 𝑟𝑒 𝑖ℓ 𝜃 | 2 ,

𝑎 ∈ C,

𝜃 ∈ 𝑆1 ,

with 𝑘 ≥ 2, 𝑟 > 0, and ℓ ∈ Z. Each fiber has 𝑘 zero-defect roots. A continuous root selection exists exactly when 𝑘 divides ℓ. How to read the assumptions The input is the circle, the deployment is a continuous single-valued complex function, the loss is uniform, 𝑘 ≥ 2, and 𝑟 > 0. The exponent ℓ determines the winding of the oracle fiber. Changing to average risk, measurable sections, multiple charts, or quotient-valued outputs changes the problem. Theorem 14.1 (Exact radical catastrophe tax). ( inf

𝐴∈𝐶 (𝑆 1 ,C)

sup | 𝐴(𝜃) 𝑘 − 𝑟𝑒 𝑖ℓ 𝜃 | 2 = 𝜃 ∈𝑆 1

0, 𝑟 2,

𝑘 | ℓ, 𝑘 ∤ ℓ.

Interpretation and boundary When 𝑘 does not divide ℓ, no continuous point-valued selector can keep the residual strictly below 𝑟 2 everywhere, and the zero deployment attains exactly that value. The result is a sharp worst-case architecture tax. It does not by itself imply a positive unrestricted nonatomic average-risk lower bound. Proof roadmap Assume a deployment has loss strictly below 𝑟 2 . Its image avoids zero, so its phase has an integer winding number. The map 𝐴 𝑘 then has winding divisible by 𝑘, but closeness inside the circle centered at the target forces winding ℓ, a contradiction unless 𝑘 | ℓ. The matching upper bound is 𝐴 ≡ 0. The full homotopy argument is in the chapter appendix. The upper witness lies on the collision stratum. The exact constant survives without strong convexity or branch separation and vanishes continuously as the target loop approaches the discriminant.

14.2. RISK LEVEL, GENERAL LOOPS, AND DEGREE

14.2

87

Risk level, general loops, and degree

Uniform versus average risk. For a nonatomic sampling law on the circle, the infimum average defect over continuous fields is zero even when the uniform tax is 𝑟 2 . A selector can follow one branch everywhere except on a narrow seam whose measure tends to zero. Thus the theorem is a uniform deterministic architecture tax. A positive average tax requires an additional regularity, lower-mass, or complexity condition preventing seam concentration.

General loops and degree. For a nonvanishing target loop 𝑧(𝜃), winding incompatibility yields two-sided catastrophe bounds controlled by radial clearance from zero. In higher dimensions, let 𝐹 : R𝑑 → R𝑑 be positively homogeneous and consider its normalized map on the sphere. If the target degree is incompatible with the degree of 𝐹/|𝐹 |, a constant-radius target sphere has an exact residual floor analogous to the radical case. The important pattern is: homotopy class incompatible with the output map =⇒ native residual at least the distance to the discriminant.

14.3

Beyond radicals: roots, eigenlines, and partial monodromy

Polynomial roots and critical values. Let 𝑝 be a complex polynomial and let a target loop avoid the critical-value discriminant. Its roots form a covering with monodromy. If the monodromy has no fixed branch, any continuous root-valued deployment must incur residual at least the target loop’s distance to the critical-value set. Near a ramification point of order 𝑚, analytic coordinates reduce the problem to the radical normal form. Using discriminant complements, monodromy, and braid structure to obstruct continuous algorithms has a classical topology-of-algorithms lineage [Smale, 1987, Vassiliev, 1988]. The additional claim here is objective-denominated: the native residual and clearance from the discriminant turn that qualitative obstruction into a numerical loss floor. Eigenline catastrophe. Consider a real symmetric 2 × 2 matrix family around a conical crossing. The top eigenline forms a projective bundle with odd holonomy. An oriented continuous unit eigenvector must fail after one loop and pays an exact Rayleigh tax. The rank-one projector 𝑃 = 𝑣𝑣 ⊤ is invariant under 𝑣 ↦→ −𝑣 and can remain continuous with zero defect. This example demonstrates that the obstruction may be a representation artifact even when the underlying geometric object is perfectly regular. It sits beside modern impossibility results for continuous canonicalization and topological bottlenecks in learned representations [Dym et al., 2024, Batson et al., 2021, Esmaeili et al., 2023]. The exact Rayleigh tax here is a more specific statement in the native spectral loss, not a consequence claimed from those adjacent results.

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88

Missing branches versus monodromy. Monodromy permutes branches that exist throughout a loop. A missing branch is undefined on part of the path. The two phenomena require partial rather than total transport maps. Forcing every edge transport to be a permutation can manufacture spurious cycles and false global inconsistency. The transport construction in Chapter 23 therefore treats transports as partial bijections and audits both fixed-point behavior and domain survival. Partial maps and their synchronization are established in multi-object matching; the distinction developed here is the oracle/task meaning assigned to disappearance and domain survival, as documented with prior art in Chapter 23.

14.4

Stability

The exact tax is stable under perturbations that preserve the homotopy class and maintain positive clearance from the discriminant. When the target approaches the discriminant, the tax can vanish. This is the correct singular behavior: topology remains incompatible, but the objective price of incompatibility becomes arbitrarily small. Boundary of the claim Topology alone does not supply a fixed numerical risk floor. The price depends on the native defect and the target’s geometric clearance from the singular set.

Exercises Exercise 14.2. Prove the radical theorem for 𝑘 = 2 and ℓ = 1 using only the argument principle. Exercise 14.3. Construct a sequence of continuous fields whose average radical defect tends to zero under the uniform measure, while the maximum defect remains at least 𝑟 2 . Exercise 14.4. For a loop of eigenlines with sign holonomy, explain why a projector-valued output is a quotient repair rather than a higher-capacity approximation.

Appendix to Chapter 14: complete proofs

89

Appendix to Chapter 14: complete proofs Proof of the exact radical catastrophe tax Dependencies. The proof uses continuity on the circle and the integer winding number of loops in C★. The lower bound is uniform; it does not imply a positive unrestricted average-risk lower bound. Proof. If 𝑘 | ℓ, write ℓ = 𝑘𝑚 and define 𝐴(𝜃) = 𝑟 1/𝑘 𝑒 i𝑚𝜃 . This is a continuous loop and satisfies 𝐴(𝜃) 𝑘 = 𝑟𝑒 iℓ 𝜃 , so the uniform defect is zero. Assume now that 𝑘 ∤ ℓ. Suppose, toward a contradiction, that a continuous field 𝐴 : 𝑆 1 → C satisfies | 𝐴(𝜃) 𝑘 − 𝑟𝑒 iℓ 𝜃 | < 𝑟 for every 𝜃. The inequality implies 𝐴(𝜃) 𝑘 ≠ 0. Consider the straight-line homotopy 𝐻𝑠 (𝜃) = (1 − 𝑠)𝑟𝑒 iℓ 𝜃 + 𝑠 𝐴(𝜃) 𝑘 ,

0 ≤ 𝑠 ≤ 1.

For every (𝑠, 𝜃), |𝐻𝑠 (𝜃) − 𝑟𝑒 iℓ 𝜃 | = 𝑠| 𝐴(𝜃) 𝑘 − 𝑟𝑒 iℓ 𝜃 | < 𝑟, so 𝐻𝑠 (𝜃) lies in the open disk of radius 𝑟 centered at 𝑟𝑒 iℓ 𝜃 ; that disk does not contain the origin. Hence the loops 𝐴 𝑘 and 𝑟𝑒 iℓ 𝜃 are homotopic in C★ and have the same winding number. But wind( 𝐴 𝑘 ) = 𝑘 wind( 𝐴)

and

wind(𝑟𝑒 iℓ 𝜃 ) = ℓ,

which would imply 𝑘 | ℓ, a contradiction. Therefore every continuous 𝐴 has some 𝜃 with residual magnitude at least 𝑟, and hence uniform squared defect at least 𝑟 2 . The constant field 𝐴 ≡ 0 has squared defect exactly |0 − 𝑟𝑒 iℓ 𝜃 | 2 = 𝑟 2 for all 𝜃. Thus the lower bound is attained.

□

Chapter 15

Atlases, Quotients, and Randomized Repair 15.1

From obstruction diagnosis to a repair contract

A positive obstruction does not imply that the original task is impossible. It means that the declared architecture contract is incompatible with the oracle geometry. A repair changes the contract in a controlled way. The principal repair families are: 1. local atlases; 2. quotient-valued outputs; 3. set-valued outputs; 4. probability-valued outputs or randomized point rules; 5. memory refinement or rectangularization; 6. nonlinear semantic projection; 7. test-time local refinement. The correct repair is the smallest contract change that removes the certified obstruction while preserving downstream information. This criterion rules out a flat menu of interchangeable fixes. Some obstructions arise because one representative cannot be chosen globally; others arise from forced sharing, an invalid semantic interface, or an insufficient deployment computation. The remainder of the chapter follows these mechanisms and then returns to the statistical problem of selecting among their repairs.

90

15.2. LOCAL AND SYMMETRY-RESPECTING REPAIR

15.2

91

Local and symmetry-respecting repair

Let {𝑈 𝑗 } 𝐾𝑗=1 cover X. On each chart 𝑈 𝑗 , suppose a continuous oracle section 𝑎★𝑗 exists. A hard atlas architecture consists of a gate 𝑔(𝑥) and local experts 𝐴 𝑗 (𝑥). The minimum number of local sections needed to cover a fibration is closely related to sectional category and Schwarz genus [Schwarz, 1961]; the theorem below uses only the existence of a declared cover and does not claim a minimal chart number. Multi-chart representation learning and chart-based autoencoders provide a separate machine-learning lineage [Schonsheck et al., 2019, Kalatzis et al., 2021, Stolberg-Larsen and Sommer, 2021]. In this chapter the equivalence between a cover by at most 𝐾 local sections and a 𝐾-chart atlas is a definitional inversion. The theorem below is only the witness lemma that such declared sections saturate the atlas class; it does not claim to invent multi-chart models or to learn the cover [Huang and Wang, 2026l]. How to read the assumptions The input space is covered by declared charts, each chart admits a legitimate local zero-defect section, and the local expert class contains that section. A measurable selector chooses one chart at deployment time. The theorem is existential and does not assert that the cover, selector, or local experts can be estimated from finite data. Theorem 15.1 (Atlas saturation). Suppose the declared charts form a measurable cover, the hard-atlas grammar contains a measurable gate 𝑔 satisfying 𝑥 ∈ 𝑈𝑔 ( 𝑥 ) , and every local expert class contains an exact oracle section on its chart. Then the hard atlas class has zero uniform obstruction: inf

(𝑔, 𝐴1 ,..., 𝐴𝐾 )

sup D 𝑥 { 𝐴𝑔 ( 𝑥 ) (𝑥)} = 0. 𝑥

Interpretation and boundary A positive obstruction of a single global point-valued architecture can disappear after changing the architecture contract to an atlas. The theorem explains why targeted multi-chart repair can be qualitatively different from merely widening one global network. It does not establish minimal chart number or gating generalization. Proof roadmap Choose on every chart its exact local oracle section and deploy the section selected by the measurable gate. At each input the selected local section has zero native defect. The chapter appendix states the short verification and its precise measurability boundary. The theorem is existential: it neither learns the cover nor proves that the chosen chart count is minimal. An atlas retains point-valued representatives but localizes where they must be chosen. When the labels themselves carry no downstream meaning, the more economical repair is instead to change the output object. Suppose a symmetry group 𝐺 acts on the auxiliary fiber and the defect is invariant: D 𝑥 (𝑔𝑎) = D 𝑥 (𝑎).

CHAPTER 15. ATLASES, QUOTIENTS, AND RANDOMIZED REPAIR

92

If the downstream task depends only on the orbit [𝑎] ∈ A/𝐺, then the quotient is the natural output. A global quotient section may exist even when no global representative section does. Examples include: • eigenprojectors instead of oriented eigenvectors; • unordered mixture components instead of labeled lists; • subspaces instead of bases; • equivalence classes of gauges instead of coordinates. For the radical family, the full root set is globally defined. A set-valued output can therefore achieve zero defect. Likewise, the uniform distribution over roots is invariant under monodromy. Probability-valued versus randomized repair. A probability-valued output returns a law 𝐾 𝑥 and therefore requires a task and loss defined on laws. A randomized point-valued deployment instead draws 𝐴 ∼ 𝐾 𝑥 and, under expected native loss, pays ∫ D 𝑥 (𝑎) 𝐾 𝑥 ( d𝑎). The two contracts ∫ remain distinct even when they use the same kernel. Neither is the deterministic ambient average 𝑎 𝐾 𝑥 ( d𝑎), when that average is defined; the average may leave the oracle set. These repairs change downstream semantics. A task that needs one labeled root may still require a chart or additional side information. A symmetric task can operate directly on the set or law. The quotient, set-valued, and probability-valued constructions also explain why a superficially simpler aggregation can be unsafe. A tempting response to multiple local branches is to average representatives. In the square-root example, averaging 𝑎 and −𝑎 gives zero, which lies on the discriminant and pays the full catastrophe tax. Ambient averaging can therefore destroy oracle information. Safe aggregation must respect the quotient or use a native barycenter under the declared defect. Thus atlas and quotient repair solve different problems: the former permits several coherent local representatives, whereas the latter removes distinctions that the task is not entitled to observe. Neither authorizes an arbitrary average in the ambient representation space.

15.3

Repairing deployment mechanisms

In sequential systems, the repair may be to enlarge the memory state so that histories requiring different local kernels no longer collide. Alternatively, one can relax shared parameters and allow independent local kernels, moving toward the rectangular hull. The benefit is measured by the coordination tax removed. The cost is increased state, memory, communication, or generalization complexity. Forced sharing is not the only mechanism-level failure. A legal linear calculation may also leave the semantic output set, in which case the repair belongs at the validity interface rather than in the memory state. The graph-CDF example uses a different repair. The raw high-order field is projected onto the valid semantic set. This changes neither the output type nor the local oracle family, but it replaces an invalid linear mechanism by a nonlinear interface.

15.4. SELECTION, VALIDATION, AND REPAIR FRONTIERS

93

The projection is justified only because it is nonexpansive in the native risk geometry. An arbitrary clipping or renormalization step requires its own theorem. Deployment computation provides a third mechanism-level intervention. A semi-amortized system starts from a shared prediction and performs local optimization at test time. This can reduce implementation error and, if the allowed iteration budget is part of the architecture, enlarge the effective class. Laboratory 24.2.1 audits this distinction on MNIST by declaring the zero-step and refined deployment classes before evaluating the held-out native objective. Test-time refinement is not equivalent to atlas repair. It may still follow the wrong labeled branch or remain trapped by a representation obstruction. The repair must be matched to the certificate. Memory refinement, semantic projection, and test-time optimization therefore alter different clauses of the contract. Their costs cannot be compared by parameter count alone; each must be charged in the resource and loss scale declared for deployment.

15.4

Selection, validation, and repair frontiers

Suppose a finite library {𝔄 (1) , . . . , 𝔄 ( 𝑀 ) } contains candidate repairs. Calibration data produce lower and upper obstruction certificates; tuning data train the candidates; test data compare their operational performance. This separation avoids selecting and certifying the same architecture on the same random fluctuations. Structural principle A repair is justified by a chain of evidence: positive baseline obstruction, mechanism-specific diagnosis, zero or smaller repaired obstruction, and independent held-out improvement. For atlas repairs, this decision can be recorded as a frontier rather than a single winning chart count. For tolerance 𝜏, define the minimum number of charts required to attain uniform defect at most 𝜏. The resulting integer-valued atlas curve can jump. Its inverse—the best loss attainable with 𝐾 charts—is often more stable under perturbation and is the preferred statistical object. The frontier keeps the structural and statistical questions separate. It records what each chart budget can realize, while sample splitting determines which point on that frontier is supported by the available evidence.

Exercises Exercise 15.2. For the square-root family, compare a two-chart repair, an unordered-pair output, and a uniform root-valued law. Which downstream tasks distinguish them? Exercise 15.3. Give a symmetric-orbit example in which ambient averaging leaves the oracle set and incurs positive native defect. Exercise 15.4. Design a sample-split protocol for choosing among three architectures: single-chart, two-chart, and quotient-valued.

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Appendix to Chapter 15: complete proofs Proof of atlas saturation Dependencies. The cover must be genuine, the gate must route only to charts containing the input, and each local expert class must contain an exact local oracle section. No learning or minimal-chart claim is involved. Proof. For each chart 𝑈 𝑗 , let 𝑎★𝑗 : 𝑈 𝑗 → A be an exact local oracle section contained in the declared expert class, so that D 𝑥 {𝑎★𝑗 (𝑥)} = 0 (𝑥 ∈ 𝑈 𝑗 ). Because the charts cover X, choose a measurable selector 𝑔(𝑥) with 𝑥 ∈ 𝑈𝑔 ( 𝑥 ) ; for a finite open cover on a standard Borel space, one may take the smallest chart index containing 𝑥. Set 𝐴 𝑗 = 𝑎★𝑗. Then for every 𝑥, D 𝑥 { 𝐴𝑔 ( 𝑥 ) (𝑥)} = 0. Therefore the supremum over 𝑥 is zero. Since defects are nonnegative, no architecture can attain a negative value, and the infimum over the atlas class is exactly zero. □

Part IV Historical and Conceptual Notes Three obstruction mechanisms, three literatures Regular transport bounds combine Lipschitz approximation, couplings, and curvature exchange. Rectangular Bellman recursion belongs to robust control and recursive multiple-priors theory [Epstein and Schneider, 2003, Iyengar, 2005, Nilim and El Ghaoui, 2005]. Loop lifting, winding, degree, discriminants, and Schwarz genus are classical topology and singularity theory [Hatcher, 2002, Milnor, 1997, Schwarz, 1961]. Part IV does not relabel these ingredients as discoveries of elimination geometry. Its organizing distinction is the deployment quantifier. A regular architecture may be unable to track an oracle that varies too quickly; a shared conditional architecture may be unable to paste locally available kernels; and a continuous point-valued output may be unable to choose one branch around a singular loop. These failures demand different witnesses and different repairs. Regime

Classical engine

Regular variation

Lipschitz comparison and native oracle- variation more regularity budget, lotransport lower certificate cal refinement, or additional charts rectangular Bellman recur- exact cost of cross-history memory refinement, rectsion and KL chain rule sharing in forward-KL units angularization, or a different sharing rule winding, monodromy, de- uniform objective- atlas, quotient, set-valued, gree, genus denominated catastrophe or randomized output tax

Coordination

Singular topology

Book/program quantity

Matched repair

The rectangular residual decomposition is a source-program architecture specialization of classical rectangularity. Atlas saturation is a direct witness lemma. The exact radical value 𝑟 2 is a useful quantitative winding corollary in native-objective units; it is not presented as a new topological mechanism.

Part IV Historical and Conceptual Notes From obstruction to resources, tasks, and composition A positive architecture obstruction is not yet an operational conclusion. Part V separates carrier information loss from decoder limitations, asks which distinctions legal downstream tasks can observe, and states when certified defect decompositions remain valid across conditioning, representations, and temperature limits.

95

Part V

Resources, Operational Semantics, and Composition Resources determine which oracle distinctions can be retained; task and context classes determine which surviving distinctions are operationally visible.

96

Chapter 16

Resource-Constrained Architecture Rate–Distortion 16.1

The architecture as carrier and decoder

Here carrier means the deployment information state passed to an action decoder. It is not the reduced slack carrier of Section 10.3, which removes representation gauge before deployment is considered. Section 10.4 supplies the conditional extraction gate from a quotient-reduced lift to a deployment carrier; without such a gate the two remain distinct. An architecture first produces a carrier 𝑍 and then decodes it: 𝜙

𝑔

𝑋− →𝑍→ − 𝐴. The terminology deliberately recalls rate–distortion theory, where a source is compressed subject to a fidelity criterion [Shannon, 1959, Cover and Thomas, 2006]. Here the distortion is not chosen independently of the learning problem: it is the native objective defect, and the resource grammar may constrain both the carrier and its decoder. Across inputs, the oracle map 𝑥 ↦→ 𝑎★ (𝑥) determines which distinctions a deployment carrier must preserve. Within each fiber, the native defect determines the price of merging those distinctions. The carrier need not retain all information in 𝑥; it need only be sufficient for the oracle signature. Conversely, width, state count, or cone size is not effective capacity unless the retained state preserves those signatures and makes them decodable. A resource grammar indexed by (𝑚, 𝑟) restricts carrier size, number of labels, communication, coherence, smoothness, or decoder complexity. Write 𝔊𝑚,𝑟 for its declared class of admissible carrier–decoder pairs (𝑍, 𝑔), and put Z𝑚,𝑟 = {𝑍 : there exists 𝑔 with (𝑍, 𝑔) ∈ 𝔊𝑚,𝑟 }. The architecture can fail in two ways. 1. The carrier erases distinctions among oracle signatures. 2. The carrier retains the information, but the decoder class cannot realize the optimal readout.

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CHAPTER 16. RESOURCE-CONSTRAINED ARCHITECTURE RATE–DISTORTION

98

16.2

Canonical Bregman carrier

Reuse the dual oracle signature 𝑆 = ∇Φ{𝑎★ (𝑋)} from Section 10.4. For a carrier 𝑍, put 𝑆 𝑍 = E(𝑆 | 𝑍). The optimal unconstrained decoder is 𝑎 †𝑍 = ∇Φ★ (𝑆 𝑍 ). This follows from conditional Bregman projection; conditional expectation as the optimal Bregman predictor and the associated centroid identity are classical [Banerjee et al., 2005a]. The resource split below specializes that identity to a carrier–decoder grammar [Huang and Wang, 2026a]. The carrier information loss is CΦ (𝑍) = E𝐷 Φ {𝑎 †𝑍 (𝑍) ∥𝑎★ (𝑋)}. For a carrier admitted by the grammar, define decoder nonsaturation by Ψ𝑚,𝑟 (𝑍) =

inf

𝑔:(𝑍 ,𝑔) ∈𝔊𝑚,𝑟

E𝐷 Φ {𝑔(𝑍) ∥𝑎 †𝑍 (𝑍)}.

The infimum is +∞ if the carrier has no admissible decoder. The native architecture distortion is the best original objective defect among admissible pairs, Dnat (𝑚, 𝑟) =

inf

(𝑍 ,𝑔) ∈𝔊𝑚,𝑟

E𝐷 Φ {𝑔(𝑍) ∥𝑎★ (𝑋)}.

The divergence orientation is chosen to match the exact Bregman Pythagorean identity. How to read the assumptions The native defect is generated by a Legendre potential on a common fiber, the oracle is unique, the dual signature is integrable, and conditional dual means are well defined. The potential and pairing terms used in the conditional calculation are integrable for every finite-cost admissible decoder. Resource constraints are encoded in the declared carrier and decoder classes. The decomposition is an identity between infima and does not require attainment; a statement about one exact zero-distortion carrier–decoder pair additionally requires that such a pair attain the infimum. Theorem 16.1 (Canonical architecture rate–distortion decomposition). Under the common-fiber Legendre assumptions,  Dnat (𝑚, 𝑟) = inf CΦ (𝑍) + Ψ𝑚,𝑟 (𝑍) . 𝑍 ∈ Z𝑚,𝑟

Interpretation and boundary The architecture distortion separates exactly into information lost by the carrier and nonsaturation of the resource-limited decoder. When the infimum is attained, zero total distortion requires both task-relative sufficiency and a decoder capable of realizing the canonical conditional oracle. A raw bit, dimension, or label count becomes meaningful only after an extraction theorem links it to these classes.

16.3. DISCRETE AND CONTINUOUS CARRIER EXTRACTION

99

Proof roadmap Condition the Bregman defect on the carrier, insert the conditional dual-mean decoder, and use the Bregman Pythagorean identity. Minimize the second term over decoders paired with that carrier and then over admissible carriers. The chapter appendix proves the decomposition and its zero-distortion criterion. The formula is a second elimination over carriers. Width or parameter count is only one possible resource coordinate; the canonical retained state is the conditional dual oracle signature. The resource grammar is upstream of the P/G/X/V/C calculus in Chapter 6: it declares the feasible architecture within which an operation is performed. Changing that grammar is not a sixth mode.

16.3

Discrete and continuous carrier extraction

Hard carrier extraction. Suppose 𝑍 takes at most 𝑚 labels and a weighted Potts coherence budget limits label changes on a graph. The inverse minimum 𝑞-cut profile gives the exact number of usable labels under the budget. On paths and trees this capacity has a closed form; related cardinalityconstrained and multiway-cut algorithms are classical [Goldschmidt and Hochbaum, 1994, Frederickson and Solis-Oba, 1998, Engelberg et al., 2007]. Precisely, for a weighted graph 𝐺 = (𝑉, 𝐸, 𝑤) let 𝑐(𝐺 \ 𝐹) denote the number of connected components after deleting 𝐹 ⊆ 𝐸, and define ( ) ∑︁ 𝜆 𝑞 (𝐺) = min 𝑤 𝑒 : 𝐹 ⊆ 𝐸, 𝑐(𝐺 \ 𝐹) ≥ 𝑞 , 𝜆1 (𝐺) = 0. 𝑒∈𝐹

The hard-label capacity under label budget 𝑚 and cut budget 𝑟 is  hard 𝜅𝐺 (𝑚, 𝑟) = min 𝑚, max{𝑞 : 𝜆 𝑞 (𝐺) ≤ 𝑟 } . It is exact for the declared Potts grammar: every feasible labeling is constant on the components left by its cut set, and conversely those components may be assigned distinct labels up to the cap 𝑚. If oracle signatures are well separated by type, the visible cardinality bound is attained. Strong convexity then converts limited label capacity into a quantization floor. Continuous carriers. For a normalized continuous conic carrier with a Lipschitz readout and quadratic graph coherence, metric entropy controls the number of distinguishable oracle signatures at resolution 𝜖. The matching lower exponent requires a co-Lipschitz readout on an interior region and a resource grammar admitting arbitrarily dense deployments in that region; the finite-site capacity cap still applies. This produces a resource-to-information map of the form   𝐶 (𝑚, 𝑟) log 𝑁 (𝜖) ≲ 𝑑 log 1 + . 𝜖 Strong defect growth converts a covering-radius lower bound into native distortion.

100

16.4

CHAPTER 16. RESOURCE-CONSTRAINED ARCHITECTURE RATE–DISTORTION

Finite frontiers and phase diagrams

The finite Bregman quantization frontier. For a finite oracle set with weights 𝑝 𝑖 and codebook size 𝑞, the optimal Bregman quantization error is min

∑︁

𝑐1 ,...,𝑐𝑞

𝑝 𝑖 min 𝐷 Φ (𝑐 𝑗 ∥𝑎★𝑖). 𝑗

𝑖

Within each assigned cell the optimal codeword is the dual-coordinate centroid. Separated signatures give a positive lower bound whenever 𝑞 is smaller than the number of distinguishable types. This is the usual vector quantization problem in the native Bregman geometry [Gray and Neuhoff, 1998, Graf and Luschgy, 2000, Friedrich et al., 2008].

Resource phase diagrams. On an 𝑛-vertex path whose edges all have the common weight 𝑤 > 0, with hard routing and quadratic local loss, the effective carrier size is 𝑞 eff = min{𝑚, 𝑛, 1 + ⌊𝑟/𝑤⌋}. The native distortion reduces to balanced scalar quantization. The operational audit distance becomes a clipped discrete covering radius. This yields an exact capacity–resolution phase diagram.

16.5

An exact data-selection frontier

The preceding examples constrain a carrier chosen from a fixed codebook. A different finite resource contract arises when the carrier must itself be a small submultiset of the observed data and the decoder is fixed in advance. This setting yields an exact distortion curve rather than only an upper or lower rate. Let 𝐷 = {𝑧1 , . . . , 𝑧 𝑁 } ⊂ R𝑑 be a finite multiset, and write 𝑁

𝐿 𝐷 (ℎ) =

𝑁

1 ∑︁ ∥ℎ − 𝑧 𝑖 ∥ 2 , 𝑁 𝑖=1

𝜇𝐷 =

1 ∑︁ 𝑧𝑖 , 𝑁 𝑖=1

𝑉𝐷 = 𝐿 𝐷 (𝜇 𝐷 ).

For an integer 𝑛 ≥ 1, an 𝑛-point carrier stores indices 𝑖1 , . . . , 𝑖 𝑛 , with repetition allowed, and the Í𝑛 decoder returns 𝜇 b = 𝑛 −1 𝑟=1 𝑧 𝑖𝑟 . For 𝑉𝐷 > 0, define its normalized worst-case native distortion by ∥𝜇 b − 𝜇𝐷 ∥2 . 𝑉𝐷 𝐷 ⊂R𝑑 𝑖1 ,...,𝑖𝑛

Dsel (𝑑, 𝑛) = sup min

The bias–variance identity is the first certified elimination: 𝐿 𝐷 (ℎ) = 𝑉𝐷 + ∥ℎ − 𝜇 𝐷 ∥ 2 . Thus the usual worst-case loss ratio is exactly 𝐹 (𝑑, 𝑛) = 1 + Dsel (𝑑, 𝑛); selection budget becomes a resource coordinate measured directly in excess squared loss.

16.5. AN EXACT DATA-SELECTION FRONTIER

101

Companion result; proof not reproduced here The exact data-selection manuscript proves 1 Dsel (𝑑, 𝑛) = 2𝑛 − 1



(1 ≤ 𝑑 ≤ 3, 𝑛 ≥ 1),

1 𝑑−1 Dsel (𝑑, 2) = max , 3 2𝑑

 (𝑑 ≥ 1).

The first formula resolves the low-dimensional mean-estimation row of the published dataselection problem; the second records its budget-two dimension transition. These statements are not counted among the book’s principal results [Hanneke et al., 2025, Huang, 2026a]. The proof exposes why this is more than a cardinality bound. A variance-nonincreasing Carathéodory reduction first preserves the mean while leaving at most 𝑑 + 1 support points. For Í Í a centered reduced law 𝑖 𝑝 𝑖 𝑥 𝑖 = 0 with variance 𝑉 = 𝑖 𝑝 𝑖 ∥𝑥𝑖 ∥ 2 , suppose the residual-capacity Í Í inequalities admit integer anchor counts 𝑠𝑖 with 𝑖 𝑠𝑖 = 𝑛 − 1 and (𝑛 − 1)𝑠𝑖 ≤ 𝑛2 𝑝 𝑖 . Put 𝑢 = 𝑖 𝑠𝑖 𝑥 𝑖 , Í 𝑆 = 𝑖 𝑠𝑖 ∥𝑥𝑖 ∥ 2 , and 𝑞𝑖 =

𝑛2 𝑝 𝑖 − (𝑛 − 1)𝑠𝑖 , 2𝑛 − 1

Γ(𝑠) = (𝑛 − 1)𝑆 − ∥𝑢∥ 2 ≥ 0.

For 𝐽 ∼ 𝑞, the remaining randomized point satisfies the exact ledger E

𝑢 + 𝑥𝐽 2 𝑉 Γ(𝑠) . = − 2 𝑛 2𝑛 − 1 𝑛 (2𝑛 − 1)

Taking the best point in the finite support of 𝑞 eliminates the auxiliary randomness and returns a legal deterministic selector. Capacity failure requires a mechanism-specific repair. At 𝑛 = 3, the remaining four-atom region is closed by a finite correlated-count certificate. For 𝑛 ≥ 4, failure forces a heavy pair; merging those two atoms releases variance 𝜆∥𝑥 1 − 𝑥2 ∥ 2 with 𝜆 ≥ (2𝑛 − 1)/(4𝑛2 ). If 𝑉 ′ is the merged variance, randomized nearest-integer splitting back to legal data points obeys 𝑉′ ∥𝑥1 − 𝑥 2 ∥ 2 𝑉 + . ≤ 2 2𝑛 − 1 2𝑛 − 1 4𝑛 The coarse-graining credit therefore pays the entire fiber-realization cost. The all-dimensional budgettwo column closes differently: a sharp pair certificate after sparse reduction meets the centered regular-simplex lower bound at the displayed dimension transition. The theorem is logically selfcontained rather than a corollary of the general calculus; what transfers is the discipline of exposing a native defect, eliminating only recoverable auxiliary objects, and closing every pushforward–recovery step in one currency.

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102

16.6

Resource limits and operational boundaries

Resource limits and foundation models. The theory suggests several resource coordinates for large AI systems: • context length and memory state; • number of experts or active routes; • communication bandwidth among modules; • test-time iteration budget; • output vocabulary or structured state dimension; • coherence imposed by parameter tying. The relevant question is not whether a model has many parameters in total, but how much oracle-relevant distinction survives the declared carrier and can be extracted by the decoder. What rate–distortion does not settle. A positive native distortion need not affect a downstream task. The task may be invariant to the lost distinctions. Operational visibility requires the transmission and exposure gates of Chapter 17.

Exercises Exercise 16.2. For squared Euclidean defect, show that the canonical decoder from a carrier 𝑍 is the conditional mean of the oracle. Exercise 16.3. Compute the optimal two-codeword quantization of three equally weighted scalar oracle values 0, 1, 3. Exercise 16.4. Propose a resource grammar for a transformer with bounded context and a fixed number of active experts. Which quantity might serve as the oracle signature?

Appendix to Chapter 16: complete proofs

103

Appendix to Chapter 16: complete proofs Proof of the canonical architecture rate–distortion decomposition Dependencies. The oracle is unique and lies in a common open convex fiber; Φ is Legendre; the dual signature 𝑆 = ∇Φ(𝑎★) is integrable; conditional means lie in the relative interior of dom Φ★; and the potential and pairing terms in the finite-cost conditional calculation are integrable. These conditions make the canonical conditional decoder and every displayed conditional expectation well defined. Proof. Fix an admissible pair (𝑍, 𝑔) ∈ 𝔊𝑚,𝑟 and write 𝐴 = 𝑔(𝑍) and 𝑎★ = 𝑎★ (𝑋). Put 𝑆 = ∇Φ(𝑎★),

𝑆 𝑍 = E(𝑆 | 𝑍),

𝑎 †𝑍 = ∇Φ★ (𝑆 𝑍 ).

Rather than subtracting two expected divergences, which could conceal an ∞ − ∞ expression, start from the finite algebraic difference of their Bregman expansions: Φ( 𝐴) − Φ(𝑎 †𝑍 ) − ⟨𝑆, 𝐴 − 𝑎 †𝑍 ⟩. Conditioning this expression on 𝑍 and using E(𝑆 | 𝑍) = 𝑆 𝑍 = ∇Φ(𝑎 †𝑍 ) gives Φ( 𝐴) − Φ(𝑎 †𝑍 ) − ⟨𝑆 𝑍 , 𝐴 − 𝑎 †𝑍 ⟩ = 𝐷 Φ ( 𝐴∥𝑎 †𝑍 ). Combining this conditional identity with the two original Bregman expansions and then integrating yields E𝐷 Φ ( 𝐴∥𝑎★) = E𝐷 Φ (𝑎 †𝑍 ∥𝑎★) + E𝐷 Φ ( 𝐴∥𝑎 †𝑍 ). The first term is the carrier information loss CΦ (𝑍) and the infimum of the second term over decoders paired with 𝑍 in 𝔊𝑚,𝑟 is Ψ𝑚,𝑟 (𝑍). Taking first the decoder infimum and then the carrier infimum yields Dnat (𝑚, 𝑟) = inf {CΦ (𝑍) + Ψ𝑚,𝑟 (𝑍)}. 𝑍 ∈ Z𝑚,𝑟

Both summands are nonnegative. If an admissible pair (𝑍, 𝑔) attains zero total defect, then CΦ (𝑍) = 0,

E𝐷 Φ {𝑔(𝑍) ∥𝑎 †𝑍 } = 0.

Strict convexity implies 𝑔(𝑍) = 𝑎 †𝑍 almost surely. Legendre duality gives CΦ (𝑍) = E𝐷 Φ★ (𝑆∥𝑆 𝑍 ), so the first equality holds exactly when 𝑆 = 𝑆 𝑍 almost surely. Conversely, an admissible pair with these two properties has zero native defect. Attainment of the pair, rather than only the outer carrier infimum, is needed for this zero characterization. □

Chapter 17

Operational Semantics and Contextual Observability 17.1

From internal difference to task-observable kernels

Two pipelines can have different internal kernels while producing the same value for every downstream task that is legally allowed to use them. Conversely, two modules that agree at one base input can become distinguishable after insertion into a larger context. Operational semantics asks when one pipeline can safely replace another under a declared task and context grammar. To make this question quantitative, represent a finite elimination pipeline by the cost it exposes between its boundary states. Such a pipeline is represented by a min-plus kernel 𝐾 (𝑦, 𝑥): the least internal cost of transforming input state 𝑥 to output state 𝑦. Given upstream potential 𝑓 and downstream terminal cost 𝑔, the optimized value is V𝐾 ( 𝑓 , 𝑔) = inf { 𝑓 (𝑥) + 𝐾 (𝑦, 𝑥) + 𝑔(𝑦)}. 𝑥,𝑦

A task contract 𝔗 is a declared collection of admissible pairs ( 𝑓 , 𝑔). This representation separates internal implementation from the values that a legal observer can extract. The next construction makes that separation canonical for a fixed task contract. legal precontext 𝐵

legal postcontext 𝐴

pipeline kernel 𝐾 or 𝐿

terminal task ( 𝑓 , 𝑔)

Figure 17.1: Contextual observability. Two internal kernels are operationally equivalent only if no legal precontext, postcontext, and terminal task can distinguish them.

104

17.2. TASK ENVELOPES AND CONTEXTUAL COMPLETION

17.2

105

Task envelopes and contextual completion

Define the task envelope ( E𝔗 𝐾) (𝑦, 𝑥) =

sup {V𝐾 ( 𝑓 , 𝑔) − 𝑓 (𝑥) − 𝑔(𝑦)}. ( 𝑓 ,𝑔) ∈𝔗

The envelope is the pointwise least kernel that preserves all declared task values. The construction is related in spirit to classical residuation and convex conjugacy, while Attouch–Wets metrics concern convergence of epigraphs on unbounded spaces rather than the finite sup-norm identity stated here [Attouch and Wets, 1986]. Proposition 17.1 is therefore proved directly from this task envelope and should not be read as an Attouch–Wets theorem [Huang and Wang, 2026j]. For two kernels of the same type, define the base task observational distance by 𝑑𝔗 (𝐾, 𝐿) =

sup V𝐾 ( 𝑓 , 𝑔) − V 𝐿 ( 𝑓 , 𝑔) . ( 𝑓 ,𝑔) ∈𝔗

How to read the assumptions The state spaces and kernels are finite, and the set of allowed tasks is nonempty and declared in advance. All kernels compared have the same input and output types, so the same task pairs apply. No context closure is needed for this result. The operational metric compares only differences visible to that task contract. Proposition 17.1 (Task-envelope isometry). The envelope preserves exactly the values V𝐾 ( 𝑓 , 𝑔) for ( 𝑓 , 𝑔) ∈ 𝔗, and the task observational distance between 𝐾 and 𝐿 equals ∥ E𝔗 𝐾 − E𝔗 𝐿 ∥ ∞ . Interpretation and boundary The task envelope is the pointwise least kernel preserving all allowed task values, and sup-norm distance between envelopes equals the worst task-value discrepancy. Internal kernel differences outside the envelope are operationally invisible. Changing the task set changes the quotient. Proof roadmap Use the variational definition of every task value to show the envelope lies below the original kernel and preserves all values. Minimality follows by comparison with any other valuepreserving kernel; the signed isometry follows by exchanging supremum and pointwise difference. Complete details are in the chapter appendix. Task equivalence at the exposed boundary is not yet a compositional notion. A module that is invisible in isolation may become visible after legal preprocessing or postprocessing, so the task contract must be closed under the deployment grammar. A grammar 𝔊 specifies legal pre- and post-compositions. The base task set must be closed under pullback through every legal context. Let Cl𝔊 𝔗

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be the least context-stable task system containing 𝔗. Define Q𝔊,𝔗 𝐾 = ECl𝔊 𝔗 𝐾.

Equivalently, if 𝐵 and 𝐴 range over type-compatible legal pre- and postcontexts, define ctx 𝑑𝔊,𝔗 (𝐾, 𝐿) = sup 𝑑𝔗 ( 𝐴 ⊗ 𝐾 ⊗ 𝐵, 𝐴 ⊗ 𝐿 ⊗ 𝐵). 𝐴,𝐵

Identity contexts are included, so this contextual distance dominates the base task distance. A pseudometric 𝜌 on type-compatible kernels is called task-adequate when 𝑑𝔗 (𝐾, 𝐿) ≤ 𝜌(𝐾, 𝐿), and grammar-nonexpansive when 𝜌( 𝐴 ⊗ 𝐾 ⊗ 𝐵, 𝐴 ⊗ 𝐿 ⊗ 𝐵) ≤ 𝜌(𝐾, 𝐿) for every legal precontext 𝐵 and postcontext 𝐴. How to read the assumptions The context grammar contains identities, is closed under composition, and acts monotonically on finite kernels. The closure operator takes the envelope over all legal contexts and terminal tasks. These grammar assumptions are essential for congruence. Theorem 17.2 (Contextual completion and full abstraction). The contextual distance is ctx 𝑑𝔊,𝔗 (𝐾, 𝐿) = ∥ Q𝔊,𝔗 𝐾 − Q𝔊,𝔗 𝐿 ∥ ∞ .

It is the least task-adequate grammar-nonexpansive pseudometric. Its zero set is the greatest grammar congruence contained in base task equivalence, and Q𝔊,𝔗 𝐾 is the pointwise least representative of the fully abstract quotient. Interpretation and boundary Two pipelines are equivalent exactly when no legal context followed by a legal task can distinguish them. The contextual quotient is fully abstract: it identifies all and only operationally indistinguishable kernels. This is task- and grammar-relative, not an absolute semantic identity. Proof roadmap Take the supremum of terminal task envelopes over the context closure. Show that equal closures imply equal values after every context; conversely use the definition of the closure to recover pointwise equality from universal contextual equivalence. Closure under composition proves congruence. The chapter appendix supplies the full argument.

17.3. FROM NATIVE DEFECT TO OPERATIONAL SEPARATION

17.3

107

From native defect to operational separation

The preceding theorem supplies the operational quotient of pipeline space. Suppose an architecture defect produces a nonnegative increment in a pipeline kernel. Two gates are needed. The first is the transmission gate: the native defect must actually increase the operational kernel rather than being absorbed by another internal choice. The second is the exposure gate: the closed task contract must contain a context that selects the affected states. Only then does a positive native architecture floor imply a positive contextual separation. Boundary of the claim Contextual observability is not an additional scalar tax to be blindly added to native distortion. It is a gate or transform determining how much native difference is operationally visible. The amount transmitted through these gates depends on the resolution of the task family. For Lipschitz task contracts, the envelope becomes a double metric erosion. If a defect spike is narrower than the task resolution, it may be clipped or smoothed away. The exact resolution profile records the observable separation as task Lipschitz budgets vary. Finite task families yield a visibility transform with an exact margin-clipped lower bound. Compact metric task classes yield a resolution coefficient determined by task and kernel Lipschitz moduli. Thus the operational image of a native defect is generally a profile rather than a binary label. At coarse task resolution a real internal seam can be invisible; as the contract becomes more discriminating, the same seam can produce a positive contextual distance.

17.4

Ordering curvature and contextual witnesses

For 𝑛 operations, each permutation produces a kernel 𝐾 𝜋 . The contextual ordering curvature is the diameter ctx max 𝑑𝔊,𝔗 (𝐾 𝜋 , 𝐾 𝜎 ). 𝜋, 𝜎

Pairwise swap budgets can bound the global curvature through an inversion-weighted Kendall sum [Kendall, 1938]. Pairwise commutation at one input does not imply contextual interchange coherence. The need for context is already visible in a minimal finite-memory witness. A two-state screening pipeline can be constructed so that every operation order is invisible under a zero-input contract, yet a legal precontext creates a value gap of one. The full resolution profile is max{0, min(1, 𝑎 − 1, 𝑏)} for task budgets (𝑎, 𝑏). The example demonstrates why one fixed audit input is insufficient for safe module substitution. This conclusion parallels contextual equivalence and full abstraction in programming-language semantics [Milner, 1977, Plotkin, 1977]. Here the boundary objects are optimization kernels and the task contract is quantitative, so the resulting quotient is both semantic and metric.

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Exercises Exercise 17.3. For a finite state space and the full task contract containing all bounded potentials, show that the task envelope reconstructs the kernel exactly up to additive normalization allowed by the contract. Exercise 17.4. Give two kernels that agree under a zero-input task but differ after precomposition with a legal context. Exercise 17.5. Give an example in which a native architecture defect is invisible to a downstream task that is permutation invariant.

Appendix to Chapter 17: complete proofs

109

Appendix to Chapter 17: complete proofs Proof of task-envelope reconstruction and isometry Dependencies. The kernel and state sets are finite, so all minima and maxima are attained. The task contract may be arbitrary; no closure under contexts is needed for this first result. Proof. For every task ( 𝑓 , 𝑔) and state pair (𝑥, 𝑦), V𝐾 ( 𝑓 , 𝑔) ≤ 𝑓 (𝑥) + 𝐾 (𝑦, 𝑥) + 𝑔(𝑦),

so ( E𝔗 𝐾) (𝑦, 𝑥) =

sup {V𝐾 ( 𝑓 , 𝑔) − 𝑓 (𝑥) − 𝑔(𝑦)} ≤ 𝐾 (𝑦, 𝑥). ( 𝑓 ,𝑔) ∈𝔗

Fix ( 𝑓0 , 𝑔0 ) ∈ 𝔗. Its own term in the supremum gives 𝑓0 (𝑥) + ( E𝔗 𝐾) (𝑦, 𝑥) + 𝑔0 (𝑦) ≥ V𝐾 ( 𝑓0 , 𝑔0 ) for every (𝑥, 𝑦). Taking the minimum over (𝑥, 𝑦) gives VE𝔗 𝐾 ( 𝑓0 , 𝑔0 ) ≥ V𝐾 ( 𝑓0 , 𝑔0 ), while E𝔗 𝐾 ≤ 𝐾 gives the reverse inequality. Hence every task value is preserved. Value preservation implies idempotence of E𝔗 , and monotonicity follows from monotonicity of the minimum. If 𝑀 has the same task values as 𝐾, then 𝑀 (𝑦, 𝑥) ≥ V 𝑀 ( 𝑓 , 𝑔) − 𝑓 (𝑥) − 𝑔(𝑦) = V𝐾 ( 𝑓 , 𝑔) − 𝑓 (𝑥) − 𝑔(𝑦) for every task. Taking the supremum proves E𝔗 𝐾 ≤ 𝑀, so the envelope is the pointwise least representative. Write 𝐾𝔗 = E𝔗 𝐾 and 𝐿 𝔗 = E𝔗 𝐿. Minimum comparison and value preservation give sup {V𝐾 ( 𝑓 , 𝑔) − V 𝐿 ( 𝑓 , 𝑔)} ≤ max{𝐾𝔗 (𝑦, 𝑥) − 𝐿 𝔗 (𝑦, 𝑥)}. ( 𝑓 ,𝑔) ∈𝔗

𝑥,𝑦

Conversely, for fixed (𝑥, 𝑦), the difference of the two suprema defining 𝐾𝔗 and 𝐿 𝔗 is bounded by the supremum of the task-value differences. Maximizing proves equality of the signed quantities. Repeating with 𝐾 and 𝐿 exchanged yields 𝑑𝔗 (𝐾, 𝐿) = ∥𝐾𝔗 − 𝐿 𝔗 ∥ ∞ . □

Proof of contextual completion and full abstraction Dependencies. The grammar contains identity or formal empty contexts and is closed under typed composition. Context closure is defined by pulling every base task back through every legal pre- and post-context.

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CHAPTER 17. OPERATIONAL SEMANTICS AND CONTEXTUAL OBSERVABILITY

Proof. For a precontext 𝐵, postcontext 𝐴, and external task (ℎ, 𝑘), both V 𝐴⊗𝐾 ⊗𝐵 (ℎ, 𝑘)

and V𝐾 ( 𝑓 𝐵,ℎ , 𝑔 𝐴,𝑘 )

are the minimum of ℎ(𝑤) + 𝐵(𝑥, 𝑤) + 𝐾 (𝑦, 𝑥) + 𝐴(𝑧, 𝑦) + 𝑘 (𝑧) over (𝑤, 𝑥, 𝑦, 𝑧). Thus contextual testing is exactly testing 𝐾 with the pullback-closed task system Cl𝔊 𝔗. Applying the preceding task-envelope isometry gives ctx 𝑑𝔊,𝔗 (𝐾, 𝐿) = ∥ Q𝔊,𝔗 𝐾 − Q𝔊,𝔗 𝐿 ∥ ∞ .

The supremum of absolute differences of scalar functionals is a pseudometric. Identity contexts show task adequacy. If fixed contexts are first placed around 𝐾 and 𝐿, any further context combines with them into another legal context; hence the metric is grammar-nonexpansive. Let 𝜌 be any task-adequate grammar-nonexpansive pseudometric. For every legal context, 𝑑𝔗 ( 𝐴 ⊗ 𝐾 ⊗ 𝐵, 𝐴 ⊗ 𝐿 ⊗ 𝐵) ≤ 𝜌( 𝐴 ⊗ 𝐾 ⊗ 𝐵, 𝐴 ⊗ 𝐿 ⊗ 𝐵) ≤ 𝜌(𝐾, 𝐿). Taking the supremum over contexts proves that the contextual metric is the least such pseudometric. Its zero set is a congruence by grammar nonexpansiveness and lies inside base task equivalence by task adequacy. Conversely, if ∼ is any grammar congruence contained in base task equivalence and 𝐾 ∼ 𝐿, then all contextual substitutions remain ∼-related and hence base-task equivalent. Every contextual value difference is zero, so 𝐾 and 𝐿 have contextual distance zero. This proves the greatest-sound-congruence property. Finally, apply task-envelope reconstruction to the closed contract. The envelope is deflationary, monotone, idempotent, preserves every closed task, and is the unique pointwise-least member of its equivalence class. Equality of contextual envelopes is equivalent to zero contextual distance, completing the fully abstract quotient statement. □

Chapter 18

Composition, Base Change, and Dequantization Here base change means passage between fine and coarse representations, memories, or architecture levels. It preserves obstruction only under a proved commuting or exact decomposition theorem; equality of one optimized objective value is not sufficient.

18.1

Conditional chain rules and architecture composition

Candidate-level identities need not survive minimization over a restricted architecture. A local certificate may also fail after conditioning, pushforward, or a zero-temperature limit. We therefore distinguish several levels of composition, each of which can fail independently. 1. Law-level or candidate-level defect identities. 2. Architecture-level infima. 3. Representation or base-change maps. 4. Operational contextual composition. 5. Temperature limits. Conditional KL chain rule.

Let 𝑄 𝑋𝑌 and 𝑃 𝑋𝑌 be joint laws. The chain rule is

KL(𝑄 𝑋𝑌 ∥𝑃 𝑋𝑌 ) = KL(𝑄 𝑋 ∥𝑃 𝑋 ) + E𝑄𝑋 KL(𝑄𝑌 | 𝑋 ∥𝑃𝑌 | 𝑋 ). The conditional term is weighted by the trial law 𝑄 𝑋 . Replacing this weight by the oracle law without a theorem changes the identity. The chain rule is the prototype of a vertical fiber decomposition: coarse defect plus expected conditional defect. Entropy-regularized and KL-control formulations of stochastic control are an established neighboring literature: path-integral and linearly-solvable control, maximum-entropy inverse control, control as inference, and soft actor–critic all exploit closely related exponential tilting or KL-penalized objectives [Kappen, 2005, Todorov, 2006, 2009, Ziebart et al., 2010, Levine, 2018, Haarnoja et al., 2018]. 111

CHAPTER 18. COMPOSITION, BASE CHANGE, AND DEQUANTIZATION

112

Accordingly, the soft chain identity below is not presented as a new control identity. Its role here is to state the exact normalization and common-loss-scale conditions under which local EG certificates may be composed. Soft kernels compose in one objective loss scale. Let X0 , . . . , X𝑚 be standard Borel spaces. For 𝑖 = 1, . . . , 𝑚, let 𝑀𝑖 (𝑑𝑥 𝑖 | 𝑥𝑖−1 ) be a reference transition kernel, let 𝐾𝑖 (𝑦, 𝑥) be the cost of a transition from input 𝑥 to output 𝑦, and let 𝐺 be a terminal cost. At inverse temperature 𝛽 > 0, define backward soft values by 𝛽

𝑉𝑚 = 𝐺, ∫ h i 1 𝛽 𝛽 𝑉𝑖−1 (𝑥) = − log exp −𝛽{𝐾𝑖 (𝑦, 𝑥) + 𝑉𝑖 (𝑦)} 𝑀𝑖 (𝑑𝑦 | 𝑥). 𝛽

(18.1)

The normalized report 𝛽

𝛽

𝛽

𝑏 𝑖 (𝑦, 𝑥) = 𝐾𝑖 (𝑦, 𝑥) + 𝑉𝑖 (𝑦) − 𝑉𝑖−1 (𝑥) induces the oracle Gibbs transition 𝛽,★

𝑃𝑖

𝛽

(𝑑𝑦 | 𝑥) = 𝑒 −𝛽𝑏𝑖 ( 𝑦, 𝑥 ) 𝑀𝑖 (𝑑𝑦 | 𝑥).

(18.2)

Î𝑚 Fix 𝑥 0 and write 𝑀 𝑥0 = 𝑖=1 𝑀𝑖 (· | 𝑋𝑖−1 ). A trial path law 𝑄 ≪ 𝑀 𝑥0 may depend on its whole past; denote its conditional laws by 𝑄 𝑖 (· | 𝑋0:𝑖−1 ). Define (𝑚 ) ∑︁ 1 H𝛽 (𝑄; 𝑥0 ) = E𝑄 𝐾𝑖 (𝑋𝑖 , 𝑋𝑖−1 ) + 𝐺 (𝑋𝑚 ) + KL(𝑄∥ 𝑀 𝑥0 ), (18.3) 𝛽 𝑖=1   1 𝛽 𝛽,★ 𝛿𝑖 (𝑄 𝑖 ; 𝑋0:𝑖−1 ) = KL 𝑄 𝑖 (· | 𝑋0:𝑖−1 ) 𝑃𝑖 (· | 𝑋𝑖−1 ) . (18.4) 𝛽 How to read the assumptions All soft partition functions are finite and strictly positive. The trial path law is absolutely continuous with respect to the reference path law, regular conditional laws exist, and the displayed costs and relative entropies are integrable. The reference kernels and inverse temperature are fixed throughout the composition. Theorem 18.1 (Soft conditional chain theorem). Let 𝛽,★

𝑃 𝑥0 (𝑑𝑥 1:𝑚 ) =

𝑚 Ö

𝛽,★

𝑃𝑖

(𝑑𝑥 𝑖 | 𝑥𝑖−1 ).

𝑖=1

Then every 𝑄 ≪ 𝑀 𝑥0 satisfies 𝛽

1 𝛽,★ KL(𝑄∥𝑃 𝑥0 ) 𝛽 𝑚 ∑︁  𝛽 = E𝑄 𝛿𝑖 (𝑄 𝑖 ; 𝑋0:𝑖−1 ) .

H𝛽 (𝑄; 𝑥0 ) − 𝑉0 (𝑥0 ) =

(18.5) (18.6)

𝑖=1

In particular, the global free-energy defect is nonnegative and vanishes exactly at the oracle path law, up to null histories.

18.2. BASE CHANGE AND DATA PROCESSING

113

Interpretation and boundary Local conditional certificates can be added because every one is measured against the same composed Gibbs objective. The trial law need not be Markov: history dependence is absorbed by its regular conditional laws. The theorem does not license adding arbitrary local losses that use different reference measures, temperatures, or target normalizations. Proof roadmap Multiply the normalized Gibbs transitions to obtain the oracle path density relative to the reference path law. The reports telescope to total path cost minus the initial soft value, yielding the first equality. Apply the relative-entropy chain rule to that product law for the second. The chapter appendix records both steps and the equality case.

18.2

Base change and data processing

Conditional chain rules live at the candidate-law level. Passing them through a representation map and an architecture infimum requires the feasible fine fiber to remain visible. The source box identifies the imported results that justify this step. Source result; proof not reproduced here The exact architecture base-change identity, canonical-fiber saturation criterion, quantitative dequantization bounds, and temperature-dependent class variants summarized in the remainder of this chapter come from COT and Elimination Geometry III. The chapter appendix proves only the soft conditional chain theorem above; these additional variants are not counted as principal book results [Huang and Wang, 2026b,h]. Within that imported scope, fix a base-change map from a fine deployment to a coarse deployment. Taking architecture infima gives an exact infimal theorem only when the feasible fine fibers are tracked as a function of the coarse law. The fine obstruction is 𝑂 fine = inf {𝑂 coarse (𝑄 coarse ) + Ψ(𝑄 coarse )} . 𝑄coarse

The fiber term depends on coarse occupancy. Replacing it by a constant can be wrong. Universal equality between fine and coarse obstructions requires canonical-fiber saturation. Objective preservation alone is insufficient. Pushforward and data processing.

A measurable map reduces relative entropy:

KL(𝑄∥𝑃) ≥ KL(𝑇# 𝑄∥𝑇# 𝑃). The chain rule used above and this data-processing inequality belong to the classical relative-entropy calculus [Cover and Thomas, 2006, Csiszár, 1975]. The difference is a data-processing gap. Equality can imply sufficiency or recoverability under additional assumptions; at architecture level, pushforward can remove an obstruction by quotienting irrelevant labels yet create a fiber-realization tax when the fine architecture cannot reconstruct the canonical conditional law.

114

18.3

CHAPTER 18. COMPOSITION, BASE CHANGE, AND DEQUANTIZATION

When conditional elimination destroys encoder semantics

The Markovity Conjecture asks whether one may impose 𝑈 − 𝑋 − 𝑉 without losing the optimum of a Marton dual problem. The useful EG question is more local: what exactly is eliminated, which observables survive, and which carrier semantics can be destroyed? For a two-receiver broadcast channel, write 𝐺 Θ ( 𝑝) = −𝛼𝐻 (𝑌 ) − (𝜆 − 𝛼)𝐻 (𝑍) + 𝐼 (𝑈; 𝑌 ) + 𝜆𝐼 (𝑉; 𝑍) − 𝐼 (𝑈; 𝑉) + E[𝑎 𝑋 ],

(18.7)

where Θ collects the channel and dual parameters. Conditional Markovization is the declared elimination ( M 𝑝) (𝑢, 𝑣, 𝑥) = 𝑝(𝑥) 𝑝(𝑢 | 𝑥) 𝑝(𝑣 | 𝑥). It removes dependence between 𝑈 and 𝑉 given 𝑋, while preserving the (𝑈, 𝑋) and (𝑉, 𝑋) marginals. Thus it preserves everything seen separately by the two receivers, but it need not preserve the semantics 𝐻 (𝑋 | 𝑈, 𝑉) = 0 of a deterministic encoder. For such an encoder, the exact native ledger is 𝐺 Θ ( M 𝑝) − 𝐺 Θ ( 𝑝) = 𝐼 𝑝 (𝑈; 𝑉 | 𝑋) − 𝐻M 𝑝 (𝑋 | 𝑈, 𝑉) . | {z } | {z } dependence credit

(18.8)

semantic ambiguity tax

The first term is the advertised reward for eliminating conditional dependence. The second is a typed realization loss: after productization, the same pair (𝑢, 𝑣) may become compatible with several input symbols. In EG language, 



M 𝑝 : 𝐻 𝑝 (𝑋 | 𝑈, 𝑉) = 0 ⊈ 𝑝 : 𝐻 𝑝 (𝑋 | 𝑈, 𝑉) = 0 .

The operation preserves visible marginals but can leave the deterministic carrier class. Rectangular symbol fibers are exactly the zero-tax case. This ledger changed the search. Instead of asking a local optimizer to find a Markov point directly, the search looked for a nonrectangular deterministic branch on which the semantic tax exceeds the dependence credit. It then separated three logically different gates, 𝐿 M ( 𝑝)

−→

Γrect (𝜋, 𝑚)

−→

Orect (Θ),

(18.9)

namely loss under direct Markovization, the best rectangular map switch at fixed cell masses, and loss after complete reoptimization over every deterministic rectangular architecture. Failure at either of the first two gates is only a candidate certificate. A counterexample requires the third. This distinction also exposed the computational blind spot. The ABCA branch is a local maximum separated from the dominant rectangular branch by a finite map-switch barrier. Convergent local searches therefore fall into the rectangular basin and report its value; sampling more starting points does not certify that the other architecture is absent. The finite-change KL/Bregman ledger priced the branch switch, while the EG gate structure routed the final task to branchwise interval certification rather than another local optimization run.

18.3. WHEN CONDITIONAL ELIMINATION DESTROYS ENCODER SEMANTICS

115

Computer-assisted companion result For an exact rational three-input dense channel, Huang [2026b] certifies 𝐺 Θ ( 𝑝 ABCA ) − sup 𝐺 Θ ( 𝑝) ≥ 2.711224394247 × 10−11 > 0. 𝑝∈ R det

Together with a strict-Jensen equality bridge, this proves that no global optimizer satisfies 𝑈 − 𝑋 − 𝑉. The result is not counted among the book’s principal results. Here R det contains all finite deterministic encoders whose nonempty fibers are rectangles. Write 𝐹 (Θ) = sup 𝐺 Θ ( 𝑝),

𝐹rect (Θ) = sup 𝐺 Θ ( 𝑝), 𝑝∈ R det

𝑝

Orect (Θ) = 𝐹 (Θ) − 𝐹rect (Θ). For every deterministic rectangular candidate 𝑞, the exact architecture ledger is 𝐹 (Θ) − 𝐺 Θ (𝑞) =

Orect (Θ) | {z }

+ 𝐹rect (Θ) − 𝐺 Θ (𝑞) . | {z }

architecture error

implementation error

(18.10)

It shows why more optimization inside the rectangular class cannot repair the result: it can remove the implementation term, never the architecture term. Only a short proof bridge is needed after the interval certificate. First, a Markov-class cardinality lemma gives finite attainment. Second, with 𝑝𝑈𝑉 fixed, 𝐺 Θ is convex in 𝐾 = 𝑝(𝑥 | 𝑢, 𝑣). Equality at a global maximum, together with the strict marginal-entropy curvature of −𝛼𝐻 (𝑌 ) − (𝜆 − 𝛼)𝐻 (𝑍), forces the original maximizing kernel itself to be deterministic whenever a positively weighted output sees every input change. This step uses neither Markovity nor full grid support. For the exact channel, 𝑇𝑌 is nonsingular. Finally, the published full-support theorem [Gohari et al., 2014, Theorem 1] completes the active grid; determinism plus 𝑈 − 𝑋 − 𝑉 then makes every symbol fiber a rectangle. Thus the interval separation excludes the literal Markov optimum as well. The rectangle-cover and generalized-AND/XOR route is not part of this proof. Boundary of the claim The word “rectangular” is used in two different directions. In Chapter 13, Rect( Arch) is an enlargement that permits independent pasting of admissible kernels. In the Marton problem, a rectangular deterministic map is a restriction on the symbol fibers of 𝑓 (𝑈, 𝑉). The common geometry is product closure, not an identification of the two constructions. The proof is classical information theory plus rigorous numerics; it does not depend on EG terminology. EG supplied the discovery and certification architecture: type the eliminated relation, measure the native semantic tax, separate candidate-level failure from architecture-level obstruction, and let the failed gate decide which certificate to build. That is the transferable lesson. The result refutes neither Marton’s achievable region nor the separate Additivity Conjecture.

CHAPTER 18. COMPOSITION, BASE CHANGE, AND DEQUANTIZATION

116

18.4

Hard limits and composition boundaries

Zero temperature.

A Gibbs defect has the form 1 KL(𝑄∥𝑃★𝛽). 𝛽

As 𝛽 → ∞, the soft objective approaches hard excess cost under uniform dequantization conditions. For a fixed finite class, minima converge. For temperature-dependent architecture classes, one needs Γ-convergence and control of inner and outer limits. A useful rate has the form Λ + osc(𝐷 ∞ )ℎ 𝛽 , 𝛽 where ℎ 𝛽 measures class convergence. Shared coordination taxes can dequantize at the sharper thermal rate when nested minima share the same perturbation. Randomized hard limits. The zero-temperature limit of a soft architecture need not be deterministic. If the architecture closure contains randomized mixtures but not pure selectors, the hard limit can remain randomized. Determinism is a property of the admissible class, not a consequence of low temperature alone. Interchange curvature. Two operations may commute individually but fail in a larger context. The Foundations program records this failure by interchange curvature. Operational semantics refines the question: which contexts and tasks can detect the noncommutation? Examples include: • conditioning before versus after representation compression; • coupling before versus after elimination; • quantization before versus after task projection; • temperature limit before versus after architecture restriction. A composition checklist.

Before composing two certificates, verify:

1. the defects use compatible orientation and units; 2. the conditioning weight is correct; 3. feasible fibers are nonempty and measurable; 4. architecture infima commute only under saturation/rectangularity conditions; 5. representation maps preserve the declared oracle object; 6. limits are uniform or controlled by epi/Γ-convergence; 7. downstream tasks expose the retained difference.

18.4. HARD LIMITS AND COMPOSITION BOUNDARIES

117

Exercises Exercise 18.2. Derive the KL chain rule and identify the trial-law weighting of the conditional term. Exercise 18.3. Give an example where a fine-to-coarse map preserves the optimal objective value but the fine architecture pays a positive realization tax. Exercise 18.4. Construct a sequence of randomized soft minimizers whose zero-temperature limit remains randomized.

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Appendix to Chapter 18: complete proofs Proof of the soft conditional chain theorem Dependencies. The normalized transitions in (18.2) must be probability kernels, the trial law must be absolutely continuous with respect to the reference path law, and all relative entropies must be well defined. Standard Borel state spaces provide the required regular conditional laws. Proof. By multiplying the transition densities in (18.2), ( ) 𝛽,★ 𝑚 ∑︁ 𝑑𝑃 𝑥0 𝛽 (𝑥1:𝑚 ) = exp −𝛽 𝑏 𝑖 (𝑥𝑖 , 𝑥𝑖−1 ) . 𝑑𝑀 𝑥0 𝑖=1 The backward values telescope: 𝑚 ∑︁

𝛽 𝑏 𝑖 (𝑥𝑖 , 𝑥𝑖−1 ) =

𝑚 ∑︁

𝛽

𝛽

{𝐾𝑖 (𝑥𝑖 , 𝑥𝑖−1 ) + 𝑉𝑖 (𝑥𝑖 ) − 𝑉𝑖−1 (𝑥𝑖−1 )}

𝑖=1

𝑖=1

=

𝑚 ∑︁

𝛽

𝐾𝑖 (𝑥𝑖 , 𝑥𝑖−1 ) + 𝐺 (𝑥 𝑚 ) − 𝑉0 (𝑥0 ).

𝑖=1

Consequently, ) (𝑚 ∑︁ 1 1 𝛽,★ 𝛽 KL(𝑄∥𝑃 𝑥0 ) = KL(𝑄∥ 𝑀 𝑥0 ) + E𝑄 𝑏 𝑖 (𝑋𝑖 , 𝑋𝑖−1 ) 𝛽 𝛽 𝑖=1 𝛽

= H𝛽 (𝑄; 𝑥0 ) − 𝑉0 (𝑥0 ), which proves (18.5). The relative-entropy chain rule for the two path laws gives 𝛽,★

KL(𝑄∥𝑃 𝑥0 ) =

𝑚 ∑︁

  𝛽,★ E𝑄 KL 𝑄 𝑖 (· | 𝑋0:𝑖−1 ) 𝑃𝑖 (· | 𝑋𝑖−1 ) .

𝑖=1

Divide by 𝛽 and use (18.4) to obtain (18.6). Each summand is nonnegative. Their sum is zero exactly when the conditional laws agree at every stage outside a 𝑄-null set; sequential factorization then gives 𝛽,★ 𝑄 = 𝑃 𝑥0 . The converse is immediate. □

Part V Historical and Conceptual Notes

119

Part V Historical and Conceptual Notes Three neighboring traditions The carrier–decoder language meets classical rate–distortion and quantization [Shannon, 1959, Cover and Thomas, 2006, Gray and Neuhoff, 1998]; its exact conditional split uses classical Bregman centroid geometry [Banerjee et al., 2005a]. Task envelopes use residuation and idempotent projection [Cohen et al., 2004], while contextual equivalence and full abstraction come from programming-language semantics [Milner, 1977, Plotkin, 1977]. The soft composition theorem uses the Gibbs variational formula and relative-entropy chain rule [Csiszár, 1975, Cover and Thomas, 2006]. Part V’s claim is not that these three traditions were previously absent. It states which interfaces must be declared before they can be combined. Question

Classical object

Part V qualification

What must the carrier source coding, quantization, condi- retain the conditional dual oracle signature retain? tional prediction needed by the native eliminated objective What can a task ob- contextual equivalence and full ab- close a declared task class under a declared serve? straction min-plus context grammar and use its canonical envelope When do local costs Gibbs normalization and KL chain use one reference family, temperature, and add? rule telescoping normalization; arbitrary local scores are not composable by declaration The architecture rate–distortion theorem and the two operational theorems are source-program realizations built on those classical cores. The soft conditional-chain theorem is a classical identity in the book’s typed notation. In particular, contextual observability is a gate or transform, not a third scalar added to carrier loss and decoder nonsaturation.

From population structure to finite-sample certification Parts I–V described what is true when the population objective, oracle family, and deployment grammar are known. Finite data do not select one population world. They generate a covered set of population states that retain inferential standing under the declared error contract; Chapter 19 formalizes this object as a confidence world. Part VI projects that set through declared queries, authorizes a resolved certificate only when all compatible worlds agree, and asks how much information is required to separate the worlds that would reverse the conclusion.

Part VI

Statistical Certification Data produce a confidence world, not one privileged population story. Its identified image is the complete finite-data answer to a declared query; a resolved certificate is authorized only when that image lies on one side of the operational boundary.

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Confidence Worlds and Statistical Elimination Geometry Population elimination geometry assumes that the defect, oracle field, architecture grammar, and operational kernel are known. In data analysis they are estimated. Finite data do not select one population world; they leave a set of worlds with inferential standing. Structural conclusions must therefore be transported through that set rather than evaluated only at a point estimate or a favored posterior story. The statistical layer has one basic chain: 𝑞 data ↦−→ 𝐶𝑛 ↦−→ 𝐽𝐶𝑛 (𝑞) ↦−→ 𝛿𝐶 ↦−→ authorized action. 𝑛

The confidence world 𝐶𝑛 supplies validity, its identified image 𝐽𝐶𝑛 (𝑞) records the full query-relevant 𝑞 uncertainty, and the typed certificate 𝛿𝐶 reports whether the declared boundary is resolved. Only after 𝑛 these objects are fixed should one ask how to acquire more information or deploy a common witness. data

𝐽𝐶𝑛 (𝑞) identified image

𝐶𝑛 confidence world

𝑞

𝛿𝐶𝑛 certificate

resolved only if all covered worlds lie on one side

Figure 19.1: Finite-data certification transports a covered set of population worlds through a declared query before producing a terminal label.

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122

19.1

Confidence worlds and finite-data permission

Let W be the universe of complete population worlds. A world specifies every population object needed by the declared query, including nuisance components that may affect its sampling law or future experiments. Definition 19.1 (Confidence world). A fixed-record confidence world is a data-dependent random set 𝐶𝑛 ⊆ W satisfying the declared coverage contract, for example the uniform guarantee inf P𝑤 {𝑤 ∈ 𝐶𝑛 } ≥ 1 − 𝛼.

𝑤∈W

(19.1)

It is the set of worlds that remain entitled to participate in the current certificate, not the set of worlds assigned the largest prior or posterior mass. The construction is a world-level confidence set followed by projection, not a new replacement for confidence-region theory. Simultaneous confidence regions and projection have a long statistical lineage [Scheffé, 1953, Dufour, 1997]; universal confidence sets for optimization solutions provide an especially close formal neighbor [Vogel, 2008]. The term confidence world records that the set must contain every population component needed by the downstream architecture query. Coverage is part of the object, not a footnote attached after a claim. The contract must say whether the guarantee is pointwise, uniform, or asymptotic and whether it protects one query, a simultaneous query class, or a data-selected query. If adaptive stopping is allowed, the appropriate object is an anytime confidence-world sequence satisfying inf P𝑤 {𝑤 ∈ 𝐶𝑡 for every 𝑡 ≥ 0} ≥ 1 − 𝛼,

𝑤∈W

(19.2)

with every protected query evaluated on the same time-uniform event. This is the confidence-sequence contract of time-uniform inference [Howard et al., 2021], lifted to the complete-world object. Table 19.1: The confidence-world construction contract. Declared item

What must be auditable

coverage regime

Fixed-record, simultaneous, or anytime validity; pointwise, uniform, or asymptotic scope. The times, queries, subgroups, model components, and selection rules covered by one event. The nuisance, regularity, sampling, and identification assumptions encoded in a complete world. Whether the reported set is exact or an inner or outer approximation, and how that approximation changes honesty or decisiveness. 𝐶𝑛 = ∅ is reported as model conflict 𝑀; it is not used for vacuous inference.

protected scope population content set status empty-set rule

Confidence worlds may be built from parametric confidence regions, distributionally robust sets, moment inequalities, partial-identification regions, valid bootstrap bands, or simultaneous module-wise envelopes. The construction method may change; the coverage and reporting contract may not be left implicit. Conformal prediction [Vovk et al., 2005] can supply a predictive set or one component of such an envelope under its own exchangeability contract; it is not an ancestor of, or substitute for, projection of a population confidence world.

19.2. IDENTIFIED IMAGES, MARGINS, AND HONEST CERTIFICATES

19.2

123

Identified images, margins, and honest certificates

Before inspecting the data, a query contract declares the population functional or truth map, its tolerance or action boundary, the observer, deployment quantifier, admissible experiment class, and stopping rule. For a real-valued functional Γ𝑞 , the confidence world induces the identified image 𝐽𝐶 (𝑞) = {Γ𝑞 (𝑤) : 𝑤 ∈ 𝐶}.

(19.3)

This image, rather than a terminal color, is the complete finite-data answer to the query. The color is its projection relative to a declared boundary. Projection of confidence regions and inference under partial identification are established ideas [Dufour, 1997, Imbens and Manski, 2004, Chernozhukov et al., 2007]. The book’s contribution is the typed architecture/deployment interpretation of the projected image, not the statistical operation of taking an image of a set. Definition 19.2 (Standard confidence certificate). Let 𝑞 : W → {𝐹, 𝐼} be a declared truth map. For any 𝐶 ⊆ W, define   𝑀, 𝐶 = ∅,    𝑞 𝛿𝐶 = 𝑑, 𝑞(𝐶) = {𝑑} for some 𝑑 ∈ {𝐹, 𝐼}, (19.4)   𝑈, |𝑞(𝐶)| ≥ 2.  Here 𝑀 records model conflict, while 𝑈 records genuine disagreement among compatible worlds. Neither state authorizes a binary population claim. For the threshold query 𝑞(𝑤) = 𝐹

⇐⇒

Γ𝑞 (𝑤) ≤ 𝜀 𝑞 ,

suppose the nonempty identified image is the interval 𝐽𝐶 (𝑞) = [𝐿 𝐶 , 𝑈𝐶 ]. Then 𝑈𝐶 ≤ 𝜀 𝑞 , 𝐿𝐶 > 𝜀𝑞 ,  𝑈, 𝐿 𝐶 ≤ 𝜀 𝑞 < 𝑈𝐶 .     𝐹,

  𝑞 𝛿𝐶 = 𝐼,  The associated certificate margin is

  𝜀 − 𝑈𝐶 ,   𝑞  𝑚 𝐶 (𝑞) = 𝐿 𝐶 − 𝜀 𝑞 ,    0, 

𝑞 𝛿𝐶 = 𝐹, 𝑞 𝛿𝐶 = 𝐼, 𝑞 𝛿𝐶 = 𝑈.

(19.5)

No margin is assigned to the model-conflict state. A positive margin records how far the entire identified 𝑞 image lies from a label change; the full map 𝜀 ↦→ 𝛿𝐶 (𝜀) is the tolerance profile. Reporting the image and margin permits another stakeholder to audit a different declared tolerance without reconstructing the population world set. The threshold readout is also continuous with the classical equivalence and noninferiority literature [Blackwelder, 1982, Berger and Hsu, 1996, Piaggio et al., 2006, Schuirmann, 1987, Lakens, 2017]. The familiar two-one-sided-tests procedure is a two-sided sibling of this set rule, not its source: here a single resolved label is authorized only when the entire covered image lies on the declared side of the boundary.

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How to read the assumptions The confidence world covers the complete population object under the declared query and stopping contract. For maximality, competing rules receive only the same realized nonempty confidence world and truth map, take values in {𝐹, 𝐼, 𝑈}, and must be correct for every world in that set; they receive no extra statistic or separate randomized error allowance. The 𝑀 output is a diagnostic for an empty set, not a population truth label. Theorem 19.3 (Honesty and maximal decisiveness). On the coverage event, every resolved output of the standard confidence certificate is correct. Among the comparator rules just declared, it is maximally decisive: it returns the common truth label whenever all compatible worlds agree and returns unresolved otherwise. Interpretation and boundary Unresolvedness is a mathematical consequence of partial identification, not a failed optimizer or a request to guess. Maximal decisiveness belongs to the exact image 𝑞(𝐶). If computation supplies only an outer enclosure of that image, resolved outputs remain honest but an unresolved output may be conservative. Proof roadmap On the coverage event, a singleton truth image contains the true label. If two labels remain possible, either binary choice is false in one compatible world, so no worldwise-correct comparator may decide. The chapter appendix proves both honesty and maximality.

Three kinds of unresolvedness. Statistical unresolvedness means that compatible population worlds imply opposite answers; computational unresolvedness means that the exact identified image is defined but has not been computed; search unresolvedness means that no witness has been found within the declared budget. The remedies differ: more computation cannot resolve the first, and more data cannot by itself repair a misspecified grammar. The compatible-world logic is rooted in partial identification [Manski, 2003], with projection confidence procedures providing a nearby inferential comparison [Kaido et al., 2019]. The book’s additional step is to require one auditable architecture certificate under the declared deployment quantifier. As a running anchor, Section 20.4.2 constructs one 95% simultaneous confidence world for the 12 department–sex admission probabilities in UCBAdmissions. Its projection through the maximum department difference is [−0.0895, 0.3109]; relative to the tolerance 0.10, the standard certificate is 𝑈. The full audit later shows why a tiny pooled 𝑝-value does not replace this worldwise conclusion.

19.3. DEFECT ENVELOPES AND ARCHITECTURE GRAMMARS

19.3

125

Defect envelopes and architecture grammars

The generic objects become structural by projecting a confidence world through the defect field and the architecture grammar. Suppose a simultaneous event provides lower and upper defect envelopes 𝐷 𝑥 (𝑎) ≤ 𝐷 ★𝑥 (𝑎) ≤ 𝐷 𝑥 (𝑎) for all relevant (𝑥, 𝑎). Suppose also that the true architecture lies between inner and outer grammars 𝔄 𝑟 ⊆ 𝔄★ 𝑟 ⊆ 𝔄𝑟 for every resource budget 𝑟. Simultaneous objective and feasible-set envelopes of this type are standard tools in stochastic and robust optimization [Vogel, 2008, Berk et al., 2013]; the point here is to preserve their common event through elimination. Use monotone 𝜌 = 𝑃𝑛 for empirical risk, or 𝜌 = 𝑃 for confidence-controlled population risk. 𝑂 (𝑟) = inf 𝜌(𝐷 𝐴), 𝐴∈𝔄𝑟

𝑂 (𝑟) = inf★ 𝜌(𝐷 ★𝐴), ★

𝐴∈𝔄𝑟

𝑂 (𝑟) = inf 𝜌(𝐷 𝐴). 𝐴∈𝔄𝑟

The directions are deliberately reversed: a larger architecture and smaller defect produce a lower bound; a smaller architecture and larger defect produce an upper bound. How to read the assumptions One simultaneous confidence event must contain pointwise lower/upper defect envelopes and inner/outer architecture-grammar inclusions for every resource budget under study. The same declared risk functional 𝜌 is used in the lower, true, and upper frontiers and is monotone. If the target is a population frontier but 𝑃 is unknown, the confidence construction must also justify the population risk operation; substituting 𝑃𝑛 silently is not sufficient. Pointwise intervals selected separately after looking at the data are not sufficient. Theorem 19.4 (Honest elimination bracket). On the simultaneous confidence event, 𝑂 (𝑟) ≤ 𝑂★ (𝑟) ≤ 𝑂 (𝑟) for all declared resource budgets 𝑟 simultaneously. Interpretation and boundary The theorem converts uncertain defects and uncertain feasible sets into a simultaneous lower and upper bracket for the entire architecture frontier defined by 𝜌. The width records statistical resolution rather than optimization failure. Honesty is inherited from the common event; no assertion is made outside it, and an empirical frontier is not relabeled as a population frontier.

CHAPTER 19. CONFIDENCE WORLDS AND STATISTICAL GEOMETRY

126 Proof roadmap

Use monotonicity to compare each defect envelope with the true defect under the same risk functional and use set inclusion to compare the three infima. Because the event is simultaneous in the budget, the deterministic sandwich holds for the whole frontier at once. The chapter appendix gives the ordered-infimum proof. At threshold 𝜏, the bracket gives the safe readout certified feasible :

𝑂 ≤ 𝜏,

certified impossible :

𝑂 > 𝜏,

unresolved :

𝑂 ≤ 𝜏 < 𝑂.

If [𝑂, 𝑂] is the exact identified image of the frontier over 𝐶𝑛 , this is the standard certificate above. If it is only an outer enclosure, the two resolved declarations remain honest, while the unresolved region can also contain conservatism from the enclosure.

19.4

Persistent atlases and operational transport

Persistent atlas number. Uniformity over 𝑟 permits data-dependent resource selection without a second pointwise argument. For tolerance 𝜏, define the minimum chart number using the convention that a “𝐾-chart deployment” has at most 𝐾 nonempty charts; an atlas with fewer charts may be padded by empty or repeated charts. Thus feasibility is monotone in 𝐾. Put 𝐾 𝐷 (𝜏) = min {𝐾 : there exists a 𝐾-chart deployment with uniform defect ≤ 𝜏} . The integer-valued curve can jump under small perturbations. Define the inverse frontier 𝐿 𝐷 (𝐾) = inf{𝜏 : 𝐾 𝐷 (𝜏) ≤ 𝐾 }. The inversion between a cover by at most 𝐾 local sections and a 𝐾-chart description is definitional. It is not the theorem below. Chart autoencoders, multi-chart flows, and atlas generative models already use multi-chart representations [Schonsheck et al., 2019, Kalatzis et al., 2021, Stolberg-Larsen and Sommer, 2021]; the theorem concerns stability of the declared loss frontier under a uniform defect perturbation. The sharp interleaving and inverse-frontier stability theorem is developed canonically in the Statistical EG companion [Huang and Wang, 2026m]. The EGML companion reuses that interface for its learning-facing atlas and one-sided certificate ledger [Huang and Wang, 2026d]; the book records the mathematical statement once, in the stronger extended-valued form below. How to read the assumptions The population and estimated defects are uniformly within 𝜖 on the complete incidence space, and the atlas number is defined from sublevel incidences using the “at most 𝐾 nonempty charts” convention above. The atlas number and inverse frontier are extended-valued, with +∞ used when no admissible finite-chart deployment exists at the declared level.

19.4. PERSISTENT ATLASES AND TRANSPORT

127

Theorem 19.5 (Atlas interleaving and stable inverse). If b ∥ ∞ ≤ 𝜖, ∥𝐷 − 𝐷 then the atlas curves are horizontally interleaved by 𝜖, and 𝐿 𝐷b (𝐾) ≤ 𝐿 𝐷 (𝐾) + 𝜖,

𝐿 𝐷 (𝐾) ≤ 𝐿 𝐷b (𝐾) + 𝜖

in the extended order for every 𝐾. Whenever either inverse frontier is finite, both are finite and therefore |𝐿 𝐷 (𝐾) − 𝐿 𝐷b (𝐾)| ≤ 𝜖 . The constant one on the finite part is sharp. Interpretation and boundary The integer-valued minimum chart number can jump, but its sublevel filtration interleaves under uniform defect perturbation. Consequently the inverse 𝐾-chart loss frontier is stable. This is a stability theorem for achievable loss, not a consistency theorem for a particular greedy atlas. Proof roadmap The uniform defect bound yields two inclusions between population and estimated sublevel incidences at thresholds shifted by 𝜖. Monotonicity of chart number under incidence inclusion gives interleaving; invert the inequalities to obtain the loss-frontier bound. Full details appear in the chapter appendix. The inverse chart-budget/loss frontier is therefore the statistically stable object. This filtrationand-confidence viewpoint is adjacent to confidence sets for persistence diagrams [Fasy et al., 2014], but the object stabilized here is an architecture loss frontier rather than a homological summary. The underlying interleaving logic is the standard stability mechanism for filtered objects [de Silva et al., 2018, Mémoli et al., 2024]. Nonexpansive operational transport. Task envelopes and contextual closure are nonexpansive in sup norm. Consequently, if internal kernels are simultaneously bracketed, observable architecture frontiers inherit the same confidence event without a new stochastic proof. This is an important design principle: prove deterministic nonexpansiveness once, then reuse the statistical envelope.

19.4.1

Pointwise feasibility versus common deployment

The standard confidence certificate asks whether each compatible population world is feasible. Deployment demands a stronger quantifier: one architecture must work throughout the confidence world. Let 𝐶 be a nonempty set of compatible worlds, let 𝔄 𝑅 be the class admitted by resource budget 𝑅, and let ℓ 𝜃 ( 𝐴) ≥ 0 be the declared audit loss. At tolerance 𝜂, put 𝜂

𝔉𝑅 (𝜃) = { 𝐴 ∈ 𝔄 𝑅 : ℓ 𝜃 ( 𝐴) ≤ 𝜂}.

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Define 𝑉lo (𝐶; 𝑅) = inf inf ℓ 𝜃 ( 𝐴),

(19.6)

𝑉pw (𝐶; 𝑅) = sup inf ℓ 𝜃 ( 𝐴),

(19.7)

𝑉cd (𝐶; 𝑅) = inf sup ℓ 𝜃 ( 𝐴).

(19.8)

𝜃 ∈𝐶 𝐴∈𝔄 𝑅 𝜃 ∈𝐶 𝐴∈𝔄 𝑅

𝐴∈𝔄 𝑅 𝜃 ∈𝐶

How to read the assumptions The confidence world and resource class are nonempty and fixed by the declared contract. The same architecture must be deployed in every world; world-specific architectures are not silently supplied by an oracle. The threshold equivalences below require finite sets or attainment of the displayed extrema. Randomized architectures count only if the declared grammar admits them. Theorem 19.6 (Common-deployment quantifier theorem). For every nonempty 𝐶 and 𝔄 𝑅 , 𝑉lo (𝐶; 𝑅) ≤ 𝑉pw (𝐶; 𝑅) ≤ 𝑉cd (𝐶; 𝑅). Under the attainment condition: 1. 𝑉lo > 𝜂 exactly when every compatible world is individually impossible at budget 𝑅; 2. 𝑉pw ≤ 𝜂 exactly when every compatible world is individually feasible; Ñ 𝜂 3. 𝑉cd ≤ 𝜂 exactly when 𝜃 ∈𝐶 𝔉𝑅 (𝜃) is nonempty. Both inequalities can be strict. In particular, pointwise feasibility in every compatible world need not provide a common deployable witness. Interpretation and boundary There are two distinct reasons not to deploy. A confidence world may mix feasible and impossible populations. Alternatively, every world may be individually feasible while the feasible sets have empty intersection. The second case is a common-witness conflict. More data can remove it by shrinking the confidence world; a router or robust architecture can remove it only by changing the declared grammar. Proof roadmap The inequalities follow from two relaxations: replacing a common architecture by a worldspecific one, and replacing a supremum over worlds by an infimum. Under attainment, each threshold statement is a direct translation of the corresponding feasible-set condition. The strict-gap example below shows why pointwise feasibility does not imply a common witness. The chapter appendix gives the full quantifier argument.

19.4. PERSISTENT ATLASES AND TRANSPORT

129

Table 19.2: Four confidence-world deployment states under attainment. State

Frontier condition

Meaning and authorized next step

common-deployable

𝑉cd ≤ 𝜂

uniformly impossible

𝑉lo > 𝜂

truth-mixed unresolved

𝑉lo ≤ 𝜂 < 𝑉pw

common-witness conflict

𝑉pw ≤ 𝜂 < 𝑉cd

The feasible sets have a common point; exhibit and deploy one common witness. Every compatible world is infeasible; report impossibility or revise the budget or contract. Some worlds are feasible and others are not; collect data that separates them. Every world is individually feasible but the feasibleset intersection is empty; shrink 𝐶 or enlarge the legal routing or robust-deployment grammar.

Example 19.7 (A minimal common-witness conflict). Let 𝐶 = {𝜃 0 , 𝜃 1 }, 𝔄 𝑅 = { 𝐴0 , 𝐴1 }, and let the loss table be 𝜃0 𝜃1 𝐴0 0 1 . 𝐴1 1 0 At tolerance 𝜂 = 0, each world has a zero-loss architecture, so 𝑉pw = 0, but every fixed architecture has worst-world loss one, so 𝑉cd = 1. Pointwise feasibility is complete while common deployment fails. Suppose now that (𝐶𝑡 )𝑡 ≥0 is an anytime-valid confidence sequence, so one time-uniform event of probability at least 1 − 𝛼 satisfies 𝜃 ∈ 𝐶𝑡 for every 𝑡. At any stopping time 𝜏, every F𝜏 -measurable witness in the feasible-set intersection is then valid in the true world. A fixed-time simultaneous confidence set alone does not justify this optional-stopping conclusion. Nor does a confidence world license choosing a different witness for each still-compatible world and presenting that family as one deployed system.

19.4.2

Distribution-free prediction under conditional and shift contracts

Prediction sets make the common-deployment quantifier visible without any architecture terminology. Under exchangeability, a conformal rule can guarantee marginal coverage for a future observation. That contract does not imply exact conditional coverage at every covariate value over a rich distribution class: distribution-free conditional guarantees can force essentially uninformative sets unless the conditioning class, distribution family, or target notion is restricted [Barber et al., 2021]. Pointwise existence of a short, well-calibrated set in each compatible world therefore does not exhibit one data-dependent set rule that is simultaneously short and conditionally valid throughout the confidence world. Covariate shift supplies a different repair only under a different declared experiment. Weighted conformal methods can recover a target-population coverage statement when the conditional law of the response given covariates is invariant and the source-to-target density ratio is known or estimated well enough for the stated guarantee [Tibshirani et al., 2019]. The ratio is an information carrier, not a free optimizer improvement. If the conditional law also shifts, or target support lies outside source support, reweighting does not close the contract. The typed repair choices are consequently distinct: enlarge the output to a set, allow abstention or an “unresolved” action, collect target-domain calibration data, or restrict the target/conditioning domain. Tuning the same point predictor cannot by itself supply missing exchangeability, overlap, or conditional information.

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Boundary of the claim This is an application of classical conformal coverage and impossibility results, not a new conformal theorem. Any use of the book’s deployment language must preserve the exact coverage level, conditioning sigma-field, shift assumptions, set-size loss, and abstention semantics.

19.4.3

Statistical ancestry: pointwise success is not uniform deployment

The common-deployment quantifier is not detached from classical statistics. The Hodges–Le Cam superefficiency example already shows that an estimator may have exceptional asymptotic risk at one fixed parameter while paying for that behavior on a neighborhood that contracts with sample size [Le Cam, 1953]. Hájek’s convolution and local asymptotic minimax theory then made regularity and locally uniform risk comparison central [Hájek, 1970, 1972]. The lesson is not that pointwise results are false. It is that their quantifiers cannot be exchanged without proof. Sparse estimation exhibits the same problem sharply. An oracle property is normally a statement at each fixed parameter: asymptotically, the procedure behaves like an estimator that knew the true submodel in advance. Leeb–Pötscher show why such pointwise asymptotics can coexist with severe nonuniformity and poor maximal risk [Leeb and Pötscher, 2005, 2008]. Thus ∀𝜃 ∃𝑛0 (𝜃)

⇏

∃𝑛0 ∀𝜃.

The deployment rule and sample size are fixed before the unknown world is revealed; the second order of quantifiers is the relevant one.

19.5 A complete quantifier case: sparse model-selection procedures with pointwise oracle guarantees The class at issue consists of support-selecting procedures whose oracle guarantees are pointwise in the underlying parameter. It includes nonconcave penalization, data-adaptive weighted penalization, and exact or algorithmic ℓ0 subset selection. SCAD, Adaptive LASSO, and BSS/BeSS below are coordinate-level representatives of this class, not its name. Consider one coordinate of an orthogonal Gaussian regression experiment, 𝜎 𝑍 𝑗 = 𝛽𝑗 + √ 𝜀 𝑗, 𝑛

𝜀 𝑗 ∼ 𝑁 (0, 1),

with every penalized objective normalized as 1 (𝑍 𝑗 − 𝑏) 2 + pen𝑛 (𝑏). 2 This coordinate model exposes the thresholds exactly. General designs add design identifiability, pilot, and optimization conditions, but do not remove the quantifier conflict.

19.5. POINTWISE-ORACLE SPARSE MODEL SELECTION

19.5.1

131

SCAD: fixed-signal unbiasedness leaves a moving transition band

For SCAD parameter 𝑎 > 2, the orthogonal coordinate solution is   sgn(𝑍 𝑗 ) (|𝑍 𝑗 | − 𝜆 𝑛 )+ ,      (𝑎 − 1)𝑍 𝑗 − sgn(𝑍 𝑗 )𝑎𝜆 𝑛  𝛽bSCAD = , 𝑗  𝑎−2     𝑍 𝑗, 

|𝑍 𝑗 | ≤ 2𝜆 𝑛 , 2𝜆 𝑛 < |𝑍 𝑗 | ≤ 𝑎𝜆 𝑛 , |𝑍 𝑗 | > 𝑎𝜆 𝑛 .

√ Under the usual oracle scaling 𝜆 𝑛 → 0 and 𝑛𝜆 𝑛 → ∞, every fixed nonzero coordinate eventually enters the unpenalized region, while a fixed zero coordinate is deleted with probability tending to one [Fan and Li, 2001]. But for the moving world 𝛽 𝑗,𝑛 = 𝑐𝜆 𝑛 , 0 < 𝑐 < 1, one has 𝑍 𝑗 /𝜆 𝑛 → 𝑐 in probability. The coordinate is therefore set to zero with probability tending to one, and 𝑛 E𝛽 𝑗,𝑛 ( 𝛽bSCAD − 𝛽 𝑗,𝑛 ) 2 ≥ (𝑐2 + 𝑜(1))𝑛𝜆2𝑛 −→ ∞. 𝑗 SCAD removes persistent shrinkage bias for fixed large signals; it relocates, rather than eliminates, the price to a threshold band moving toward zero.

19.5.2

Adaptive LASSO: the conflict moves into a random weight

Use the same score as pilot and let 𝑤 𝑗 = |𝑍 𝑗 | −𝛾 , 𝛾 > 0. The coordinate solution is  𝛽bAL 𝑗 = sgn(𝑍 𝑗 )

 𝜆𝑛 . |𝑍 𝑗 | − |𝑍 𝑗 | 𝛾 +

Its effective threshold is 1/(𝛾+1)

𝑡𝑛 = 𝜆𝑛

,

𝛽bAL 𝑗 ≠ 0 ⇐⇒ |𝑍 𝑗 | > 𝑡 𝑛 .

In the present normalization, the classical pointwise oracle conditions can be written √

𝑛𝜆 𝑛 → 0,

𝑛 (𝛾+1)/2 𝜆 𝑛 → ∞.

The first makes the penalty negligible at a fixed nonzero coordinate; the second is equivalent to √ 𝑛𝑡 𝑛 → ∞ and deletes a fixed zero coordinate [Zou, 2006]. At the moving world 𝛽 𝑗,𝑛 = 𝑐𝑡 𝑛 , 0 < 𝑐 < 1, however, 𝑍 𝑗 /𝑡 𝑛 → 𝑐, so deletion occurs with probability tending to one and 2 2 2 𝑛 E𝛽 𝑗,𝑛 ( 𝛽bAL 𝑗 − 𝛽 𝑗,𝑛 ) ≥ (𝑐 + 𝑜(1))𝑛𝑡 𝑛 −→ ∞.

Adaptive weighting resolves an important fixed-zero/fixed-nonzero conflict of ordinary LASSO, but near zero the pilot amplifies its own noise. The nonuniform region becomes a pilot-dependent lockout region.

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BSS and BeSS: a method-specific repair cannot create information

The one-dimensional ℓ0 objective 1 (𝑍 𝑗 − 𝑏) 2 + 𝜇 𝑛 1{𝑏 ≠ 0} 2 has the global hard-threshold solution 𝛽bBSS = 𝑍 𝑗 1{|𝑍 𝑗 | > 𝑢 𝑛 }, 𝑗

𝑢𝑛 =

√︁

2𝜇 𝑛 .

√ If 𝑢 𝑛 → 0 and 𝑛𝑢 𝑛 → ∞, it has the same fixed-parameter oracle pattern and the same failure of uniformity at 𝛽 𝑗,𝑛 ≍ 𝑢 𝑛 . For one already frozen Adaptive-LASSO rule, choosing 𝑢 𝑛 /𝑡 𝑛 → 0 lets BSS recover signals in the corridor 𝑢 𝑛 ≪ |𝛽 𝑗,𝑛 | ≲ 𝑡 𝑛 . That is a genuine rule-specific repair. It is not a new information bound: the Adaptive-LASSO sequence could itself be retuned, and both rules still use the same observation. For a general design, exact BSS additionally needs a subset-identifiability margin, while an approximate solver must spend part of that margin on optimization error. BeSS searches subsets by a primal–dual active-set algorithm and selects model size by an additional sequential or golden-section rule [Wen et al., 2020]. Even if BeSS is assumed to return the exact global BSS solution, it removes only failures caused by pilot weights, convex relaxation, or optimization. Exact support recovery still requires signal separation and design identifiability [Guo et al., 2020].

19.5.4

The information-limited core and the honest action

Compare the local worlds 𝛽 (0) 𝑗 = 0,

ℎ 𝛽 (1) 𝑗,𝑛 = √ . 𝑛

Their Kullback–Leibler divergence is ℎ2 /(2𝜎 2 ) and does not diverge with 𝑛. Every support selector 𝑆b𝑛 therefore obeys the two-point testing bound 𝑃 𝛽 (0) ( 𝑗 ∈ 𝑆b𝑛 ) + 𝑃 𝛽 (1) ( 𝑗 ∉ 𝑆b𝑛 ) ≥ 1 − TV{𝑃 𝛽 (0) , 𝑃 𝛽 (1) } > 0. 𝑛

𝑛

No support-selecting sparse procedure with only a pointwise oracle guarantee can uniformly recover support on a parameter class containing these local worlds; this includes all three representative mechanisms above. In a high-dimensional √︁ multiple-coordinate problem, the corresponding separation scale is typically enlarged to order 𝜎 log 𝑝/𝑛, with constants depending on the design and error criterion. There are consequently two different blind regions. A method-induced region, such as pilot-weight lockout, can be narrowed by BSS/BeSS. An information-theoretic indistinguishability region cannot be removed by any selector using the same data. A uniform support claim must either restrict the parameter domain, for example to   Θ𝑛 (𝑏 𝑛 ) = 𝛽 : min |𝛽 𝑗 | ≥ 𝑏 𝑛 , 𝑗 ∈𝑆 (𝛽)

with its design margin, model-size rule, and optimization error declared, or report a three-way action: stably nonzero, stably zero, or unresolved.

19.6. OBSERVATIONAL OVERLAP AND DEPLOYMENT CONFLICT

133

Boundary of the claim This section proposes no new sparse-estimation theorem. Pointwise-oracle sparse model selection and the representative mechanisms analyzed here—SCAD, Adaptive LASSO, hard thresholding/BSS, and BeSS—have classical antecedents; so do the two-point lower bound and BSS identifiability margins. The book-level contribution is to place method-induced repair and information-limited nonrepair in one typed common-deployment interface, thereby stating when a method change alters an algorithmic output and when it can actually change an authorized deployment conclusion.

19.6

Main interface theorem family: observational overlap and deployment conflict

The sparse example raises a general question. If every world has its own low-risk architecture, can the data reliably select the required one at deployment time? Let the finite world set be W = {0, . . . , 𝑚 − 1} and 𝑌 ∼ 𝑃𝑤 . Put 𝑚−1 1 ∑︁ d𝑃𝑤 𝜇= 𝑃𝑤 , 𝑝𝑤 = . 𝑚 𝑤=0 d𝜇 Let the deployment action space A be compact metric, let each native loss 𝐷 𝑤 (𝑎) be lower semicontinuous in 𝑎, and define 𝐹𝜂 (𝑤) = {𝑎 ∈ A : 𝐷 𝑤 (𝑎) ≤ 𝜂},

W (𝑦) = {𝑤 : 𝑝 𝑤 (𝑦) > 0}.

How to read the assumptions The experiment laws, native losses, deployment action class, and tolerance are declared together. The exact statement concerns one measurable rule chosen before the unknown world is revealed. The quantitative statement below concerns the declared binary action class {𝑎 0 , 𝑎 1 }; enlarging the action class can alter its conclusion. Theorem 19.8 (Deployment conflict under observational overlap). Under the preceding conditions: 1. A measurable deployment rule 𝛿(𝑌 ) satisfying 𝐷 𝑤 {𝛿(𝑌 )} ≤ 𝜂

𝑃𝑤 -a.s. for every 𝑤

𝐹𝜂 (𝑤) ≠ ∅

for 𝜇-a.e. 𝑦 with W (𝑦) ≠ ∅.

exists if and only if Ù

(DC-E)

𝑤 ∈ W ( 𝑦)

2. For two worlds and two candidate architectures, suppose Δ0 = 𝐷 0 (𝑎 1 ) − 𝐷 0 (𝑎 0 ) > 0,

Δ1 = 𝐷 1 (𝑎 0 ) − 𝐷 1 (𝑎 1 ) > 0.

Every measurable rule taking values in {𝑎 0 , 𝑎 1 } satisfies max E𝑖 [𝐷 𝑖 {𝛿(𝑌 )} − 𝐷 𝑖 (𝑎 𝑖 )] ≥

𝑖 ∈ {0,1}

Δ0 Δ1 {1 − TV(𝑃0 , 𝑃1 )}. Δ0 + Δ1

(DC-R)

CHAPTER 19. CONFIDENCE WORLDS AND STATISTICAL GEOMETRY

134

When Δ0 = Δ1 = Δ, the right side is Δ{1 − TV(𝑃0 , 𝑃1 )}/2. The same number is the exact equal-prior mean-regret Bayes value, attained by a likelihood-ratio rule. The maximum-regret lower bound is not claimed sharp for every asymmetric experiment. Proof roadmap The exact condition says that every observational fiber must contain one action acceptable in all worlds that can produce that observation. For two actions, expected architecture regret is the cross-loss gap times the corresponding testing error. Le Cam’s identity lower-bounds the sum of the two errors by 1 − TV; balancing the two gap-weighted errors gives the displayed harmonic factor. The chapter appendix supplies the complete argument. Corollary 19.9 (Repair trichotomy and invariance). For the problem in Theorem 19.8: 1. changing only the optimizer, while preserving 𝑃𝑤 , 𝐷 𝑤 , and A, changes neither (DC-E) nor the bound (DC-R); 2. an information or exposure repair changes the experiment 𝑃𝑤 , and may reduce overlap by increasing total variation; 3. an architecture or class repair changes A and the feasible sets 𝐹𝜂 (𝑤), and may create a common acceptable action without changing the observations. An empirical case can therefore claim that this interface changed an architecture choice only if it froze the experiment/information carrier, native loss, action class, and compute contract before confirmation, and the mechanism-matched repair beats equal-information, equal-compute controls on a separately declared endpoint. Interpretation and boundary √ Between the local worlds 0 and ℎ/ 𝑛, all three representative procedure families rewrite a support action using the same observation. They do not change 𝑃0 or 𝑃1 and hence cannot remove the information core of (DC-R). BSS can still repair an extra Adaptive-LASSO lockout corridor for a frozen rule. That is a method-level repair, not an erasure of observational overlap. Boundary of the claim The mathematical ingredients of Theorem 19.8 are a measurable common-witness condition and Le Cam’s two-point testing identity [Le Cam, 1953, Tsybakov, 2009]; the information-repair interpretation is adjacent to Blackwell comparison of experiments [Blackwell, 1953]. They are not claimed here as a new decision-theoretic lower bound. The book-level increment is the typed interface that forces solver, information, and architecture repair to leave different auditable traces.

19.6.1

Treatment policies under observational overlap and deployment constraints

Let 𝑌 denote the observational calibration record used to select a policy, let 𝜋 map deployment covariates to a treatment in {0, 1}, and let 𝐷 𝑤 (𝜋) be negative policy welfare or regret in causal world 𝑤. Each compatible world may have its own optimal policy. If two calibration-data laws 𝑃0 , 𝑃1

19.7. ACTIVE-SET CONFLICT AND REPAIR DEPTH

135

overlap while the preferred policies 𝜋0 , 𝜋1 reverse and have positive cross-gaps, then Theorem 19.8 lower-bounds the regret of every selector that must choose from {𝜋0 , 𝜋1 } using only 𝑌 . The statement concerns the frozen observational experiment and policy class; it is not a causal identification theorem. Orthogonal nuisance estimation, double/debiased machine learning, heterogeneous-effect learners, and observational policy learning can improve effect or value estimation under their identification and regularity assumptions [Chernozhukov et al., 2018, Athey and Wager, 2021]. They do not by themselves create positivity, measure an unobserved confounder, or transport effects to a population unsupported by the data. When opposite treatment preferences remain observationally indistinguishable, replacing the learner while preserving (𝑃𝑤 , 𝐷 𝑤 , A) leaves the deployment-conflict certificate unchanged. The certificate instead points to typed repairs. Randomization, an additional effect modifier, or a valid proxy changes the information experiment; a referral/abstention option or a richer dynamic policy changes the action class; restricting to an overlap population changes the deployment domain. Any claim that the repair changed policy architecture must freeze these choices and compare the mechanism-matched intervention against equal-information, equal-compute controls on independent units. Boundary of the claim The potential-outcome identification conditions and policy-learning methods cited here are classical. This example only instantiates the book’s observational-overlap interface. Without consistency, an appropriate exchangeability condition, positivity, and a declared transport population, the native welfare 𝐷 𝑤 is not identified and the displayed audit cannot be estimated from observational data alone.

19.7 Flagship candidate family: active-set conflict, saturation, and repair depth The preceding theorem separates information repair from architecture repair abstractly. The next family makes the architecture side computable for one important inverse problem. Fix a dictionary 𝐷 ∈ R𝑚×𝑘 of full column rank and 𝜆 > 0, and consider nonnegative sparse inference 𝑧★ (𝑥) = arg min 𝐽 𝑥 (𝑧), 𝑧 ≥0

𝐽 𝑥 (𝑧) =

1 ∥𝐷𝑧 − 𝑥∥ 22 + 𝜆1⊤ 𝑧. 2

(19.9)

Put 𝐻 = 𝐷 ⊤ 𝐷, 𝜅 = 𝜆 min (𝐻), 𝐿 = 𝜆 max (𝐻), and 𝑞 = 1 − 𝜅/𝐿 ∈ [0, 1). Full column rank is stronger than is needed for local active-set formulas, but it makes the oracle unique and the global native loss strongly convex. At an input 𝑥 with strict KKT support 𝑆, write 𝐻𝑆 = 𝐷 ⊤ 𝑆 𝐷 𝑆 and 𝑧★𝑆 (𝑥) = 𝐻𝑆−1 (𝐷 ⊤ 𝑆 𝑥 − 𝜆1), 𝑄 𝑆 = 𝐻𝑆−1 𝐷 ⊤ 𝑆, 𝑠ℓ (𝑥) = 𝐷 ℓ⊤ {𝐷 𝑆 𝑧★𝑆 (𝑥) − 𝑥} + 𝜆,

𝑧★𝑆𝑐 (𝑥) = 0, 𝑃𝑆 = 𝐷 𝑆 𝑄 𝑆 ,

(19.10)

ℓ ∉ 𝑆.

Strictness means 𝑧★𝑗 (𝑥) > 0 for 𝑗 ∈ 𝑆 and 𝑠ℓ (𝑥) > 0 for ℓ ∉ 𝑆. Let 𝑉 ⊆ R𝑚 be a declared nonzero linear perturbation subspace, let Π𝑉 be its orthogonal projector, and let 𝑞 ⊤ 𝑆, 𝑗 denote row 𝑗 of 𝑄 𝑆 . Define the

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CHAPTER 19. CONFIDENCE WORLDS AND STATISTICAL GEOMETRY

fully data-computable 𝑉-restricted radius ) ( 𝑧★𝑗 (𝑥) 𝑠ℓ (𝑥) 1 , min , 𝑟 𝑆,𝑉 (𝑥) = min min ⊤ ℓ∉𝑆 ∥𝐷 ⊤ (𝑃𝑆 − 𝐼)Π𝑉 ∥ 2 𝑗 ∈𝑆 ∥𝑞 2 𝑆, 𝑗 Π𝑉 ∥ 2 ℓ

(19.11)

where a positive numerator divided by zero is +∞ and the minimum over an empty set is +∞. Write 𝑟 𝑆 = 𝑟 𝑆,R𝑚 for the unrestricted case, and put 𝐵𝑉 (𝑥, 𝑟) = {𝑥 + 𝑢 : 𝑢 ∈ 𝑉, ∥𝑢∥ 2 ≤ 𝑟 }. How to read the assumptions The oracle objective, nonnegative code domain, dictionary, penalty, and native loss are fixed. The shallow comparator class is exactly F1 = {𝑥 ↦→ [𝑊𝑥 + 𝑏] + }, with one shared affine row per output coordinate. Two certificate centers have strict KKT supports and positive radii no larger than (19.11). The perturbation subspace and any declared input domain containing both restricted balls are fixed before evaluation. A broader ReLU network, a router, or an unrolled optimizer is a different architecture class rather than a member of F1 . Theorem 19.10 (Certified active-set conflict modulus and one-pass saturation). Let 𝑥 1 , 𝑥2 be strict KKT inputs with supports 𝑆1 , 𝑆2 . Fix finite radii 0 < 𝑟 𝑖 ≤ 𝑟 𝑆𝑖 ,𝑉 (𝑥𝑖 ) and write 𝐵𝑖 = 𝐵𝑉 (𝑥𝑖 , 𝑟 𝑖 ). Then: 1. on the closed restricted ball 𝐵𝑖 , the support remains 𝑆𝑖 and 𝑧★𝑆𝑖 (𝑥) = 𝑧★𝑆𝑖 (𝑥𝑖 ) + 𝑄 𝑆𝑖 (𝑥 − 𝑥𝑖 ),

𝑧★𝑆𝑐 (𝑥) = 0; 𝑖

⊤ 2. if a coordinate 𝑗 ∈ 𝑆1 ∩ 𝑆2 has local oracle rows 𝑞 ⊤ 𝑖 = 𝑞 𝑆𝑖 , 𝑗 , define

𝜌𝑖 = 𝑧★𝑗 (𝑥𝑖 ) − 𝑟 𝑖 ∥𝑞 ⊤ 𝑖 Π𝑉 ∥ 2 ,

ℎ = (𝑟 1−1 + 𝑟 2−1 ) −1 ,

Γ12, 𝑗,𝑉 = min{𝜌1 , 𝜌2 , ℎ∥ (𝑞 1 − 𝑞 2 ) ⊤ Π𝑉 ∥ 2 }.

(19.12)

Every one-pass thresholded-affine encoder 𝑓 ∈ F1 obeys sup

𝑥 ∈ 𝐵1 ∪𝐵2

∥ 𝑓 (𝑥) − 𝑧★ (𝑥) ∥ 2 ≥ Γ12, 𝑗,𝑉 .

(19.13)

Consequently its worst-case native-loss gap satisfies sup {𝐽 𝑥 ( 𝑓 (𝑥)) − 𝐽 𝑥 (𝑧★ (𝑥))} ≥

𝑥 ∈ 𝐵1 ∪𝐵2

𝜅 2 Γ . 2 12, 𝑗,𝑉

(19.14)

The certificate is nonvacuous exactly when the displayed conflict modulus is positive. Interpretation and boundary The obstruction is not merely that two active sets differ. It is the quantitative incompatibility of two oracle Jacobian rows along perturbations that the deployment contract actually permits, on neighborhoods where the shared coordinate must stay positive. A single ReLU coordinate is then forced to use one affine row on both neighborhoods, while the oracle uses two. The harmonic radius converts that mismatch into a code-error floor; strong convexity converts the code floor into the declared native loss. The choice 𝑉 = R𝑚 recovers ordinary Euclidean balls. A smaller 𝑉 can freeze categorical or otherwise inadmissible coordinates, but the restricted balls must still lie inside the declared input domain.

19.7. ACTIVE-SET CONFLICT AND REPAIR DEPTH

137

Define one nonnegative proximal-gradient repair step by 

 1 ⊤ 𝑇𝑥 (𝑧) = 𝑧 − (𝐻𝑧 − 𝐷 𝑥 + 𝜆1) . 𝐿 +

(19.15)

This is a legal structural repair only when inference-time recurrence is admitted by the deployment grammar and charged to its compute ledger. Theorem 19.11 (Repair contraction, native-loss upper envelope, and crossing budget). For every 𝑥 and 𝑧 ≥ 0, ∥𝑇𝑥𝑡 (𝑧) − 𝑧★ (𝑥) ∥ 2 ≤ 𝑞 𝑡 ∥𝑧 − 𝑧★ (𝑥) ∥ 2 .

(19.16)

Let 𝐾 ⊂ R𝑚 be nonempty and compact, let 𝑓 : 𝐾 → R+𝑘 be bounded, and put 𝐸 0 = sup ∥ 𝑓 (𝑥) − 𝑧★ (𝑥) ∥ 2 , 𝑥 ∈𝐾

𝐵 𝐾 = sup ∥𝐻𝑧★ (𝑥) − 𝐷 ⊤ 𝑥 + 𝜆1∥ 2 . 𝑥 ∈𝐾

Then the repaired architecture 𝑓𝑡 (𝑥) = 𝑇𝑥𝑡 { 𝑓 (𝑥)} satisfies sup {𝐽 𝑥 ( 𝑓𝑡 (𝑥)) − 𝐽 𝑥 (𝑧★ (𝑥))} ≤ 𝐵 𝐾 𝑞 𝑡 𝐸 0 +

𝑥 ∈𝐾

𝐿 2𝑡 2 𝑞 𝐸0 . 2

(19.17)

For a native-loss tolerance 𝜂 > 0, set √︃ 𝑑𝜂 =

𝐵2𝐾 + 2𝐿𝜂 − 𝐵 𝐾 . 𝐿

(19.18)

Define the sufficient crossing depth for all boundary cases by  0,       𝑡 𝜂 := 1,    log(𝐸 0 /𝑑 𝜂 )   ,   − log 𝑞

𝐸0 ≤ 𝑑 𝜂 , 𝐸 0 > 𝑑 𝜂 and 𝑞 = 0,

(19.19)

𝐸 0 > 𝑑 𝜂 and 0 < 𝑞 < 1.

Every integer depth 𝑡 ≥ 𝑡 𝜂 guarantees native-loss gap at most 𝜂 on 𝐾. The contraction factor 𝑞 is sharp for this fixed proximal mechanism whenever the projection is locally inactive along a 𝜅-eigenvector of 𝐻. Corollary 19.12 (Certificate-driven architecture change). Take 𝐾 ⊇ 𝐵1 ∪ 𝐵2 and suppose 0 < 𝜂 < 2 𝜅Γ12, 𝑗,𝑉 /2. No architecture in F1 can meet the native-loss tolerance uniformly on 𝐾. If a declared initializer 𝑓 ∈ F1 has finite 𝐸 0 and the grammar admits at least 𝑡 𝜂 applications of (19.15), then 𝑓𝑡𝜂 does meet it. Under the same information carrier and native loss, the certificate therefore changes the admissible architecture choice from a one-pass thresholded-affine encoder to a recurrent or unrolled repair of sufficient depth.

CHAPTER 19. CONFIDENCE WORLDS AND STATISTICAL GEOMETRY

138 Proof roadmap

Strict KKT margins survive the certified perturbation radius, yielding the two affine oracle cells. If a one-pass encoder approximated both restricted balls more closely than Γ12, 𝑗,𝑉 , its shared coordinate would be positive on both balls. Symmetric perturbations inside 𝑉 force the restriction of its one affine row to lie within 𝑒/𝑟 𝑖 of each restricted oracle row, contradicting the Jacobian conflict. Strong convexity yields the native floor. For repair, ReLU is nonexpansive and ∥𝐼 − 𝐻/𝐿 ∥ 2 = 𝑞; expanding the quadratic objective around the KKT point gives the native upper envelope and then the explicit crossing depth. The chapter appendix supplies all steps and boundary cases. Boundary of the claim This family is an architecture-specific quantitative synthesis, not a claim that shallow sparse encoders or proximal repair were previously unknown. Gregor–LeCun introduced learned finite-depth approximations to sparse coding [Gregor and LeCun, 2010]; unfolded ISTA has established convergence theory [Chen et al., 2018]; proximal-gradient and FISTA rates are classical [Beck and Teboulle, 2009]; local sparse/piecewise-linear radii have close neighbors in sparse local Lipschitz analysis [Muthukumar and Sulam, 2023]. Most importantly, O’Neill et al. [2025] already prove a global amortisation gap for a one-layer linear–nonlinear sparse autoencoder and show that inference-time optimization can improve sparse inference. The candidate increment here is narrower: an input-computable two-neighborhood Jacobian conflict modulus, its native-loss floor, and a matched sufficient repair-depth formula. The exact combination was not located in the recorded priority search, but independent priority is not established; the family is therefore tagged O, not advertised as an independently confirmed original theorem.

19.7.1

A no-free-lunch theorem for hidden winding

Source result; proof not reproduced here The hidden-winding impossibility, exact mixed-integer degree-identification formulation, localspacing recovery condition, and matching random-design rates summarized in this and the next section are established in the Statistical Elimination Geometry manuscript. They are not principal formal results of this book, and their proofs are not in the chapter appendix [Huang and Wang, 2026m]. In the radical architecture, the one-chart obstruction is zero or one according to a divisibility condition on the target degree after normalization. With finitely many sampled angles and no regularity or coverage condition, a winding can be inserted into an unsampled interval. The observed data can then be identical under feasible and impossible population worlds. Integer-wrap and phase-unwrapping formulations provide the classical computational background [Itoh, 1982, Goldstein et al., 1988, Costantini, 1998, Yokoyama, 1997]. Information-based complexity results for topological degree of multidimensional Lipschitz maps provide a further neighboring lower-bound tradition [Boult and Sikorski, 1986]. That cube problem is not identical to the present one-dimensional circle-winding

19.8. IDENTIFICATION, HIDDEN WINDING, AND DATA-DEPENDENT ARCHITECTURE 139 contract. Therefore no uniformly valid binary procedure can be informative in general. An honest procedure must sometimes return unresolved.

19.8

Identification, hidden winding, and data-dependent architecture

Degree identification under regularity. Assume the target phase is 𝐿-Lipschitz and observed with bounded angular error. The set of degrees compatible with the data can be described exactly by a finite mixed-integer program. Divisibility of the entire identification set yields a valid radical certificate. A local spacing condition can collapse the identification set to one degree. Under uniform random design, upper and lower sample-complexity bounds match at order 𝐿{log 𝐿 + log(1/𝛼)} for the square-root parity problem. The rate is governed by coverage of locations where an obstruction can hide, not only by pointwise estimation accuracy. The limited novelty claim is the noisy exact mixed-integer identification set together with the random-design logarithmic coverage surcharge. The bare fact that a degree-𝐿 loop requires order-𝐿 spatial resolution is not claimed as new. Sample splitting and data-dependent architecture. A practical OALI analysis can use three independent levels: 1. calibration data estimate local oracles, transports, and obstruction certificates; 2. tuning data choose among a finite repair library and fit architectures; 3. held-out biological units evaluate final operational performance. This design avoids the need for a fully general random-set theorem in the first implementation, though such theory remains valuable for data efficiency.

Exercises Exercise 19.13. Verify the direction of the inner/outer architecture bracket. Why does the lower bound use the outer grammar? Exercise 19.14. Show that a sup-norm perturbation of the defect shifts the inverse 𝐾-chart loss frontier by at most the same amount. Exercise 19.15. Construct two circle maps that agree on a finite sample but have different winding parity.

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Appendix to Chapter 19: complete proofs Proof of honesty and maximal decisiveness Dependencies. The confidence world must cover the true population world. The query and its truth map are declared, and the exact truth image 𝑞(𝐶) is available. Compactness is unnecessary for the set-inclusion rule; it is needed only if one replaces the identified image by attained lower and upper endpoints. Proof. Fix a realized nonempty confidence world 𝐶. If 𝑞(𝐶) = {𝑑}, then every compatible world has truth label 𝑑. On the event that the true world belongs to 𝐶, the resolved output is therefore correct. Thus any false resolved declaration implies failure of confidence-world coverage, whose probability is at most 𝛼. For maximal decisiveness, suppose 𝑞(𝐶) contains both 𝐹 and 𝐼. An 𝐹 declaration is incorrect in a compatible 𝐼-world, while an 𝐼 declaration is incorrect in a compatible 𝐹-world. Hence no rule required to be correct for every world in 𝐶 may make either terminal declaration. The unresolved output is forced. If 𝑞(𝐶) is a singleton, the standard certificate reports its common label. It therefore declares in every and only every case in which a worldwise-correct comparator can declare. When 𝐶 = ∅, the separate output 𝑀 records model conflict and makes no population claim. □

Proof of the honest elimination bracket Dependencies. One simultaneous event must contain both the pointwise defect envelopes and the inner/outer grammar inclusions for every resource budget under consideration. The same declared risk functional is used for all three frontiers and must be monotone; translation equivariance is useful for perturbation bounds but not needed for the bracket itself. Proof. Fix a resource budget 𝑟. On the simultaneous event, for every 𝐴 ∈ 𝔄★ 𝑟, 𝜌{𝐷 ( 𝐴, ·)} ≤ 𝜌{𝐷 ★ ( 𝐴, ·)} ≤ 𝜌{𝐷 ( 𝐴, ·)} by monotonicity. Moreover, 𝔄 𝑟 ⊆ 𝔄★ 𝑟 ⊆ 𝔄𝑟 . Enlarging a feasible set can only decrease an infimum, while restricting it can only increase an infimum. Hence inf 𝜌{𝐷 ( 𝐴, ·)} ≤ inf★ 𝜌{𝐷 ★ ( 𝐴, ·)} ≤ inf 𝜌{𝐷 ( 𝐴, ·)}. 𝐴∈𝔄𝑟

𝐴∈𝔄𝑟

𝐴∈𝔄𝑟

The middle term is 𝑂★ (𝑟). Because the event is simultaneous in 𝑟, the same deterministic inequalities hold for every declared budget at once. If the event has probability at least 1 − 𝛼 uniformly over population worlds, the complete frontier bracket is uniformly honest at that level. □

Proof of atlas interleaving and inverse stability Dependencies. The defects are continuous only to make the local-section formulation standard; the numerical interleaving itself uses the uniform sup-norm bound.

Appendix to Chapter 19: complete proofs

141

Proof. If 𝐷 (𝑥, 𝑎) ≤ 𝜏 − 𝜖, then b (𝑥, 𝑎) ≤ 𝐷 (𝑥, 𝑎) + 𝜖 ≤ 𝜏, 𝐷 so 𝐸 𝐷 (𝜏 − 𝜖) ⊆ 𝐸 𝐷b (𝜏). Similarly, 𝐸 𝐷b (𝜏) ⊆ 𝐸 𝐷 (𝜏 + 𝜖). A local section of a smaller incidence remains a local section after inclusion into a larger incidence. Since the minimum chart number decreases when the incidence enlarges, 𝐾 𝐷 (𝜏 + 𝜖) ≤ 𝐾 𝐷b (𝜏) ≤ 𝐾 𝐷 (𝜏 − 𝜖), with the convention 𝐾 𝐷 (𝑠) = +∞ for 𝑠 < 0. For the inverse frontier, consider any 𝐾-chart deployment. Its maximum defect under 𝐷 and under b differs by at most 𝜖. Taking the infimum over all 𝐾-chart deployments gives 𝐷 𝐿 𝐷b (𝐾) ≤ 𝐿 𝐷 (𝐾) + 𝜖 . b gives Interchanging 𝐷 and 𝐷 𝐿 𝐷 (𝐾) ≤ 𝐿 𝐷b (𝐾) + 𝜖 . Both inequalities are meaningful in the extended order. If either frontier is finite, the two inequalities make the other finite as well, and subtraction then gives |𝐿 𝐷 (𝐾) − 𝐿 𝐷b (𝐾)| ≤ 𝜖 . Sharpness on the finite part follows on singleton input and action spaces with constant defects 𝐷 ≡ 𝑐 b ≡ 𝑐 + 𝜖. and 𝐷 □

Proof of the common-deployment quantifier theorem Dependencies. The inequality chain itself needs only nonempty world and architecture sets. Finiteness or attainment is used for the exact threshold equivalences, because an unattained infimum at the threshold need not supply a feasible witness. Proof. For the first inequality, write 𝑚(𝜃) = inf ℓ 𝜃 ( 𝐴). 𝐴∈𝔄 𝑅

Then 𝑉lo (𝐶; 𝑅) = inf 𝑚(𝜃) ≤ sup 𝑚(𝜃) = 𝑉pw (𝐶; 𝑅). 𝜃 ∈𝐶

𝜃 ∈𝐶

For every fixed 𝐴 ∈ 𝔄 𝑅 and every 𝜃 ∈ 𝐶, inf ℓ 𝜃 ( 𝐴′ ) ≤ ℓ 𝜃 ( 𝐴).

𝐴′ ∈𝔄 𝑅

Taking the supremum over 𝜃 and then the infimum over 𝐴 yields 𝑉pw (𝐶; 𝑅) ≤ inf sup ℓ 𝜃 ( 𝐴) = 𝑉cd (𝐶; 𝑅). 𝐴∈𝔄 𝑅 𝜃 ∈𝐶

Under attainment, 𝑉lo > 𝜂 says precisely that ℓ 𝜃 ( 𝐴) > 𝜂 for every pair (𝜃, 𝐴), which is equivalent 𝜂 to 𝔉𝑅 (𝜃) = ∅ for every 𝜃. Likewise, 𝑉pw ≤ 𝜂 says that each worldwise minimum is at most 𝜂, so

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each feasible set is nonempty. Finally, 𝑉cd ≤ 𝜂 says that one attained architecture has loss at most 𝜂 Ñ 𝜂 simultaneously for every world, exactly the condition 𝜃 ∈𝐶 𝔉𝑅 (𝜃) ≠ ∅. For strictness of the second inequality, take two worlds and two deterministic architectures with loss table 𝜃0 𝜃1 𝐴0 0 1 . 𝐴1 1 0 Then 𝑉lo = 𝑉pw = 0 but 𝑉cd = 1: each world has a zero-loss architecture, yet there is no common zero-loss witness. With the singleton architecture class { 𝐴0 }, one instead has 𝑉lo = 0 < 𝑉pw = 𝑉cd = 1, proving that the first inequality can also be strict. Adding randomized architectures may change these values, but only when randomization belongs to the declared grammar. □

Proof of deployment conflict under observational overlap and the repair trichotomy Dependencies. The world set is finite. The experiment laws, native losses, action class, and tolerance are common to the statement. The exact condition uses one measurable deployment rule. The regret bound restricts that rule to the two declared candidate actions. Proof. Suppose first that a deployment rule 𝛿 satisfies the tolerance in every world. For each 𝑤, 𝛿(𝑦) ∈ 𝐹𝜂 (𝑤) on {𝑝 𝑤 > 0} outside a 𝜇-null set. There are only finitely many worlds, so the union of these exceptional sets is still 𝜇-null. At every remaining 𝑦, the chosen action belongs to 𝐹𝜂 (𝑤) for every 𝑤 ∈ W (𝑦), proving necessity of (DC-E). Conversely, suppose the intersections in (DC-E) are nonempty almost everywhere. The support pattern W (𝑦) takes only finitely many values. For every nonempty pattern 𝑆 ⊆ W that occurs off the Ñ exceptional set, choose one action 𝑎 𝑆 ∈ 𝑤 ∈𝑆 𝐹𝜂 (𝑤) and set 𝛿(𝑦) = 𝑎 W ( 𝑦) . Each set {𝑦 : W (𝑦) = 𝑆} is measurable because the densities 𝑝 𝑤 are measurable, so 𝛿 is measurable. For a fixed world 𝑤, whenever 𝑝 𝑤 (𝑦) > 0 one has 𝑤 ∈ W (𝑦) and hence 𝛿(𝑦) ∈ 𝐹𝜂 (𝑤), outside the common null set. This proves sufficiency. For the binary bound, put 𝑒 0 = 𝑃0 {𝛿(𝑌 ) = 𝑎 1 },

𝑒 1 = 𝑃1 {𝛿(𝑌 ) = 𝑎 0 }.

Le Cam’s two-point testing identity gives 𝑒 0 + 𝑒 1 ≥ 1 − TV(𝑃0 , 𝑃1 ) =: 𝑐. The expected architecture regrets in the two worlds are respectively Δ0 𝑒 0 and Δ1 𝑒 1 . If their maximum is 𝑀, then 𝑒 0 ≤ 𝑀/Δ0 and 𝑒 1 ≤ 𝑀/Δ1 . Therefore   1 1 𝑐≤𝑀 + , Δ0 Δ1 which rearranges to (DC-R). With equal gaps, the likelihood-ratio test minimizes (𝑒 0 + 𝑒 1 )/2 and attains {1−TV(𝑃0 , 𝑃1 )}/2, proving the stated equal-prior Bayes claim. The maximum-regret statement used only the preceding lower bound and does not require the two errors to be equalizable. For the corollary, changing only the optimizer preserves 𝑃𝑤 , 𝐹𝜂 (𝑤), the gaps, and total variation. Information repair changes the experiment laws and can therefore change total variation. Architecture repair changes the action space and feasible-set intersections. The frozen empirical requirements are an identification contract keeping these three interventions from being conflated, not an additional probabilistic inequality. □

Appendix to Chapter 19: complete proofs

143

Proof of the active-set conflict and repair-depth family Dependencies. The dictionary has full column rank, so 𝐻 = 𝐷 ⊤ 𝐷 is positive definite and the constrained oracle is unique. The lower bound is specific to the one-pass class [𝑊𝑥 + 𝑏] + and to two symmetric balls in one declared perturbation subspace on which a shared oracle coordinate is strictly positive. The repair bound charges each proximal application and uses the fixed step 1/𝐿. Proof. We first prove the certified-cell statement. At a strict KKT point with support 𝑆, stationarity on the active coordinates gives 𝐻𝑆 𝑧★𝑆 − 𝐷 ⊤ 𝑆 𝑥 + 𝜆1 = 0, and hence the formula in (19.10). For a perturbation 𝑢 ∈ 𝑉, consider the candidate e 𝑧 𝑆 = 𝑧★𝑆 (𝑥) + 𝑄 𝑆 𝑢,

e 𝑧 𝑆 𝑐 = 0.

It still satisfies active stationarity at 𝑥 + 𝑢. For every 𝑗 ∈ 𝑆, e 𝑧 𝑗 ≥ 𝑧★𝑗 (𝑥) − ∥𝑞 ⊤ 𝑆, 𝑗 Π𝑉 ∥ 2 ∥𝑢∥ 2 > 0 whenever ∥𝑢∥ 2 ≤ 𝑟 𝑆,𝑉 (𝑥); the factor 1/2 in (19.11) makes the inequality strict. For ℓ ∉ 𝑆, the inactive KKT slack becomes e 𝑠ℓ = 𝑠ℓ (𝑥) + 𝐷 ℓ⊤ (𝑃𝑆 − 𝐼)𝑢 ≥ 𝑠ℓ (𝑥) − ∥𝐷 ℓ⊤ (𝑃𝑆 − 𝐼)Π𝑉 ∥ 2 ∥𝑢∥ 2 > 0. The zero-denominator convention covers a slack or active coordinate that is unchanged by 𝑢. Thus the candidate satisfies the KKT conditions with the same strict support. Strict convexity makes it the unique oracle throughout the closed restricted ball. The conclusion remains valid for every smaller positive radius. Now fix a shared coordinate 𝑗 and abbreviate Γ = Γ12, 𝑗,𝑉 . Suppose, for contradiction, that some 𝑓 (𝑥) = [𝑊𝑥 + 𝑏] + has uniform code error 𝑒 < Γ on 𝐵1 ∪ 𝐵2 . On 𝐵𝑖 the oracle coordinate obeys 𝑧★𝑗 (𝑥) ≥ 𝑧★𝑗 (𝑥𝑖 ) − 𝑟 𝑖 ∥𝑞 ⊤ 𝑖 Π𝑉 ∥ 2 = 𝜌𝑖 . Because 𝑒 < 𝜌𝑖 , the 𝑗th output of 𝑓 is strictly positive throughout both balls. It is therefore the same affine function 𝑤 ⊤ 𝑥 + 𝑏 𝑗 on both balls, where 𝑤 ⊤ is row 𝑗 of 𝑊. On 𝐵𝑖 , the oracle coordinate is another affine function with row 𝑞 𝑖 . For any unit vector 𝑣 ∈ 𝑉, evaluate their difference at 𝑥 𝑖 + 𝑟 𝑖 𝑣 and 𝑥 𝑖 − 𝑟 𝑖 𝑣. Both absolute errors are at most 𝑒, so their difference gives 2𝑟 𝑖 |(𝑤 − 𝑞 𝑖 ) ⊤ 𝑣| ≤ 2𝑒. Taking the supremum over unit 𝑣 ∈ 𝑉 yields ∥ (𝑤 − 𝑞 𝑖 ) ⊤ Π𝑉 ∥ 2 ≤ 𝑒/𝑟 𝑖 . The triangle inequality now implies ∥ (𝑞 1 − 𝑞 2 ) ⊤ Π𝑉 ∥ 2 ≤ 𝑒(𝑟 1−1 + 𝑟 2−1 ), or 𝑒 ≥ ℎ∥ (𝑞 1 − 𝑞 2 ) ⊤ Π𝑉 ∥ 2 , contradicting 𝑒 < Γ. Hence (19.13) holds, including the vacuous case Γ = 0. The function 𝑧 ↦→ 𝐽 𝑥 (𝑧) + 𝜄R+𝑘 (𝑧) is 𝜅-strongly convex. At its minimizer, the subgradient inequality gives 𝜅 𝐽 𝑥 (𝑧) − 𝐽 𝑥 (𝑧★ (𝑥)) ≥ ∥𝑧 − 𝑧★ (𝑥) ∥ 22 (𝑧 ≥ 0). 2

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Combining this pointwise inequality with the code-error floor proves (19.14). For the repair result, write   𝑇𝑥 (𝑧) = (𝐼 − 𝐻/𝐿)𝑧 + 𝐷 ⊤ 𝑥/𝐿 − 𝜆1/𝐿 + . The KKT conditions are equivalent to the fixed-point relation 𝑇𝑥 (𝑧★ (𝑥)) = 𝑧★ (𝑥). Coordinatewise projection onto the nonnegative orthant is nonexpansive, while the spectrum of 𝐻 lies in [𝜅, 𝐿]. Therefore ∥𝑇𝑥 (𝑧) − 𝑧★ (𝑥) ∥ 2 ≤ ∥ (𝐼 − 𝐻/𝐿) (𝑧 − 𝑧★ (𝑥)) ∥ 2 ≤ (1 − 𝜅/𝐿) ∥𝑧 − 𝑧★ (𝑥) ∥ 2 . Iteration proves (19.16). Let 𝑒 𝑡 = 𝑓𝑡 (𝑥) − 𝑧★ (𝑥) and let 𝑠(𝑥) = 𝐻𝑧★ (𝑥) − 𝐷 ⊤ 𝑥 + 𝜆1 be the KKT slack vector. Direct expansion of the quadratic objective gives 𝐽 𝑥 ( 𝑓𝑡 (𝑥)) − 𝐽 𝑥 (𝑧★ (𝑥)) =

1 ⊤ 𝑒 𝐻𝑒 𝑡 + 𝑠(𝑥) ⊤ 𝑒 𝑡 . 2 𝑡

The slack is zero on active coordinates; on inactive coordinates 𝑧★ = 0, 𝑓𝑡 ≥ 0, and hence 𝑠⊤ 𝑒 𝑡 ≥ 0. Cauchy–Schwarz and the largest eigenvalue bound give 𝐽 𝑥 ( 𝑓𝑡 (𝑥)) − 𝐽 𝑥 (𝑧★ (𝑥)) ≤

𝐿 ∥𝑒 𝑡 ∥ 22 + ∥𝑠(𝑥) ∥ 2 ∥𝑒 𝑡 ∥ 2 . 2

Taking suprema and using ∥𝑒 𝑡 ∥ 2 ≤ 𝑞 𝑡 𝐸 0 proves (19.17). The positive root of 𝐵 𝐾 𝑑 + 𝐿𝑑 2 /2 = 𝜂 is exactly 𝑑 𝜂 . Thus 𝑞 𝑡 𝐸 0 ≤ 𝑑 𝜂 is sufficient. If 𝐸 0 ≤ 𝑑 𝜂 , depth zero already works. If 𝐸 0 > 𝑑 𝜂 and 𝑞 = 0, the contraction inequality makes the first repaired iterate exact. If 𝐸 0 > 𝑑 𝜂 and 0 < 𝑞 < 1, solving the same inequality for integer 𝑡 gives the third branch of (19.19). Compactness and continuity of the unique strongly convex solution map make 𝑧★ and 𝐵 𝐾 bounded; boundedness of the declared initializer makes 𝐸 0 finite. Finally, suppose a KKT point is interior and a sufficiently small error is parallel to a unit eigenvector 𝑣 of 𝐻 with eigenvalue 𝜅, with the entire segment remaining inside the positive orthant. Projection is then inactive and 𝑇𝑥 (𝑧★ + 𝑎𝑣) − 𝑧★ = (𝐼 − 𝐻/𝐿)𝑎𝑣 = 𝑞𝑎𝑣. Hence no smaller uniform contraction coefficient is valid for this fixed mechanism. For the corollary, the initializer is explicitly restricted to F1 . Applying the lower bound to 𝐵1 ∪ 𝐵2 ⊆ 𝐾 rules out that initializer class, while the upper envelope at the well-defined depth 𝑡 𝜂 certifies its recurrent repair. □

Chapter 20

Certificate Statistics and Resolution Complexity 20.1

The certificate record and four nonexchangeable objects

An estimate or posterior alone does not record which population worlds remain compatible with the data, which decision boundary is being audited, or why the answer remains unresolved. Certificate statistics treats the primary finite-information output as a typed record. Companion framework and proof provenance The confidence-world record, resolution complexity, evidence-slack criterion, and recursive quotient developed in this chapter are organized in the Certificate Statistics companion manuscript [Huang and Wang, 2026c]. The chapter appendices give the complete proofs counted by this book; the source citation records provenance rather than replacing those proofs.

A certificate record contains: • a declared query contract and operational boundary 𝐵; • a confidence world 𝐶𝑛 with an explicit coverage contract; • the identified image 𝐽𝐶𝑛 (𝑞) and current certificate color; • the certificate margin or tolerance profile when the query is ordered; • evidence or witnesses supporting the color; • a resolution profile indicating what information is still needed. The record connects four objects without identifying their validity contracts. 145

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Object

Evaluated on

What it can guarantee

population truth 𝑞(𝑤)

one fixed population world 𝑤

𝑞 confidence certificate 𝛿𝐶 𝑛

a covered confidence world 𝐶𝑛

posterior color 𝜒(𝑏 𝑛 )

a prior-relative belief state 𝑏 𝑛

deployment action 𝑎 𝑛

observer-visible information and the declared deployment grammar

the task cell that is true at the population level worldwise error control for every resolved finite-data declaration a summary of current posterior belief, not uniform validity over worlds operational loss control only when one admissible action or witness works across the required worlds

The four objects can constrain one another, but agreement of their displayed labels does not make them interchangeable. A resolved confidence certificate is not yet a common deployment witness, and high posterior probability is not by itself a worldwise certificate. Reject-option classification and three-way decision theory are important neighbors [Chow, 1970, Yao, 2010]. They optimize or summarize actions under a probabilistic or loss-based contract. Proposition 20.1 isolates a different question: whether one terminal label is correct simultaneously for every world retained by a confidence set. The distinction, not abstention itself, is the point. How to read the assumptions A prior and posterior are well defined for the positive statement. For the separation, the model permits two observationally indistinguishable worlds with different truth labels. No amount of repeated observation can identify those labels under the available experiment.

Proposition 20.1 (Posterior credibility is not worldwise validity). For any fixed posterior credibility level below one, there exist a prior and two observationally indistinguishable worlds with opposite certificate colors such that the posterior assigns high probability to one color at every time, while the minority world is reported incorrectly with probability one. Under the available experiment family the corresponding worldwise resolution complexity is infinite. Interpretation and boundary Small posterior error is a prior-average statement. It does not imply a uniform frequentist guarantee for every possible world: a low-prior but observationally indistinguishable world can be reported incorrectly with probability one. This does not invalidate Bayesian credibility; it identifies a different validity contract. Proof roadmap Use iterated expectation for the prior-average guarantee. For the counterexample, place small prior mass on one of two observationally identical worlds with opposite labels; the posterior never moves, so the majority report remains credible but fails pointwise in the minority world. The chapter appendix gives the construction.

20.1. THE CERTIFICATE RECORD AND FOUR NONEXCHANGEABLE OBJECTS

20.1.1

147

Truth-map notation for the standard certificate

The preceding proposition warns against reading posterior mass as an honest architecture certificate. Chapter 19 defined the standard confidence certificate for an arbitrary truth map. Here we record the compact notation needed by the resolution theory. For a confidence world 𝐶𝑛 , write 𝑐(𝐶𝑛 ) = {𝑐(𝑤) : 𝑤 ∈ 𝐶𝑛 } ⊆ {𝐹, 𝐼}, and set   𝑀, 𝐶𝑛 = ∅,    𝑐 𝛿𝐶𝑛 = 𝑑, 𝑐(𝐶𝑛 ) = {𝑑} for 𝑑 ∈ {𝐹, 𝐼},   𝑈, |𝑐(𝐶𝑛 )| > 1.  How to read the assumptions The nonempty confidence world has uniform coverage, and a declared truth map is evaluated on that world. An empty set produces the model-conflict state 𝑀, not a population declaration. Simultaneous post-selection validity requires one confidence event for the entire query class, not separately chosen pointwise intervals. Corollary 20.2 (Truth-map form of three-way honesty). If 𝐶𝑛 covers the true world with probability at 𝑐 is honest at level 1 − 𝛼. On a realized nonempty confidence least 1 − 𝛼, every resolved output of 𝛿𝐶 𝑛 world, no other rule based on the same world can make more declarations while preserving worldwise correctness. Interpretation and boundary This is Theorem 19.3 written in the notation used below. Unresolvedness means that the truth image is not a singleton; model conflict means that no compatible world survived the declared model and confidence construction. The two states must not be conflated. Proof roadmap Apply Theorem 19.3 to the two cells of the truth partition. One simultaneous coverage event gives the post-selection extension; the chapter appendix records this reduction.

CHAPTER 20. CERTIFICATE STATISTICS AND RESOLUTION COMPLEXITY

148

20.2

Resolution complexity

Confidence sequences make the same statement valid under adaptive querying and stopping. Datadependent hypothesis sets and random-set generalization bounds address a related selection problem [Dupuis et al., 2024]; the object here is different because the set represents covered population worlds rather than a learned hypothesis class. A current unresolved state contains compatible worlds of opposite certificate colors. The relevant information is therefore not global parameter information but separation from the nearest world that would reverse the declared conclusion. Let Θ be a finite world set, let 𝑞 : Θ → Q be the truth partition, and let E be a finite experiment set. Experiment 𝑒 has law 𝑃 𝜃 ,𝑒 in world 𝜃. Define 𝑑 𝑒 (𝜃, 𝜆) = KL(𝑃 𝜃 ,𝑒 ∥𝑃𝜆,𝑒 ), I𝑋𝑞,∗ (𝜃) = max min 𝑤 ∈Δ( E ) 𝜆∈Alt ( 𝜃 ) 𝑞

∑︁

𝑤 𝑒 𝑑 𝑒 (𝜃, 𝜆),

Alt𝑞 (𝜃) = {𝜆 : 𝑞(𝜆) ≠ 𝑞(𝜃)},

(20.1)

𝑇𝑋𝑞,∗ (𝜃) = {I𝑋𝑞,∗ (𝜃)} −1 .

(20.2)

𝑒∈ E

We use the convention 1/0 = +∞. The allocation 𝑤 is an experimental frequency, whereas the inner minimum is chosen by the least separated opposite-label world. A sequential rule (𝜏, 𝑞b) is 𝛼-worldwise valid when sup P 𝜃 {b 𝑞 ≠ 𝑞(𝜃), 𝜏 < ∞} ≤ 𝛼.

(20.3)

𝜃 ∈Θ

How to read the assumptions The experiment and world sets are finite; adaptive experiment choices are nonanticipating; the pairwise KL divergences are defined on common measurable spaces; and the standard sequential change-of-measure identity applies. At the world under audit, E 𝜃 𝜏 < ∞. Pairwise identifiability is not assumed: if every allocation leaves an opposite label at zero KL rate, the theorem correctly returns an infinite lower bound. Theorem 20.3 (Worldwise resolution lower bound). For 0 < 𝛼 < 1/2, every 𝛼-worldwise-valid rule with E 𝜃 𝜏 < ∞ satisfies E 𝜃 𝜏 ≥ 𝑇𝑋𝑞,∗ (𝜃) kl(1 − 𝛼, 𝛼).

(20.4)

Consequently, lim inf 𝛼↓0

E𝜃 𝜏 ≥ 𝑇𝑋𝑞,∗ (𝜃). log(1/𝛼)

Interpretation and boundary The certificate logic determines the alternative set, and that set determines the sampling geometry. The bound concerns expected observations needed for a worldwise terminal statement; it is not a generalization bound, posterior entropy, or deployment loss. If the characteristic information is zero, additional observations through the same experiment family cannot produce a finite-time worldwise certificate.

20.2. RESOLUTION COMPLEXITY

149

Proof roadmap Compare the stopped transcript under 𝜃 with each opposite-label world 𝜆. Sequential change of measure lower-bounds the accumulated expected KL by the binary information needed to separate correct and incorrect terminal events. Normalized expected experiment counts form one allocation in Δ(E); the max–min definition then supplies the sharp information rate. The chapter appendix carries out these steps. The max–min geometry is the established partition-identification lower-bound mechanism [Kaufmann et al., 2016, Garivier and Kaufmann, 2016, Degenne and Koolen, 2019]. Certificate statistics contributes the typed truth partition and connects this rate to evidence and recursive carriers; it does not rename generic pure exploration as a new theorem.

20.2.1

AND/OR phase change

In a Gaussian multichannel threshold problem, feasibility may require verifying that every channel satisfies a constraint, while impossibility can be proved by one violated channel. The optimal information geometry changes accordingly. • To certify feasibility, allocation must cover all near-boundary channels: an AND geometry. • To certify impossibility, allocation may concentrate on one strongest obstruction: an OR geometry. For example, sampling channel 𝑖 from 𝑁 (𝜇𝑖 , 𝜎 2 ) and declaring feasibility when max𝑖 𝜇𝑖 ≤ 𝜀 gives, away from the boundary, ∑︁  2  2𝜎 (𝜀 − 𝜇𝑖 ) −2 , max𝑖 𝜇𝑖 < 𝜀,     𝑞,∗ 𝑖 𝑇𝑋 (𝜇) = (20.5) 2𝜎 2   , max𝑖 𝜇𝑖 > 𝜀.   max 2 𝑖:𝜇𝑖 > 𝜀 (𝜇𝑖 − 𝜀)  On the feasible side the optimal allocation equalizes the coordinate-wise KL rates. On the impossible side it concentrates on a strongest violation. Thus the asymmetry is intrinsic to the certificate, not to a particular algorithm. We retain this closed form as a worked interpretation; the general exponentialfamily and partition-identification setting belongs to the pure-exploration literature [Kaufmann et al., 2018, Juneja and Krishnasamy, 2019].

150

20.3

CHAPTER 20. CERTIFICATE STATISTICS AND RESOLUTION COMPLEXITY

Evidence carriers and task-relative sufficiency

For each experiment 𝑒, let 𝑋𝑒 denote the raw observation and let a parameter-independent Markov kernel produce retained evidence 𝑍 𝑒 = 𝑇𝑒 (𝑋𝑒 ). Write 𝑄 𝜃 ,𝑒 for the retained law and define 𝑑 𝑒𝑋 (𝜃, 𝜆) = KL(𝑃 𝜃 ,𝑒 ∥𝑃𝜆,𝑒 ),

𝑑 𝑒𝑍 (𝜃, 𝜆) = KL(𝑄 𝜃 ,𝑒 ∥𝑄 𝜆,𝑒 ),

ℓ𝑒 (𝜃, 𝜆) = E𝑄 𝜃 ,𝑒 KL{𝑃 𝜃 ,𝑒 (· | 𝑍 𝑒 ) ∥𝑃𝜆,𝑒 (· | 𝑍 𝑒 )}.

(20.6) (20.7)

The conditional KL chain rule gives the exact pairwise KL decomposition 𝑑 𝑒𝑋 (𝜃, 𝜆) = 𝑑 𝑒𝑍 (𝜃, 𝜆) + ℓ𝑒 (𝜃, 𝜆).

(20.8)

Consequently, if I𝑍𝑞,∗ is defined from retained KL rates, I𝑍𝑞,∗ (𝜃) ≤ I𝑋𝑞,∗ (𝜃),

𝑇𝑍𝑞,∗ (𝜃) ≥ 𝑇𝑋𝑞,∗ (𝜃).

(20.9)

Pairwise KL loss is not yet the right sufficiency test. Certificate sufficiency need not preserve every likelihood ratio; it need only avoid lowering the max–min floor. At a fixed world 𝜃 and allocation 𝑤, put ∑︁ ∑︁ 𝐺 𝑋 (𝑤, 𝜆) = 𝑤 𝑒 𝑑 𝑒𝑋 (𝜃, 𝜆), 𝐿 (𝑤, 𝜆) = 𝑤 𝑒 ℓ𝑒 (𝜃, 𝜆), (20.10) 𝑒

𝑒

𝑠 𝑋 (𝑤, 𝜆) = 𝐺 𝑋 (𝑤, 𝜆) − I𝑋𝑞,∗ (𝜃).

(20.11)

At a raw-optimal allocation, 𝑠 𝑋 measures how far an alternative lies above the binding certificate floor. How to read the assumptions The raw and retained characteristic-information maxima are attained, the evidence kernel is the same in every world, and the displayed KL quantities are finite. Attainment is needed only to choose an optimizer in the converse direction; an explicit allocation satisfying the inequality below is sufficient without a retained optimizer. Theorem 20.4 (Exact certificate preservation by slack). The retained carrier preserves the raw characteristic information at 𝜃, I𝑍𝑞,∗ (𝜃) = I𝑋𝑞,∗ (𝜃), if and only if there is an allocation 𝑤★ ∈ Δ(E) such that 𝐿 (𝑤★, 𝜆) ≤ 𝑠 𝑋 (𝑤★, 𝜆)

for every 𝜆 ∈ Alt𝑞 (𝜃).

(20.12)

Every such 𝑤★ is raw-optimal. In particular, every raw binding alternative at 𝑤★ has zero weighted KL loss. Interpretation and boundary Compression may discard nuisance information when nonbinding alternatives have enough excess separation to absorb the loss. It may not discard any likelihood direction belonging to a binding opposite-label world. This is weaker than classical parametric sufficiency and stronger than preserving only a posterior mean or present decision.

20.3. EVIDENCE CARRIERS AND TASK-RELATIVE SUFFICIENCY

151

Proof roadmap Sum the chain-rule decomposition over experiments to obtain 𝐺 𝑍 = 𝐺 𝑋 − 𝐿. Slack domination keeps every retained alternative above the raw max–min floor, while data processing prevents it from exceeding that floor. Conversely, a retained-optimal allocation at equality must keep every retained alternative above the common optimum; rearranging the same identity gives the slack inequalities. The complete argument is in the chapter appendix.

Locally, the score map projected to the nuisance-quotiented tangent space gives a task-relative information operator. Its minimum modulus determines stability; inverse-square modulus controls sample inflation. This modulus language follows the established analysis of ill-posed and semiparametric inverse problems [Blundell et al., 2007, Chen and Reiss, 2011]; certificate statistics uses it only after restricting the tangent directions to those that can reverse the declared truth label.

20.3.1

Average Bayesian loss versus worldwise complexity

Let 𝑄 = 𝑞(Θ) be the task label, let 𝑋 be the raw record, and let 𝑍 be retained evidence generated from 𝑋. Write Π𝑋𝑄 = L (𝑄 | 𝑋) and Π 𝑍𝑄 = L (𝑄 | 𝑍). Under the Markov relation 𝑄 → 𝑋 → 𝑍, the standard conditional-information identity gives the exact prior-predictive identity

E KL(Π𝑋𝑄 ∥Π 𝑍𝑄 ) = 𝐼 (𝑄; 𝑋 | 𝑍).

(20.13)

The left side measures how much the retained representation distorts the full-data task posterior on average under the chosen prior. It is therefore a legitimate loss scale for Bayesian representation fidelity and expected experimental utility [Cover and Thomas, 2006, Lindley, 1956]. It is not a worldwise certificate loss scale. For a sharp separation, take two worlds with 𝑋 | 𝜃 0 ∼ Bernoulli(1/4) and 𝑋 | 𝜃 1 ∼ Bernoulli(3/4), let 𝑍 be constant, and assign prior mass 𝑟 to 𝜃 1 . The posterior loss in (20.13) is at most the binary entropy ℎ2 (𝑟) and therefore tends to zero as 𝑟 ↓ 0. Yet the retained laws are identical, so I𝑍𝑞,∗ = 0 and the worldwise characteristic time is infinite, while the raw pair remains distinguishable. Boundary of the claim The Bayesian identity averages over worlds selected by a prior. The certificate criterion minimizes over opposite-label worlds and maximizes over experiment allocations. Both use KL, but their quantifiers make them nonexchangeable: small posterior distortion neither authorizes stopping nor bounds worldwise sample inflation.

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152

Table 20.1: Three statistical objects that must not be conflated. Object

Typical form

Quantifier, guarantee, and proper use

Bayesian belief

posterior, Bayes risk, expected task information worldwise likelihood, max–min KL, characteristic time

Prior-average; evaluates expected task value and helps select the next experiment. Worst binding opposite-label world under fixed confidence; authorizes stopping and a terminal claim. Closed under every future observation context; determines which states may be safely merged.

Certificate evidence

Recursive state

colored belief quotient, labeled update kernels, 𝑁 ℎ

Posterior belief can guide design, worldwise evidence authorizes a certificate, and recursive state preserves what future design will need.

20.4

Recursive certificate posterior quotient

The current posterior probability of certificate colors may be insufficient to choose the next experiment. Two beliefs with the same current color probabilities can imply different optimal future measurements. Assume now that hypotheses, experiments, observation alphabets, and the reachable belief set R ⊆ Δ(Θ) are finite. The set R is closed under every positive-probability Bayes update. Let 𝜒 : R → C be the declared contract color. It may record a present certificate permission, observer gate, or vector of immediate costs; fields omitted from 𝜒 are not protected by the quotient. For experiment 𝑒 and observation 𝑦, define 𝑝 𝑒 (𝑦 | 𝑏) =

∑︁

𝑏(𝜃)𝑃 𝜃 ,𝑒 (𝑦),

(20.14)

𝜃

𝑏(𝜃)𝑃 𝜃 ,𝑒 (𝑦) 𝐵𝑒,𝑦 𝑏(𝜃) = , 𝑝 𝑒 (𝑦 | 𝑏)
𝐾𝑒 𝑏; {𝑦} × 𝐴 = 𝑝 𝑒 (𝑦 | 𝑏)1{𝐵𝑒,𝑦 𝑏 ∈ 𝐴}.

(20.15) (20.16)

The label 𝑦 is retained: two states are not equivalent merely because they have the same unlabeled aggregate transition. Start with 𝑏 ∼0 𝑏 ′ when 𝜒(𝑏) = 𝜒(𝑏 ′ ). Given ∼ℎ , define 𝑏 ∼ℎ+1 𝑏 ′ when 𝑏 ∼ℎ 𝑏 ′ and 𝐾𝑒 (𝑏; {𝑦} × 𝐴) = 𝐾𝑒 (𝑏 ′ ; {𝑦} × 𝐴)

(20.17)

for every experiment 𝑒, observation 𝑦, and ∼ℎ -class 𝐴. Put ∼∞ =

Ù

∼ℎ ,

𝑁 ℎ = |R/∼ℎ |.

ℎ≥0

A recursive carrier 𝜙 : R → S is exact if the color factors through 𝜙 and, for every experiment, the joint law of the observation label and successor carrier state depends on 𝑏 only through 𝜙(𝑏).

20.4. RECURSIVE CERTIFICATE POSTERIOR QUOTIENT

153

How to read the assumptions The model, observation alphabets, and reachable belief set are finite, and zero-probability Bayes branches are omitted. Exactness preserves labeled observations and successor states for every declared experiment. The result does not claim that the full posterior is minimal, nor that a color protecting only the present terminal action is sufficient for future acquisition. Theorem 20.5 (Recursive quotient and minimal exact state). The relations ∼ℎ are decreasing equivalence relations and 𝑁0 ≤ 𝑁1 ≤ · · · ≤ 𝑁∞ . If 𝑏 ∼ℎ 𝑏 ′ , every fixed observation-history policy using at most ℎ further observations induces the same law of observation labels and contract colors from 𝑏 and 𝑏 ′ . The limit ∼∞ is the largest color-respecting equivalence relation stable under every labeled kernel 𝐾𝑒 . The quotient map 𝑏 ↦→ [𝑏] ∞ is exact, and every exact recursive carrier refines it: 𝜙(𝑏) = 𝜙(𝑏 ′ )

=⇒

𝑏 ∼∞ 𝑏 ′ .

(20.18)

Thus every exact deterministic implementation has at least 𝑁∞ reachable states. If each of 𝐿 charts distinguishes at most 𝐾 states, then   𝑁∞ . (20.19) 𝐿≥ 𝐾 Signature refinement terminates after at most 𝑁∞ − 𝑁0 strict block increases, and every strict split returns an experiment, observation, and successor block witnessing the missing distinction. The count 𝑁∞ concerns only the recursive belief carrier specified in the theorem. A concrete GLR test, deficit tracker, or D-tracking implementation may keep likelihood sums, visit counts, or time indices in separate and potentially unbounded auxiliary memory. Those counters are not included in 𝑁∞ unless the declared carrier grammar explicitly folds them into the protected recursive state. Interpretation and boundary 𝑁0 is the state capacity needed to preserve the present contract color. 𝑁1 also preserves which labeled observation can expose the next relevant distinction, and 𝑁∞ preserves every finite future acquisition context. A recurrent state that is adequate now can therefore be structurally inadequate one observation later. Proof roadmap Refinement signatures give decreasing equivalence relations. Induction on the remaining horizon proves policy invariance. Any stable color-respecting relation survives every refinement and is therefore contained in the limit. Conversely, finiteness makes the refinement stabilize, so the limit itself defines a reduced labeled kernel. The kernel relation induced by any exact carrier is stable and color-respecting, forcing it to refine the limit quotient; counting yields the state and chart bounds. This construction specializes strong belief bisimulation [Castro et al., 2009, Larsen and Skou, 1991] and realizes the finite quotient by color-respecting partition refinement, connecting model minimization

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CHAPTER 20. CERTIFICATE STATISTICS AND RESOLUTION COMPLEXITY

and classical refinement algorithms [Givan et al., 2003, Paige and Tarjan, 1987]. Its role here is architectural: the quotient cardinality is a task-relative recurrent capacity, and a split witness tells a repair procedure which state distinction must be installed.

20.4.1

CertTrack and certificate-aware design

A certificate-aware tracker assigns two different jobs to two different loss scales. At current belief 𝑏, the Bayesian task gain from experiment 𝑒 is 𝐺 𝑞𝑒 (𝑏) = 𝐼𝑏 {𝑞(Θ); 𝑌𝑒 }. This prior-average quantity may decide which informative experiment to run. It cannot by itself decide when every opposite-color world has been excluded. For retained evidence, define the 𝜌-optimal worldwise face ( ) ∑︁ 𝑞,∗ 𝑍 𝑍 W𝜌 (𝜃) = 𝑤 ∈ Δ(E) : min (20.20) 𝑤 𝑒 𝑑 𝑒 (𝜃, 𝜆) ≥ (1 − 𝜌)I𝑍 (𝜃) . 𝜆∈Alt𝑞 ( 𝜃 )

𝑒

Í A posterior-guided design may maximize 𝑒 𝑤 𝑒 𝐺 𝑞𝑒 (𝑏) inside this face. At 𝜌 = 0 it only breaks ties among certificate-optimal allocations; with 𝜌𝑡 ↓ 0 it may trade a vanishing fraction of the leading worldwise rate for prior-relevant information. Stopping remains separate. It occurs only when a simultaneous confidence world has one truth color, or when an anytime-valid likelihood-ratio process crosses its declared boundary. Such validity holds under any nonanticipating design; it does not rely on posterior odds. Forced exploration and deficit tracking can then make empirical allocations follow the selected face [Garivier and Kaufmann, 2016, Kaufmann et al., 2018]. Boundary of the claim Bayesian task information chooses where to look; a confidence sequence or likelihood-ratio martingale authorizes what may be said. Adding the two objectives would invent an exchange rate between prior-average utility and least-favorable-world progress. Under finite-model regularity, the resulting tracker attains the optimal almost-sure leading constant. Expected-time equality additionally requires uniform integrability; no posterior-odds optional-stopping argument is being used.

20.4.2

A compressed UCBAdmissions audit

The R UCBAdmissions table records 4526 applicants to six large UC Berkeley departments in 1973, classified by department, sex, and admission outcome [R Core Team, 2026]. It is a standard example in which aggregation changes the apparent association [Bickel et al., 1975]. Pooling departments gives 1198 admissions among 2691 male applicants and 557 among 1835 female applicants: a male-minus-female admission-rate difference of 0.1416. A two-sided Fisher exact test of that pooled association gives 𝑝 = 4.84 × 10−22 . That calculation does not answer every department-level question. As a worked certificate, declare 𝑑 𝑗 = 𝑝 𝑗, 𝑀 − 𝑝 𝑗,𝐹 ,

Γ=

max

𝑗 ∈ { 𝐴,...,𝐹 }

𝑑 𝑗,

Γ ≤ 0.10.

(20.21)

20.4. RECURSIVE CERTIFICATE POSTERIOR QUOTIENT

155

The ten-percentage-point tolerance is illustrative, not an ethical or causal criterion. Construct two-sided exact Clopper–Pearson intervals for the 12 department–sex cell probabilities at marginal error 0.05/12, and project their Bonferroni rectangle onto each difference 𝑑 𝑗 . This gives a 95% simultaneous confidence world. Table 20.2: Department admission-rate differences and 95% simultaneous exact intervals, in percentage points.

Dept.

𝑛𝑀

𝑛𝐹

Estimate

Simultaneous interval

A B C D E F

825 560 325 417 191 373

108 25 593 375 393 341

−20.35 −4.96 2.86 −1.84 3.83 −1.14

[−34.32, −2.96] [−33.14, 31.09] [−10.47, 16.35] [−15.67, 11.99] [−11.57, 19.83] [−8.95, 6.58]

Every department point estimate is below ten points, but the simultaneous intervals for B–E cross that boundary, and no lower endpoint exceeds it. The honest certificate for (20.21) is therefore 𝑈. Department B is the resolution bottleneck: its upper excursion is largest, driven by only 25 female applicants. The same confidence world gives the entire tolerance profile. Because it is rectangular, the identified image of Γ has endpoints 𝐿 Γ = max 𝐿 𝑗 = −0.0895, 𝑗

𝑈Γ = max 𝑈 𝑗 = 0.3109, 𝑗

to the displayed precision. Hence, for the query Γ ≤ 𝜀,   𝐼,    𝛿(𝜀) = 𝑈,    𝐹, 

𝜀 < −0.0895, −0.0895 ≤ 𝜀 < 0.3109, 𝜀 ≥ 0.3109,

(20.22)

up to endpoint rounding. Thus a pooled 𝑝-value near 10−22 coexists with unresolved department tolerances across a forty-point range. readout

authorized action

pooled association

report the pooled disparity for that declared query; do not transport it into a department-level verdict return 𝑈, withhold a binary department verdict, and direct further resolution effort toward the widest binding interval

department certificate

All displayed quantities are deterministic functions of the public contingency table. No regression model, asymptotic normal approximation, or prior enters this audit. A more structured model may narrow the confidence world, but it would constitute an additional declared assumption rather than a reinterpretation of the same certificate.

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Interpretation and boundary The pooled test and the department certificate are not contradictory: they have different truth maps. The certificate retains the pooled association, withholds an unsupported department-level verdict, and directs new information toward the bottleneck beginning with Department B female applicants. The table alone does not identify causal discrimination, qualifications, selection, or historical mechanisms; those require a larger world and additional assumptions.

Exercises Exercise 20.6. Construct a confidence world containing both feasible and impossible parameter values. Show that every honest binary rule must abstain or make an error on some compatible world. Exercise 20.7. Compute the max–min information allocation for a two-channel Gaussian OR certificate. Exercise 20.8. Construct a raw-optimal allocation with one binding and one nonbinding alternative. Determine how much KL the evidence carrier may discard in each direction without increasing characteristic time. Exercise 20.9. Give two posterior distributions with the same current probability of feasibility but different optimal next experiments. Identify the first refinement signature that separates them.

Appendix to Chapter 20: complete proofs

157

Appendix to Chapter 20: complete proofs Proof that posterior credibility is not worldwise validity Dependencies. The positive statement is an application of iterated expectation. The negative statement requires no identifiability: two worlds may have identical observation laws but different truth labels. Proof. If P𝑏0 {𝑞(Θ) ≠ 𝑞b | 𝐻 𝜏 } ≤ 𝜂

almost surely,

then taking expectation with respect to the prior-predictive law yields P𝑏0 {𝑞(Θ) ≠ 𝑞b} = E𝑏0 P𝑏0 {𝑞(Θ) ≠ 𝑞b | 𝐻 𝜏 } ≤ 𝜂. This is a prior-average statement. For the separation, take two worlds 𝜃 𝐹 and 𝜃 𝐼 with different truth labels and identical laws for every observation under every available experiment. Assign prior masses 1 − 𝜂 and 𝜂, respectively. Since the likelihood ratio is identically one, the posterior never changes. Reporting the majority label has posterior error 𝜂 at every time and therefore satisfies the credibility criterion. Conditional on the minority world, however, the report is wrong with probability one. Thus no nontrivial uniform worldwise error guarantee follows from posterior credibility alone. Moreover, every experiment has zero KL divergence between the two opposite-label worlds, so the characteristic information for their truth partition is zero and the corresponding worldwise resolution complexity is infinite. □

Proof of the truth-map form of three-way honesty Dependencies. Theorem 19.3 proves the claim for an arbitrary truth map on a nonempty confidence world. The simultaneous-query clause requires one coverage event for the whole query class; the empty-set output 𝑀 is diagnostic and is not a resolved truth claim. Proof. Apply Theorem 19.3 to the two cells 𝑐 −1 (𝐹) and 𝑐 −1 (𝐼). A singleton image 𝑐(𝐶𝑛 ) is the agreeing case of that theorem, and a two-label image is its unresolved case; honesty and maximal decisiveness follow immediately. If one confidence world covers the complete population object simultaneously over a query class, the same reduction holds for every query on that event and hence for any data-selected query. □

Proof of the worldwise resolution lower bound Dependencies. We use the standard sequential change-of-measure inequality: for a stopped transcript, any event 𝐴 measurable at the stopping time, and any pair of worlds 𝜃, 𝜆, ∑︁ E 𝜃 𝑁𝑒 (𝜏) KL(𝑃 𝜃 ,𝑒 ∥𝑃𝜆,𝑒 ) ≥ kl{P 𝜃 ( 𝐴), P𝜆 ( 𝐴)}. 𝑒

This follows by applying data processing to the likelihood ratio of the stopped experiment.

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Proof. Fix 𝜆 ∈ Alt𝑞 (𝜃) and take 𝐴 𝜃 = {𝜏 < ∞, 𝑞b = 𝑞(𝜃)}. Because E 𝜃 𝜏 < ∞, the rule stops almost surely under 𝜃. Worldwise validity therefore gives P 𝜃 ( 𝐴 𝜃 ) ≥ 1 − 𝛼. Since 𝑞(𝜆) ≠ 𝑞(𝜃), the same event is an error under 𝜆, so P𝜆 ( 𝐴 𝜃 ) ≤ 𝛼. For 0 < 𝛼 < 1/2, monotonicity of binary relative entropy on these two ranges and sequential change of measure yield ∑︁ E 𝜃 𝑁𝑒 (𝜏)𝑑 𝑒 (𝜃, 𝜆) ≥ kl(1 − 𝛼, 𝛼). (20.23) 𝑒

Define the normalized expected counts 𝑤¯ 𝑒 =

E 𝜃 𝑁𝑒 (𝜏) . E𝜃 𝜏

They are nonnegative and sum to one. Divide (20.23) by E 𝜃 𝜏 and minimize over 𝜆 ∈ Alt𝑞 (𝜃): min

𝜆∈Alt𝑞 ( 𝜃 )

∑︁

𝑤¯ 𝑒 𝑑 𝑒 (𝜃, 𝜆) ≥

𝑒

kl(1 − 𝛼, 𝛼) . E𝜃 𝜏

The left side is no larger than its maximum over all allocations, namely I𝑋𝑞,∗ (𝜃). Rearrangement gives E𝜃 𝜏 ≥

kl(1 − 𝛼, 𝛼) = 𝑇𝑋𝑞,∗ (𝜃) kl(1 − 𝛼, 𝛼). I𝑋𝑞,∗ (𝜃)

If the characteristic information is zero, the preceding inequality forces E 𝜃 𝜏 = +∞, contradicting the assumed finite expectation; hence no such finite-time rule exists. Finally, kl(1 − 𝛼, 𝛼)/log(1/𝛼) → 1, which proves the asymptotic statement. □

Proof of exact certificate preservation by slack Dependencies. The evidence kernel is parameter-independent, so the KL chain rule for (𝑋𝑒 , 𝑍 𝑒 ) has raw KL on the left. The converse uses attainment of the retained max–min problem. Proof. For every allocation 𝑤 and alternative 𝜆, summing (20.8) gives 𝐺 𝑍 (𝑤, 𝜆) = 𝐺 𝑋 (𝑤, 𝜆) − 𝐿 (𝑤, 𝜆).

(20.24)

Because 𝐿 ≥ 0, every retained alternative rate is no larger than its raw counterpart. Taking the alternative minimum and allocation maximum proves I𝑍𝑞,∗ (𝜃) ≤ I𝑋𝑞,∗ (𝜃). Suppose first that 𝑤★ satisfies (20.12). For every opposite-label 𝜆, 𝐺 𝑍 (𝑤★, 𝜆) = 𝐺 𝑋 (𝑤★, 𝜆) − 𝐿 (𝑤★, 𝜆) ≥ 𝐺 𝑋 (𝑤★, 𝜆) − 𝑠 𝑋 (𝑤★, 𝜆) = I𝑋𝑞,∗ (𝜃).

Appendix to Chapter 20: complete proofs

159

Thus I𝑍𝑞,∗ (𝜃) ≥ I𝑋𝑞,∗ (𝜃). Data processing supplies the reverse inequality, so equality holds. Moreover, min 𝐺 𝑋 (𝑤★, 𝜆) ≥ min 𝐺 𝑍 (𝑤★, 𝜆) ≥ I𝑋𝑞,∗ (𝜃). 𝜆

𝜆

By definition of the raw maximum, equality must hold and 𝑤★ is raw-optimal. Conversely, assume I𝑍𝑞,∗ (𝜃) = I𝑋𝑞,∗ (𝜃) and choose a retained-optimal allocation 𝑤★. Then every opposite-label alternative obeys 𝐺 𝑍 (𝑤★, 𝜆) ≥ I𝑍𝑞,∗ (𝜃) = I𝑋𝑞,∗ (𝜃). Rearranging (20.24) gives 𝐿 (𝑤★, 𝜆) ≤ 𝐺 𝑋 (𝑤★, 𝜆) − I𝑋𝑞,∗ (𝜃) = 𝑠 𝑋 (𝑤★, 𝜆), which is the required slack domination. The preceding raw-optimality argument applies again. Finally, if 𝜆 is binding in the raw problem at 𝑤★, then 𝑠 𝑋 (𝑤★, 𝜆) = 0. Since 𝐿 ≥ 0, slack domination forces 𝐿(𝑤★, 𝜆) = 0. □

Proof of the recursive quotient and minimal exact state Dependencies. All sets are finite. This avoids measurable-quotient issues and makes the decreasing partition sequence stabilize after finitely many strict refinements. Policies are fixed observation-history rules, not rules separately tailored to the two initial beliefs. Proof. Equality of 𝜒 is an equivalence relation, so ∼0 is an equivalence relation. Suppose ∼ℎ is one. Within each ∼ℎ -class, associate to 𝑏 the finite signature   sigℎ (𝑏) = 𝐾𝑒 (𝑏; {𝑦} × 𝐴) 𝑒,𝑦, 𝐴∈ R/∼ . ℎ

The relation ∼ℎ+1 is equality of these signatures inside a ∼ℎ -class. It is therefore an equivalence relation and refines ∼ℎ . The quotient cardinalities are consequently nondecreasing. We prove finite-horizon invariance by induction. At horizon zero, related beliefs have the same color. Suppose the claim holds at horizon ℎ and let 𝑏 ∼ℎ+1 𝑏 ′ . A fixed observation-history policy selects the same first experiment at the two empty histories. Equation (20.17) gives the same probability to every observation label and successor ∼ℎ -class. Conditional on such a label and class, the induction hypothesis gives the same law for the remaining ℎ observations and colors. Summing over the first step proves the claim at horizon ℎ + 1. Call an equivalence relation 𝑅 stable when it respects 𝜒 and gives equal mass, for every 𝑒 and 𝑦, to every 𝑅-class. We show that every such 𝑅 is contained in every ∼ℎ . Color respect gives 𝑅 ⊆∼0 . If 𝑅 ⊆∼ℎ , each ∼ℎ -class is a union of 𝑅-classes. Stability on the finer 𝑅-classes can therefore be summed to give equality on every ∼ℎ -class, so 𝑅 ⊆∼ℎ+1 . Induction yields 𝑅 ⊆∼∞ . Because R is finite, the decreasing equivalence relations stabilize: for some 𝐻, ∼ 𝐻 =∼ 𝐻+1 =∼∞ . The defining equality at the fixed point makes ∼∞ itself stable. Hence it is the largest stable color-respecting equivalence relation. Stability also makes the color decoder and every reduced labeled kernel on R/∼∞ well defined, proving exactness of the quotient map. Let 𝜙 be any exact recursive carrier and define 𝑏𝑅 𝜙 𝑏 ′ when 𝜙(𝑏) = 𝜙(𝑏 ′ ). Color factorization makes 𝑅 𝜙 color-respecting. Factorization of the joint observation–successor law makes it stable.

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Maximality therefore implies 𝑅 𝜙 ⊆∼∞ , which is exactly (20.18). Thus distinct ∼∞ -classes require distinct carrier states, giving at least 𝑁∞ states. If 𝐿 charts expose at most 𝐾 states each, they expose at most 𝐿𝐾 distinct chart–state pairs, and 𝐿𝐾 ≥ 𝑁∞ proves (20.19). Starting from 𝑁0 blocks, every strict signature-refinement round increases the block count by at least one and no round can exceed 𝑁∞ . Hence there are at most 𝑁∞ − 𝑁0 strict increases. When a block splits, two of its beliefs have unequal signatures; one coordinate of that inequality supplies the promised experiment 𝑒, observation 𝑦, and successor block 𝐴. □

Part VI Historical and Conceptual Notes Confidence world: the name is local, the confidence-set principle is not A confidence world 𝐶𝑛 is a random confidence set for a complete population world. Its mathematical coverage contract is not a new species of confidence set. In particular, Dufour [1997] observed that a covered joint set can be mapped through any function 𝑔: the image 𝑔(𝐶𝑛 ) remains a valid confidence set because membership of the true joint parameter implies membership of its image. The book’s identified image 𝐽𝐶𝑛 (𝑞) is this projection principle applied to a declared structural query. The closest neighboring objects nevertheless live on different axes. Lineage

Primary object

Main question

Relation to 𝐶𝑛

Manski partial iden- population identified set tification [Manski, Θ𝐼 (𝑃) 2003] Dufour projection covered joint confidence [Dufour, 1997] set and its image 𝑔(𝐶𝑛 )

what is determined even a population identification with the population law object, not the finite-sample known? random set itself how is coverage preserved direct antecedent of the for a transformation or sub- identified-image validity arvector? gument Set inference [Cher- estimators and confidence when do set estimators and supplies model-specific nozhukov et al., 2007, regions in partially identi- confidence procedures have construction and asympAndrews and Soares, fied or moment-inequality consistency, coverage, and totic theory that the abstract 2010] models useful power? coverage definition alone does not provide Vogel random op- simultaneous random ob- how can feasible sets, opti- direct neighbor of the hontimization [Vogel, jective and constraint infor- mal values, and solution sets est inner/outer architecture 2008] mation be covered together? bracket Confidence world complete population which structural conclusions an enriched bookkeeping (this book) object, including nuisance and actions are authorized and interface convention and the sampling/future- on one coverage event? built on the preceding experiment structure confidence-set principles needed by the declared query Coverage alone gives no contraction theorem: the vacuous choice 𝐶𝑛 ≡ W satisfies a coverage lower bound. Convergence to an identified set, Hausdorff consistency, nonconservativeness, or power requires additional assumptions and a concrete construction, as in the set- and moment-inequality literatures just cited. Part VI therefore claims transport of a declared coverage event, not automatic consistency of every confidence world.

The three-way rule and the model-conflict state For a nonempty 𝐶𝑛 , the rule is set unanimity: 𝑞(𝐶𝑛 ) = {𝐹} ⇒ 𝐹,

𝑞(𝐶𝑛 ) = {𝐼} ⇒ 𝐼,

|𝑞(𝐶𝑛 )| > 1 ⇒ 𝑈.

Part VI Historical and Conceptual Notes

161

This follows in one line from confidence-set projection and is historically adjacent to equivalence and noninferiority testing, where a confidence set must lie wholly inside or outside a declared margin [Blackwelder, 1982, Berger and Hsu, 1996, Piaggio et al., 2006]. The book does not claim that the 𝐹/𝐼/𝑈 threshold logic is new. Its increment is to type the query as a structural architecture claim, preserve the whole identified image and tolerance profile, connect 𝑈 to resolution complexity, and keep authorization separate from posterior belief and deployment. The symbol 𝑀 is book terminology for an empty confidence world. Empty-set diagnostics are not without precedent: Dufour [1997] explicitly notes that an empty valid confidence set can be interpreted as rejection of the model or overidentifying restrictions. The book’s contribution is to prevent that diagnostic from being confused with either a feasible or impossible population truth label.

What the theorem-level audit leaves as genuine increments The worldwise resolution lower bound and recursive quotient import partition-identification and probabilisticbisimulation machinery. The posterior/worldwise counterexample is standard. The atlas inverse and commondeployment results are program-specific specializations. The exact evidence-slack criterion asks when compression preserves one max–min certificate floor rather than an entire parametric experiment. Its proof is a short KL data-processing and max–min argument, so the book treats it as a useful specialization rather than a new information-theoretic mechanism; see Appendix G.1.

From finite-sample certification to structural intervention Part VI established confidence sets, three-way decisions, resolution limits, and the evidence that must be retained. A certificate is not yet a deployable repair. Part VII requires one common deployment witness, turns a typed structural failure into a controlled repair workflow, develops a partial-transport implementation for multibranch oracles, and ends with independent validation. Chapter 22 introduces the general intervention protocol and its terminology. Five reductions and what survives Operation

Removes

Must preserve

Vertical fiber optimized-away native defect: original-objective height quotient auxiliary state above the oracle fiber Source-side oracle target-invisible labels, oracle signatures and target-visible quotient gauge, and padding in distinctions the raw carrier Sink-side kernel differences canonical operational behavior contextual invisible to every quotient legal context and task Statistical coordinates its entire identified image over all projection suppressed by the data-compatible populations declared audit map Recursive quotient only distinctions future-closed certificate and action invisible to color and distinctions every labeled future update

It is not an arbitrary parameter distance raw parameter count native carrier sufficiency a structural quotient or a point estimate compression by current posterior or current action alone

Part VII

Obstruction-Aware Learning and Structural Repair A structural certificate becomes actionable only when it localizes the failed interface, authorizes a compatible intervention, and predicts an improvement that survives independent validation and matched controls.

162

Chapter 21

Toward a Certified Structural Learning Theory A structural learning theorem should not hide distinct mechanisms inside one undifferentiated excessrisk bound. It should report lower or upper certificates for the risk components that the data and structural analysis can actually identify. b be a possibly data-dependent architecture selected on calibration data, and let 𝐴 b be trained Let 𝔄 on an independent training split. On a held-out test split, the desired statement has the form b − 𝑅★ ≤ Δmodel + Δarch + Δgen + Δopt . Δmodel + Δarch ≤ 𝑅( 𝐴) Every term should have a declared source. Companion interface and proof provenance The typed-carrier synthesis and no-compensation interface are developed in the second EGML companion [Huang and Wang, 2026e]. This chapter gives the complete book proofs of its principal interface statements and keeps the statistical, recursive, deployment, and operational loss scales separate.

21.1

Three nonexchangeable carrier contracts

Chapter 20 has already fixed the evidence-carrier contract in Section 20.3 and the recursive carrier contract in Section 20.4. We reuse those definitions unchanged. The additional interface here is the deployment carrier 𝐻dep : in the carrier–decoder factorization of Chapter 16, it maps operating inputs to the state used by an action decoder. When 𝐻dep is induced by a nonnative lift, it enters this chapter only after passing the quotient-faithful extraction gate of Section 10.4. Placed side by side, the three exactness obligations have different domains and loss scales: deployment:

𝑎★ = 𝑑dep ◦ 𝐻dep ,

evidence: 𝑇𝑍𝑞,∗ (𝜃) = 𝑇𝑋𝑞,∗ (𝜃), recursion:

the declared color and every observation-labeled update factor through 𝜙rec . 163

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The first is charged in native loss, the second in expected observations, and the third in reachable states or charts. They form a typed audit vector, not three terms of one excess-risk sum. Figure 21.1 makes the three chains and their common terminal gate visible before the no-compensation theorem is stated. The surrounding foundations are classical: statistical decision functions, sufficiency and comparison of experiments, treatment rules, and MDP state abstraction all precede this synthesis [Wald, 1950, Bahadur, 1954, Blackwell, 1951, 1953, Manski, 2004, Li et al., 2006]. The claimed contribution is the typed interface and its explicit common-witness gate, not those component theories.

Resource grammar and operating inputs

Architecture worlds and raw records

Reachable beliefs and experiment menu

Deployment carrier 𝐻dep

Evidence carrier 𝑇ev

Recursive carrier 𝜙rec

Native loss contextual consequence

Resolution time worldwise certificate

Joint quotient future-closed action

Common certified action: one witness over every compatible world, with all three interfaces passed Figure 21.1: The three architecture chains after finite-information closure. Each has its own domain, exactness obligation, and reporting scale; they meet only at a common certified action.

How to read the assumptions The evidence and recursive statements use a finite world set, finite experiment menu, and finite reachable belief set closed under positive-probability Bayes updates. The evidence carrier is independent of the unknown world, the terminal task 𝑞 is declared in advance, and 0 < 𝛼 < 1/2. The deployment branch uses a finite positive-mass input problem with a common compact action space. Its native defects are finite and lower-semicontinuous in the action and vanish only at their respective oracle actions. Theorem 21.1 (Typed no-compensation). Each of the following failures survives unlimited resources in the other two carrier branches. 1. Evidence blindness. If opposite-label worlds 𝜃 and 𝜆 satisfy 𝑄 𝜃 ,𝑒 = 𝑄 𝜆,𝑒 for every experiment 𝑒, no retained-evidence rule can stop almost surely and have worldwise error at most 𝛼 at both worlds, even with exact recursive and deployment carriers. 2. Recursive collision. If 𝜙rec (𝑏) = 𝜙rec (𝑏 ′ ) for beliefs in different classes of the minimal recursive quotient from Theorem 20.5, a finite labeled future context distinguishes them. No decoder on that fixed recurrent state can realize the exact reduced process, even with raw evidence and an unrestricted deployment decoder. 3. Deployment collision. If 𝐻dep (𝑥) = 𝐻dep (𝑥 ′ ) while 𝑎★𝑥 ≠ 𝑎★𝑥′ , no decoder on that fixed carrier

21.1. THREE NONEXCHANGEABLE CARRIER CONTRACTS

165

realizes both oracle actions. Under the compact-separation assumptions above, when both inputs have positive mass the best such decoder has strictly positive native defect even with raw evidence and identity recursive state. Consequently, none of the three exactness predicates by itself implies either of the other two. This refutes bare implications and automatic compensation; it does not rule out cross-branch reasoning from a conjunction of hypotheses, a proved factorization theorem, or an optimization problem with an explicitly declared exchange rate. The evidence-blind branch is the zero-information endpoint of the classical two-world change-ofmeasure method [Kaufmann et al., 2016]; the theorem keeps that statistical obstruction typed separately from deployment and recursive-state failures. A minimal memory-split stress test makes the boundary especially sharp. Let a fair history bit 𝐻 be erased by a constant baseline memory before a binary action is chosen. In one world the binary target is an independent fair coin; in another it is 𝐻 passed through a binary symmetric channel. Every baseline-executable policy then has the same retained action–reward law in both worlds, although retaining 𝐻 improves value only in the second world. If a temporary memory split is forbidden, the architecture query has zero characteristic information. If the split is allowed diagnostically, choosing the action 𝐴 = 𝐻 reduces the query to testing Bern(1/2) against Bern(1 − 𝜂). Charging for the diagnostic changes information per unit cost, not the statistical object. Thus operational exposure is an experiment-menu gate: under a resettable finite contract it is absorbed by active sequential experiment design [Chernoff, 1959, Naghshvar and Javidi, 2013], while the history quotient remains within state-abstraction theory [Li et al., 2006]. The companion go/no-go audit records the exact finite calculation; it is not an additional principal result. Interpretation and boundary More parameters in a deployment head cannot recover a likelihood ratio erased by the sensor; more samples after a recurrent collision cannot restore the two histories that were merged; and a perfect population certificate does not manufacture a missing action decoder. A failed gate therefore authorizes repair only on its own typed branch. Proof roadmap Evidence blindness follows because every adaptive retained transcript has the same law in the two opposite worlds. Recursive collision is witnessed at the first finite refinement depth separating the two beliefs. Deployment collision follows because one decoder value cannot equal two distinct oracle actions; lower semicontinuity and compactness turn this incompatibility into a strictly positive separation margin. Identity choices on the remaining branches establish non-compensation. The chapter appendix supplies all three arguments.

21.1.1

Routed experts and external memory under bounded observability

In a sparsely routed expert system, let 𝐻dep (𝑥) be exactly the state visible to the router and let the action be an expert index, or a legal top-𝑘 expert set. A per-input oracle can choose the best expert for every token or query, yet this pointwise family need not factor through one measurable gate on 𝐻dep . If two positive-mass query classes collide in the router-visible state and reverse their preferred experts, the

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166

deployment-collision branch of Theorem 21.1 gives a positive native defect. More router optimization with the same carrier and action grammar cannot remove that collision. Sparse expert routing is the representative mechanism, not the name of the problem class [Shazeer et al., 2017, Fedus et al., 2022]. The possible repairs act on different contracts. Giving the router an additional feature or retrieved evidence changes its information carrier; top-𝑘 routing, a shared residual expert, or an abstention/fallback path changes the action class; adding experts without making the distinguishing information visible may change neither. A larger context window changes a charged resource budget and must be analyzed as such. Retrieval-augmented generation is particularly revealing: retrieval changes the experiment and external-memory carrier available to the generator, rather than merely optimizing the same closed-book decoder [Lewis et al., 2020]. A credible architecture-choice claim therefore freezes the router-visible information, retrieval corpus and index, expert/action library, native task loss, and compute/latency budget before confirmation. It then tests the certificate-directed carrier or class repair against equal-information and equal-compute controls on an independent endpoint. Without those controls, an improved routed or retrieval system does not identify which typed repair caused the gain. Boundary of the claim This is a theorem-interface example, not a new MoE or RAG theorem and not a new experiment. The obstruction applies only to the frozen carrier and legal action class; changing either may be the correct repair and also changes the proposition being tested.

21.2

Typed architecture realization

The no-compensation theorem is the negative half of the interface. The positive half explains how the deterministic deployment chain, statistical authorization, and recursive continuation meet without identifying their loss scales. Let 𝑞 arch : Θ → Qarch be an architecture-action color. It may record feasibility, a witness class, or a repair permission, but it must distinguish worlds requiring different terminal deployments. Let 𝐶𝑡 be an anytime confidence world for the same population model. Apply the raw and retained characteristic-information definitions of Chapter 20 to this color, and denote the resulting characteristic times by 𝑇𝑋arch,∗ (𝜃)

and 𝑇𝑍arch,∗ (𝜃).

For recursive control, color each reachable retained-evidence state 𝑢 by
𝜁 (𝑢) = 𝑐 dep (𝑢), 𝑐 cert (𝑢), 𝑐 ctx (𝑢) ,

(21.1)

where the entries record the deployment output, certificate permission, and sound contextual equivalence class. Applying the labeled-kernel refinement of Section 20.4 to 𝜁 gives a stable joint quotient; write 𝑁joint for its number of classes.

21.2. TYPED ARCHITECTURE REALIZATION

167

How to read the assumptions The world, experiment, observation, and reachable-state sets are finite; the retained-evidence channel is independent of the unknown world; and the confidence worlds are simultaneously valid under adaptive stopping. The architecture color is declared before the terminal decision, the contextual component of 𝜁 is a sound operational congruence, and the deployment loss, resource grammar, and task contract are the same ones used in Chapters 16, 17, and 19. The terminal time 𝜏 is a stopping time for the retained-evidence filtration, and 𝐴 𝜏 is one F𝜏 -measurable selector taking values in the declared resource class. It may use the full retained certificate state; if the grammar instead requires the action to depend only on 𝑞 arch , the corresponding color-to-action factorization is an additional hypothesis. The evidence-slack equivalence additionally uses the attainment conditions of Theorem 20.4. No common-witness membership is implied by these standing conditions; confidence-world authorization requires the separate gate displayed in the theorem below. All positive-composition conclusions are asserted on one declared time-uniform event of probability at least 1 − 𝛼. If the retained-evidence and confidence-world guarantees are constructed on separate events with errors 𝛼ev and 𝛼cw , use their intersection and require 𝛼ev + 𝛼cw ≤ 𝛼. Theorem 21.2 (Typed architecture realization interface). Under the preceding conditions, the evidence and recursive branches impose the following necessary bounds. The confidence-world conclusion is a separate conditional interface. 1. Every 𝛼-worldwise-valid system that resolves 𝑞 arch from retained evidence has expected stopping time satisfying E 𝜃 𝜏 ≥ 𝑇𝑍arch,∗ (𝜃) kl(1 − 𝛼, 𝛼). (21.2) Evidence processing gives 𝑇𝑍arch,∗ ≥ 𝑇𝑋arch,∗ , with equality exactly under the certificate-slack criterion for the binding opposite-action worlds. 2. Every exact shared recursive carrier preserving 𝜁 and every observation-labeled update has at least 𝑁joint reachable states. If each of 𝐿 charts distinguishes at most 𝐾 states, then   𝑁joint . (21.3) 𝐿≥ 𝐾 3. For an adapted, measurable terminal selector 𝐴 𝜏 , the declared confidence-world authorization gate is the common-witness condition Ù 𝜂 𝔉𝑅 (𝜆). 𝐴𝜏 ∈ (21.4) 𝜆∈𝐶 𝜏

If this gate is verified, then on the simultaneous coverage event the witness satisfies the declared tolerance in the true world. Items 1 and 2 do not imply the gate. 4. Conditional on the same common-witness gate, resource capacity and native loss remain governed by the carrier–decoder decomposition of Chapter 16; a claimed operational consequence follows only through the transmission and exposure gates of Chapter 17. For the positive composition direction, an anytime-valid retained-evidence certificate, an exact joint recursive carrier, an adapted witnessed common deployment, and the declared native and contextual gates compose into an anytime-valid common certified action.

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168

Interpretation and boundary The deployment chain answers what can be executed, the evidence chain what can be authorized from finite information, and the recursive chain what must remain distinguishable after future updates. They meet at one action but do not become one loss: extra state cannot repair a blind sensor, extra samples cannot manufacture a decoder, and pointwise feasibility cannot replace a common witness. The common-witness membership is a declared authorization hypothesis, not a consequence of worldwise correctness of the architecture color or of the recursive state bound. The 𝑁joint bound concerns the recursive carrier; it does not assert that separately feasible evidence, recursive, and deployment carriers fit simultaneously inside one resource budget. That assertion belongs to a declared joint grammar. Likewise, the common-witness route may select 𝐴 𝜏 from the full retained certificate state; a rule of the form 𝐴 𝜏 = 𝐷 act {𝑞 arch , 𝐻dep } requires a separate color-to-action factorization and is not implied by the other gates. Moreover, 𝑁joint counts the protected recursive carrier classes, not auxiliary likelihood sums, clocks, experiment counts, or tracking variables; those may require separate or unbounded memory unless the joint grammar explicitly includes them. Proof roadmap Apply the resolution lower bound and evidence-slack theorem of Chapter 20 to 𝑞 arch . Apply its recursive quotient theorem to the joint color 𝜁. For the conditional deployment branch, assume the displayed common-witness gate and apply Theorem 19.6; the native and contextual clauses are the interfaces of Chapters 16 and 17. Apply the common-witness implication at the adapted stopping time on the single time-uniform coverage event. Their conjunction proves the positive composition without adding the four loss scales. The chapter appendix gives the formal reduction.

21.2.1

Radical instance: a genuine common witness

The radical loop makes the common-witness quantifier concrete. Let 𝐶𝑡 be any confidence world of continuous target loops 𝑧𝜆 : 𝑆 1 → C satisfying inf inf |𝑧 𝜆 (𝜃)| ≥ 𝜌 > 0.

𝜆∈𝐶𝑡 𝜃 ∈𝑆 1

Freeze the repaired operating contract before the terminal decision: the deployed input contains 𝑢 = 𝑧𝜆 (𝜃), or a carrier from which 𝑢 is exactly recoverable, and the native defect of a point output is 𝐷 𝑢 (𝑎) = |𝑎 𝑘 − 𝑢| 2 . For a symmetric task, the full-root rule 𝑊 𝑘 (𝑢) = {𝑎 ∈ C : 𝑎 𝑘 = 𝑢} is one world-independent set-valued architecture. If its loss is the largest native defect of a returned 0 (𝜆) for the zero-tolerance feasible set under this repaired grammar. Then element, write 𝔉set

21.3. FINITE-LIBRARY COMPARISON AND DATA-DEPENDENT SELECTION

𝑊𝑘 ∈

Ù

169

0 𝔉set (𝜆).

𝜆∈𝐶𝑡

If a local point representative is required, cover C \ {0} by the two slit domains 𝑈0 = C \ (−∞, 0],

𝑈1 = C \ [0, ∞).

Each domain has a continuous logarithm and hence a local root 𝜓 𝑗 (𝑢) = exp{Log 𝑗 (𝑢)/𝑘 }. A fixed measurable router chooses an available domain and retains its chart label. The resulting two-chart
0 (𝜆) for the corresponding rule 𝐴atl (𝑢) = 𝑗 (𝑢), 𝜓 𝑗 (𝑢) (𝑢) is again independent of 𝜆. Writing 𝔉atl zero-tolerance feasible set, Ù 0 𝐴atl ∈ 𝔉atl (𝜆). 𝜆∈𝐶𝑡

Thus the confidence world need not collapse to a singleton before either repaired grammar has an actual common witness. Degree identification still decides whether the original 𝜃-indexed one-chart baseline is ruled out. It does not, by itself, exhibit a common one-chart root: even if every compatible loop has degree divisible by 𝑘, their worldwise root fields may be different. This construction does not silently solve the original contract of Theorem 14.1. If the deployed input is only 𝜃 and does not expose 𝑧 𝜆 (𝜃), neither rule above is admissible. If the task requires one globally labeled root, the set-valued rule changes the output semantics and the atlas rule changes the routing grammar. In that case one must instead shrink 𝐶𝑡 and exhibit a single admissible 𝜃 ↦→ 𝑎(𝜃) in the intersection. The distinction is exactly the difference between a mechanism-matched repair and a world-specific oracle family disguised as one deployment. This interface closes the statistical upgrade of the architecture theory. Chapter 22 turns the first failed gate into a typed return path for data acquisition, evidence repair, state refinement, deployment expansion, or contextual revision.

21.3

Finite-library comparison and data-dependent selection

A finite-library version.

Suppose calibration data choose a finite candidate library L = {𝔄 (1) , . . . , 𝔄 (𝐾 ) },

b is selected without using the test sample. Fix 𝑀 predeclared from which one trained architecture 𝐴 baselines 𝐴1 , . . . , 𝐴 𝑀 , also without inspecting the test sample. Let the test-unit loss lie in [0, 𝐵]. For 𝑏 = 1, . . . , 𝑀 define 𝑛te 1 ∑︁ b b 𝑍𝑖 )}. Δ𝑏 = {ℓ( 𝐴𝑏 ; 𝑍𝑖 ) − ℓ( 𝐴; 𝑛te 𝑖=1 Hoeffding’s inequality and a union bound give, with probability at least 1 − 𝛿, simultaneously for all 𝑏 = 1, . . . , 𝑀, √︄ Δ𝑏 ≥ b Δ𝑏 − 𝐵 for the stated range convention.

2 log(𝑀/𝛿) 𝑛te

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How to read the assumptions b and all baselines are fixed before the independent test All fitted candidates, the selected 𝐴, sample is inspected. The test observations are independent at the true experimental-unit level, and each paired loss difference lies in [−𝐵, 𝐵]. The candidate library has size 𝐾, while 𝑀 is the finite number of predeclared baseline comparisons covered by the union bound. Theorem 21.3 (Finite-library held-out improvement). With probability at least 1 − 𝛿, simultaneously for every 𝑏 = 1, . . . , 𝑀, √︄ Δ𝑏 ≥ b Δ𝑏 − 𝐵

2 log(𝑀/𝛿) . 𝑛te

Consequently, on that event, if √︄ min b Δ𝑏 > 𝐵 𝑏

2 log(𝑀/𝛿) , 𝑛te

then Δ𝑏 > 0 for every baseline, so the selected architecture has smaller population loss than every predeclared baseline. Interpretation and boundary The theorem turns held-out paired improvements into simultaneous population lower bounds against all predeclared baselines. In multi-cell or repeated-measure applications the biological unit, not the cell, is the concentration unit. The result does not cover adaptively added baselines without an adjusted selection argument. Proof roadmap Apply Hoeffding’s inequality to each paired loss difference using range length 2𝐵, choose a tolerance with tail probability 𝛿/𝑀, and take a union bound over the library. The chapter appendix verifies the constant and conditional-on-development formulation. The independent unit is the biological sample, task, or trajectory, not the individual cell or time point when those observations are dependent within units. Architecture certificate plus held-out gain. ments. Structural statement.

A convincing OALI conclusion combines two state-

Calibration data yield 𝔒(𝔄0 ) > 0

for the baseline architecture and a small upper witness for the repair. Predictive statement. Independent test data show lower operational loss for the repair. Either statement alone is weaker. A held-out gain without a certificate may be generic architecture search. A certificate without held-out gain may be operationally irrelevant.

21.4. CONFIDENCE, STOPPING, AND THE LEARNING-THEORY FRONTIER

171

Data-dependent classes. If the same data define the atlas and fit the experts, uniform fixed-class theory is insufficient. Three routes are available. 1. Sample splitting. Simple, transparent, but less data efficient. 2. Finite-library conditioning. Condition on calibration output and validate a predeclared finite set. 3. Random-set complexity. Bound generalization of the data-dependent set of architectures or optimizer trajectories. PAC–Bayesian bounds for random hypothesis sets provide one general route to the third option [Dupuis et al., 2024]. The current OALI program uses the first two routes. A theorem specialized to learned atlases and their structural certificates remains open. Certification-aware architecture selection. Ordinary model selection minimizes estimated predictive risk. Certification-aware selection uses a two-dimensional criterion:
held-out risk, unresolved obstruction interval . An architecture with slightly lower empirical loss but an unresolved or unstable certificate may be less scientifically auditable than a robustly certified alternative.

21.4

Confidence, stopping, and the learning-theory frontier

Structural and operational confidence. A population obstruction can be positive but hidden from the chosen task. Conversely, a small native defect can be amplified by a sensitive downstream decision. A complete confidence statement may therefore bracket both native and observable frontiers. The deterministic nonexpansiveness of task envelopes allows one confidence event to propagate to the operational audit when kernel errors are controlled. Stopping rules for architecture research. Certificate statistics suggests a principled stopping rule. Continue data collection while the confidence world contains both: • a world where the baseline is saturated; and • a world where the baseline has a practically meaningful obstruction. Stop when the certificate becomes decisive or when the cost of resolution exceeds the value of the architecture decision. What would constitute a new learning theory. A mature structural learning theory would provide: 1. objective-native architecture lower bounds; 2. resource-indexed upper witnesses; 3. honest finite-data transport; 4. data-dependent generalization; 5. optimization residual certificates;

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6. task-relative operational interpretation; 7. automated, mechanism-specific repair. The current program closes several of these interfaces in finite, convex, graph, and singular normal forms. It does not yet supply one universal theorem for arbitrary deep networks.

Exercises Exercise 21.4. Derive the finite-library held-out bound with a two-sided confidence interval and compare constants. Exercise 21.5. Explain why treating individual cells as independent test units can invalidate a cytometry held-out claim. Exercise 21.6. Design a stopping rule for collecting new calibration instances near a suspected monodromy seam.

Appendix to Chapter 21: complete proofs

173

Appendix to Chapter 21: complete proofs Proof of typed no-compensation Dependencies. The evidence branch is a finite adaptive experiment with a parameter-independent retained channel. The recursive branch uses the finite labeled-kernel refinement of Chapter 20. The deployment branch has positive mass on the colliding inputs, a common compact action space, and finite lower-semicontinuous native defects whose zero sets are their distinct oracle actions. Proof. For evidence blindness, fix opposite-label worlds 𝜃 and 𝜆 with 𝑄 𝜃 ,𝑒 = 𝑄 𝜆,𝑒 for every experiment. Induction on time shows that every retained transcript has the same law in the two worlds: conditional on a common transcript, a nonanticipating rule chooses the same distribution over experiments, and the next retained observation has the same conditional law. The conclusion remains true at a stopping time. If 𝐴 is the event that the rule reports 𝑞(𝜃), almost-sure stopping and worldwise correctness would require P 𝜃 ( 𝐴) ≥ 1 − 𝛼, P𝜆 ( 𝐴) ≤ 𝛼. Equality of transcript laws makes the two probabilities equal, contradicting 𝛼 < 1/2. For a recursive collision, let 𝑏 and 𝑏 ′ be merged by 𝜙rec but separated by the stable relation ∼∞ of Theorem 20.5. Because the reachable set is finite, the refinement stabilizes after finitely many steps, so there is a least ℎ with 𝑏 ≁ℎ 𝑏 ′ . If ℎ = 0, their declared colors differ. Otherwise they agree through depth ℎ − 1, but some experiment, observation label, and ∼ℎ−1 successor block has different probability from the two beliefs. This is a finite future context with different reduced laws. A decoder receiving the same recurrent state cannot reproduce both laws. For a deployment collision, every decoder 𝑑 satisfies 𝑑{𝐻dep (𝑥)} = 𝑑{𝐻dep (𝑥 ′ )}. Write 𝑝 𝑥 , 𝑝 𝑥 ′ > 0 for the two input masses and define, on the common action space, 𝛿 𝑥, 𝑥 ′ = min{𝑝 𝑥 D 𝑥 (𝑎) + 𝑝 𝑥 ′ D 𝑥 ′ (𝑎)}. 𝑎

The minimum is attained by compactness and lower semicontinuity. If it were zero, nonnegativity would force one action to have zero defect at both inputs, hence to equal both distinct oracle actions. Therefore 𝛿 𝑥, 𝑥 ′ > 0, and every decoder on the collided carrier has expected native defect at least this margin. This conclusion is unchanged if the evidence carrier is the identity and the recursive state retains the full belief. In each construction, choose identity carriers on the other two branches. Conversely, finite product components can place any one collision beside exact versions of the other interfaces. Thus exactness of either remaining branch cannot compensate for the failed one, and the three predicates are pairwise logically independent. □

Proof of the typed architecture realization interface Dependencies. The architecture action color is a specialization of the truth map in Chapter 20; the joint color uses the same finite labeled-kernel refinement. The terminal deployment statement uses the single time-uniform confidence event of Chapter 19; the stopping time is adapted and the terminal

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architecture is an F𝜏 -measurable selector in the declared resource class. Membership in the displayed common-witness intersection is a separate authorization hypothesis, not a consequence of worldwise color resolution or recursive-state preservation. The native and contextual conclusions retain their original contracts. No joint resource-budget conclusion or color-to-action factorization is inferred. If statistical components are constructed separately, their error budgets are combined by a union bound before this common event is invoked. Proof. Use 𝑞 = 𝑞 arch in Theorem 20.3. Because the terminal rule is worldwise valid under the retained laws, that theorem gives (21.2) with 𝑇𝑍arch,∗ . Applying data processing and Theorem 20.4 to the same opposite-action partition gives 𝑇𝑍arch,∗ ≥ 𝑇𝑋arch,∗ and its equality criterion. Now use 𝜁 as the contract color in Theorem 20.5. Its stable quotient is the joint quotient by definition, so the universal refinement property gives at least 𝑁joint states; counting chart–state pairs gives (21.3). For the conditional deployment branch, assume the terminal selector passes the authorization gate (21.4). On the time-uniform coverage event, the true world belongs to 𝐶𝑡 for every 𝑡, hence to 𝐶 𝜏 at the adapted stopping time. Measurability of the selector makes 𝐴 𝜏 a legitimate terminal action, and membership in (21.4) places 𝐴 𝜏 in the true world’s feasible set. This is exactly the commondeployment implication of Theorem 19.6. The carrier–decoder decomposition then evaluates the witness in the declared native loss scale, and the transmission and exposure gates determine which part is visible under the declared contextual contract. Thus simultaneous passage of the evidence, recursive, common-witness, native, and contextual gates produces the claimed common certified action on the coverage event, whose complement has probability at most 𝛼. □

Proof of finite-library held-out improvement Dependencies. The candidate architectures and baselines must be fixed before the independent test sample is inspected. The independent units, not lower-level observations nested inside them, are the concentration units. The loss difference must be bounded in an interval of length 2𝐵. Proof. For baseline 𝑏, let b 𝑍𝑖 }, 𝑋𝑖(𝑏) = ℓ{ 𝐴𝑏 ; 𝑍𝑖 } − ℓ{ 𝐴; and write

𝑛te 1 ∑︁ b 𝑋 (𝑏) , Δ𝑏 = 𝑛te 𝑖=1 𝑖

−𝐵 ≤ 𝑋𝑖(𝑏) ≤ 𝐵,

Δ𝑏 = E𝑋𝑖(𝑏) .

Hoeffding’s inequality for a variable with range length 2𝐵 gives 

 2 𝑛 𝑡 te P{Δ𝑏 < b Δ𝑏 − 𝑡} ≤ exp − 2 . 2𝐵 Set

√︄ 𝑡=𝐵

2 log(𝑀/𝛿) . 𝑛te

Appendix to Chapter 21: complete proofs

175

The right-hand side becomes 𝛿/𝑀. A union bound over the 𝑀 predeclared baselines yields, with probability at least 1 − 𝛿, √︄ 2 log(𝑀/𝛿) for every 𝑏. Δ𝑏 ≥ b Δ𝑏 − 𝐵 𝑛te If the smallest empirical improvement exceeds the common tolerance, every population improvement is positive. □

Chapter 22

Obstruction-Aware Learning and Inference The preceding chapters separate population structure from finite-sample evidence. Obstruction-Aware Learning and Inference (OALI) is proposed as a falsifiable experimental and audit protocol, not as an established general architecture-selection theory. Its premise is deliberately restrictive: a structural repair should be attempted only after a baseline deployment contract has been declared, a mechanism-specific obstruction has been diagnosed, and the available data support a certificate strong enough to justify changing that contract.

22.1

From diagnosis to controlled intervention

The basic workflow is local oracle estimation → structural audit → certificate → mechanism-matched repair → training → independent validation

local oracle

structural audit

certificate

independent validation

training

matched repair

Figure 22.1: The proposed OALI workflow. Arrows denote audit stages rather than theorem implications. A certificate motivates a repair candidate but does not establish saturation, mechanism exclusivity, or held-out improvement.

Before fitting the repaired system, one declares: • the local objective and native defect; • the baseline deployment class and output semantics; • the population aggregation and statistical unit; 176

22.2. THE REPAIR MUST MATCH THE DIAGNOSED MECHANISM

177

• the resource and regularity constraints; • the downstream endpoint and independent validation protocol. The local oracle stage estimates the relevant fiber, branch multiplicity, and uncertainty. Related work on amortized inference separates per-instance optimization from the limitations of a shared inference mechanism [Cremer et al., 2018, Margossian and Blei, 2024, Marino et al., 2018]. OALI uses a more restrictive intervention rule: it changes the deployment contract only after the failure has been typed and certified. Some gaps therefore call for better local optimization rather than structural repair. The structural audit then determines how local oracle objects correspond across inputs: through point distances, permutations, partial bijections, quotient relations, or set-valued correspondences. Only after this representation has been stabilized does the procedure compute a structural certificate such as a transport lower bound, a coordination obstruction, a cycle holonomy, a resource lower bound, or a three-way feasible/impossible/unresolved certificate. The distinction between OALI and OAI is operational. OALI is the general workflow above. OAI is a concrete instance designed for multibranch oracle families, using partial transport, cycle consistency, and atlas construction.

22.2

The repair must match the diagnosed mechanism

A positive certificate does not by itself say “use a larger model.” The diagnosed mechanism determines which part of the deployment contract should change. Table 22.1: Structural mechanisms and corresponding repair families. Diagnosed mechanism

Candidate structural intervention

excess oracle variation label monodromy missing branches coordination obstruction carrier information loss decoder nonsaturation probability or shape invalidity operational invisibility optimization/reachability only

additional charts, local experts, or test-time refinement atlas, quotient, set-valued, or randomized output partial transport, abstention, or variable-cardinality output rectangularization, memory refinement, or revised sharing richer carrier or task-relative sufficient state stronger decoder with the carrier audit retained projection or nonlinear validity repair no repair for the declared downstream task increase the optimization budget without a structural impossibility claim

The design principle is minimal intervention. Change only the interface named by the certificate whenever possible, while keeping parameter count, computation, training protocol, and data usage comparable to the original baseline. This makes it possible to distinguish a structural repair from a generic increase in flexibility. A repaired class can still be implemented badly. Atlas experts can collapse onto the same branch; a quotient decoder can reintroduce an arbitrary canonicalization; a gate can use information unavailable at deployment. Training diagnostics should therefore retain the structural audit and report, as applicable, chart occupancy, within-chart native defect, transport consistency, gate stability, resource use, and numerical residuals.

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CHAPTER 22. OBSTRUCTION-AWARE LEARNING AND INFERENCE

A falsifiable saturation–repair–validation chain

Training saturation is not an architecture theorem and is not implied by a positive certificate. Misspecification, optimization failure, data scarcity, regularization, or an insensitive metric can all produce plateaus. A stronger structural interpretation requires the following sequence. 1. A calibration-stage certificate predicts a positive obstruction under a fixed deployment contract. 2. Increasing capacity, sample size, or optimization effort without changing that contract does not remove the diagnosed native gap once those ordinary errors have been controlled. 3. A minimal mechanism-matched change reduces the obstruction or native defect in the direction predicted by the theory. 4. On independent statistical units, the repaired system improves the predeclared endpoint relative to parameter- and compute-matched baselines. 5. Random partitions, generic mixtures, or other negative controls do not stably reproduce the improvement; in no-obstruction controls the structural procedure does not trigger unnecessary complexity. This sequence is intentionally falsifiable. Failure of any step prevents the full architecture-choice conclusion, although earlier steps may retain their narrower meaning. No experiment in the present edition completes all five steps; the sequence is a proposed validation standard for future work. Laboratory 24.2.1 illustrates only the certificate and repair-comparison stages for amortized variational inference. Its data, width, and training-effort curves continue to improve at their tested endpoints and therefore do not establish saturation. Its Monte Carlo reference and evaluation draws are reused, and it lacks compute-matched and negative-control repairs. It consequently does not validate the complete chain or justify a general architecture choice. Its endpoint is the sampled native variational objective, not a downstream classification benefit.

22.4

When a certificate implies population improvement

The final comparison requires a common loss scale. Suppose the baseline class has a certified lower b relative to the local oracle, the repaired system has oracle and implementation error at most bound 𝑂 𝜖 rep , and the required finite-sample tolerances sum to Δ. How to read the assumptions The baseline lower certificate and the repaired-system upper certificate must hold on one declared event, refer to the same population target, and be expressed in the same native or explicitly converted loss scale. The statistical and implementation terms may not be counted twice. Theorem 22.1 (Population improvement template). Under the stated certificate event, the population improvement of the repaired system over the baseline architecture class is at least b − 𝜖 rep − Δ. 𝑂 In particular, the improvement is strictly positive whenever the right-hand side is positive.

22.4. WHEN A CERTIFICATE IMPLIES POPULATION IMPROVEMENT

179

Interpretation and boundary A structural diagnosis becomes an actionable guarantee only when the certified baseline floor exceeds the combined cost of learning and implementing the repair. A nonpositive right-hand side does not prove that the repair fails; it means that the current evidence is insufficient to prove strict population improvement. Nor does monodromy or any other internal obstruction alone imply improvement in an unrelated downstream metric. Proof roadmap Write the population difference between the best baseline and the repaired system as the baseline-to-oracle gap minus the repair-to-oracle gap, then substitute the certified lower and upper bounds. The chapter appendix gives the exact subtraction. The theorem also fixes the role of independent validation. OALI is not identified by the use of experts, charts, or routing; those devices are common architectural choices. Its distinguishing claim is conditional: a declared baseline contract is first shown to have a typed structural failure, and the repair is selected because it targets that failure. Capacity-matched random partitions, wider single models, additional test-time computation, and alternative repairs are therefore required controls rather than optional ablations.

Exercises Exercise 22.2. Write a complete OALI contract for amortized inference in a symmetric two-component mixture. State which observation units are reserved for diagnosis and which for independent validation. Exercise 22.3. Design a parameter- and compute-matched negative control for an atlas repair. What result would weaken the claim that cycle structure is the source of the gain? Exercise 22.4. Give three ways an implementation can destroy a theoretically valid zero-obstruction repair.

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Appendix to Chapter 22: complete proofs Proof of the population improvement template Dependencies. The baseline certificate must be a valid lower bound for every baseline architecture on the same event. The repaired-risk upper bound must charge oracle approximation, implementation, and generalization errors only once. Proof. Let 𝑅★ = inf 𝐴∈𝔄base 𝑅( 𝐴) and let 𝑅oracle be the local-oracle risk. The certified baseline event base states b 𝑅★ base − 𝑅oracle ≥ 𝑂. Suppose the repaired architecture satisfies brep ) − 𝑅oracle ≤ 𝜖rep + Δ, 𝑅( 𝐴 where 𝜖 rep contains the declared oracle/implementation residuals and Δ contains the independent generalization tolerances. Subtracting the second inequality from the first gives ★ b b 𝑅★ base − 𝑅( 𝐴rep ) = {𝑅base − 𝑅oracle } − {𝑅( 𝐴rep ) − 𝑅oracle } b − 𝜖 rep − Δ. ≥𝑂

The population improvement is therefore positive whenever the final lower bound is positive.

□

Chapter 23

Partial Transport, Cycle Holonomy, and Recovery The OALI workflow in Chapter 22 treats structural audit and mechanism-matched repair at the procedural level. This chapter specializes that interface to finite branch sets and fixes the partial-bijection model used by the recovery and holonomy theorems.

23.1

Local branch sets

At graph vertex 𝑣, let B𝑣 = {1, . . . , 𝐾 𝑣 } index estimated local oracle branches. Cardinalities may differ because components appear, disappear, split, or remain unresolved. For edge 𝑒 = (𝑢, 𝑣), a transport is a partial bijection 𝑇𝑒 : 𝐷 𝑒 ⊆ B𝑢 −→ 𝑅𝑒 ⊆ B𝑣 . The inverse transport is used on the reverse orientation. When all branch sets have the same size and every transport is total, this problem is related to angular and permutation synchronization [Singer, 2011, Pachauri et al., 2013]. Partial correspondences, cycle-consistent multi-matching, and partial permutation synchronization are also established topics [Chen et al., 2014, Zhou et al., 2015, Bernard et al., 2019, Li et al., 2022]. The book does not claim to invent partial maps. Its distinction is semantic: disappearance and unresolved oracle branches remain observable, and domain survival is carried into the task-level audit rather than being filled by dummy permutations. That semantic distinction is not, by itself, a novelty claim. On finite fibers, every partial bijection can be extended to a total permutation on a larger fiber while a validity mask forbids dummy deployment states. This preserves the original global sections exactly and places the resulting statistical problem next to masked synchronization and finite constraint satisfaction. A partial-domain learning theorem would need an additional operational or statistical ingredient that is not preserved by this totalization.

181

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23.2

CHAPTER 23. PARTIAL TRANSPORT, CYCLE HOLONOMY, AND RECOVERY

Partial matching objective

Let 𝑐𝑢𝑣 𝑖 𝑗 be the cost of matching branch 𝑖 at 𝑢 to branch 𝑗 at 𝑣. An unmatched penalty allows branches to disappear. A structural partial matching minimizes total matched and unmatched cost. The relevant assignment margin is the gap between the optimal structural matching and the second-best distinct structural matching. Dummy-label permutations are not distinct structures. Definition 23.1 (Structural assignment margin). If 𝐶𝑒 (𝑇) is the edge objective and 𝑇𝑒★ is its unique optimal partial bijection, define Γ𝑒 = min★ {𝐶𝑒 (𝑇) − 𝐶𝑒 (𝑇𝑒★)}. 𝑇≠𝑇𝑒

An implementation that excludes only one optimal matched edge can overestimate this margin by missing alternatives that keep all optimal pairs and add another pair. Exact second-best enumeration or an equivalent optimization is required.

23.3

Noisy recovery

Assume true branch representatives 𝑎★𝑣𝑘 and estimates b 𝑎 𝑣, 𝜋𝑣 (𝑘 ) satisfy 𝑑 (b 𝑎 𝑣, 𝜋𝑣 (𝑘 ) , 𝑎★𝑣𝑘 ) ≤ 𝜀 𝑣 for unknown local relabelings 𝜋 𝑣 . For metric matching costs, changing representatives perturbs the cost of any matched pair by at most 𝜖 𝑢 + 𝜖 𝑣 . A matching uses at most min(𝐾𝑢 , 𝐾 𝑣 ) pairs. How to read the assumptions Each local oracle fiber is finite and estimated up to an unknown local permutation. The ★ ★ matched-pair cost is the ambient metric, 𝑐𝑢𝑣 𝑖 𝑗 = 𝑑 (𝑎 𝑢𝑖 , 𝑎 𝑣 𝑗 ). The population partial matching has a unique structural optimum after dummy permutations are quotiented out, the unmatched penalty is fixed, and representative errors are bounded by 𝜀 𝑣 . The strict assignment margin must dominate twice the worst total cost perturbation. Theorem 23.2 (Noisy partial-transport recovery). If Γ𝑒 > 2 min(𝐾𝑢 , 𝐾 𝑣 ) (𝜀 𝑢 + 𝜀 𝑣 ), then the estimated optimal partial matching is 𝑇b𝑒 = 𝜋 𝑣 𝑇𝑒★ 𝜋𝑢−1 . The matched, unmatched, domain, and range structures are recovered exactly up to local relabeling. Interpretation and boundary Under margin separation, all matched pairs, missing branches, domains, and ranges are recovered exactly up to local relabeling. The theorem deliberately outputs no forced match when the margin condition fails. It applies to richer costs once a uniform structural-cost perturbation radius replaces the metric bound.

23.4. PATH TRANSPORT, HOLONOMY, AND GLOBAL SECTIONS

183

Proof roadmap Bound the perturbation of every matched-pair distance by the two endpoint radii, sum over at most min(𝐾𝑢 , 𝐾 𝑣 ) pairs, and invoke a finite-class argmin stability lemma. The chapter appendix includes the lemma and the exact structural proof.

23.4

Path transport, holonomy, and global sections

For a path 𝑝 = 𝑒 1 · · · 𝑒 𝑚 , define the partial composition 𝑇 𝑝 = 𝑇𝑒𝑚 ◦ · · · ◦ 𝑇𝑒1 on the branches for which every intermediate image is defined. For a cycle 𝑐 based at root 𝑟, the holonomy is 𝐻𝑐 = 𝑇𝑐 : 𝐷 (𝐻𝑐 ) ⊆ B𝑟 → B𝑟 . A branch is: • fixed if 𝐻𝑐 (𝑘) = 𝑘; • moved if 𝐻𝑐 (𝑘) is defined and different from 𝑘; • undefined if it disappears along the cycle. Global sections.

A global labeled section is a tuple 𝑠 = (𝑠 𝑣 ) 𝑣 ∈𝑉 ,

𝑠 𝑣 ∈ B𝑣 ,

such that, for every oriented edge 𝑒 = (𝑢, 𝑣), one has 𝑠𝑢 ∈ 𝐷 𝑒 and 𝑇𝑒 (𝑠𝑢 ) = 𝑠 𝑣 . Thus the definition is intrinsic to the edge transports and does not depend on a root or spanning tree. Fix a spanning tree and let 𝑝 𝑣 be the unique tree path from root 𝑟 to vertex 𝑣. The next theorem characterizes the global labeled sections by the root branches that survive every path and are fixed by every fundamental-cycle holonomy. How to read the assumptions The graph is finite and connected, reverse transports are partial inverses, a root and spanning tree are fixed, and partial compositions are evaluated on their natural domains. A global section must use one common root branch that survives every path and satisfies every chord constraint. Theorem 23.3 (Cycle-space characterization). The set of global labeled sections is in bijection with ( S𝑟 = 𝑘 : 𝑘 ∈

) Ù 𝑣

𝐷 (𝑇 𝑝𝑣 ),

𝐻𝑐 (𝑘) = 𝑘 for every fundamental cycle 𝑐 .

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Interpretation and boundary Global labeled sections are exactly the common fixed branches of the fundamental holonomies that survive all root paths. The count and the missing-versus-moved distinction are invariant under local label permutations. Checking each cycle for some fixed point is insufficient because different cycles may fix different branches. Proof roadmap Transport a candidate root branch along the spanning tree and use each chord holonomy to verify the remaining edges. Conversely, recover the root branch from any global section. Conjugate all path transports and holonomies under local relabeling. The complete bijection and recovery proof is in the chapter appendix. Checking each cycle only for the existence of some fixed branch is insufficient. Different cycles may fix different branches while the common intersection is empty. Cycle-space diagnostics are naturally related to combinatorial Hodge methods [Jiang et al., 2011]; the theorem above additionally carries the domains of partial maps, so an undefined branch is not conflated with nontrivial permutation holonomy. Local-to-global consistency and global sections have a broad applied-sheaf and cellular-sheaf lineage [Curry, 2014, Robinson, 2017, Hansen and Ghrist, 2019]. Constraint-network theory likewise long predates this example and already separates local consistency from global satisfiability [Montanari, 1974, Freuder, 1978]. The theorem’s scoped contribution is the spanning-tree/fundamental-cycle characterization for the declared partial-bijection domains and its gauge-invariant branch accounting. This local-to-global consistency question is also adjacent to phase unwrapping, where locally observed phase increments must integrate around cycles [Itoh, 1982, Goldstein et al., 1988, Costantini, 1998]. Partial branch transport has the extra possibility that a label becomes undefined, so phase closure alone cannot represent the full obstruction here. Gauge invariance.

Under local relabeling 𝜋 𝑣 , 𝑇b𝑢𝑣 = 𝜋 𝑣 𝑇𝑢𝑣 𝜋𝑢−1 .

Path transports transform similarly, and root holonomies satisfy b𝑐 = 𝜋𝑟 𝐻𝑐 𝜋𝑟−1 . 𝐻 Therefore fixed/moved/undefined counts, existence of global sections, and the number of global sections are gauge invariant.

23.5. MULTI-CYCLE WITNESS, ATLAS CONSTRUCTION, AND UNCERTAINTY

185

Figure 23.1: Synthetic multi-cycle graph illustrating partial transports, cycle-space obstruction, and atlas coverage.

23.5

Multi-cycle witness, atlas construction, and uncertainty

The OAI v0.5 companion construction has 28 vertices, 30 edges, and cycle rank three. Each fundamental cycle individually has at least one fixed branch, yet the common global-section set is empty. Two cycles also contain undefined branches. A two-chart atlas covers all graph cells. The specific instance is a diagnostic illustration; the logical phenomenon that individually consistent local constraints may have no common global solution is not claimed as novel. Atlas construction on a graph. This is an analytic companion construction, not a real-data scientific result. A chart is a connected subgraph on which a consistent local labeling exists. A greedy construction removes obstructed cycle edges until the remaining subgraphs admit sections and checks vertex and edge coverage; it does not claim globally minimal chart number. Unlike the radical-loop audit below, this release ships no executable artifact for the multi-cycle example. Statistical uncertainty. The exact recovery theorem reduces the statistical bridge to local representative radii and assignment margins. If the margin condition fails, the correct output is unresolved transport rather than a forced edge match.

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186

23.6

Deterministic code-path audit of the radical repair

Section 21.2.1 exhibited the object that the generic workflow still owed: one fixed repaired architecture that works throughout a confidence world, rather than a different root field for every compatible loop. A small executable audit checks the corresponding formulas and code paths. It is not an independent fitted-model or held-out population validation. The frozen square-root family is

𝑧(𝜃) = 𝑟 exp[𝑖{ℓ𝜃 + 𝛽 + 𝛾 sin(𝜃 + 𝜔)}] ,

𝜃 ∈ [0, 2𝜋].

The audit draws ℓ uniformly from {−3, . . . , 3} and draws 𝑟, 𝛽, 𝛾, 𝜔 independently and uniformly from

[0.75, 1.25],

[−𝜋, 𝜋],

[−0.35, 0.35],

[0, 2𝜋],

respectively. Writing 𝜙(𝜃) = ℓ𝜃 + 𝛽 + 𝛾 sin(𝜃 + 𝜔) gives

|𝜙′ (𝜃)| = |ℓ + 𝛾 cos(𝜃 + 𝜔)| ≤ |ℓ| + |𝛾| ≤ 3.35,

so the stated public constant is 𝐿 max = 3.35 throughout the sampling family. The deployed input contains 𝑢 = 𝑧(𝜃), the loss is |𝑎 2 − 𝑢| 2 , and a repaired point output retains its chart label. The two slit-plane charts and their router are fixed before any loop is drawn. They receive 𝑢, not the hidden tuple (ℓ, 𝑟, 𝛽, 𝛾, 𝜔), so the same atlas is evaluated in every world. For the certificate check, 1,024 equally spaced contexts are observed with exact complex target values. The public uniform bound |𝜙′ | ≤ 𝐿 max = 3.35 keeps every adjacent phase increment below 𝜋; the certificate code does not receive the generating values of ℓ or 𝛾. Summing principal increments therefore gives a singleton degree identification set. One seeded pool of 240 generated loops exercises the routine, the fixed atlas, and the full-root-set rule. The script also constructs a worldwise oracle one-chart witness. It reads the generator’s degree and phase, returns an exact root in even degree, and returns zero in odd degree. It is therefore an analytic benchmark for the zero and 𝑟 2 identities, not a deployable common one-chart baseline. No parameter is fitted, so splitting the generated loops into calibration and test subsets would create a false impression of statistical independence doing inferential work.

23.6. DETERMINISTIC CODE-PATH AUDIT OF THE RADICAL REPAIR

187

Table 23.1: Radical-loop common-witness deterministic code-path audit. Check

Audit units

Result

degree certificate

240 loops

fixed two-chart common witness full-root-set common witness even-degree oracle identity

240 loops 240 loops 101 loops

odd-degree oracle identity

139 loops

collapsed-root ablation

240 loops

router trace

240 loops

all singleton sets recovered the generating degree; the certified adjacent-increment bound was 0.02056 < 𝜋 maximum recorded native defect 1.79 × 10−30 maximum recorded native defect 1.79 × 10−30 maximum worldwise oracle-witness defect 5.55 × 10−31 maximum residual from the analytic 𝑟 2 identity 6.67 × 10−16 averaging the two roots returns zero; maximum residual from the analytic 𝑟 2 identity 6.67 × 10−16 chart identifiers are aggregated from the per-context router outputs, not asserted as a literal constant

The numerical residuals in Table 23.1 are floating-point checks of analytic identities. There is no Hoeffding bound: the script fits nothing, and the odd-degree benchmark already uses generator information to instantiate its worldwise oracle witness. The run therefore audits implementation paths and artifact integrity; it neither estimates a population improvement nor replaces the structural proofs. A random seam is deliberately not called a negative control. Moving a valid branch cut is a gauge choice in this exact family and yields another correct atlas. The mechanism-destroying ablation is instead to erase the retained root distinction: averaging 𝑎 and −𝑎 produces zero and pays the full 𝑟 2 catastrophe tax. The even-winding control checks the other direction: the procedure must not report a one-chart obstruction when continuation closes. The complete standard-library-only entry point, unit tests, per-loop output, summary, and content hashes are under reproducibility/radical_loop/. This is a deterministic code-path audit with exact target values and a known regularity bound. It is not independent evidence for a fitted architecture and does not validate noisy degree recovery, a learned router, neural optimization, population generalization, or an application effect. Those remain separate gates rather than being inferred from the synthetic closure.

CHAPTER 23. PARTIAL TRANSPORT, CYCLE HOLONOMY, AND RECOVERY

188

23.7

Closing the radical loop: from native defect to validated repair

The radical example is the book’s longitudinal audit, whereas the six laboratories in Chapters 4 and 24 are a basis and its application-level recombinations. Table 23.2 records the logical dependency order: Part III closes the defect-consistency and lift-validity gates, and Part IV then formulates the covering-space obstruction. The remaining rows add task meaning, finite-data permission, and certified intervention. Every row therefore passes forward a typed object together with the gate that keeps it valid. Table 23.2: The radical loop as a seven-step certification chain. Stage and location

Certified object

Handoff and non-negotiable boundary

1. Native defect Parts I–IV; Chs. 1, 5, and 14

With continuous single-valued point output and uniform loss frozen, 𝐷 𝜃 (𝑎) = |𝑎 𝑘 − 𝑧(𝜃)| 2 is the objective-native loss scale and inf 𝑎 𝐷 𝜃 (𝑎) = 0.

The same defect prices constructive upper witnesses and converse lower certificates. A convenient parameter distance cannot replace it.

2. Lift Parts III–IV; Chs. 10 and 14

The radical question is whether 𝑧 : 𝑆 1 → C★ lifts through 𝑝 𝑘 (𝑎) = 𝑎 𝑘 ; a lift exists exactly when 𝑘 | deg 𝑧.

This covering lift is not permission for a target-calling auxiliary coordinate. Any carrier that exposes 𝑢 = 𝑧(𝜃) must be declared and pass the lift-admissibility audit.

3. Architecture obstruction Part IV; Ch. 14

The failed lift becomes an exact native floor: for the round loop, Theorem 14.1 gives 0 when 𝑘 | ℓ and 𝑟 2 otherwise.

The claim is for the declared continuous, one-chart, point-valued class under uniform risk. Average risk, measurable output, an atlas, or a quotient is a different contract.

4. Visibility Parts IV–V; Chs. 15 and 17

A labeled root exposes monodromy; the full root set or an appropriate quotient can erase it. Legal contexts decide whether that distinction remains task-observable.

Visibility is a semantic gate, not another scalar tax. A change of output meaning or context class must be declared before certification.

5. Confidence world Part VI; Ch. 19

Finite data determine a compatible degree set and hence a feasible–impossible–unresolved certificate; regularity and coverage can collapse that set to one degree.

Hidden winding may not be guessed. Without adequate regularity or coverage, unresolved is the honest result, and every conclusion must hold throughout 𝐶𝑡 .

6. Common witness Part VII; Ch. 21

When 𝐶𝑡 stays away from zero and deployment exposes 𝑢, the fixed rule 𝑊 𝑘 (𝑢), or the fixed two-chart atlas with retained chart label, belongs to every world’s repaired feasible set.

Worldwise divisibility alone does not produce one common 𝜃 ↦→ 𝑎(𝜃). If 𝑢 is hidden or a global labeled root is required, these witnesses are inadmissible.

7. OAI code-path audit Part VII; Chs. 22 and 23

The failed one-chart gate returns the atlas repair; its charts and router are fixed and exercised on one seeded loop pool, with even-degree, odd-degree, and collapsed-root identity checks.

This exact synthetic audit checks formulas and code paths. It is not a held-out performance experiment and does not validate noisy recovery, learned routing, neural optimization, or application efficacy.

The loop closes only if Step 7 tests the same common witness authorized in Step 6. That witness is evaluated in the native loss scale fixed in Step 1, against the obstruction diagnosed in Step 3, after the lift and visibility gates of Steps 2 and 4 and the finite-data permission of Step 5. No later step may retroactively change an earlier contract. Chapter 24 recombines these mechanisms in broader systems; it is not an eighth arrow in the audit.

23.7. CLOSING THE RADICAL LOOP

189

Part VIII is not an eighth inferential step. It stress-tests the seven certification arrows. The chain reopens at a named gate if uniform risk is replaced by average risk, clearance from the discriminant vanishes, regularity or coverage is absent, output semantics change, or a deterministic identity audit is relabeled as held-out evidence. In those cases the correct response is a narrower claim, a new certificate, or unresolvedness—not a declaration that the loop has closed.

Exercises Exercise 23.4. Construct three cycle holonomies on four root branches such that every cycle has a fixed point but their common fixed-point set is empty. Exercise 23.5. Prove gauge invariance of the global-section count. Exercise 23.6. For two local branch sets of sizes two and three, enumerate all structural partial matchings and compute the exact second-best margin for a chosen cost matrix.

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Appendix to Chapter 23: complete proofs A finite-class stability lemma Dependencies. The lemma uses only finiteness of the candidate set and a strict population margin. It is the deterministic device that turns a uniform cost perturbation bound into exact structural recovery. Lemma 23.7 (Finite-class argmin stability). Let T be finite. Suppose 𝐶 : T → R has a unique minimizer 𝜏★ and margin Γ = min★ {𝐶 (𝜏) − 𝐶 (𝜏★)} > 0. 𝜏≠𝜏

If b(𝜏) − 𝐶 (𝜏)| ≤ 𝑟 sup |𝐶

and

Γ > 2𝑟,

𝜏∈T

b then 𝜏★ is the unique minimizer of 𝐶. Proof. For any 𝜏 ≠ 𝜏★, b(𝜏) − 𝐶 b(𝜏★) = {𝐶 b(𝜏) − 𝐶 (𝜏)} + {𝐶 (𝜏) − 𝐶 (𝜏★)} + {𝐶 (𝜏★) − 𝐶 b(𝜏★)} 𝐶 ≥ −𝑟 + Γ − 𝑟 = Γ − 2𝑟 > 0. Thus every competitor has strictly larger estimated cost.

□

Proof of noisy partial-transport recovery Dependencies. The local fibers are finite; the population structural optimum is unique after dummy-label permutations are quotiented out; each estimated representative lies within its declared radius after a local relabeling; and the pair cost is the ambient metric while the unmatched penalty is held fixed. Boundary used in the proof. The conclusion is exact only under a strict margin. When the margin is not separated from the perturbation radius, the theorem requires an unresolved edge rather than a forced matching. Proof. Fix an edge 𝑒 = {𝑢, 𝑣} and align the estimated labels with the population labels through 𝜋𝑢 and 𝜋 𝑣 . Let 𝜏 be any injective partial matching. Its population cost is ∑︁ 𝐶𝑒 (𝜏) = 𝑑 (𝑎★𝑢𝑘 , 𝑎★𝑣, 𝜏 (𝑘 ) ) + 𝜆 𝑒 {𝐾𝑢 + 𝐾 𝑣 − 2|𝐷 (𝜏)|}. 𝑘 ∈𝐷 ( 𝜏 )

The estimated version of the same structural matching, written in the estimated labels, is 𝜋 𝑣 𝜏𝜋𝑢−1 . The unmatched term is identical in the two costs. For each matched pair, the triangle inequality gives 𝑑 (b 𝑎 𝑢, 𝜋𝑢 (𝑘 ) , b 𝑎 𝑣, 𝜋𝑣 ( 𝜏 (𝑘 ) ) ) − 𝑑 (𝑎★𝑢𝑘 , 𝑎★𝑣, 𝜏 (𝑘 ) ) ≤ 𝑑 (b 𝑎 𝑢, 𝜋𝑢 (𝑘 ) , 𝑎★𝑢𝑘 ) + 𝑑 (b 𝑎 𝑣, 𝜋𝑣 ( 𝜏 (𝑘 ) ) , 𝑎★𝑣, 𝜏 (𝑘 ) ) ≤ 𝜀𝑢 + 𝜀 𝑣 . Because a partial matching contains at most 𝑚 𝑒 = min(𝐾𝑢 , 𝐾 𝑣 ) pairs, b𝑒 (𝜋 𝑣 𝜏𝜋𝑢−1 ) − 𝐶𝑒 (𝜏) ≤ 𝑚 𝑒 (𝜀 𝑢 + 𝜀 𝑣 ) for every 𝜏. 𝐶

Appendix to Chapter 23: complete proofs

191

Apply the finite-class stability lemma with 𝑟 = 𝑚 𝑒 (𝜀 𝑢 + 𝜀 𝑣 ). The assumed inequality Γ𝑒 > 2𝑚 𝑒 (𝜀 𝑢 + 𝜀 𝑣 ) preserves the strict ordering of the population optimum against every structural competitor. Hence b 𝜏𝑒 = 𝜋 𝑣 𝜏𝑒★ 𝜋𝑢−1 is the unique estimated optimum. Conjugation by the two local permutations transports the graph of the matching bijectively, so its domain, range, matched pairs, and unmatched branches are all recovered up to the unavoidable local labels. □

Proof of the cycle-space characterization Dependencies. The graph is finite and connected; reverse edge transports are partial inverses; a root and spanning tree have been fixed; and all compositions are interpreted on their natural partial domains. Boundary used in the proof. It is not enough that each fundamental cycle possess some fixed branch. A global section requires one root branch that survives every tree path and is fixed by every chord holonomy simultaneously. Proof. Let T be the fixed spanning tree and 𝑟 its root, and let 𝑝 𝑣 be the unique tree path from 𝑟 to 𝑣. Take 𝑘 ∈ S𝑟 and define

From an admissible root branch to a global section. 𝑠 𝑣 = 𝑇 𝑝𝑣 (𝑘),

𝑣 ∈ 𝑉.

The domain condition in the definition of S𝑟 guarantees that every 𝑠 𝑣 is defined. Compatibility on every tree edge is automatic because the 𝑇 𝑝𝑣 are tree-path compositions. Now let 𝑒 = (𝑢, 𝑣) be a chord, oriented from 𝑢 to 𝑣, and let 𝑐 𝑒 be its fundamental cycle. The root holonomy is 𝐻𝑐𝑒 = 𝑇 𝑝−1 ◦ 𝑇𝑒 ◦ 𝑇 𝑝𝑢 . 𝑣 Since 𝐻𝑐𝑒 (𝑘) = 𝑘, all terms in this composition are defined and 𝑇 𝑝−1 {𝑇𝑒 (𝑇 𝑝𝑢 (𝑘))} = 𝑘. 𝑣 Applying 𝑇 𝑝𝑣 on its domain yields 𝑇𝑒 (𝑠𝑢 ) = 𝑠 𝑣 . Thus the tuple (𝑠 𝑣 ) 𝑣 ∈𝑉 is compatible with every edge and is a global labeled section. From a global section to an admissible root branch. Conversely, let 𝑠 = (𝑠 𝑣 ) 𝑣 ∈𝑉 be a global section. Compatibility along the unique tree path from 𝑟 to 𝑣 implies inductively that 𝑠 𝑣 = 𝑇 𝑝𝑣 (𝑠𝑟 ), so 𝑠𝑟 survives every tree path. Compatibility on a chord (𝑢, 𝑣) gives 𝑇𝑒 {𝑇 𝑝𝑢 (𝑠𝑟 )} = 𝑇 𝑝𝑣 (𝑠𝑟 ). Applying 𝑇 𝑝−1 shows 𝐻𝑐𝑒 (𝑠𝑟 ) = 𝑠𝑟 . Hence 𝑠𝑟 ∈ S𝑟 . The two constructions are inverse because a global 𝑣 section is uniquely determined by its root value. This proves the bijection. Because the set of global sections is defined without reference to a root or spanning tree, its existence and cardinality are independent of those choices.

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Gauge invariance.

Let local relabelings 𝜋 𝑣 act on the edge transports by 𝑇b𝑒 = 𝜋 𝑣 𝑇𝑒 𝜋𝑢−1 .

For a one-edge tree path the corresponding relation for path transport is immediate. If it holds for the path to 𝑢, composition with the next edge gives 𝑇b𝑝𝑣 = 𝑇b𝑒𝑇b𝑝𝑢 = 𝜋 𝑣 𝑇𝑒 𝜋𝑢−1 𝜋𝑢𝑇 𝑝𝑢 𝜋𝑟−1 = 𝜋 𝑣 𝑇 𝑝𝑣 𝜋𝑟−1 . Induction proves this relation for every vertex. Substitution into the root-holonomy formula yields b𝑐𝑒 = (𝑇b𝑝𝑣 ) −1𝑇b𝑒𝑇b𝑝𝑢 𝐻 = 𝜋𝑟 𝑇 𝑝−1 𝜋 −1 𝜋 𝑣 𝑇𝑒 𝜋𝑢−1 𝜋𝑢𝑇 𝑝𝑢 𝜋𝑟−1 𝑣 𝑣 = 𝜋𝑟 𝐻𝑐𝑒 𝜋𝑟−1 . Conjugation by 𝜋𝑟 is a bijection between domains, undefined complements, fixed-point sets, and moved sets. It also sends S𝑟 to the estimated admissible set. Therefore all their cardinalities and the existence of a global section are gauge invariant. Finally, on the simultaneous edge-recovery event, every estimated edge transport is exactly a gauge conjugate of its population counterpart. The preceding path and holonomy identities then show that the complete estimated cycle audit—including undefined branches, fixed branches, moved branches, and the common global-section set—is recovered exactly up to root relabeling. □

Chapter 24

Four Applied Laboratories 24.1

How the laboratories close the loop

Chapter 4 isolated four pure mechanism directions: oracle survival under coarse-graining, semantic repair for graph-indexed probability fields, coordination under shared conditional deployment, and resource-indexed expansion from one-pass prediction to local refinement. Chapters 5 and 18 supplied exact and soft base-change decompositions; Chapters 22 and 23 supplied a workflow and its transport machinery. The examples below are neither one-to-one sequels nor new theories. Each recombines several mechanisms inside a more complex scientific contract and occupies a different point on the certification ladder. From pure mechanisms to composite laboratories Amortized and sparse inference: one-pass sharing plus test-time refinement. The MNIST study is a five-run held-out audit whose scaling curves do not establish saturation. A frozen digits control upgrade compares repairs at equal native information and equal computation. A separately frozen patient-level diabetes benchmark carries the same contract to a scientific dataset, preserves a failed WDBC precursor in the evidence ledger, and changes the deployed architecture only after all confirmatory gates pass. Posterior-family Monte Carlo: carrier reuse plus certified base change. A shared augmentation or particle cloud may discard target-specific simulation work only if the requested posterior functional remains recoverable. Uncorrected sharing pays stationary-law defect, exact kernel correction pays rejection and mixing variance, and exact-ratio reuse pays finite-particle error through overlap. Current evidence is an exactly enumerable Gibbs family plus a fixed analytic Gaussian-posterior pilot. Multi-sample cytometry: carrier survival plus graph compatibility. Local summaries must retain rare populations while partial transports align the surviving branches and pass cycle audits. Current evidence is a synthetic smoke test plus a confirmatory protocol. Sequential systems: dynamic coarse-graining plus coordination. Memory merges histories while one shared system must coordinate the distinctions that remain across occupancy and downstream contexts. Exact finite interfaces are available; broad systems extensions remain open.

193

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CHAPTER 24. FOUR APPLIED LABORATORIES

The comparison enforces one editorial rule: an analytic mechanism check, a working software path, and a completed real-data claim are different kinds of evidence. It also keeps the operation label separate from the scientific contract: P/G/X/V/C records how a declared field, surrogate, objective, or joint law is handled, while an atlas or quotient changes the contract itself.

24.2

Laboratory I: one-pass inference audits

24.2.1

MNIST amortization: frozen but incomplete

This laboratory asks a deliberately narrow real-data question. Fix a trained VAE decoder and the diagonal-Gaussian variational family. For image 𝑥, let 𝐽 𝑥 (𝑎) be the negative evidence lower bound at variational parameter 𝑎, and let e 𝑎 𝑥 be a high-accuracy imagewise refit. The audited native defect of a deployed encoder 𝐴 is b 𝑥 { 𝐴(𝑥)} = 𝐽 𝑥 { 𝐴(𝑥)} − 𝐽 𝑥 (e D 𝑎 𝑥 ).

The baseline contract 𝔄0 is a frozen finite library of trained one-pass encoders. The repaired contract 𝔄𝑇 starts from the selected encoder and permits 𝑇 imagewise optimization steps. Because test-time steps are declared as a resource coordinate, the comparison 𝔄0 ⊂ 𝔄𝑇 is a deployment-contract expansion rather than a claim that the same computation was merely implemented more carefully.

Frozen design and ordinary-error controls. Before the confirmatory results were generated, the protocol froze five decoder-training seeds, a 55,000-image fitting split, a 5,000-image validation split, and the same 512 held-out official test images per run. Validation alone selects among seven one-pass recipes varying training sample size, epoch count, and hidden width; the official test images do not select the encoder. The imagewise reference starts from both the best library output and the standard-normal variational parameter, continues each start by 800 Adam and 30 L-BFGS steps, and retains the lower objective. Every run passed the predeclared reference-dominance and gradient-residual gates, with no fallback invocation. Table 24.1 compares changes inside the one-pass contract with a test-time resource expansion. Increasing data, epochs, and width reduces the mean gap throughout the tested ranges. The curves therefore do not establish a plateau or structural saturation; observing a positive gap at the largest tested setting is not enough. The validation-selected encoder is the width-512 recipe in four runs and the width-256 recipe in one.

24.2. LABORATORY I: ONE-PASS INFERENCE

195

Table 24.1: Prospectively frozen MNIST audit. The first three rows report mean test amortization gaps in nats across five independently trained decoders. The final two rows are the seed-level primary endpoints; parentheses contain the seed standard deviation and confidence bounds are one-sided 95% Student 𝑡 bounds. Audit component

Frozen settings

Held-out result

Interpretation

Training data

10,000 / 30,000 / 55,000 images

5.084/3.016/2.642 nat

Training effort

4 / 12 / 24 epochs

4.296/3.053/2.642 nat

Encoder width

64 / 256 / 512 hidden units

3.778/2.642/2.527 nat

Selected one-pass gap

validation-selected finite library

Eight-step residual ratio

uniform 𝑇 = 8 refinement

2.493 (0.126) nat; lower bound 2.373 0.354 (0.027); upper bound 0.380

continued improvement; no saturation claim continued improvement; no saturation claim continued improvement; no saturation claim passes > 0.5-nat gate passes < 0.80 in 5/5 runs

Observed gap and refinement comparison. The selected one-pass gap averages 2.493 nats, with a one-sided 95% lower bound of 2.373. Uniform refinement leaves residual-gap ratios 0.728/0.479/0.354/0.243 after 𝑇 = 1, 4, 8, 16 steps. At the predeclared eight-step endpoint it therefore removes about 64.6% of the measured defect. Of 320 audited encoder–reference segments, 311 have positive minimum curvature on the declared seven-node grid, and all 311 such paths satisfy the local curvature bracket. This is a local diagnostic, not a global-convexity claim. These results establish that additional imagewise optimization lowers the sampled variational objective relative to the selected one-pass encoder. They do not identify one-pass sharing as the dominant mechanism. The same Monte Carlo draw collection is used to construct the numerical reference, guide refinement and curvature calculations, and evaluate the final gaps; hence the reported endpoint is a fixed sample-average objective rather than an independently evaluated population ELBO. The study also lacks a compute-matched generic optimizer, random or generic mixture controls, and an independent downstream endpoint. It is therefore an incomplete OALI laboratory, not a successful certificate–saturation–repair validation. It is not an OAI atlas experiment: no branch transport, holonomy, or chart selection is involved. Adaptive allocation has mean adaptive/uniform ratio 0.982, but its one-sided upper bound 0.991 does not establish a stable one-percent advantage over uniform allocation; that secondary claim is withheld.

Reproducibility record. The book snapshot and compact machine-readable summaries are under reproducibility/ch24/; within that directory, verify_mnist_audit.py checks the archived seed summaries, primary decisions, generated TeX numbers, frozen-source hashes, and shipped checksums. Full retraining remains a separate, explicitly requested run; it is not hidden inside the normal book build.

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This result is conditional on the frozen decoder, diagonal-Gaussian local family, seven-recipe one-pass library, optimization protocol, fixed MNIST audit set, and reused Monte Carlo sample-average objective. It establishes a positive measured gap for the validation-selected encoder and shows that imagewise refinement reduces that same measured objective. It does not establish saturation of the one-pass class, a mechanism-exclusive structural obstruction, independent population-ELBO improvement, superiority to compute-matched generic alternatives, improvement in classification or another downstream task, or a population confidence interval for all handwritten digits. Accordingly it cannot by itself justify an architecture choice.

24.2.2

Handwritten digits: equal-information, equal-compute sparse repair

The preceding MNIST result must remain incomplete: its archived outputs cannot retroactively supply independent Monte Carlo evaluation or a saturation certificate. The R1 digits protocol first closed a narrower frozen workflow, but its generic control did not receive the native gradient used by the proximal update. It matched multiply–accumulates, not information. Protocol DIGITS-SPARSE-OALI-R2-EQUAL-INFO-2026-08-17 froze a new control, the common carrier, compute budget, fitted models, numerical gates, code, environment record, and hashes before the control saw confirmatory outputs. A fit-free acceptor then scored the frozen artifacts. R2 reuses the R1 image split, so it is a strict control upgrade rather than a fresh replication. The data are the 1,797 handwritten 8 × 8 images shipped with scikit-learn. A nonnegative 32-atom dictionary 𝐷, learned on the 60% training split and then frozen, defines the imagewise oracle 𝐽 𝑥 (𝑧) =

1 ∥𝐷𝑧 − 𝑥∥ 22 + 0.02∥𝑧∥ 1 , 2

𝑧 ≥ 0.

The baseline architecture is one thresholded-affine pass 𝑧0 (𝑥) = [𝑥𝑊 + 𝑏] + . Ridge regularization and training-set compensation are selected on a disjoint 20% development split. Both repairs receive the same native carrier {𝑧0 , 𝐿 −1 ∇𝐽 𝑥 (𝑧 0 ), 𝐷 ⊤ 𝐷, 𝜆, 𝐿}. The mechanism-matched repair performs two proximal-gradient steps. The learned control first computes the same native gradient and then applies a 64 → 32 → 32 residual MLP. Two native-gradient evaluations for the targeted repair and one gradient plus the MLP for the generic control each cost exactly 6,144 multiply–accumulates. A two-step fixed-coordinate permutation of the gradient supplies an equal-information, equal-compute mechanism-destroying control. Why the saturation statement is structural. For a strict active set 𝑆, the nonnegative Lasso oracle is locally affine, with active-coordinate Jacobian −1 ⊤ (𝐷 ⊤ 𝑆 𝐷𝑆) 𝐷𝑆 .

In contrast, coordinate 𝑗 of a thresholded-affine encoder has one fixed derivative whenever it is active. The frozen audit searched for two strict KKT neighborhoods sharing an active coordinate but having incompatible oracle Jacobian rows. It found 359 strict points among 360 confirmatory images and a maximum shared-row conflict of 0.869. Hence no one fixed thresholded-affine map can agree with the

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oracle on both certified neighborhoods. This conclusion is about the declared one-pass family; it is not a lower bound for arbitrary ReLU networks. Table 24.2: Digits R2 equal-information, equal-compute control upgrade. Bounds are one-sided 95% paired bootstrap bounds over 360 confirmatory images. The table groups all 12 frozen gates; every gate passed. Link

Frozen gate

Confirmatory result

Decision role

Protocol integrity (3)

protocol ID, source hashes, and confirmatory completeness gap lower bound > 0.004, oracle dominance, proximal residual near zero strict-KKT Jacobian conflict > 0.05

all three passed

the contract predates scoring

gap 0.00831; lower 0.00768

Targeted repair (1)

residual-ratio upper bound < 0.25

ratio 0.134; upper 0.141

Equalinformation/equalcompute control (1) Negative control (1)

repair/generic upper bound < 0.50

ratio 0.218; upper 0.233

permuted-gradient lower bound > 1.05

ratio 3.049; lower 2.887

Operational endpoints (2)

SSE upper change < 0; accuracy change > −0.02

SSE -0.00871 (upper -0.00780); accuracy 0.9500 to 0.9528

positive fitted-baseline defect; not a class infimum one fixed pass cannot hit both certified neighborhoods two-step mechanism-matched reduction rules out extra native information or computation alone destroying the mechanism reverses the gain no fidelity or recognition trade

Operational gap (3)

Structural saturation (1)

conflict 0.869

The frozen acceptor returned ACCEPT_ONE_PASS_PLUS_TWO_STEP_PROXIMAL_EQUAL_INFO. It trained no model and read only the predeclared thresholds, frozen hashes, raw per-image table, and sealed summary. Under the same information carrier and deployment computation, the decision changes from a one-pass encoder to that encoder plus two proximal steps. The baseline gap is an operational defect of one fitted rule, not the class infimum 𝔒; class-level failure of exact saturation comes from the Jacobian conflict, not from the positive gap alone. Boundary of the claim This closure is restricted to the frozen digits split, dictionary, nonnegative ℓ1 objective, thresholded-affine grammar, and 6,144-MAC envelope. R2 removes R1’s clearest extrainformation explanation but uses images already scored in R1. It is a frozen real- benchmark control upgrade, not a fresh replication, external-person audit, or scientific-domain validation. Proximal gradient, active-set geometry, sparse coding, and algorithm unrolling remain classical.

24.2.3

Scientific case: patient-level diabetes progression repair

To test whether the same structural certificate can change an architecture choice on independent scientific units, protocol DIABETES-SPARSE-OALI-R1-2026-08-17 was frozen prospectively. The

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data contain ten baseline variables and one-year disease-progression responses for 442 patients. Fixed 60/20/20 patient splits separate fitting, development, and confirmation. The fitting patients alone determine the min–max transform, an eight-atom nonnegative dictionary, one-pass ridge encoder, equal-information generic control, and downstream ridge progression predictor. The 89 confirmatory patients were scored once, after all source files, fitted models, thresholds, environment records, and ten hashes were frozen. The native objective is again nonnegative ℓ1 sparse reconstruction. Here 𝑚 = 10 and 𝑘 = 8, so one native gradient costs 𝑚𝑘 + 𝑘 2 = 144 multiply–accumulates and two proximal steps cost 288 MAC. The generic control receives the identical (𝑧 0 , 𝐿 −1 ∇𝐽) carrier and applies a 16 → 6 → 8 MLP; its one gradient plus MLP also costs 288 MAC. The two-step permuted-gradient control has the same budget. The scientific endpoint is not the reconstruction objective under another name: it is diseaseprogression MSE from a predictor fitted only on the training patients. Table 24.3: Frozen patient-level diabetes confirmation. One-sided 95% paired bootstrap bounds treat the patient as the independent unit. All 12 frozen gates passed. Link

Frozen gate

Confirmatory result

Architecture role

Protocol and numerical quality (5)

ID, ten hashes, 89 rows, oracle residual, oracle dominance baseline-gap lower bound > 10−4 strict-KKT conflict > 0.02

all five passed

residual-ratio upper bound < 0.30 target/generic upper bound < 0.50

ratio 0.100; upper 0.149

excludes post-hoc refitting and a numerical pseudo-oracle positive native defect of the fitted one-pass rule one thresholded-affine pass cannot hit both certified neighborhoods large reduction in native defect carrier and MAC alone do not explain the gain

target/permuted upper bound < 0.50 SSE-change upper bound < 0; progression-MSE change upper bound < 100

ratio 0.0456; upper 0.0670

Operational gap (1) Structural saturation (1) Targeted repair (1) Equalinformation/equalcompute control (1) Mechanism negative control (1) Independent endpoints (2)

gap 0.000478; lower 0.000304 89 points; conflict 0.601

ratio 0.118; upper 0.187

SSE -0.000256 (upper -0.0000672); MSE 3035.46 to 3011.38 (change upper -2.10)

correct gradient coordinates matter native improvement without sacrificing the frozen scientific endpoint

The fit-free acceptor returned ACCEPT_DIABETES_ONE_PASS_PLUS_TWO_STEP_PROXIMAL_ EQUAL_INFO. In the language of Theorem 19.8, the observation law and information carrier are held fixed while the deployable mechanism class is enlarged inside the same compute envelope. The theorem family is not a post-hoc explanation of a favorable number: it dictated which actions, information, regret scale, and controls had to be frozen before the final choice changed. Post-confirmation alignment with the quantitative depth theorem. The preceding R1 decision predates Theorems 19.10 and 19.11; it cannot be relabeled as a prospective test of them. A separate fit-free audit now recomputes their quantities from the sealed dictionary, oracle, one-pass encoder, and confirmatory records. The post-confirmation geometry file freezes the binary sex coordinate, admits

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199

perturbations only in the other nine coordinates, restricts them to the training-minmax box, and requires the two centers to have the same frozen value. Of the exact KKT centers, 77 have positive restricted radii inside that box. The deterministic rule selects, among those pairs with different supports and a shared active coordinate, the pair maximizing the restricted theorem floor. It selects confirmatory rows 17 and 68 (dataset rows 61 and 172), shared atom 6, with radii 0.0410702 and 0.120586. The restricted Jacobian-row conflict is 0.389454, giving Γ = 0.0119313,

𝜅 2 Γ = 3.55849 × 10−5 , 2

𝜅 = 0.499944.

Thus every one-pass thresholded-affine encoder—not merely the fitted baseline—exceeds native-loss tolerance 𝜂 = 3.20264 × 10−5 somewhere on the union of the two certified restricted balls. For the frozen baseline, an analytic restricted-ball envelope gives 𝑞 = 0.737882 and a sufficient repair depth of 24 proximal steps at that tolerance. This is a literal theorem-driven architecture recommendation: one pass is ruled out, while the declared recurrent mechanism at the certified depth is guaranteed to cross the native-loss threshold. The new calculation also prevents two attractive overclaims. The restricted balls respect the declared box grammar and freeze the binary coordinate, but the audit does not prove that every point in them lies in the support of the patient population. The lower bound is therefore uniform for the declared input grammar, not a distributional lower bound over attainable patients. The R1 two-step architecture was accepted on patient-average endpoints and equal-information/equal-compute controls; it is not guaranteed to cross this stronger uniform two-ball tolerance. Conversely, the 24-step guarantee was not a frozen R1 endpoint and is not charged against the 288-MAC comparison. The two conclusions answer different deployment contracts and are reported separately. A failed precursor remains in the evidence ledger. The earlier frozen WDBC-R1 experiment passed 11 of 12 gates but failed its preregistered requirement that a permuted gradient be worse than no repair. The residual-ratio one-sided 95% lower bound was 0.615, below 1.05. Although the targeted repair beat the permuted repair, the all-gates acceptor returned REJECT_ARCHITECTURE_REPAIR_CLAIM. Before touching its fresh confirmatory patients, the diabetes protocol replaced that mechanism question by a direct targeted/permuted comparison. The WDBC failure was not deleted or retrospectively reclassified. Boundary of the claim This is a prospectively frozen patient-level validation on a scientific dataset, with a fit-free hash-checking acceptor separated from model fitting. It is not an external-person or externalinstitution replication. Disease- progression MSE is a benchmark endpoint, not evidence of clinical utility, treatment benefit, or causality. The case instantiates the theorem’s typed contract and changes one frozen architecture decision; it does not prove the Le Cam/Blackwell ingredients or establish a universal architecture-choice theory. The active-set depth certificate is a post-confirmation theorem- alignment audit, not an independently frozen validation of the new theorem family. A future claim that the family itself changed a prospectively accepted architecture must freeze its pair-selection rule, uniform tolerance, repair depth, and compute budget before opening new confirmation units.

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24.3

Laboratory II: certified Monte Carlo reuse across posterior families

24.3.1

The reuse contract

Let {𝜋𝑡 : 𝑡 ∈ T } be posterior laws on a common latent or parameter space, and let ∫ 𝜇𝑡 ( 𝑓 ) = 𝑓 (𝑧) 𝜋𝑡 ( d𝑧) be the declared oracle. Independent simulation at every 𝑡 preserves the contract but ignores family structure. A shared Monte Carlo carrier is useful only when it retains what is needed to recover 𝜇𝑡 ( 𝑓 ): the relevant functional, the source identity, and either a certified conditional-law approximation or an evaluable source-to-target density ratio. A large bank without those oracle distinctions is storage, not effective capacity. This is the applied counterpart of the first laboratory in Chapter 4. Coarse-graining may safely forget the random-number history and other target-irrelevant details of a source cloud, but it may not forget the source law or information needed for target base change. The horizontal posterior family says which distinctions must survive; the vertical Monte Carlo defect says what is paid when they do not.

24.3.2

Three contracts and three diagnostic scales

For a two-block Gibbs family, write ∫ 𝑃𝑡 (𝑥, 𝐴) = 𝑞 𝑡 ( d𝑧 | 𝑥)𝑟 𝑡 ( 𝐴 | 𝑧),

𝑞¯ 𝑡 =

∑︁

𝛼𝑡 𝑠 𝑞 𝑠 ,

𝑠

e𝑡 insert 𝑞¯ 𝑡 while retaining the target update 𝑟 𝑡 . With and let 𝑃 𝜖 𝑡 = sup TV{ 𝑞¯ 𝑡 (· | 𝑥), 𝑞 𝑡 (· | 𝑥)},

𝛿(𝑃𝑡 ) ≤ 𝛾𝑡 < 1,

𝑥

the elementary kernel-perturbation recursion gives TV(e 𝜇𝑡 , 𝜇𝑡 ) ≤

𝜖𝑡 , 1 − 𝛾𝑡

|e 𝜇𝑡 𝑓 − 𝜇𝑡 𝑓 | ≤ osc( 𝑓 )

𝜖𝑡 1 − 𝛾𝑡

whenever the two invariant laws exist [Mitrophanov, 2005]. If the conditional field is twice differentiable Í Í and 𝑚 1,𝑡 = 𝑠 𝛼𝑡 𝑠 (𝑠 − 𝑡), then 𝜖 𝑡 ≤ 𝐿 1 |𝑚 1,𝑡 | + (𝐿 2 /2) 𝑠 𝛼𝑡 𝑠 (𝑠 − 𝑡) 2 . One-sided reuse is therefore generically first order in the sharing radius; a centered operator cancels the first-order term and is second order under the stated smoothness. The cancellation is a property of the sharing operator, not a generic consequence of a smooth posterior functional. An exact Metropolis–Hastings correction changes the relevant component. It preserves the target law when the acceptance ratio is exact, but stronger coupling can appear as rejection and autocorrelation. It should not be ranked against an uncorrected kernel using stationary defect alone. The third contract uses a particle bank. From independent source clouds 𝑍 𝑠 𝑗 ∼ 𝜋 𝑠 , define d𝜋𝑡 𝑤𝑡 𝑠 = , d𝜋 𝑠

Í 𝜇 b𝑡←𝑠 ( 𝑓 ) =

𝑗 𝑤 𝑡 𝑠 (𝑍 𝑠 𝑗 ) 𝑓 (𝑍 𝑠 𝑗 )

Í

𝑗 𝑤 𝑡 𝑠 (𝑍 𝑠 𝑗 )

,

Í ( 𝑗 𝑤𝑡 𝑠 𝑗 )2 eff b 𝑁𝑡 𝑠 = Í 2 . 𝑗 𝑤𝑡 𝑠 𝑗

24.3. LABORATORY II: MONTE CARLO REUSE

201

Exact ratios remove deterministic localization bias. Finite self-normalized importance sampling still has 𝑂 (𝑁 𝑠−1 ) bias, while its leading variance is controlled by 𝜌𝑡 𝑠 = E𝑠 𝑤 2𝑡 𝑠 = 1 + 𝜒2 (𝜋𝑡 ∥𝜋 𝑠 ),

−1 b𝑡eff 𝑁 𝑠 /𝑁 𝑠 −→ 𝜌 𝑡 𝑠 .

Thus an ESS gate screens usable overlap and requests a fresh target cloud when no source passes; ESS is a diagnostic and pooling heuristic, not a test-function-specific oracle [Agapiou et al., 2017, Elvira et al., 2022].

24.3.3

What the fixed pilots establish

The Gibbs pilot uses an exactly enumerable binary augmentation family. Its 2 × 2 kernels provide invariant laws without Monte Carlo error over nine sharing radii. Log–log slopes are fitted on the six smallest radii; 300 independent chains at radius 0.1 serve only as an implementation check. The observed slopes in Table 24.4 recover the predicted first- and second-order regimes, and every exact stationary bias lies below its declared perturbation bound. The particle pilot generates 240 covariates and 41 analytic local Gaussian posteriors. The oracle is Pr𝑡 (𝜃 > 0); 21 alternating targets supply independent anchor clouds. Across 300 replications with seed 20260714, the low-cost bank uses 96 particles per anchor, whereas the independent baseline uses 96 at every target. The equal-fresh-draw bank uses 187 particles per anchor, 3,927 draws in total, and attains MSE 0.132 × 10−3 . The low-cost and equal-budget banks additionally perform 82,656 and 161,007 density-ratio evaluations per replication. Mean empirical ESS tracks {1 + 𝜒2 (𝜋𝑡 ∥𝜋 𝑠 )} −1 with correlations 0.9896 in the smooth family and 0.9891 after an imposed mean discontinuity. Under that negative control, no inspected cross-discontinuity source passes the mean ESS gate and all equally local same-side sources pass. Table 24.4: Three diagnostic scales for posterior-family Monte Carlo reuse. The final column reports only the fixed analytic pilots described above. Mechanism

Target contract

Coupling price and certificate

Reproducible evidence

Uncorrected shared Gibbs

perturbed unless 𝜖𝑡 = 0

stationary defect 𝜖 𝑡 /(1 − 𝛾𝑡 ); conditional and kernel discrepancy

exact binary slopes: 1.006 one-sided, 2.006 centered; all 18 evaluated perturbation bounds pass

MH-corrected coupled move

exact under the declared ratio

rejection and mixing variance; acceptance and autocorrelation

comparator only; no numerical claim in this laboratory

Exact-ratio particle bank

population base change is exact; finite-𝑁 self-normalization remains

𝜒2 overlap, ratio evaluations, memory, and ESS-triggered refresh

smooth-family MSE 0.221 × 10−3 versus 0.658 × 10−3 , with 2,016 versus 3,936 fresh draws

Reproducibility record. The single entry point is reproducibility/ch24/run_all.py. The exact source snapshot, tests, frozen protocol, and claim audit are vendored under vendor/mcmc_aux/ beside that entry point; the source-of-truth result is results/monte_carlo_reuse/summary.json. Seed, configuration, costs, all 300-replication summaries, and pass/fail thresholds are machine-readable,

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and SHA256SUMS.txt identifies every artifact. The default run uses no file outside the reproduction directory. Boundary of the claim The Gibbs pilot validates a finite-state localization calculation; the particle pilot validates exact-ratio reuse and its overlap gate in a low-dimensional conjugate family. Fresh draws and ratio evaluations are reported separately, so lower MSE per fresh draw is not a wall-clock claim. No nonlinear state-space, long-trajectory, adaptive-source, or application-scale superiority result is asserted.

24.4

Laboratory III: multi-sample cytometry

24.4.1

Alignment with missing populations

In multi-sample cytometry, cells are nested inside biological samples and organized into local populations. The problem is not only clustering but whether those populations correspond across samples and whether one global labeling is scientifically coherent. Cardinalities can change through biological absence, rare-population sampling failure, local split or merge, marker drift, or ambiguous matching. Full permutation synchronization assumes away these events; the partial transports of Chapter 23 retain them as explicit outcomes. The OALI pipeline is: 1. estimate local populations and uncertainty on calibration samples; 2. build a scientifically justified sample graph; 3. estimate partial edge transports and their assignment margins; 4. audit cycle holonomy and global-section existence; 5. construct charts only when the obstruction is stable; 6. fit within-chart transfer rules and evaluate them on held-out biological samples.

24.4.2

Reproducible smoke test and numerical limits

The fixed smoke test generates ten samples, 39 marker coordinates, and 24 designed populations. Biological absence is simulated by dropping populations sample by sample, with a higher drop probability for four rare populations. The sample graph has 25 edges and hence cycle rank 16. After within-sample standardization, rectangular Hungarian assignments are retained only below the predeclared distance threshold 2.40; retained partial transports are then audited on the non-tree edges and propagated to the reference sample. The local components and their labels are generated, so this check does not evaluate cell clustering. Table 24.5 prevents the phrase “recovers the designed labels” from hiding two different conclusions. Edge matching is nearly exact, but two false accepted matches produce two inconsistent fundamental cycles and substantially weaken rare-population recovery. The pipeline path is executable; the rare-population endpoint is not yet strong.

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203

Table 24.5: Synthetic cytometry smoke test with 95% intervals. Sample-level endpoints use 5,000 bootstrap resamples of the ten samples; discrete edge and cycle endpoints use Wilson intervals. Endpoint Sample-level macro-F1 Rare-population recall Unresolved-cell rate Edge-match precision Edge-match recall Missing-branch detection recall Cycle-transport agreement

Estimate

95% interval

0.9304 0.4683 0.0129 0.9959 1.0000 0.9623 0.9920

(0.9171, 0.9443) (0.3419, 0.5709) (0.0104, 0.0155) (0.9851, 0.9989) (0.9921, 1.0000) (0.9070, 0.9852) (0.9714, 0.9978)

Unit and method 10 samples; bootstrap 10 samples; bootstrap 10 samples; bootstrap 486 matches; Wilson 484 shared branches; Wilson 106 missing branches; Wilson 251 transports; Wilson

Reproducibility record. The code snapshot is identified by SHA-256 prefix f1c69873bf997112; the full hash, configuration, package versions, counts, and intervals are in reproducibility/ ch24/results/cytometry/cytometry_run.json. The checked environment is Python 3.11.5, NumPy 1.24.3, and SciPy 1.11.1. Seed 20260806 generates the actual data entry reproducibility/ ch24/results/cytometry/generated_components.csv; seed 20260807 fixes the bootstrap. The directory-level reproducibility/ch24/SHA256SUMS.txt checks every source and result artifact. No real-data cytometry result is reported. The predeclared entry point for a future confirmation is the Samusik_all_SE() object in HDCytoData (or its corresponding flowSet), with original files at FlowRepository accession FR-FCM-ZZPH [Samusik et al., 2016, Weber and Soneson, 2019]. That resource contains 841,644 cells from ten mice, 39 markers, and 24 manually gated populations. These matching dimensions do not turn the synthetic table into a Samusik result. The confirmatory protocol therefore separates biological samples into calibration, tuning, and final test sets. Tuning selects graph density, unmatched penalties, and a predeclared repair. Final endpoints include macro-F1, rare-population recall, unresolved-branch rate, and runtime. Baselines include pooled clustering, one-reference Hungarian matching, tree propagation, full synchronization, generic mixtures or random atlases, and OAI partial transport with cycle audit.

24.4.3

The scientific claim boundary

A structural result would require a stable nontrivial obstruction or unresolved region on calibration data. The same analysis should predict a plateau for the global labeling, identify the cycles or missing branches to be repaired, and yield held-out improvement that random cuts, wider pooled models, or tree-only propagation do not reproduce. Required negative controls include sample-level bootstrap, graph perturbation, cluster-number and penalty sensitivity, transport margins, and label or graph randomization. If the certificate is unstable, the conclusion remains unresolved. The same contract can arise in multi-site extremes, hospital demand states, ecological communities, single-cell transcriptomics, and drifting sensor populations. Each application still needs its own native oracle and scientific endpoint; an atlas is not a result merely because it can be fit.

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24.5

Laboratory IV: policies, world models, and memory

24.5.1

Compressed histories and the four-component decomposition

At a history ℎ𝑡 , a local oracle may be a conditional policy, posterior belief, predictive kernel, planning auxiliary, or control variate. A deployed system compresses history to 𝑧 𝑡 = 𝜌(ℎ𝑡 ) and reuses one policy, transition, or value architecture. Subject to a justified exchange into the control objective, the decomposition has the form control regret = local model error + state/architecture obstruction + generalization + planning/optimization error. Local predictive error is not automatically control regret; an operational exposure theorem is required.

24.5.2

Coordination, recursion, and context

A shared policy couples decisions across histories. Rectangularization compares it with independent local conditional choices, but occupancy is endogenous: an early shared action changes which later histories are seen. A memory state is sufficient only when histories it merges have identical task-relevant conditional futures. Preserving the present action or one-step prediction need not preserve the next experiment, a changed reward, multi-step planning, an intervention, or a safety constraint. The minimal recursive quotient of Chapter 20 supplies the finite exact model of this requirement. Test-time computation is itself a resource coordinate. With 𝔄𝑇 = {deployments realizable with at most 𝑇 test-time steps}, the frontier 𝑇 ↦→ 𝔒 𝑃 (𝔄𝑇 ) separates an enlarged effective class from better optimization inside a fixed class.

24.5.3

Witness-directed memory refinement

An obstruction-aware procedure estimates local conditional signatures, finds histories merged by the present memory, computes a coordination or carrier certificate, splits only the binding states, and validates on held-out trajectories. A failure to certify preservation of a safety-relevant distinction is reported as unresolved together with the experiment that could separate it; it is not silently converted into a state merge. Exact finite-history results support this design. Continuous states and actions, learned occupancies, nonstationarity, approximate local kernels, data-dependent representations, nonconvex training, and long-horizon exposure remain outside the present closure. What the four laboratories jointly show. The MNIST audit records a positive sample-average amortization gap for a selected finite library and shows that additional imagewise computation reduces it. Because saturation, independent Monte Carlo evaluation, matched generic baselines, and negative controls are absent, it does not isolate a structural mechanism or complete the proposed OALI chain. The posterior-family interface tests carrier reuse under approximate and exact base change; its finite-state and conjugate pilots establish the declared mechanisms, but not application-scale speed or superiority. Cytometry couples carrier survival to graph compatibility; its smoke test verifies the software path

Part VII Historical and Conceptual Notes

205

while exposing a rare-branch weakness that independent scientific validation must address. The sequential interface makes coarse-graining dynamic and couples it to shared coordination. Together they show how the Chapter 4 mechanisms compose without claiming that one architecture or scalar score is universally best.

Exercises Exercise 24.1. For a contracting Gibbs kernel with a one-step perturbation bounded by 𝜖, derive the finite-time recursion and its stationary 𝜖/(1 − 𝛾) limit. State exactly where existence of an invariant law is used. Exercise 24.2. Design an ESS-gated particle bank for a one-parameter posterior family. Report fresh draws, density-ratio evaluations, memory, refreshes, integrated MSE, and a discontinuity control; explain why fresh draws alone do not prove a runtime gain. Exercise 24.3. Explain why a cell-level train/test split can overstate generalization across biological samples, and give the corresponding sample-level split. Exercise 24.4. Propose a negative control that distinguishes a biologically meaningful atlas from a graph-partition artifact. Exercise 24.5. Construct two histories with identical one-step action distributions but different optimal two-step decisions. Identify the recursive refinement witness. Exercise 24.6. Propose a task-relative oracle signature for a latent world model shared by several reward functions, and state which changed contexts it must preserve.

Part VII Historical and Conceptual Notes OALI is a typed workflow, not a new generic architecture family Mixtures of experts, routing, recurrent memories, active acquisition, matching, and held-out comparison all have independent literatures. OALI does not claim ownership of those components. Its methodological position is that a repair should follow a typed obstruction witness and should be judged on an independent population endpoint after the mechanism-specific change. Neighbor

Shared operation

OALI distinction

Architecture search or compare or route among candi- first identify whether the failed branch is deploymixture of experts date models ment, evidence, recursive state, or contextual exposure; expand that branch only Active learning and ex- select the next informative obser- resolve a declared population certificate with periment design vation worldwise authorization; posterior utility may guide acquisition but does not authorize stopping Graph synchronization align local labels across edges preserve partial domains, missing branches, and and assignment [Singer, common fixed-section requirements rather than 2011, Pachauri et al., force a total permutation 2013] Held-out model compar- concentration on an independent predeclare the experimental unit and baselines, ison test sample and combine gain with the separate structural certificate

206

CHAPTER 24. FOUR APPLIED LABORATORIES

Ownership of the formal results Typed no-compensation and typed realization are book/program interface syntheses. The finite-library theorem is Hoeffding plus a union bound, and the population-improvement theorem is certificate subtraction. Noisy partial-transport recovery is finite-class argmin stability. The cycle-space theorem specializes classical holonomy and graph consistency to partial maps. Their distinctive value lies in the typed end-to-end contract and the missing-versus-moved branch semantics, not in rebranding their classical engines.

From worked systems to stress tests and limits The applications show how structural realizability depends on the local oracle, observation unit, representation, and deployment contract. They also expose where the theory still depends on finite-state structure, stable transports, or independent validation. Part VIII subjects the framework to a noncommutative stress test, records its claim boundaries, and closes with a falsifiable research program.

Part VIII

Stress Tests, Boundaries, and Research Program A structural theory must survive foreign geometries, expose its boundaries, and make its claims falsifiable. Its final question is what knowledge a learning system can coherently realize.

207

Chapter 25

Quantum Elimination Geometry and Noncommutative Rigidity Classical elimination relies heavily on disintegration: a joint law splits into a marginal law and conditional kernels, and relative entropy obeys an exact chain rule. In a quantum system, partial trace plays the role of marginalization, but no universal classical conditional state has the same properties. The quantum theory therefore tests which parts of elimination geometry are genuinely structural and which depend on commutativity. Strong-coupling thermodynamics and Hamiltonians of mean force are established subjects [Talkner and Hänggi, 2020, Cresser and Anders, 2021, Trushechkin et al., 2022]. This chapter does not claim the mean-force construction itself. It asks the narrower intervention question: when can one field-independent coarse Hamiltonian reproduce the reduced Gibbs operator for every declared external field?

25.1

Universal Gibbs base change

Let 𝑅 = 𝑅★ act on a finite-dimensional bipartite Hilbert space H 𝐴 ⊗ H𝐵 . Fix 𝛽 > 0. Ask whether there exist 𝐾 = 𝐾 ★ on 𝐴 and 𝑐 > 0, independent of every external Hermitian field 𝐹, such that Tr 𝐵 𝑒 −𝛽 (𝐹 ⊗𝐼𝐵 +𝑅) = 𝑐𝑒 −𝛽 (𝐹+𝐾 )

for all 𝐹.

Completely positive dilations and the sufficiency/recovery theory for quantum channels provide the structural background [Stinespring, 1955, Petz, 1988, Mosonyi and Petz, 2004, Jenčová and Petz, 2006]. Equality structures for quantum entropy inequalities supply a closely related rigidity tradition [Hayden et al., 2004]. The result below is a finite-dimensional Hamiltonian-language specialization of the established factorization and exponential-family sufficiency theory, with a self-contained proof included here; it is not claimed as a new quantum-information theorem. How to read the assumptions The system and environment are finite dimensional, 𝛽 > 0, all Gibbs states are faithful, and one fixed system Hamiltonian correction 𝐾 and scalar 𝑐 must work for every Hermitian system field 𝐹. The quantifier includes noncommuting fields; restricting it to a commuting sector leads to a weaker controlled-interaction theorem. 208

25.1. UNIVERSAL GIBBS BASE CHANGE

209

Theorem 25.1 (Universal Gibbs base-change rigidity). The identity holds for every Hermitian 𝐹 if and only if 𝑅 = 𝐾 ⊗ 𝐼 𝐵 + 𝐼 𝐴 ⊗ 𝐿 + 𝜆𝐼 𝐴𝐵 for some Hermitian 𝐿 and scalar 𝜆. Interpretation and boundary Universal exact Gibbs elimination is rigid: the bipartite Hamiltonian must contain no genuine interaction, apart from system, environment, and scalar terms. One field or one commuting family is not enough to force this conclusion. The scalar partition-function shadow for all fields already contains the same rigidity. Proof roadmap Pass from operator base change to scalar pressure equality, take convex conjugates to obtain universal constrained relative-entropy saturation, use one Petz recovery channel for every marginal to force a product Gibbs reference, and take the Hermitian logarithm to recover an additive Hamiltonian. The chapter appendix proves each intermediate lemma and the full equivalence. Jenčová and Petz characterize sufficiency for quantum exponential families and the factorization of sufficient coarse-grainings [Jenčová and Petz, 2006, Theorems 5 and 7]. Taking the coarse observable map to be 𝑋 ↦→ 𝑋 ⊗ 𝐼 𝐵 and allowing a basis of all Hermitian system fields gives the product-state core of the theorem. The proof below records the same implication directly through pressure duality and the Petz map; its scalar-shadow formulation is a convenient corollary-level packaging. Proof architecture. Thus universal exact base change over the full noncommutative system algebra excludes genuine interaction. The proof proceeds through several rigidities. 1. Equality of partition functions for all 𝐹 identifies the constrained Gibbs–Fenchel dual. 2. Equality in Umegaki data processing yields one recovery channel for the entire system-state family. 3. A recovery extension that leaves every input state unchanged has product form. 4. The Hamiltonian must therefore separate into system, environment, and scalar terms. The scalar partition-function shadow already suffices: it remembers the interaction when tested against all noncommuting fields.

CHAPTER 25. QUANTUM ELIMINATION GEOMETRY

210

25.2

Commuting sectors

The universal theorem used all Hermitian interventions and therefore forced product structure. Restricting interventions to a commuting algebra changes the answer; the source box marks that shift and the provenance of the sector classification. Source result; proof not reproduced here The commuting-sector classification, replacement and recoverability identities, and losslessclassicalization variants summarized in the remainder of the chapter are established in the Quantum Elimination Geometry manuscript. The chapter appendix gives a complete proof only of the universal noncommutative rigidity theorem above [Huang and Wang, 2026k]. Under this restricted contract, if admissible external fields lie in a commuting sector algebra, genuine controlled interaction can be compatible with exact base change. The necessary and sufficient condition is block diagonal structure over sectors, with each block’s partial Gibbs operator scalar on the corresponding system sector. For a maximal abelian algebra, this is a controlled Hamiltonian. The contrast is sharp: • full noncommutative field variation forces product structure; • classical sector variation permits controlled interaction.

25.3

Replacement, measurement, and operational meaning

The replacement identity.

For a general interaction, a trial-state Umegaki defect decomposes into:

1. a coarse Gibbs defect; 2. a nonnegative data-processing gap; 3. a signed Hamiltonian-of-mean-force anomaly. The anomaly records external-field dependence of the effective coarse Hamiltonian. Universal recovery maps convert the data-processing gap into a quantitative recoverability certificate. In the rigid product case, the anomaly vanishes and the gap becomes an exact conditional relative entropy. Measurement and lossless classicalization.

A fixed measurement 𝑀 satisfies data processing,

𝐷 (𝜌∥𝜎) ≥ KL(𝑀 𝜌∥ 𝑀𝜎). Entropy monotonicity under completely positive maps and its equality/recovery theory provide the classical background [Lindblad, 1975, Petz, 1986, 1988, Mosonyi and Petz, 2004]. The measurement preserves the Umegaki defects of an entire faithful family exactly if and only if the family is jointly recoverable. One recovery channel followed by copying the classical outcome would broadcast the family. By the no-broadcasting theorem [Barnum et al., 1996], pairwise commutativity is necessary; it is also sufficient through a common eigenbasis measurement.

25.3. REPLACEMENT, MEASUREMENT, AND OPERATIONAL MEANING

211

Boundary of the claim Measured KL is generally a lower certificate, not an exact replacement for quantum relative entropy. Classicalization is lossless for a faithful family only under a commutativity condition.

Why this is relevant to AI. The quantum results provide a general warning. A coarse or measured representation can preserve one scalar objective while failing to preserve an entire family of defects under intervention. Exact base change under all contexts is a strong rigidity statement. The same logic appears classically in world models and representation learning: preserving predictions at one task is weaker than preserving all downstream conditional decompositions.

Exercises Exercise 25.2. Verify the universal base-change identity in the product Hamiltonian case. Exercise 25.3. Explain why one recovery channel for all states in a family, combined with classical copying, produces a broadcasting channel. Exercise 25.4. Construct a controlled two-qubit Hamiltonian that satisfies exact base change for diagonal system fields but not for arbitrary noncommuting fields.

CHAPTER 25. QUANTUM ELIMINATION GEOMETRY

212

Appendix to Chapter 25: complete proofs Preparatory Gibbs–Fenchel duality Dependencies. All Hilbert spaces are finite dimensional and all Gibbs references are faithful. These hypotheses ensure that Umegaki relative entropy is finite on every trial state, constrained minima are attained, and ordinary finite-dimensional Fenchel–Moreau duality applies. Let 𝜌 be a faithful state on H 𝐴 ⊗ H𝐵 . Define 𝑃𝜌 (𝑋) = log Tr 𝐴𝐵 exp(log 𝜌 + 𝑋 ⊗ 𝐼 𝐵 ) and 𝐼𝜌 (𝛼) =

min

𝐷 (𝜔∥ 𝜌).

𝜔:Tr 𝐵 𝜔=𝛼

For a faithful state 𝜎 on 𝐴, put 𝑃 𝜎 (𝑋) = log Tr 𝐴 exp(log 𝜎 + 𝑋). Lemma 25.5 (Gibbs variational identity). For a faithful state 𝜌 and Hermitian 𝑌 on the same space, log Tr 𝑒 log 𝜌+𝑌 = max{Tr(𝜔𝑌 ) − 𝐷 (𝜔∥ 𝜌)}. 𝜔

The maximizer is unique and equals 𝜔𝑌 = 𝑒 log 𝜌+𝑌 /Tr 𝑒 log 𝜌+𝑌 . Proof. Let 𝑍 = Tr 𝑒 log 𝜌+𝑌 . Since log 𝜔𝑌 = log 𝜌 + 𝑌 − log 𝑍, 𝐷 (𝜔∥𝜔𝑌 ) = 𝐷 (𝜔∥ 𝜌) − Tr(𝜔𝑌 ) + log 𝑍. The nonnegativity of Umegaki relative entropy, with equality only at 𝜔 = 𝜔𝑌 , gives the claimed variational formula and uniqueness. □ Proposition 25.6 (Constrained pressure duality). For every faithful bipartite state 𝜌, 𝑃𝜌 (𝑋) = max{Tr(𝛼𝑋) − 𝐼𝜌 (𝛼)}, 𝛼

𝐼𝜌 (𝛼) = sup{Tr(𝛼𝑋) − 𝑃𝜌 (𝑋)}. 𝑋

For a faithful system state 𝜎, 𝐷 (𝛼∥𝜎) = sup{Tr(𝛼𝑋) − 𝑃 𝜎 (𝑋)}. 𝑋

Proof. Apply the Gibbs variational identity with 𝑌 = 𝑋 ⊗ 𝐼 𝐵 and group trial states by their 𝐴-marginal: 𝑃𝜌 (𝑋) = max{Tr(𝜔 𝐴 𝑋) − 𝐷 (𝜔∥ 𝜌)} 𝜔

= max{Tr(𝛼𝑋) − 𝐼𝜌 (𝛼)}. 𝛼

The constrained functional 𝐼𝜌 is proper, convex, and lower semicontinuous on the finite-dimensional state space, so Fenchel–Moreau duality gives the second formula. Applying the same Gibbs variational identity directly on 𝐴 gives the formula for 𝐷 (𝛼∥𝜎). □

Appendix to Chapter 25: complete proofs

213

Universal saturation of the constrained extension problem Dependencies. The converse uses equality in Umegaki data processing for the fixed channel Tr 𝐵 and the fixed faithful reference 𝜌. The associated Petz recovery map is therefore the same for every marginal 𝛼. Lemma 25.7 (A universal right inverse of partial trace is product). Let E : L (H 𝐴) → L (H 𝐴 ⊗ H𝐵 ) be a quantum channel satisfying Tr 𝐵 E (𝑋) = 𝑋

for every 𝑋.

Then there exists a fixed state 𝜏𝐵 such that E (𝑋) = 𝑋 ⊗ 𝜏𝐵

for every 𝑋.

Proof. Use the Choi representation. Let 𝐽 E ≥ 0 be the Choi matrix of E on 𝐴out ⊗ 𝐵 ⊗ 𝐴in . The condition Tr 𝐵 ◦E = id 𝐴 is equivalent to Tr 𝐵 𝐽 E = 𝐽id = |Ω⟩⟨Ω|, where |Ω⟩ is a maximally entangled vector between 𝐴out and 𝐴in . The partial trace of the positive operator 𝐽 E has rank one. Hence the support of 𝐽 E is contained in span{|Ω⟩} ⊗ H𝐵 . Indeed, if ΠΩ is the support projection of |Ω⟩⟨Ω| and 𝑃 = 𝐼 − ΠΩ , then Tr{(𝑃 ⊗ 𝐼 𝐵 )𝐽 E } = Tr{𝑃 Tr 𝐵 𝐽 E } = 0. Positivity implies (𝑃 ⊗ 𝐼 𝐵 )𝐽 E (𝑃 ⊗ 𝐼 𝐵 ) = 0 and also eliminates the off-diagonal support blocks. Therefore 𝐽 E = |Ω⟩⟨Ω| ⊗ 𝜏𝐵 for some positive 𝜏𝐵 . Trace preservation of E normalizes 𝜏𝐵 to unit trace. The Choi inversion formula now gives E (𝑋) = 𝑋 ⊗ 𝜏𝐵 . □ Proposition 25.8 (Universal constrained saturation). For faithful 𝜌 with 𝜌 𝐴 = Tr 𝐵 𝜌, the following are equivalent: 1. 𝐼𝜌 (𝛼) = 𝐷 (𝛼∥ 𝜌 𝐴) for every state 𝛼 on 𝐴; 2. 𝜌 = 𝜌 𝐴 ⊗ 𝜏𝐵 for some state 𝜏𝐵 . Proof. If 𝜌 = 𝜌 𝐴 ⊗ 𝜏𝐵 , choose the feasible extension 𝜔 = 𝛼 ⊗ 𝜏𝐵 . Additivity of relative entropy gives 𝐼𝜌 (𝛼) ≤ 𝐷 (𝛼 ⊗ 𝜏𝐵 ∥ 𝜌 𝐴 ⊗ 𝜏𝐵 ) = 𝐷 (𝛼∥ 𝜌 𝐴). Data processing under partial trace gives the reverse inequality, so equality holds. Conversely, for every 𝛼 choose a constrained minimizer 𝜔 𝛼 . The assumed equality says 𝐷 (𝜔 𝛼 ∥ 𝜌) = 𝐷 (𝛼∥ 𝜌 𝐴). Equality in data processing for partial trace and the fixed faithful reference 𝜌 implies exact recovery by the Petz map R 𝜌 : 𝜔 𝛼 = R 𝜌 (𝛼). Since Tr 𝐵 𝜔 𝛼 = 𝛼, the same channel satisfies Tr 𝐵 R 𝜌 (𝛼) = 𝛼 for every state and hence, by linearity, for every operator. The preceding lemma gives R 𝜌 (𝑋) = 𝑋 ⊗ 𝜏𝐵 . Evaluating at 𝑋 = 𝜌 𝐴 and using R 𝜌 (𝜌 𝐴) = 𝜌 yields 𝜌 = 𝜌 𝐴 ⊗ 𝜏𝐵 . □

CHAPTER 25. QUANTUM ELIMINATION GEOMETRY

214

Proof of universal Gibbs base-change rigidity Dependencies. Fix 𝛽 > 0, a bipartite Hermitian 𝑅, and a system Hermitian 𝐾. The quantifier is over every Hermitian system field 𝐹, not merely a commuting subalgebra. This universal noncommutative quantifier is the source of rigidity. Boundary used in the proof. The result does not assert that an interacting Hamiltonian can never have a simple reduced Gibbs state at one field. It asserts that one field-independent correction 𝐾 cannot work after every Hermitian intervention unless the Hamiltonian is additive. Proof. Define 𝑍 𝑅 (𝐹) = Tr 𝐴𝐵 𝑒 −𝛽 (𝐹 ⊗𝐼𝐵 +𝑅) , with normalized references 𝜌=

𝑒 −𝛽𝑅 , 𝑍 𝑅 (0)

𝑍 𝐾 (𝐹) = Tr 𝐴 𝑒 −𝛽 (𝐹+𝐾 ) ,

𝜎=

𝑒 −𝛽𝐾 . 𝑍 𝐾 (0)

We prove the equivalence through six formulations. Operator base change implies normalized and scalar base change.

Suppose

Tr 𝐵 𝑒 −𝛽 (𝐹 ⊗𝐼𝐵 +𝑅) = 𝑐𝑒 −𝛽 (𝐹+𝐾 ) for all 𝐹. Taking the full trace gives 𝑍 𝑅 (𝐹) = 𝑐𝑍 𝐾 (𝐹). Dividing the operator identity by that scalar identity gives equality of the normalized reduced Gibbs state and the 𝐾-shifted system Gibbs state. Normalized base change implies scalar base change. Assume only equality of the normalized reduced states. Put 𝜙(𝐹) = log 𝑍 𝑅 (𝐹) − log 𝑍 𝐾 (𝐹). For a Hermitian direction 𝐺, differentiation of the trace exponential and cyclicity of trace give 𝑑 log 𝑍 𝑅 (𝐹) [𝐺] = −𝛽 Tr 𝐴 (𝛼𝐹 𝐺),

𝑑 log 𝑍 𝐾 (𝐹) [𝐺] = −𝛽 Tr 𝐴 (𝜎𝐹 𝐺),

where 𝛼𝐹 is the reduced bipartite Gibbs state and 𝜎𝐹 is the system Gibbs state. The normalized identity gives 𝛼𝐹 = 𝜎𝐹 , hence 𝑑𝜙(𝐹) [𝐺] = 0 for all 𝐹, 𝐺. The real vector space of Hermitian matrices is connected, so 𝜙 is constant. Thus 𝑍 𝑅 (𝐹) = 𝑐𝑍 𝐾 (𝐹) with 𝑐 = 𝑒 𝜙 (0) . Scalar base change implies equality of constrained duals. For Hermitian 𝑋, set 𝐹 = −𝑋/𝛽. The scalar identity and its value at zero imply 𝑃𝜌 (𝑋) = log

𝑍 𝑅 (−𝑋/𝛽) 𝑍 𝐾 (−𝑋/𝛽) = log = 𝑃 𝜎 (𝑋). 𝑍 𝑅 (0) 𝑍 𝐾 (0)

Taking convex conjugates in the constrained pressure-duality proposition gives 𝐼𝜌 (𝛼) = 𝐷 (𝛼∥𝜎)

for every 𝛼.

Appendix to Chapter 25: complete proofs

215

Equality of constrained duals forces a product reference. The state 𝜌 is feasible for the constraint 𝛼 = 𝜌 𝐴, so 𝐼𝜌 (𝜌 𝐴) = 0. The dual identity yields 𝐷 (𝜌 𝐴 ∥𝜎) = 0, hence 𝜌 𝐴 = 𝜎. We therefore have 𝐼𝜌 (𝛼) = 𝐷 (𝛼∥ 𝜌 𝐴)

for every 𝛼.

Universal constrained saturation gives a faithful state 𝜏𝐵 such that 𝜌 = 𝜎 ⊗ 𝜏𝐵 . A product reference forces an additive Hamiltonian. 𝜏𝐵 = with 𝐿 = −𝛽 −1 log 𝜏𝐵 . From

we obtain

Write

𝑒 −𝛽𝐿 Tr 𝑒 −𝛽𝐿

𝑒 −𝛽𝑅 𝑒 −𝛽𝐾 𝑒 −𝛽𝐿 = ⊗ , 𝑍 𝑅 (0) 𝑍 𝐾 (0) Tr 𝑒 −𝛽𝐿 𝑒 −𝛽𝑅 = 𝑞 𝑒 −𝛽 (𝐾 ⊗𝐼𝐵 +𝐼 𝐴 ⊗𝐿)

for a positive scalar 𝑞. Both sides are positive definite. Uniqueness of the Hermitian logarithm gives 𝑅 = 𝐾 ⊗ 𝐼 𝐵 + 𝐼 𝐴 ⊗ 𝐿 + 𝜆𝐼 𝐴𝐵 for a real scalar 𝜆. Additivity implies operator base change. Conversely, if the last display holds, the two tensor summands commute and   Tr 𝐵 𝑒 −𝛽 (𝐹 ⊗𝐼𝐵 +𝑅) = 𝑒 −𝛽𝜆 Tr 𝐵 𝑒 −𝛽 (𝐹+𝐾 ) ⊗ 𝑒 −𝛽𝐿 = 𝑒 −𝛽𝜆 Tr(𝑒 −𝛽𝐿 )𝑒 −𝛽 (𝐹+𝐾 ) . This is universal operator base change with 𝑐 = 𝑒 −𝛽𝜆 Tr(𝑒 −𝛽𝐿 ). All formulations, and hence the stated equivalence, follow. □

Chapter 26

Boundaries of Structural Explanation A structural theory becomes uninformative if every failure can be labeled an obstruction after the fact. The claims in this book therefore depend on five declared contracts. The local objective must generate the defect; the representation must be intrinsic to the target; the deployment class must match the system actually used; the relevant risk level must be stated; and finite data must support the structural conclusion being reported.

26.1

Failures that invalidate the structural object

The first failure occurs before architecture analysis. A reported discrepancy may not arise from a certified elimination. Locally assembled increments can be nonintegrable, oriented incorrectly, or measured in an arbitrary parameter metric. In that case the architecture analysis must stop until the defect system has been derived globally or repaired by an integrability theorem. A second failure is an artificial representation. Unrestricted exact lifts permit dummy and targetcalling coordinates, so geometric complexity can be created by representation rather than by the declared target. Slack anchoring, quotient reduction, and an admissible lift language are therefore prerequisites for intrinsic complexity claims. A third failure is architecture misspecification. A positive obstruction is conditional on the deployment contract. A lower bound for continuous point-valued outputs is irrelevant if the deployed system is allowed external anchors, discontinuities, additional memory, or variable-cardinality outputs. Finally, a locally optimal variational family need not be a scientifically adequate model. A zero local defect only establishes optimality inside the declared local family. Model misspecification remains a separate source of population risk.

216

26.2. INTERFACES THAT REQUIRE ADDITIONAL THEOREMS

26.2

217

Interfaces that require additional theorems

Several stronger conclusions do not follow automatically from a valid architecture obstruction. 1. A positive worst-case topological cost need not imply a positive average population risk; mass and regularity assumptions are required. 2. A native defect need not be visible to the downstream task; an exposure or risk-transfer theorem is required. 3. A deterministic population obstruction need not be identifiable from finite data; a valid statistical certificate is required. 4. Lift or extension size does not imply runtime complexity without an explicit computation model, encoding, conditioning, and precision analysis. 5. A zero-temperature limit need not become deterministic if the closure of the admissible class still contains randomized laws. 6. Classical conditional identities do not automatically pass through noncommutative marginalization; quantum base change requires additional product or commuting structure. These are not technical footnotes. They define the levels at which claims in the book are valid.

26.3

Alternative explanations and falsification

Poor training can imitate structural saturation. Additional capacity can imitate a mechanism-matched repair. An unstable branch match can imitate monodromy. A rare topological seam can be irrelevant to average risk. Consequently, a proposed structural explanation should be weakened or rejected when any of the following persists under appropriate controls: 1. the structural lower certificate is zero or statistically unresolved; 2. stronger optimization removes the gap without changing the deployment contract; 3. parameter- and compute-matched generic enlargements reproduce the claimed repair gain; 4. the diagnosed structure is unstable under resampling or reasonable changes in the audit graph; 5. the native defect is operationally invisible to the predeclared downstream task; 6. the independent scientific endpoint does not improve; 7. the claimed obstruction disappears after replacing an arbitrary label or lift by an intrinsic representation. Boundary of the claim Performance saturation is a diagnostic signal, not a proof of architecture obstruction. The structural claim requires an independent native-risk certificate and a repair whose effect follows the diagnosed mechanism.

CHAPTER 26. BOUNDARIES OF STRUCTURAL EXPLANATION

218

26.4

Conditions that must remain visible

The current theorem system depends on several assumptions that should not be hidden by streamlined exposition: compact metric structure where balls and Lipschitz maps are used; explicit form-domain conditions in continuous-spectrum arguments; well-defined certified convex sets on general cochain complexes; exact second-best structural matching margins; biological-unit rather than cell-level independence in confirmatory applications; and full-rank assumptions for finite-temperature quantum identities. These conditions define where the present theory stops. The next chapter treats extensions beyond them as research problems rather than completed results.

Exercises Exercise 26.1. For three results in the book, identify the smallest change in assumptions that would make the stated conclusion invalid or uninterpretable. Exercise 26.2. Explain why a positive uniform radical obstruction does not imply a positive average risk without additional mass or regularity conditions. Exercise 26.3. Design an ablation that separates atlas structure from the benefit of extra capacity.

Chapter 27

A Falsifiable Research Program for Structural Learning The long-term value of elimination geometry does not depend on adoption of its full vocabulary. It depends on reusable results for a recurring problem: local optima are available, but a shared deployment contract may not realize them simultaneously. A mature theory should quantify the resulting floor, determine whether finite data can certify it, and predict a minimal intervention whose benefit survives independent validation. No theorem in the present book achieves that program in complete generality. The proved results close specific interfaces in convex, graph, conditional, singular, finite-sample, and quantum settings. The remaining questions are best organized by the points at which the current certification chain can still fail.

27.1

Data-dependent and computation-dependent deployment

The statistical learning literature already provides tools for data-dependent classes, including PAC– Bayesian bounds for random hypothesis sets [Dupuis et al., 2024]. Lifelong and multitask representation learning also develop complexity measures for shared representations [Wang et al., 2026]. The open question here is narrower: how should those tools interact with a class whose structure was itself selected because a structural certificate diagnosed a specific failure? A repaired deployment class is often generated from data: b b ↦−→ 𝔄. 𝑆cal ↦−→ C The relevant complexity is then not that of a fixed class but of a random atlas, quotient, memory refinement, or routing structure. A useful theory should control chart number, overlap, transport uncertainty, gate stability, and selection effects without discarding the structural information that generated the class. Computation creates a related ambiguity. For a family 𝔄 𝑚,𝑇 ,Alg , additional computation can have three distinct effects: 1. it can reduce implementation error inside the same effective class; 2. it can enlarge the class of functions realized at test time; 219

220

CHAPTER 27. STRUCTURAL LEARNING RESEARCH PROGRAM

3. it can restrict the reachable subset through optimizer dynamics or implicit bias. A computation-dependent architecture theory should separate these effects rather than treating test-time iteration as either pure optimization or pure capacity by convention.

27.2

Population effects of singularity and composition

Singular normal forms often give exact worst-case costs, while average-risk lower bounds require additional control of how failure can concentrate near a seam. A central open problem is a phase diagram relating regularity of the deployment class, lower-mass assumptions on the population, and the smallest achievable average defect. Coordination theory faces a complementary extension problem. The current exact results are strongest for finite conditional structures. Extending rectangularity, memory obstruction, and coordination bounds to continuous states, learned occupancies, world models, and function approximation requires joint control of the deployment restriction and the state distribution induced by the learned system. Both directions ultimately require task-relative semantics. A representation should retain distinctions that are necessary for the local oracle and visible to the downstream task, while quotienting distinctions that are irrelevant to both. The goal is not maximal reconstruction fidelity but minimal structural information sufficient for realizability and decision.

27.3

Scientific AI and foundation-model contracts

Scientific applications should be chosen where forced global reuse has a substantive consequence. The structural object must change a scientific or decision endpoint; an atlas, quotient, or certificate is not a contribution by itself. Candidate domains include multi-site environmental distributions, multi-batch cell populations, sensor systems with branch-specific drift, and shared latent-state models across institutions or operating regimes. At foundation-model scale, the same questions arise through routing, retrieval, memory, shared representations, and structured outputs. The hypotheses should remain concrete. A shared representation may force incompatible local solutions. A memory bottleneck may merge histories that require different actions. A routing contract may create coordination obstruction. A quotient or set-valued output may remove an artificial labeling problem. Each claim must be tied to a native objective and a falsifiable structural prediction.

27.4. SUCCESS CRITERIA AND STOPPING RULES

27.4

221

Success criteria and stopping rules

The program would become a useful branch of learning theory if it yields tools that are valuable independently of the surrounding terminology. Examples would include: 1. computable architecture-obstruction certificates used before model scaling; 2. sharp saturation theorems under explicit representation or memory contracts; 3. automatic atlas, quotient, or memory repairs with reproducible held-out gains against matched controls; 4. statistically valid unresolved certificates that guide additional data collection; 5. resource–risk frontiers that determine when more representation, memory, or communication is worth its cost. The failure criteria are equally important. The program should be narrowed if most certificates are vacuous or uncomputable. It should also be narrowed when the examples reduce to known approximation or topology without new objective-level consequences; when generic mixtures or extra parameters reproduce OALI gains; when realistic samples leave the certificates unresolved; or when the proposed structural distinctions do not change independent scientific decisions. Partial domains supply one concrete stopping-rule example. For finite branch sets, a partial bijection can be completed to a total permutation on an enlarged fiber, with masks excluding dummy deployment states. The global section and repair problems then become masked synchronization or finite constraint problems. In the minimal noisy cycle experiment, deciding whether one anomalous domain forces a seam costs order log |𝐸 | observations per edge, but the matching lower bound is exactly sparse anomalous-coordinate detection. Therefore partiality, domain survival, and a chart decision do not alone constitute a new learning-theoretic mechanism. Operational exposure supplies a second stopping-rule example rather than an escape. In the minimal memory-conflict experiment, every policy executable through the baseline memory is exactly blind to two worlds with opposite resource-adjusted architecture choices. Forbidding a diagnostic split makes worldwise certification impossible. Allowing a temporary split turns it into a controlled sensing action, after which the lower bound is the standard max–min KL resolution bound of Chapter 20; a priced split gives the same information calculation per unit cost. Thus, for finite resettable diagnostics, operational exposure changes the experiment menu but does not create a new learning-theoretic rate. This closes the partial-domain and operational-memory flagship routes. Reopening either requires an explicit hard family with a minimax term that cannot be absorbed into controlled sensing, partial observability, automata discrimination, or ordinary representation selection, not merely a larger memory example. Chapter 19 opens a different, narrower route. For nonnegative sparse inference it turns strict active-set Jacobian conflict into a computable one-pass native-loss floor and pairs that lower certificate with a sufficient proximal repair depth. This result can change an architecture choice at a declared uniform tolerance, but it is not yet a settled flagship: its ingredients have close antecedents, including the global sparse- autoencoder amortisation gap of O’Neill et al. [2025], independent priority is unresolved, and the current diabetes alignment was computed after confirmation. The next stopping rule is therefore exact: obtain an independent proof/priority review and freeze the conflict-pair rule, uniform tolerance, repair depth, and compute ledger before opening new scientific confirmation units.

CHAPTER 27. STRUCTURAL LEARNING RESEARCH PROGRAM

222

These criteria suggest a practical research sequence: local oracle → native loss → deployment contract → structural certificate → finite-sample certification → minimal repair → independent validation The sequence should stop when the native defect has no demonstrated relation to the downstream task, when the diagnosed obstruction occupies negligible population mass, when matched generic baselines explain the gain, or when the certificate is too unstable to support a structural decision. The resulting mission is deliberately narrower than a universal theory of learning: When local optima are available, determine whether a shared deployment contract can realize them, quantify the native risk forced when it cannot, certify that conclusion from finite data, and identify the smallest structural change whose benefit survives independent validation.

Research exercises Exercise 27.1. Formulate a computation-dependent architecture obstruction for a fixed-point network with 𝑇 test-time iterations, separating class enlargement from within-class optimization. Exercise 27.2. Propose an average-risk theorem for a singular oracle family under a Lipschitz deployment constraint and a lower-mass population condition. Exercise 27.3. Design a nonresettable certificate-aware experiment for deciding whether a shared memory state should be split. State which diagnostic interventions are reversible, include a negative control and a stopping rule, and identify the proposed lower-bound term that is not already a controlled-sensing cost.

Part VIII Historical and Conceptual Notes Quantum antecedents and the rigidity specialization Umegaki relative entropy, data processing, Petz equality/recovery, quantum sufficiency, and Hamiltonians of mean force are mature theories [Umegaki, 1962, Lindblad, 1975, Petz, 1986]. Part VIII uses them in their established roles. It does not claim a new data-processing equality theorem, a new recovery map, or a new general theory of open-system equilibrium. The displayed Hamiltonian statement is narrower: if one field-independent unnormalized Gibbs base-change law must hold for every Hermitian external field, then the bipartite Hamiltonian is additive and contains no genuine interaction. A result-level priority audit found that this content is already implied by the factorization of sufficient quantum coarse-grainings and the explicit sufficiency criterion for quantum exponential families [Jenčová and Petz, 2006, Theorems 5 and 7]. The exact Hamiltonian wording is useful, but it is a direct finite-dimensional specialization rather than a defensible standalone originality claim.

Part VIII Historical and Conceptual Notes

Statement

Status

223 Boundary

Universal Gibbs change rigidity

base- direct sufficiency/factorization spe- finite dimensions, faithful Gibbs states, cialization with a complete book one fixed correction, and all noncomproof muting external fields Commuting-sector classifi- source result summarized, proof allows controlled interactions because cation left in the companion paper the intervention algebra is smaller Recoverability and mea- classical quantum-information core measurement generally yields a lower surement limits with source specializations certificate, not an exact classical defect ledger The final boundary chapters are therefore part of the contribution audit: they distinguish a transparent specialization proved under a sharp noncommutative quantifier from analogies, imported sector results, and open extensions.

Appendices: Technical Background, Terminology, and Crosswalks

224

Appendix A

Convex Analysis and Bregman Geometry This appendix collects the convex-analytic facts used throughout the book. It is not intended to replace a full text on convex analysis [Rockafellar, 1970].

A.1

Convex conjugacy

For a proper lower-semicontinuous convex function Φ on a finite-dimensional vector space, Φ★ (𝑢) = sup{⟨𝑢, 𝑎⟩ − Φ(𝑎)}. 𝑎

Fenchel–Young gives Φ(𝑎) + Φ★ (𝑢) ≥ ⟨𝑢, 𝑎⟩, with equality exactly when 𝑢 ∈ 𝜕Φ(𝑎). If Φ is Legendre, ∇Φ and ∇Φ★ are inverse maps on the interiors of their domains.

A.2

Bregman divergence

For differentiable Φ, 𝐷 Φ (𝑎∥𝑏) = Φ(𝑎) − Φ(𝑏) − ⟨∇Φ(𝑏), 𝑎 − 𝑏⟩. It is nonnegative but generally asymmetric and does not satisfy the triangle inequality. The three-point identity is 𝐷 Φ (𝑎∥𝑐) = 𝐷 Φ (𝑎∥𝑏) + 𝐷 Φ (𝑏∥𝑐) + ⟨𝑎 − 𝑏, ∇Φ(𝑏) − ∇Φ(𝑐)⟩. This identity underlies Bregman Pythagorean identities and centroid formulas.

225

APPENDIX A. CONVEX ANALYSIS AND BREGMAN GEOMETRY

226

A.3

Conditional Bregman projection

Let 𝐴★ be an integrable random oracle and 𝑍 a carrier. Under regularity,
arg min E𝐷 Φ {𝑔(𝑍) ∥ 𝐴★ } = ∇Φ★ E[∇Φ( 𝐴★) | 𝑍] , 𝑔 (𝑍 )

where Φ★ denotes the convex conjugate and the right-hand side is interpreted through ∇Φ★. This is the canonical carrier decoder used in Chapter 16. The reverse orientation has a different centroid. When the conditional primal mean lies in the admissible domain, arg min E𝐷 Φ { 𝐴★ ∥𝑔(𝑍)} = E( 𝐴★ | 𝑍). 𝑔 (𝑍 )

A.4

Strong convexity and smoothness

If Φ is 𝜇-strongly convex, 𝐷 Φ (𝑎∥𝑏) ≥

𝜇 ∥𝑎 − 𝑏∥ 2 . 2

If ∇Φ is 𝐿-Lipschitz, 𝐿 ∥𝑎 − 𝑏∥ 2 . 2 These inequalities exchange native defects and geometric distances. 𝐷 Φ (𝑎∥𝑏) ≤

A.5

Metric projection

For a nonempty closed convex set 𝐶 in a Hilbert space, the metric projection 𝑃𝐶 is firmly nonexpansive: ∥𝑃𝐶 𝑢 − 𝑃𝐶 𝑣∥ 2 ≤ ⟨𝑃𝐶 𝑢 − 𝑃𝐶 𝑣, 𝑢 − 𝑣⟩. In particular, ∥𝑃𝐶 𝑢 − 𝑣∥ ≤ ∥𝑢 − 𝑣∥ for 𝑣 ∈ 𝐶.

A.6

Primal–dual gaps

For a convex program with strong duality, a primal feasible point 𝑥 and dual feasible point 𝑦 satisfy 0 ≤ 𝑓 (𝑥) − 𝑓 ★ ≤ 𝑓 (𝑥) − 𝑔(𝑦). The primal–dual gap is therefore a certified objective defect. Turning the gap into runtime requires a numerical complexity model.

Appendix B

Graph Operators, Min-Plus Algebra, and Partial Maps B.1

Weighted graph Laplacians

For an undirected weighted graph with weights 𝑤 𝑖 𝑗 ≥ 0, the Laplacian is ∑︁ 𝐿 𝑖𝑖 = 𝑤𝑖 𝑗 , 𝐿 𝑖 𝑗 = −𝑤 𝑖 𝑗 (𝑖 ≠ 𝑗). 𝑗

It is positive semidefinite and 𝑓 ⊤𝐿 𝑓 =

B.2

1 ∑︁ 𝑤 𝑖 𝑗 ( 𝑓𝑖 − 𝑓 𝑗 ) 2 . 2 𝑖, 𝑗

Spectral filters

For a Borel function 𝑠 on the spectrum of 𝐿, define 𝑆 = 𝑠(𝐿) by spectral calculus. The effective dimension for filtered squared error is often tr(𝑆 2 ). A Markov filter must preserve constants and positivity. Positive semidefiniteness alone does not imply entrywise nonnegativity.

B.3

Min-plus algebra

On R ∪ {+∞} define 𝑎 ⊕ 𝑏 = min(𝑎, 𝑏),

𝑎 ⊗ 𝑏 = 𝑎 + 𝑏.

Kernel composition is (𝐾 ⊗ 𝐿) (𝑧, 𝑥) = inf {𝐾 (𝑧, 𝑦) + 𝐿 (𝑦, 𝑥)}. 𝑦

Associativity is the algebraic form of Bellman recursion and obstruction-tower composition.

227

228

B.4

APPENDIX B. GRAPH OPERATORS, MIN-PLUS ALGEBRA, AND PARTIAL MAPS

Cycle rank

For a connected finite graph, 𝛽1 = |𝐸 | − |𝑉 | + 1 is the cycle-space dimension. A spanning tree plus one fundamental cycle for every non-tree edge gives a basis.

B.5

Partial bijections

A partial bijection 𝑇 : 𝐷 ⊆ 𝐴 → 𝑅 ⊆ 𝐵 is a bijection between its domain and range. Composition is defined only when intermediate values survive. The inverse is a partial bijection 𝑇 −1 : 𝑅 → 𝐷. Partial transports form an inverse semigroup rather than a permutation group. This distinction is essential when branches disappear.

Appendix C

Covering Spaces, Degree, and Monodromy This appendix summarizes the topology used in the singular and atlas chapters. Standard references include Hatcher [2002] and Schwarz [1961].

C.1

Covering spaces

A map 𝑝 : 𝐸 → 𝑋 is a covering if every 𝑥 ∈ 𝑋 has a neighborhood 𝑈 whose inverse image is a disjoint union of sheets, each mapped homeomorphically to 𝑈. Local oracle branches in the regular separated regime form such sheets. A loop in 𝑋 lifts to a path in 𝐸. The endpoint of the lifted path defines a permutation of the fiber: the monodromy action. A global labeled section exists only if the monodromy fixes a branch consistently.

C.2

Winding number

For a nonvanishing loop 𝑧 : 𝑆 1 → C★, 1 wind(𝑧) = 2𝜋𝑖

𝑧 ′ (𝜃) d𝜃 𝑆 1 𝑧(𝜃)

∫

when differentiable, with the usual topological extension. Winding is invariant under homotopies avoiding zero and satisfies wind(𝑧 𝑘 ) = 𝑘 wind(𝑧).

229

APPENDIX C. COVERING SPACES, DEGREE, AND MONODROMY

230

C.3

Topological degree

For a continuous map 𝑓 : 𝑆 𝑑−1 → 𝑆 𝑑−1 , the degree is an integer invariant under homotopy. Composition multiplies degrees. Degree incompatibility gives higher-dimensional analogues of the radical obstruction.

C.4

Schwarz genus and atlas number

For a fibration 𝑝 : 𝐸 → 𝑋, the Schwarz genus is the minimum number of open sets covering 𝑋 on each of which a continuous section exists. It is a topological lower bound on chart number in exact atlas repair. Statistical atlas numbers depend additionally on the native defect and tolerance. They need not equal the topological genus away from zero tolerance.

C.5

Discriminants

A discriminant is the set where the oracle fiber changes type: roots collide, eigenvalues cross, or the vertical Hessian loses rank. Away from the discriminant, branches may form a covering. Near it, local ramification normal forms replace separated-cover geometry.

Appendix D

Statistical Tools for Honest Certification D.1

Uniform concentration

If a loss class is bounded in [0, 𝐵] and finite with size 𝑀, Hoeffding’s inequality gives √︂ log(2𝑀/𝛿) sup |𝑃𝑛 𝑓 − 𝑃 𝑓 | ≤ 𝐵 2𝑛 𝑓 ∈F with probability at least 1 − 𝛿. For infinite classes, Rademacher complexity, covering numbers, stability, or PAC-Bayes tools may be used. The correct class is the actual deployment class, potentially conditional on calibration data.

D.2

Sample splitting

Let calibration, training, tuning, and test samples be independent at the level of the scientific unit. Conditioning on earlier splits turns a data-dependent candidate into a fixed object for the test analysis. This is the simplest route to honest OALI validation.

D.3

Confidence sets and three-way decisions

If 𝐶𝑛 covers the true world with probability 1 − 𝛼, then any declaration shared by all worlds in 𝐶𝑛 is valid on the coverage event. If compatible worlds disagree, the maximally decisive honest output is unresolved. If 𝐶𝑛 = ∅, the output is model conflict rather than a vacuous declaration.

231

232

D.4

APPENDIX D. STATISTICAL TOOLS FOR HONEST CERTIFICATION

Confidence sequences

A confidence-world sequence {𝐶𝑡 }𝑡 ≥1 satisfies P𝑤 {𝑤 ∈ 𝐶𝑡 for all 𝑡} ≥ 1 − 𝛼. It permits adaptive stopping and querying. Likelihood-ratio martingales and nonnegative supermartingales are standard constructions [Howard et al., 2021, Ville, 1939].

D.5

Testing lower bounds

Le Cam, Fano, and Assouad arguments convert indistinguishable alternatives into risk lower bounds. Certificate resolution uses a related but boundary-specific alternative set: only opposite-certificate worlds are binding.

D.6

Clustered data

When cells, time points, or repeated observations are nested within a biological unit, the independent test sample size is the number of units, not the number of lower-level observations. Resampling and confidence intervals should respect this hierarchy.

Appendix E

Validation and Dependency Map The book is ordered by logical validation. A structural lower bound is not interpreted until the native defect, the integrability of its defect system, and the admissibility of every nonnative carrier have been established. The main spine is an audit workflow, not a chain of theorem implications: define elimination ⇝ derive a native defect ⇝ audit integrability and carrier admissibility ⇝ choose an architecture-obstruction theorem ⇝ add resource and operational assumptions ⇝ construct a statistical certificate ⇝ test saturation, repair, and validation. Here ⇝ means “next audit gate.” It does not mean that the object on the left mathematically implies the object on the right. Each transition may require a new theorem, a new modeling assumption, or an independent experiment. In particular, a positive obstruction does not imply empirical saturation, and neither one implies that a proposed repair improves a held-out endpoint.

E.1

Constructive spine

1. Conjugate defect identity. 2. P/G/X/V/C distinctions. 3. Exactification and approximate-jet certificates. 4. Graph-CDF validity repair and CRPS risk decomposition.

233

APPENDIX E. VALIDATION AND DEPENDENCY MAP

234

E.2

Validity spine

1. Flat integrability audits assembled local defect reports. 2. Hodge and period terms diagnose and repair nonintegrable defect fields. 3. Lift Complexity excludes dummy and target-calling carriers. 4. Target-visible reduction and quotient-faithful extraction connect an admissible lift to deployment capacity when a model-specific gate is proved.

E.3

Architecture spine

1. Architecture obstruction is the second elimination. 2. EOT gives regular metric and flow lower bounds. 3. COT gives rectangularity and coordination decompositions. 4. Singular EG gives exact catastrophe taxes when regular assumptions fail. 5. Atlas, quotient, set-valued, and randomized repairs change the declared deployment contract.

E.4

Resources, semantics, and composition

1. Resource rate–distortion separates carrier loss from decoder nonsaturation. 2. Operational semantics determines contextual visibility. 3. Foundations and Part V control composition, base change, and limits.

E.5

Statistical and intervention spine

1. Simultaneous defect and grammar envelopes bracket architecture frontiers. 2. Persistent atlas inverse frontiers are stable. 3. Three-way decisions encode honest unresolvedness. 4. Certificate statistics controls resolution effort and evidence compression. 5. Typed realization joins finite-information authorization, recursive closure, and one common deployment witness without identifying their different units. 6. Sample-split held-out bounds validate repaired architectures.

E.6. NO CIRCULARITY IN THE GRAPH-CRPS RESULT

E.6

235

No circularity in the graph-CRPS result

The book-proved CDF validity repair and exact CRPS risk decomposition do not use the graph-universal probability-validity classification. The sharp path upper and lower minimax theorem is summarized from Elimination Geometry I and audited in Elimination Geometry II; its full proof is not reproduced here and is logically independent of the probability-validity classification.

E.7

No circularity in exactification

The constructive exactification theorem restores target tangency without using the converse theorem. Target descent follows only after the separate candidate-wise defect-remainder or acceptance condition is verified. The converse later shows that first-order target preservation forces the defect-jet form modulo a flat term.

Appendix F

Controlled Glossary of Structural Terms This glossary fixes the book’s working vocabulary. It is distinct from the front-matter symbol table, which records notation, and from the thematic index, which records every substantive occurrence. The boundary clause in each definition prevents a nearby concept from being silently substituted. The gateway is the first systematic treatment, not necessarily the first mention. Table F.1: Controlled glossary of book-level terms. Term

Working definition and boundary

Gateway

A declared set 𝔄 of deployable fields satisfying the output, measurability, sharing, regularity, and resource contracts. It is not the training algorithm or merely the subset that one optimizer happens to reach. The rules that generate legal representations, fields, compositions, and resource budgets. In finite-data work the grammar itself may be uncertain; it is not automatically fixed by a model name or parameter count. The infimum of native defect over a declared architecture class under a stated risk aggregation rule. It is a class-level realizability floor, not the implementation gap of one trained deployment. A cover of the instance space by domains admitting legal local sections or experts, together with an admissible routing rule. Adding an atlas changes the deployment contract; it is not merely widening one global model. Passage between fine and coarse representations, memories, or architecture levels. It preserves obstruction only under a proved commuting or exact decomposition theorem; information loss alone does not make the change exact. The oriented Bregman divergence generated by a conjugate elimination. Its orientation and scale come from the objective, so an arbitrary symmetric distance is not an interchangeable substitute.

Ch. 2

A–C Architecture class

Architecture grammar

Architecture obstruction

Atlas

Base change

Bregman defect

Ch. 2

Ch. 11

Ch. 15

Ch. 18

Ch. 5

Continued on next page

236

E.6. NO CIRCULARITY IN THE GRAPH-CRPS RESULT

237

Table F.1 continued Term

Working definition and boundary

Gateway

Carrier

The deployment information state 𝑍 passed to a decoder or readout. It is distinct from the quotient-reduced slack carrier used to audit a conic lift; identifying the two requires an interface theorem. A typed finite-data record containing a confidence world, a declared boundary, a structural color, supporting evidence, and a resolution profile. It is more than a point estimate, posterior probability, or diagnostic score. The distance from the complete identified image to the declared action boundary when the certificate is resolved. A positive margin quantifies stability of the color; the tolerance profile records the color across all candidate boundaries. An exact triple (𝐻, 𝐽, D) with 𝐽 (𝑥) = inf 𝑎 𝐻 𝑥 (𝑎) and D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) ≥ 0. “Certified” refers to identity in the original objective; attainment of the infimum is not required. The complete declared package of population, auxiliary fibration, objective, defect, architecture, representation/resource grammar, task contract, and statistical experiment. No obstruction claim is absolute outside this package. One legal architecture that works simultaneously across the indexed inputs or population worlds. It is stronger than pointwise feasibility ∀𝑥 ∃𝐴 𝑥 and cannot be inferred by exchanging quantifiers. A data-dependent set 𝐶𝑛 ⊆ W that contains the true population world under its declared fixed-record, simultaneous, or anytime coverage contract. It is the set of worlds retaining inferential standing, not a set of high-posterior stories. Closure of the declared task set under all legal pre- and post-compositions. It yields a task- and grammar-relative fully abstract quotient, not an absolute identity between internal pipelines. The excess obstruction caused by requiring one shared conditional mechanism, measured relative to the rectangular hull that may paste all locally available kernels independently. A quantified failure of local operations, transports, or elimination orders to commute or integrate. Square, ordering, and interchange curvatures live at different interfaces and are not one universal scalar tax.

Ch. 16

Certificate

Certificate margin

Certified elimination

Certified learning system

Common deployment

Confidence world

Context closure

Coordination tax

Curvature

Ch. 20

Ch. 19

Ch. 2

Ch. 2

Ch. 19

Ch. 19

Ch. 17

Ch. 13

Ch. 9

D–H Continued on next page

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APPENDIX E. VALIDATION AND DEPENDENCY MAP

Table F.1 continued Term

Working definition and boundary

Gateway

Decoder nonsaturation

The residual defect remaining after a carrier 𝑍 is fixed because the legal decoder class cannot realize the optimal readout. It is separate from information already erased by the carrier. Two nonnegative quantities may be added only when an exact identity, infimal decomposition, or exchange theorem converts them into the same declared objective loss scale. The exact residual D 𝑥 (𝑎) = 𝐻 𝑥 (𝑎) − 𝐽 (𝑥) produced by elimination. A metric, regularizer, or surrogate discrepancy is not a defect unless a theorem links it to this residual. Simultaneous lower and upper bounds on the population defect over a confidence world. Together with an architecture-grammar envelope it brackets an entire obstruction frontier, not just one fitted architecture. Optimization of an auxiliary object while retaining the optimized value and its exact residual. It is not automatically marginalization, projection, or conditioning unless the declared objective makes those operations coincide. A nested sequence of fine-to-coarse eliminations whose realization costs compose by exact or infimal decompositions. It separates choice of a coarse state from the cost of realizing its fine fiber. A retained statistic used for future certification. Sufficiency is relative to the certificate truth partition and its binding alternatives, not necessarily to the full parametric model. Subtraction of the relevant value or defect jet to restore target touching and derivatives of a surrogate. Exactification restores the interface; target descent still needs a separate acceptance or remainder condition. The smallest size of a representation in a declared lift family. Without an exchange theorem and computational model, it is neither native defect nor an unrestricted runtime lower bound. The least vertical defect required to realize a chosen coarse state by an admissible fine object. It is the fine-fiber term in a coarse/fine obstruction decomposition. The separation of model approximation, architecture obstruction, generalization, and optimization or implementation components relative to compatible reference risks. The components do not become interchangeable merely because they appear in one algebraic identity. Invariance of a structural conclusion under legal changes of coordinates, factor gauges, or local branch labels. Raw representatives may change while cycle type, section count, or quotient information remains fixed.

Ch. 16

Declared-loss scale rule

Defect

Defect envelope

Elimination

Elimination tower

Evidence carrier

Exactification

Extension complexity

Fiber realization tax

Four-component risk decomposition

Gauge invariance

Ch. 2

Ch. 2

Ch. 19

Ch. 2

Ch. 6

Ch. 20

Ch. 7

Ch. 10

Ch. 2

Ch. 3

Ch. 23

Continued on next page

E.6. NO CIRCULARITY IN THE GRAPH-CRPS RESULT

239

Table F.1 continued Term

Working definition and boundary

Gateway

Global section

A tuple (𝑠 𝑣 ) 𝑣∈𝑉 with 𝑠 𝑣 ∈ B𝑣 such that every oriented edge transport is defined at 𝑠𝑢 and satisfies 𝑇𝑢𝑣 (𝑠𝑢 ) = 𝑠 𝑣 . Root-path survival and common fixedness under fundamental holonomies characterize this edgewise definition; a separate fixed branch for each cycle does not establish one common global section. Projection of an observed edge defect field into exact, flat, and certified components, separating local curvature, harmonic periods, and the final nonnegative/touching constraint. The composite branch transport around a cycle. For partial transports it may fix, move, or lose a branch; under relabeling it changes by conjugacy rather than as an absolute labeled permutation.

Ch. 23

The query-relevant projection 𝐽𝐶 (𝑞) = {Γ𝑞 (𝑤) : 𝑤 ∈ 𝐶} of a confidence world. It is the complete finite-data range for the declared functional; a certificate color is only its projection relative to an action boundary. The defect of a chosen deployment above the infimum within its declared architecture class. Better training may reduce this term; it does not by itself reduce the architecture obstruction. The condition that locally reported elimination increments arise from one global potential. Face flatness suffices only on the appropriate simply connected complex; global periods must otherwise also be audited. Target-faithfulness of a nonnative lifted representation after slack, minimal-face, quotient, and gauge reduction. Finite auxiliary dimension by itself does not make a lift intrinsic. Existence and well-posedness of the pointwise oracle problem at each input, including the native cost of deviating from that oracle. It does not imply that one shared deployment can realize all local optima simultaneously. Existence of one rule in the declared deployment class that realizes the relevant local oracle family simultaneously. It is conditional on the representation, sharing, memory, regularity, and resource contracts. The ability of the declared statistical experiment to distinguish realizability, nonrealizability, and unresolvedness with stated error control. It is a property of the evidence available about the population structure, not of the population obstruction alone. The typed output 𝑀 produced when the declared confidence world is empty. It signals conflict among data, uncertainty construction, and model restrictions; it does not authorize vacuous feasible or impossible claims.

Ch. 19

Hodge repair

Holonomy

Ch. 9

Ch. 23

I–O Identified image

Implementation gap

Integrability

Lift admissibility

Local solvability

Global realizability

Finite-sample certifiability

Model conflict

Ch. 2

Ch. 9

Ch. 10

Ch. 1

Ch. 1

Ch. 1

Ch. 19

Continued on next page

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APPENDIX E. VALIDATION AND DEPENDENCY MAP

Table F.1 continued Term

Working definition and boundary

Gateway

Monodromy

The permutation or partial transformation of local oracle branches induced by lifting loops around a singular or multiply connected region. It is distinct from a branch that genuinely disappears. The units of the objective-generated defect. Parameter distance, topological degree, communication, runtime, and task regret remain different units until an exchange theorem connects them. The concrete prototype and software line used for flow certificates, partial transport, cycle audit, and atlas construction. OAI v0.1–v0.5 is an implementation path inside the broader OALI workflow, not a second name for that workflow. The workflow that declares the contract, estimates local oracles, builds transports, computes typed certificates, selects a mechanism-matched repair, and validates it independently. It is not generic mixture-of-experts routing. The portion of an internal difference that some legal task in some legal context can expose. A positive native defect need not produce positive downstream separation without transmission and exposure gates. The zero-defect set O (𝑥) at one instance; the oracle field is the family of these fibers over 𝑥. A fiber may be set-valued even when no legal global point-valued selection exists.

Ch. 14

A partial bijection between local branch sets that permits unmatched branches to appear, disappear, or remain unresolved. It avoids manufacturing a forced correspondence when no branch match is justified. The minimum chart count needed to achieve a stated uniform defect tolerance. The integer count may jump, while its inverse chart-budget/loss frontier is stable under uniform perturbation. A contract-based classification of pointwise elimination, external globalization, defect-jet exactification, variational coupling, and fixed-marginal coupling. The label is determined by what changes and what is held fixed, not by the numerical algorithm alone. Infima of expected defect and worst-case defect, respectively. A small population seam can make the first zero while the second remains positive; neither aggregation contract may replace the other silently.

Ch. 23

Native loss scale

Obstruction-Aware Inference (OAI)

Obstruction-aware learning and inference (OALI) Operational visibility

Oracle fiber

Ch. 5

Chs. 22–23

Ch. 22

Ch. 17

Ch. 2

P–W Partial transport

Persistent atlas number

P/G/X/V/C calculus

Population and uniform obstruction

Ch. 19

Ch. 6

Ch. 2

Continued on next page

E.6. NO CIRCULARITY IN THE GRAPH-CRPS RESULT

241

Table F.1 continued Term

Working definition and boundary

Gateway

Quotient repair

Replacement of labeled representatives by the task-relevant equivalence class, such as an eigenprojector or unordered mixture. It removes label artifacts only when the downstream contract is invariant to the quotient. A deployment kernel used either as a probability-valued output or to draw a point-valued action. These are different task contracts and neither is the deterministic average of representatives; the applicable loss must be declared on the law or on the sampled action. The conditional architecture formed by independently pasting every locally available kernel. It is a comparison class used to isolate coordination, not automatically a legal shared deployment. The information or sample effort needed to separate a world from the nearest compatible world with the opposite certificate label. It is not a generic generalization bound or posterior entropy. The best native defect attainable under a declared carrier/decoder resource budget, often decomposed into carrier information loss and decoder nonsaturation. Parameter count alone does not define the frontier. Zero architecture obstruction under a declared risk contract. Exact saturation additionally requires an attained zero-defect section; a zero unattained infimum gives approximation but not exact realization. An oracle whose correct output is a set, orbit, projector, subspace, or law rather than one labeled point. Forcing a representative can introduce an artificial obstruction. The objective gap between the best partial branch matching and the second-best structurally distinct matching after dummy-label permutations are removed. It controls exact recovery under perturbation. An exact lift that appends coordinates computed directly from the target or an arbitrarily prescribed field. It can manufacture fake coherence and is excluded by intrinsic lift admissibility. The conditional dual oracle signature 𝑅𝑈 = E(𝑆 | 𝑈) retained by a deployment carrier. It is the coarsest quotient preserving optimal native Bregman defect; its visible cardinality is not determined by cone size without an extraction theorem. The pointwise least operational kernel preserving every value in the declared task family. Internal differences removed by the envelope are invisible to that task contract, though another contract may expose them.

Ch. 15

Randomized repair

Rectangular hull

Resolution complexity

Resource rate-distortion

Saturation

Set-valued oracle

Structural assignment margin

Target-calling lift

Target-visible quotient

Task envelope

Ch. 15

Ch. 13

Ch. 20

Ch. 16

Ch. 11

Ch. 2

Ch. 23

Ch. 10

Ch. 10

Ch. 17

Continued on next page

242

APPENDIX E. VALIDATION AND DEPENDENCY MAP

Table F.1 continued Term

Working definition and boundary

Gateway

Three-way certificate

The feasible/impossible/unresolved rule obtained by evaluating a structural truth map over one simultaneous confidence world. It declares a color only when every compatible world agrees. An empty confidence world is handled separately as model conflict 𝑀. Vanishing of the native defect, and when required its relevant derivatives, at a declared oracle witness. Nonnegativity without touching does not certify the intended eliminated target. A declared rule for comparing or propagating local oracle states across inputs, graph edges, or representations. It may be partial, set-valued, or gauge-dependent; path consistency must be proved rather than assumed. The honest finite-data state in which the confidence world contains opposite structural truth labels. It is an authorized conclusion about present resolution, not a failed optimizer or an instruction to guess. Uniform correctness of a certificate for every population world under the declared experiment. It is stronger than prior-average posterior credibility and can require infinite effort when opposite labels are observationally indistinguishable.

Ch. 19

Touching

Transport

Unresolved

Worldwise validity

Ch. 2

Ch. 23

Ch. 19

Ch. 20

Appendix G

Source Manuscript Crosswalk This monograph reorganizes the source manuscripts by concept rather than publication order. Source

Primary book locations

Elimination Geometry I

Chapters 2 and 5–8; native defect, P/G/X/V/C, exactification, graphCDF/CRPS, and model-specific examples Elimination Geometry II Chapters 7–9 and boundary material in Chapter 26; exactification converse, probability-validity gates, integrability, and representation limits Elimination Geometry III Chapter 18; soft conditional chains, pushforward, base change, conditioning, and zero temperature Foundations Chapters 2, 6, and 18; defect fibrations, obstruction towers, and interchange coherence EOT Chapters 11–12; second elimination, regular obstruction transfer, flow duality, and population thickening COT Chapters 6, 13, and 18; rectangularity, coordination tax, architecture base change, and dequantization Singular EG Chapters 14–15 and Appendix C; radical tax, degree/discriminant mechanisms, and atlas/quotient repair Lift Complexity Chapter 10; target-calling no-go, slack factorization, quotient reduction, and complexity boundaries Resource-Constrained Ar- Chapters 10, 16, 19, and 21; target-visible extraction, carrier– chitectures decoder rate–distortion, common deployment, and typed realization Exact Data Selection Chapter 16; exact low-dimensional and budget-two selection frontiers, auxiliary-law elimination, and merge–split recovery Operational Semantics Chapter 17; task envelopes, context closure, full abstraction, and ordering curvature Marton counterexample Chapter 18; credit–semantic-tax identity, rectangular-switch frontier, and a certified positive Markov-architecture obstruction; exact verification package shipped with the book Statistical EG Chapter 19; confidence worlds, identified images, honest frontier brackets, persistent atlases, typed architecture certificates, and the active-set depth-separation candidate family 243

APPENDIX E. VALIDATION AND DEPENDENCY MAP

244 Certificate Statistics EGML EGML II OAI packages Quantum EG Chapter 24 reproducibility packages

G.1

Chapters 20–21; resolution complexity, evidence carriers, recursive state, and typed no-compensation Chapters 1, 3, 11, and 21–24; four-component risk theory, integrated ML positioning, OALI, and application protocols Chapters 20–22; typed carriers, non-compensation, and adaptive learning interfaces Chapters 22–24; transport recovery, cycle audit, atlas construction, and synthetic validation Chapter 25; Gibbs base-change rigidity, sectors, recoverability, and measurement limits Chapter 24; posterior-family and cytometry proof-of-mechanism checks

Principal-result audit ledger

This section inventories the 37 formal-result families selected for complete proof in Version 1.9. The number is a proof-coverage count, not a count of independent contributions: the rows range from classical identities and direct workflow lemmas to book-level interfaces and a small number of narrow priority candidates. Calling all 37 “principal results” would overstate the density of original mathematics. The classification concerns intellectual provenance, not correctness. C

A classical or explicitly imported result supplies the mathematical core.

S

The statement is a direct specialization, corollary, or workflow lemma; the application may be useful but is not a priority claim.

P

A possible narrow program-specific increment. Close antecedents remain, so this is not a general originality claim.

O

A priority-audit candidate. This code records an unresolved literature question, not an established original theorem.

B

A book-level synthesis or interface theorem that organizes several results and should not be cited as one wholly new mathematical mechanism.

No. Formal result

Class Antecedent and audited increment

1

Theorem 3.1: exact four- B component risk identity

Algebraic telescoping over the classical approximation, estimation, and optimization ledger. The book-level increment is the contract-relative split of approximation into local-model and shared- architecture accounts; compare amortization-gap work [Cremer et al., 2018, Margossian and Blei, 2024].

2

Theorem 3.2: certified learn- S ing bound

Standard uniform convergence plus approximate empirical-risk minimization. The specialization preserves the model/architecture floor; it adds no new concentration inequality. continued on next page

E.7. NO CIRCULARITY IN EXACTIFICATION

No. Formal result

245

Class Antecedent and audited increment

3

Theorem 5.1: conjugate defect C identity

Fenchel–Young/Bregman identity [Bregman, 1967, Rockafellar, 1970]. The book uses it to fix the native defect and its direction.

4

Theorem 6.1: tower associa- C tivity

Classical infimal-convolution/Bellman associativity [Bellman, 1957, Baccelli et al., 1992]; the typed tax interpretation is the book’s organizational layer.

5

Theorem 7.1: jet exactification S

Elementary Taylor subtraction, adjacent to first-order surrogate touching [Mairal, 2013]. The specialization uses the exact elimination defect rather than an arbitrary correction.

6

Theorem 7.2: converse exacti- P fication normal form

Within the declared frozen-lift class, target-jet preservation forces defect-jet subtraction modulo a flat term. The source claims this normal form, not tangent-surrogate methodology in general.

7

Theorem 8.1: CDF validity C repair

Hilbert projection and coordinatewise nonexpansiveness, specialized to squared-CDF/CRPS risk and graph roughness.

8

Theorem 9.1: flat elimination S criterion

Classical potential/cycle integrability [Monderer and Shapley, 1996, Jiang et al., 2011]; nonnegativity and fiberwise touching specialize the potential to partial-minimum defects.

9

Theorem 10.1: finite- S complexity target-calling no-go

An elementary augmentation argument specialized to targetcalling lift geometry. Its value is diagnostic: unrestricted lifts make finite-overhead complexity vacuous. No new extensioncomplexity mechanism is claimed.

10

Theorem 10.2: conic lift– C factorization gate

Explicitly imported from Gouveia et al. [2013], extending Yannakakis [1991].

11

Theorem 10.4: visible quo- P tient and lift-to-carrier transfer

Conditional Bregman prediction is classical [Banerjee et al., 2005a]. The claimed increment is the coarsest dual-signature quotient in the reduced-lift interface and the factorized resourceto-native-defect transfer.

12

Theorem 12.1: oracle- P variation transport bound

Coupling, Lipschitz, and strong-growth ingredients are classical. EOT claims the stated sharp native-objective lower certificate; this audit does not assert exhaustive priority over equivalent metric forms.

13

Theorem 13.2: rectangu- P lar Bellman and coordination residual

Robust-control rectangularity is classical [Epstein and Schneider, 2003, Iyengar, 2005, Nilim and El Ghaoui, 2005]. COT’s increment is to rectangularize the deployable architecture and retain failed sharing as an exact forward-KL tax.

14

Theorem 14.1: exact radical P catastrophe tax

Winding and loop lifting are classical. The exact uniform value 𝑟 2 is a useful quantitative corollary for the declared objective, not evidence of a new topological mechanism.

15

Theorem 15.1: atlas saturation S

Direct witness lemma once a covering by exact local sections is given; sectional category and Schwarz genus are classical [Schwarz, 1961]. continued on next page

APPENDIX E. VALIDATION AND DEPENDENCY MAP

246 No. Formal result

Class Antecedent and audited increment

16

Theorem 16.1: architecture P rate–distortion decomposition

Conditional Bregman decomposition and rate–distortion are classical neighbors [Banerjee et al., 2005b, Shannon, 1959]. The source claims the native carrier-erasure/decoder-nonsaturation ledger under a declared resource and extraction grammar.

17

Theorem 17.1: task-envelope P isometry

Classical residuation and conjugacy isometries precede it [Cohen et al., 2004, Attouch and Wets, 1986]. The Attouch–Wets result concerns Legendre–Fenchel/epigraph geometry, not the finite task-envelope sup-norm proposition. The source claims the latter’s task-restricted, two-sided min-plus kernel realization, which the book proves directly.

18

Theorem 17.2: contextual P completion and full abstraction

Contextual equivalence and full abstraction are classical [Milner, 1977, Plotkin, 1977]. The claimed increment is their simultaneous realization by one explicit closed min-plus task envelope.

19

Theorem 18.1: soft condi- C tional chain theorem

Gibbs variational identity, telescoping normalization, and the relative-entropy chain rule [Csiszár, 1975, Cover and Thomas, 2006], with the book’s common-reference typing.

20

Theorem 19.3: honesty and C maximal decisiveness

Confidence-set projection [Dufour, 1997], partial identification [Manski, 2003], and equivalence/noninferiority threshold logic [Blackwelder, 1982, Berger and Hsu, 1996]. The 𝐹/𝐼/𝑈/𝑀 record is book terminology, not a claim to invent the projection rule.

21

Theorem 19.4: honest elimi- S nation bracket

Monotone projection through simultaneous objective and feasible-set uncertainty; closest explicit neighbor is universal confidence sets for random optimization [Vogel, 2008].

22

Theorem 19.5: atlas interleav- P ing and stable inverse

The canonical companion source is Statistical EG [Huang and Wang, 2026m]; EGML reuses the interface in its learning-facing atlas ledger [Huang and Wang, 2026d]. Interleaving stability is classical [de Silva et al., 2018]. The claimed increment is the sharp inverse 𝐾-chart loss frontier in native objective units.

23

Theorem 19.6: common- P deployment quantifier theorem

The minimax inequality is elementary. Statistical antecedents include Hodges–Le Cam superefficiency, Hájek local asymptotic minimaxity, and Leeb–Pötscher analyses of oracle-property nonuniformity [Le Cam, 1953, Hájek, 1970, 1972, Leeb and Pötscher, 2005, 2008]. The source-program increment is only the explicit separation of pointwise feasibility from one common deployable witness over a confidence world.

24

Theorem 20.1: posterior credi- C bility is not worldwise validity

Standard prior-average versus uniform- frequentist distinction, illustrated by observationally equivalent opposite- label worlds; retained as an expository typed counterexample.

25

Theorem 20.2: truth-map C form of three-way honesty

A notational corollary of row 20 and Dufour’s projection principle; it has no independent novelty claim.

26

Theorem 20.3: worldwise res- C olution lower bound

Imported sequential change-of-measure and partition- identification geometry [Kaufmann et al., 2016, Garivier and Kaufmann, 2016]; the certificate truth map selects the alternative set. continued on next page

E.7. NO CIRCULARITY IN EXACTIFICATION

No. Formal result

247

Class Antecedent and audited increment

27

Theorem 20.4: exact certifi- S cate preservation by slack

A short specialization of KL data processing and max–min rearrangement to preservation of one certificate floor. It is not presented as a new information-theoretic mechanism.

28

Theorem 20.5: recursive quo- S tient and minimal exact state

Strong probabilistic bisimulation and partition refinement are classical [Larsen and Skou, 1991, Givan et al., 2003, Paige and Tarjan, 1987]. Certificate color and the state/chart reading are the specialization.

29

Theorem 21.1: compensation

typed no- B

Synthesizes a zero-information lower bound, a bisimulation collision, and a decoder collision. The contribution is their typed logical independence, not new proofs of each component impossibility.

30

Theorem 21.2: typed architec- B ture realization interface

Collects earlier sample-time, evidence, state, common-witness, native-loss, and contextual gates. It is an end-to-end interface theorem rather than one new underlying bound.

31

Theorem 21.3: finite-library C held-out improvement

Hoeffding’s inequality plus a finite union bound, with the book’s independent-unit and predeclaration contract.

32

Theorem 22.1: population im- S provement template

Direct subtraction of a baseline lower certificate and repair upper certificate on one loss scale and event.

33

Theorem 23.2: noisy partial- C transport recovery

Finite-class argmin stability under a strict margin, with a metric perturbation bound. The partial matched/unmatched semantics are the application-specific contract.

34

Theorem 23.3: cycle-space S characterization

Classical spanning-tree holonomy and graph consistency [Singer, 2011, Jiang et al., 2011]; the specialization retains domains of partial maps and requires a common surviving fixed branch.

35

Theorem 25.1: universal S Gibbs base-change rigidity

Direct finite-dimensional Hamiltonian specialization of the factorization and quantum-exponential-family sufficiency criteria [Jenčová and Petz, 2006, Theorems 5 and 7]. The book supplies an independent pressure-duality/Petz proof and makes no standalone priority claim.

36

Theorem 19.8: deployment B conflict under observational overlap, with repair trichotomy

The exact intersection condition expands the existence of a common measurable witness. The quantitative bound is Le Cam’s two-point testing/total-variation identity [Le Cam, 1953, Tsybakov, 2009]; the information-repair interpretation is adjacent to Blackwell comparison [Blackwell, 1953]. The book-level increment is the typed placement of solver, information, and architecture repair in one deployment interface. No new decisiontheoretic lower bound is claimed. continued on next page

APPENDIX E. VALIDATION AND DEPENDENCY MAP

248 No. Formal result 37

Class Antecedent and audited increment

Theorems 19.10 and 19.11: O active-set conflict modulus, one-pass native-loss saturation, and sufficient repair depth

Active-set affine sensitivity, shallow sparse encoders, LISTA/unrolling, and proximal contraction are established [Gregor and LeCun, 2010, Chen et al., 2018, Beck and Teboulle, 2009, Muthukumar and Sulam, 2023]. Most importantly, O’Neill et al. [2025] already prove a global one-layer linear–nonlinear sparse-autoencoder amortisation gap and study inference-time optimization. The narrower candidate increment is the inputcomputable two-neighborhood Jacobian conflict modulus along a declared perturbation subspace, its quantitative native-loss floor, and a sufficient crossing depth on the same compact set. The exact combination was not located, but independent priority is not established; see the dated dedicated audit shipped with the book.

The ledger should not be summarized as four original and nine program-specific theorems. The finite target-calling construction is an elementary augmentation argument; the radical value is a quantitative winding corollary; and the evidence-slack criterion is a short KL data-processing/max–min consequence. They are retained for their fit with the audit framework, not presented as established new mechanisms. The dedicated result-level audit of universal Gibbs base-change rigidity closes it as a direct specialization, not an originality candidate. Several P rows may contain narrow useful formulations, but their engines are close to established approximation, control, information-theoretic, or semantic results. The main book-level contribution claimed here is the typed, native-loss, audit-oriented synthesis.

G.2

Editorial principle

The source manuscripts remain the record of model-specific variants and advanced results not selected for the book. Every formal result inventoried here has a complete proof in its chapter-local appendix. The monograph’s claimed increment is its terminology, typed interfaces, audit order, and cross-domain interpretation. Results retained for context without a complete book proof are identified in place by a Source status box and are excluded from the 37-result proof-coverage inventory.

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Thematic Index repair, 91 Schwarz genus, 230 attainment, 9

active-set conflict modulus, 135 Adaptive LASSO, 130 aggregation department audit, 154 amortization gap, 135 amortized inference finite-grammar audit, 194 resource frontier, 20 applied laboratories, 193 comparative lessons, 204 approximate selection, 9 architecture certificate-aware selection, 171 class, 236 comparison, 70 data-dependent, 139, 171 deployment contract, 7 finite library, 169 grammar, 125, 236 output contract, 71 regularity contract, 71 risk contract, 71 typed realization, 166 architecture obstruction Marton dual, 114 architecture repair compute-matched validation, 196 information-matched, 133 scientific validation, 197 architecture saturation native loss, 135 atlas, 236 graph construction, 185 persistent frontier, 93 persistent number, 126, 240

barrier method defect exchange, 63 base change, 111, 236 architecture, 113 exact, 82 memory compression, 82 quantum, 208 Bayesian design versus worldwise stopping, 153 Bayesian risk versus worldwise complexity, 151 belief state recursive quotient, 152 Bellman residual, 81 BeSS, 130 best subset selection, 130 bias–variance decomposition, 46 bisimulation belief state, 152 Bregman geometry, 225 canonical carrier, 98 conditional projection, 226 defect, 236 divergence, 25, 225 pooling, 28 Pythagorean identity, 35 quantization, 100 relative entropy, 26 calibration graph, 74 carrier, 237 coarsening tax, 17 263

264 continuous, 99 deployment, 97, 163 evidence, 149, 163, 238 hard, 99 information loss, 98 joint recursive, 166 recursive, 152, 163 target-visible quotient, 61, 241 catastrophe tax, 86 causal inference deployment contract, 134 CDF set, 45 certificate, 237 adaptive design, 153 AND/OR phase change, 149 architecture plus gain, 170 auditable record, 145 confidence certificate, 145 feasible, 123 impossible, 123 margin, 123, 237 population truth, 145 resolution complexity, 148, 241 three-way rule, 123, 147, 231, 242 historical comparison, 160 tolerance profile, 123, 154 truth map, 147 unresolved, 123, 242 certificate statistics, 145 certificate-guided intervention, 176 certification error, 41 summable, 41 certification ladder radical loop, 188 CertTrack, 153 characteristic information, 148 characteristic time, 148 claim boundary, 216 cytometry, 203 fixed-marginal coupling, 35 quantum analogy, 208 rate–distortion, 102 regular transfer, 76 clustered data, 232 coarse-graining oracle-preserving, 17

THEMATIC INDEX common deployment, 127 radical instance, 168 radical validation, 186 typed realization, 166 composition checklist, 116 soft kernels, 112 theorem requirement, 111 computational complexity conditioning, 63 concentration uniform, 231 conditional architecture, 78 conditional policy shared deployment, 19 shared kernel, 81 world model, 204 confidence structural versus operational, 171 confidence sequence, 148, 232 confidence world, 121, 231, 237 anytime, 122 definition, 122 historical comparison, 160 simultaneous, 122 conformal prediction conditional coverage, 129 conic lift, 59 conjugate transmission, 42 contextual closure, 105, 237 continuous ranked probability score, 18, 46 convex analysis, 225 coupling fixed-marginal, 32, 35 variational, 32 covariate shift prediction sets, 129 coverage contract, 122 critical value, 87 curvature interchange, 116 operational ordering, 56 ordering, 107 square, 52 structural, 237 cycle space, 185, 228

THEMATIC INDEX cytometry missing populations, 202 multi-sample alignment, 202 data processing, 113 data selection, 100 declared-loss scale rule, 9, 238 decoder, 97 nonsaturation, 98, 238 defect confidence envelope, 125, 238 exchange inequality, 28 fibration, 8 fine-to-coarse decomposition, 8 local consistency, 52 native, 4, 6, 238 objective-generated, 25 set-valued, 29 touching, 242 defect jet, 38 deployability, 2 deployment common, 237 deployment action, 145 deployment conflict, 133 dequantization, 116 descent identity candidate-wise, 39 diabetes patient-level architecture repair, 197 discriminant, 85, 230 effective sample size posterior-family gate, 201 eigenline catastrophe, 87 elimination, 238 certified, 6, 27, 237 pointwise, 31 saddle, 28 second, 69 tower, 33, 238 elimination geometry research program, 219 epigraph geometry, 59 exactification, 32, 238 algorithm, 42 converse normal form, 40

265 jet, 38 expectation–maximization, 27 extension complexity, 60, 238 versus runtime, 63 falsification local integrability, 55 structural explanation, 5 Fenchel conjugacy, 25, 225 fiber auxiliary, 6 fiber realization tax, 8, 238 finite-sample certifiability, 3, 239 flow duality, 74 foundation models resource limits, 102 four-component risk decomposition, 238 architecture obstruction component, 10 historical comparison, 23 sequential decisions, 204 full abstraction, 105 historical comparison, 119 Gaussian field, 35 generative model density versus transport contract, 55 global realizability, 3, 239 global section, 2, 183, 239 globalization loss of touching, 38 plug-in, 32 graph Laplacian, 227 graph smoothing, 44 path rates, 48 graph theory, 227 graph-indexed learning distributional prediction, 18, 44 held-out validation finite library, 169 independent, 178 predictive gain, 170 hidden winding, 138 Hodge decomposition, 54 identified image, 123, 239 historical comparison, 160

266 implementation gap, 7, 239 infimal convolution, 33 integrability, 52, 239 approximate flatness, 54 audit ordering, 55 historical comparison, 67 Hodge repair, 54, 239 KL divergence conditional chain rule, 111 information loss, 149 learning bound four-component target, 163 learning system certified, 7, 237 Legendre conjugacy, 25 lift admissibility, 58, 239 fidelity–coherence–complexity frontier, 60 gauge reduction, 60 historical comparison, 67 slack carrier, 60 target-calling, 58, 241 target-visible reduction, 61 Lipschitz architecture, 73 local solvability, 3, 239 LogDet divergence, 26 Markovity Conjecture, 114 Marton’s inner bound, 114 memory compressed history, 204 compression, 82 finite-memory screening, 107 witness-directed refinement, 204 metric entropy, 99 metric projection, 226 metric resolution, 107 min-plus algebra, 227 min-plus kernel, 104 mixture of experts routing carrier, 165 MNIST amortization audit, 194 model conflict, 123, 239 monodromy

THEMATIC INDEX historical comparison, 94 Monte Carlo reuse across posterior families, 200 three diagnostic scales, 200 mutual information task posterior, 151 native loss scale, 4, 25, 240 composition, 112 historical comparison, 49 negative control, 178 no-compensation theorem, 163 no-free-lunch theorem hidden winding, 138 OALI historical comparison, 205 observational overlap, 133 obstruction architecture, 4, 69, 236 class-level floor, 69 lower witness, 70 population, 7 population versus uniform, 240 representation admissibility, 63 scaling, 63 theorem template, 72 uniform, 7 upper witness, 70 versus approximation error, 71 versus optimization error, 71 zero floor, 70 obstruction transfer historical comparison, 94 quarter transport bound, 73 regular regime, 73 singular regime, 85 obstruction-aware inference, 240 obstruction-aware learning and inference, 176, 240 applications, 193 closed-loop validation, 188 one-pass prediction, 20 operational semantics, 104 operational visibility, 104, 240 exposure gate, 107 native-to-operational transfer, 107

THEMATIC INDEX nonexpansive transport, 127 transmission gate, 107 optimization certificate, 38 optional stopping, 148 oracle approximate, 41 fiber, 6, 240 field, 2 missing branch, 88 set-valued, 8, 29, 75, 241 oracle geometry finite cover, 8 regular, 8, 73 singular, 8, 85 oracle property pointwise versus uniform, 130 P/G/X/V/C calculus, 31, 240 classification test, 33 partial identification common witness, 127 policy learning observational overlap, 134 polynomial roots, 87 population thickening, 75 posterior credibility, 145 posterior family shared simulation, 200 primal–dual gap, 226 principal result provenance audit, 244 probability validity, 45 barrier, 47 pushforward, 113 quantum theory commuting sector, 210 Gibbs base change, 208 historical comparison, 222 lossless classicalization, 210 measurement, 210 noncommutative boundary, 208 relative entropy, 27 replacement identity, 210 rigidity proof, 209 radical normal form

267 OAI validation, 186 common witness, 168 seven-step closure, 188 randomization hard limit, 116 rate–distortion historical comparison, 119 rectangularity, 19 coordination decomposition, 79 coordination tax, 79, 237 historical comparison, 94 rectangular hull, 78, 241 repair, 92 relative entropy, 26 repair atlas, 91 historical comparison, 205 mechanism-matched, 90, 177 memory refinement, 92 nearest certified, 55 probability-valued, 92 projection, 45 quotient, 91, 241 randomized, 92, 241 selection, 93 semantic projection, 92 set-valued, 92 test-time refinement, 92 unsafe averaging, 92 repair depth, 135 representation quotient reduction, 60 quotient-valued, 91 resource constraints carrier–decoder factorization, 97 data-selection budget, 100 foundation models, 102 phase diagram, 100 quantization frontier, 100 rate–distortion, 102, 241 retrieval-augmented generation information carrier, 165 risk observable estimate, 46 uniform versus average, 87 risk decomposition

THEMATIC INDEX

268 four-component, 10 running laboratories, 17 cross-laboratory comparison, 22

sufficiency task-relative, 149 superefficiency, 130

sample splitting, 139, 231 sampling unit, 232 saturation, 70, 82, 241 SCAD, 130 scientific AI quantum boundary, 211 score field integrability, 55 shared deployment, 2 singularity radical normal form, 86 stability, 88 slack factorization, 59 smoothness, 226 software semantics, 107 source manuscript crosswalk, 243 editorial principle, 248 sparse coding frozen architecture repair, 196 sparse model selection pointwise oracle guarantee, 130 spectral filter, 227 spectral output, 87 statistical certification, 121 technical tools, 231 statistical elimination geometry, 121 statistical experiment, 7 stopping rule architecture research, 171 worldwise authorization, 153 strong convexity, 226 structural explanation boundary, 216 structural learning theory, 163 completion criteria, 171 research program, 219 structural realizability, 5 structural saturation, 178

task contract, 7, 105 task envelope, 105, 241 test-time refinement MNIST audit, 194 quadratic laboratory, 20 testing lower bound, 232 topology, 229 covering space, 229 cycle consistency, 183 degree, 87, 230 degree identification, 139 loop lifting, 87 monodromy, 85, 88, 240 winding number, 229 touching identity, 38 transport, 242 assignment margin, 182, 241 cycle holonomy, 183, 239 dual flow, 74 gauge invariance, 184, 238 local branch set, 181 multi-cycle witness, 185 noisy recovery, 182 partial bijection, 181, 202, 228, 240 partial matching, 182 path transport, 183 statistical uncertainty, 185 UCBAdmissions audit, 154 validation and dependency map, 233 visible capacity, 61 world model memory, 204 worldwise validity, 123, 147, 151, 242 zero-temperature limit, 82, 116

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