arXiv CSopen access

Contextual Utility of Quantization Moves in Extreme Low-Bit LLMs

Preprint. Under review.

Contextual Utility of Quantization Moves in Extreme Low-Bit LLMs Xu Cao∗ Astrmira Tech. [email protected]

Wenxuan Xiao∗ Astrmira Tech. [email protected]

Abstract

arXiv:2609.09867v1 [cs.DB] 9 Sep 2026

Post-training quantizers accept a finite change of codes when it lowers a reconstruction proxy, and loss-aware variants score the change by the loss gradient at the current weights. We show that the utility of such a move to the deployed model has to be read in the state context the move traverses, and that this context has two parts. The first is the move’s own displacement: the gradient at the move’s midpoint integrates the curvature along the move that the current-state gradient omits. On 97 two-bit moves frozen before any endpoint was computed, 98 standard-GPTQ moves on Llama-3.2-1B and 392 replacements of an activation-ordered GPTQ solution on Llama-3.2-3B, it predicts the direction of the loss change on 96, 97 and 390 moves where the current-state gradient predicts 68, 85 and 153. The second is the other moves: on exhaustive lattices of legal states the utility of a set of moves is a nearly quadratic pseudo-Boolean function whose small pairwise term, 0.01–13% of the energy, decides the Pareto front, is exactly what a common-center search omits, and differs between functionals, which disagree in direction on 46 of the 98 GPTQ moves and 183 of the 392. A reconstruction-optimal code re-selection on Qwen3-4B improves the proxy in every group and leaves 2 of 255 lattice states improving both declared functionals, against 197 for the preceding level refit; a current-state linearization orders its states correctly on 2.75%, the midpoint on 99.6%, and the solver’s output loses 6.9 zero-shot HellaSwag points. Reading each move at its own midpoint and choosing additively repairs that step, gaining 2.7–2.9 accuracy points on three tasks never used in selection, and improves GPTQ without activation ordering on Llama-3.2-1B by 11% in unread test perplexity. At larger support the additive read stops: an exact-endpoint beam at the same budget finds a two-projection change that improves both functionals (−19% perplexity) and dominates the 61-change state; on the activation-ordered solution the 37-change one-shot state raises the loss where the beam’s two-projection state does not; and re-pricing the moves from the state reached after 16 changes reverses the ARC sign of 121 of 208. Utility is contextual at the granularity of a few moves, and composition must judge at exact endpoints from the state reached.

1

Introduction

A post-training quantization solver accepts a move between low-bit hard states because the move lowers a reconstruction proxy. The deployment judges the same move by next-token loss on the domains it serves, by agreement with a stronger teacher, or by the option geometry of the tasks it answers, and the two judgements come apart at every scale we have measured. A free two-bit codebook recovered on Qwen3-4B enters the NVFP4 grid at a cost of 0.84% in perplexity when its assignments are kept and only the codebook is snapped, and at 72.8% when a pipeline first re-runs Lloyd quantization, changing 9.74M assignments, and then applies the same snap. On the same model a standard solver step, re-selecting every group’s codes by exhaustive reconstruction-optimal assignment, improves the proxy in every group ∗

Equal contribution.

1

Preprint. Under review.

and destroys the function (Section 5). Whether a move preserves the function is decided by its context, in two senses, and both are structured enough to build on. The first context is the move’s own displacement. A finite move carries curvature that the gradient at the current state, where loss-aware criteria evaluate candidates, cannot see; the gradient at the move’s midpoint integrates that curvature exactly for quadratics (Proposition 1) and, on the moves solvers propose, almost exactly in practice. On 97 legal same-rate moves proposed by solvers on 0.6B and 4B models, the midpoint gradient predicts the sign of the loss change on 96 and orders the moves with Spearman 0.999. The currentstate gradient predicts 68, and the reconstruction proxy, which accepts every proposed move, is right on the 54 that improve. Along one 4B solver step 22 of 26 proxy-improving batches raise the loss and the midpoint orients every one. The ordering transfers to standard GPTQ. On Llama-3.2-1B, 10 of 75 moves that improve even the full layer reconstruction raise held-out loss and 35 worsen ARC option-KL, and the midpoint predicts 97/98 and 96/98 of the signs where the current-state gradient predicts 85 and 86; on Llama-3.2-3B with activation ordering, 390/392 and 391/392 against 153 and 357. The second context is the other moves. On exhaustive 256-state lattices of legal moves scored under three functionals, the utility of a set of moves is close to a quadratic pseudo-Boolean function (R2 0.97–1.00) whose linear coefficients are the moves’ own-midpoint effects and whose pairwise coefficients are mixed curvatures (Proposition 2). The pairwise term carries 0.01–13% of the energy and draws the Pareto front: an additive model recovers 37% of the exact front and a quadratic 87–90%, and six of eight moves improve ARC option-KL in one context and worsen it in another, which no monotone transform of an additive score admits. The same term is what a common-center search omits, with all nine drift intervals on a 1.7B bank excluding zero, and each functional has its own: held-out NLL and ARC option-KL disagree in direction on 46 of the 98 GPTQ moves and on 183 of the 392. The 4B solver step is these two structures at full scale. On exhaustive lattices of its moves, 2 of 255 states improve both declared functionals against 197 for the level-refit step that precedes it in the same solver, and a current-state linearization orders the re-selection states correctly on 2.75% of them where the midpoint does on 99.6%: the solver composes millions of proxy-improving moves judged at one stale center in one shot. The same structures say what a constructor has to evaluate. One backward at each move’s own midpoint and an additive choice repair that step: the selected state improves both functionals over the solver’s output on untouched test blocks, matches two 199-endpoint searches at half their cost, and gains 2.7–2.9 zero-shot accuracy points on three tasks never used in selection, where the solver’s output loses 6.9 on HellaSwag. On Llama-3.2-1B the same rule gives a state 11% lower in unread test perplexity than GPTQ without activation ordering. There the additive read stops. At support 61 of 112 the pairwise term is no longer small: an exact-endpoint beam at the same budget finds a two-projection change that improves both functionals and dominates the one-shot state; on the activation-ordered solution the 37-change one-shot state raises the loss where the beam’s two-projection state does not; and re-pricing the moves from the state reached after 16 changes reverses more than half of the ARC prices (Section 6).

Contributions. (i) A prospective measurement of where a move’s utility is read, with the curvature mechanism behind it (Sections 2, 3). (ii) Exhaustive lattice evidence that the utility of a set of same-rate moves is nearly quadratic, with the pairwise term deciding the Pareto front and its omission the measured drift of common-center search (Section 4). (iii) Exhaustive lattices on two consecutive steps of a 4B solver: a reconstruction-optimal code re-selection destroys the function through a narrow jointly improving region and current-state misordering (Section 5). (iv) Constructors: one-shot midpoint selection repairs that step with zero-shot accuracy gains and improves GPTQ without activation ordering by 11% perplexity on unread units; at larger support an exact-endpoint beam at matched budget beats it on both GPTQ solutions, and re-pricing from the reached state measures the price reversals that set the boundary (Section 6). 2

Preprint. Under review.

2

Legal Moves, Functionals, and Three Places to Measure

Hard states and legal moves. A low-bit hard state q assigns every weight group a codebook entry and per-group magnitudes; every state we compare is a legal state of the same code at identical bitrate, verified by decoding from packed form. Our primary format is a two-bit code (sign bit, strong/weak bit, two magnitudes per group of 128); Section 3 adds asymmetric INT3 with one scale and zero point per group of 128 as produced by standard GPTQ. A solver proposes a move d = q1 − q0 between two such states because it improves a local reconstruction proxy P ; along a recovery trajectory or between Pn two solvers the full proposal is large, so we split it into n disjoint primitive moves d = i=1 di on non-overlapping groups. Any subset x ∈ {0, 1}n then yields a legal state q0 + Dx with D = [d1 · · · dn ], so the bank spans an exact Boolean lattice of 2n hard states. Banks are frozen before any functional is measured. Declared functionals. A deployment cares about functionals Fj , j ∈ T : next-token NLL on a domain, full-vocabulary KL to a floating-point master, restricted option-KL or gold margin on a multiple-choice task. Each is an average over statistical units (2048-token blocks or item microbatches) whose per-unit values we keep for paired intervals. Write fj (x) = Fj (q0 + Dx) − Fj (q0 ),

x ∈ {0, 1}n ,

(1)

for the exact utility of a subset of moves under functional j. Three places to measure a move. For a single move d from q0 , three first-order quantities cost one gradient each: d⊤ ∇F (q0 ) at the current state, which is what loss-aware criteria use; 1 d⊤ ∇F (q0 + 12 d) at the move’s own midpoint; and d⊤ i ∇F (q0 + 2 d) at the common center of the full proposal, which one backward returns for every primitive at once. Proposition 1 (Midpoint identity). If ∇2 F is LH -Lipschitz on [q0 , q1 ] and m = (q0 + q1 )/2, then F (q1 ) − F (q0 ) = d⊤ ∇F (m) + r with |r| ≤ LH ∥d∥3 /24; the current-state expansion has remainder 12 d⊤ ∇2 F (ξ) d for some ξ on the segment, of order ∥d∥2 . The midpoint expansion is exact for quadratics because the even terms cancel; the currentstate one is not, and on the moves solvers propose the omitted curvature decides the sign about a third of the time (Section 3). The lattice as a quadratic pseudo-Boolean function. Every set function on {0, 1}n has a unique Möbius expansion X X f (x) = si xi + βik xi xk + R≥3 (x), si = f (ei ), βik = f (ei +ek ) − f (ei ) − f (ek ), i

i<k

(2) with f (0) = 0 and R≥3 collecting degree-three and higher terms; Section 4 measures how much of f lives in R≥3 . Proposition 2 (Coefficients under constant curvature). If ∇2 F ≡ H on the lattice’s convex hull, then R≥3 ≡ 0, ⊤ 1 1 ⊤ si = d⊤ i ∇F (q0 + 2 di ) = di ∇F (q0 ) + 2 di Hdi ,

βik = d⊤ i Hdk ,

(3)

and the common-center response of primitive i at the full-proposal midpoint m = q0 + 12 D1 P 1 is vi = d⊤ i ∇F (m) = si + 2 k̸=i βik . Consequently, for any subset x, X vi xi − f (x) = 21 x⊤ β (1 − x), (4) i

where β is the symmetric matrix with entries βik and zero diagonal. P P Along a path of moves the identity telescopes: F (qT ) − F (q0 ) = t d⊤ t ∇F (mt ) + t rt with each rt cubic in ∥dt ∥, and P under constant curvature the exact marginal of adding move i to a selected set S is si + k∈S βki (Proposition 3, Appendix A). Pricing each move at its own midpoint from the state actually reached therefore accumulates the pairwise term without 3

Preprint. Under review.

estimating β; Section 6 measures how far that carries. Section 3 reads the linear part of (2) off one gradient, Section 4 measures β and the front it sets, and Section 4.2 measures (4), a functional-specific term because Without constant curvature the common-center minus R 1 H is. 2 own-midpoint response is 12 0 d⊤ (m − 2t dS̄ ) dS̄ dt (Appendix A), the remaining gap to ∇ F S the exact endpoint being the cubic remainder of Proposition 1.

3

A Move’s Utility Is Read at Its Own Midpoint

We froze 97 solver-proposed moves before computing any endpoint and asked three predictors to order them. The moves come from four sources: 22 and 23 partial transitions in two 0.6B contexts (WA , WC ), and 26 dose batches along each of the two consecutive 4B solver steps of Section 5 (a level refit; a code re-selection). Each is a legal same-rate change of between one and 11.6M groups, dose batches doubling in size, and proxy-improving by construction. Predictors use calibration blocks and exact differences disjoint held-out blocks; the functional is next-token NLL. The midpoint response d⊤ ∇F (q0 + 12 d), estimated on this bank by a central difference at the midpoint (two forwards; the banks below use one hooked backward), recovers the exact sign on 96 of 97 moves with Spearman 0.9994. The current-state gradient d⊤ ∇F (q0 ) recovers 68 (Spearman 0.65), and the reconstruction proxy, which accepts every proposed move, is right on the 54 that improve (Table 5, Figure 2, Appendix B). Along the re-selection step 22 of 26 proxy-improving batches raise the loss, and 9 of 26 along the level refit; the midpoint orients all 52 moves of the two steps, the current-state gradient 12 and 20 of 26.

Standard GPTQ on Llama-3.2-1B. The same measurement on standard GPTQ (asymmetric INT3, groups of 128, block 128, sequential calibration, per-group scale and zero point, 1% damping; our implementation matches the reference to zero decoded-weight difference) gives Table 1. The 98 legal moves, frozen before any functional was read, are 75 layer-, family- and dose-batched interventions from RTN to the GPTQ codes and 23 from the 1% to a 0.1%-damping solution (Appendix B). Two reconstruction proxies are recorded, GPTQ’s group-local objective and the full layer reconstruction with its cross-group terms. All 75 RTN→GPTQ moves improve both, yet 10 raise held-out NLL (7 resolved), 35 worsen ARC option-KL (29 resolved), and the two functionals move in opposite directions on 37. The 23 low-damping moves improve the group objective and worsen the full-layer one on all 23, and 17 raise held-out NLL: the stronger proxy predicts that harm but, being one scalar, no functional conflict. On the calibration units the midpoint response recovers 97/98 NLL and 96/98 ARC signs (Spearman 0.9996 and 0.987) and predicts both directions jointly on 95/98, against 76 for the current-state gradient and 78 for a current-state Gauss–Newton estimate. Its NLL integration error is 30× below the current-state gradient and 39× below the Gauss–Newton estimate (10× and 11× for ARC). Across the data split every predictor loses: on held-out NLL the midpoint keeps 86/98 with Spearman 0.90, the current-state estimates 82 and 81 with 0.63–0.65. The midpoint integrates the move’s effect on the data it sees almost exactly; the change of data no calibration-side predictor escapes. Activation ordering changes the base model and not the result. With act-order and static groups the GPTQ(1%) model beats RTN on Llama-3.2-3B (NLL −0.23 [−0.27, −0.17]). Over all 392 single-projection replacements of it by RTN or by the 0.1%-damping codes, the midpoint recovers 390/392 NLL and 391/392 ARC signs (Spearman 0.9996 and 0.9993), the current-state gradient 153 and 357 (Spearman 0.29 on NLL); on the 1B act-order bank the counts are 223/224 and 222/224 against 50 and 179 (Appendix E). One backward at a common center returns the responses of all n primitives of a bank at once; Section 4 measures what that center omits. 4

Preprint. Under review.

Table 1: Sign agreement with the exact endpoint for 98 frozen standard-GPTQ moves on Llama-3.2-1B at INT3 g128 (Spearman in parentheses). NLL on the calibration units and across the held-out split; ARC-Easy option-KL on 592 label-free items; resolved: moves whose endpoint interval excludes zero (81 held-out NLL, 74 ARC); joint: both exact directions predicted. NLL, same Predictor

units

group-local reconstruction 70 (0.01) full-layer reconstruction 83 (0.45) current-state gradient 85 (0.70) current-state Gauss–Newton 84 (0.64) midpoint 97 (0.9996)

NLL, held-out

ARC option-KL

joint

all 98

resolved

all 98

resolved

98

71 (0.08) 82 (0.47) 82 (0.63) 81 (0.65) 86 (0.90)

58 71 69 69 75

49 (0.04) 54 (0.08) 86 (0.83) 89 (0.82) 96 (0.987)

36 39 69 73 74

— — 76 78 95

Table 2: Two exhaustive 256-state lattices, three functionals: Walsh energy by degree (share of the non-constant energy), R2 of the full quadratic fit, participation rank of its 8 × 8 pairwise matrix β, and four-fold cross-validated recall of the exact Pareto front. The stratified lattice was selected without any functional outcome. Walsh energy

Pareto recall 2

Lattice

Functional

deg. 2 deg. ≥ 3 R quad. p-rank β additive quadratic low-rank 4

witness witness witness

D0 NLL ARC KL Hella KL

1.70% 6.67% 11.43%

0.07% 1.86% 2.85%

0.9992 0.9802 0.9688

2.91 3.47 2.66

54.5%

63.6%

72.7%

stratified D0 NLL stratified ARC KL stratified Hella KL

1.20% 7.01% 12.74%

0.03% 0.92% 1.10%

0.9997 0.9907 0.9889

2.24 3.42 2.45

36.7%

86.7%

90.0%

4

The Utility of a Set of Moves Is Nearly Quadratic, and the Small Pairwise Term Draws the Front

Section 3 settles the linear coefficients of (2); this section measures the rest of the expansion on two exact Boolean lattices. 4.1

Two exhaustive lattices

From a 23-primitive 0.6B bank we selected two sublattices of eight primitives each: a witness lattice containing two pairs earlier runs had found to compose plus four primitives chosen by manifest strata, and a stratified lattice chosen from manifest metadata alone. All 256 legal states of each were scored under held-out next-token NLL (64 blocks), ARC-Easy option-KL to the floating-point master (592 items) and HellaSwag option-KL (400 items), with one backward at each of 14 centers returning all eight responses per statistical unit (Appendix C). Table 2 gives the structure (Figure 3). Degree-two terms carry 1.2–1.7% of the non-constant Walsh energy for D0 and 6.7–12.7% for the task functionals; degree-three-and-higher terms carry 0.03–1.1% on the stratified lattice and 0.07–2.9% on the witness lattice. On the stratified lattice the quadratic explains 98.9–99.97% of the variance and cuts the additive model’s cross-validated RMSE 2.5–5.6×. Four controls locate this structure (Table 9). Freeing the additive model’s intercept recovers 30% of the stratified front but ties the quadratic on the witness lattice (63.6%), whose front separates the two under no metric. A monotone transform of an additive score h(a⊤ x), fitted by isotonic regression with Isotron direction updates, recovers 45% and 30% of the two exact fronts against 73% and 90% for the rank-4 quadratic, and declares 8 and 11 states safe that are not. Reversals exclude the class itself. Under any monotone h the same move cannot improve in one context and worsen in another, yet on the stratified lattice 6 of 8 moves do so for ARC option-KL and 4 of 8 for HellaSwag (simultaneous intervals), and 14 and 11 of the 28 pairs reverse their order; the witness lattice 5

Preprint. Under review.

gives 5, 4, 14 and 12. The quadratic also extrapolates. Fitted only on the 93 states of support at most three, it recovers 92% of the exact front among the 163 states of support four to eight on the stratified lattice with no false-safe state, against 52% for the additive model with or without intercept and 32% for the single-index model, and halves the ARC and HellaSwag RMSE on those states; on the witness lattice the additive, affine and quadratic models all recover 80%. The pairwise term is small in energy and decisive in geometry. The stratified lattice has 30 exact Pareto states, of which the additive model recovers 11 and the rank-4 quadratic 27, and the gap persists under paired resampling (36% against 84–88% mean recall). Set recall does not order single-state choice, however. Under a fixed utility (least worst standardized functional among states predicted to improve all three) the full quadratic selects a state with regret 0.040 against 0.072 for both additive models, while the rank-4 quadratic, best on membership, selects worse (0.086). Section 6 tests construction at exact endpoints. 4.2

The omitted term, measured as drift

P Proposition 2 says a common-center linear score i vi xi overshoots the exact utility by 1 ⊤ 2 x β(1 − x), half the interaction between the selected and the omitted moves. On a 1.7B model with a 25-primitive bank and three functionals (two NLL domains, a CommonsenseQA option-KL target) we recomputed, for three frozen candidates on the same units, the response at the full-proposal midpoint and at the candidate’s own midpoint. All nine drift intervals exclude zero and seven of nine reverse the sign of the prediction (Table 14). For the support-4 candidate the same selected–omitted interaction drifts D0 by −0.301 [−0.374, −0.228] and the target by +0.073 [+0.066, +0.080]: opposite signs that no scalar property of the candidate produces and a matrix βj differing by functional does. 4.3

The functional is part of the context

The coefficients of (2) carry a functional index, and the orderings they induce differ by about half. On the 98 standard-GPTQ moves of Table 1, held-out NLL and ARC-Easy option-KL move in opposite directions on 46, and on 31 of the 62 moves where both effects are resolved. The functional-specific midpoint responses classify conflict on 95/98 moves on the calibration units (59/62 resolved across the split), where a single reconstruction direction is right on 52/98. On the act-order banks the two functionals conflict on 81 of 224 replacements (1B) and 183 of 392 (3B), and the midpoint classifies conflict on 389/392 where the current-state gradient does on 164. On Qwen3-4B a re-selection batch of 65,536 groups that improved held-out NLL by 0.00259 lowered the MMLU and ARC-Challenge gold margins by 0.054 and 0.065, and a state certified on HellaSwag improved teacher-KL, gold margin and option cross-entropy on 1,820 items while changing no top-1 prediction (Appendix F). For an undeclared functional the sign is unconstrained by the declared responses (Appendix A.4). 4.4

What gradient probes recover

With r(z) = D⊤ ∇F (q0 + 12 Dz), constant curvature gives 2[r(z) − r(0)] = Bz with B = D⊤ HD, so β could be read from gradients alone. Not quite: the operator fitted to all thirteen non-zero centers reproduces the sign of the exact β on 64–93% of pairs, and the operator from the eight unit centers mispredicts the response at the full and random centers on 31–39 of 40 components. A function can agree with a quadratic on every vertex and carry a different gradient field inside the cube (any xi (1 − xi )h(x) vanishes at the vertices), so exhaustive endpoints recover what gradient probes cannot. As a screen the response quadratic still reproduces 92–100% of exact signs.

5

What Destroys the Function Is the Solver’s Move, Not the Grid

The 4B moves of Section 3 are two consecutive steps of one solver acting on a Qwen3-4B master trained inside the grid: from the grid-legal hard state it deploys, the solver first 6

Preprint. Under review.

frees each group’s two magnitudes with codes fixed (the level-refit step), then re-selects each group’s codes by exhaustive magnitude-ranked search with free levels (the re-selection step); both lower the reconstruction objective in every group they touch, 21.8M groups in the second. A third move snaps the classic arm’s free two-bit codebook onto the NVFP4 grid with its code planes kept. Each move was split into 26 proxy-ranked dose batches, eight selected by manifest strata alone, and all 256 states of each lattice scored under D2 NLL and ARC-Challenge option-KL with 18 response centers (Appendix D, Table 10 and Figure 4). After the level refit, 197 of 255 states improve both functionals and the full eight-batch state is safe (−0.0042 D2, −0.068 ARC option-KL). After the re-selection, two states do, a single batch and one pair. The full eight-batch state, which is what accepting every proxy-improving move produces, worsens D2 by 0.0081 and ARC option-KL by 0.214, and the complete 26-batch output lowers zero-shot HellaSwag accuracy by 6.9 points and MMLU by 0.8 (Table 3). A current-state linearization summed over the selected moves predicts the D2 sign of the re-selection states on 2.75% of them, declares 33 states safe that are not, and selects a state whose exact D2 change is +0.0026 against a prediction of −0.0056. The same moves scored at their own midpoints give 99.6%, one false-safe state, and the safe pair. Interaction is secondary here (degree-two Walsh energy 2.4% and 4.6% on the re-selection lattice against 0.014% and 0.24% on the refit lattice) and its harm concentrates at high support. The grid snap is a third regime. Its lattice is additive to within 0.12% of its Walsh energy, no state improves both functionals with an interval excluding zero, and the full snap costs +0.0018 in D2 (interval spanning zero) and +0.0098 in ARC option-KL with all 586 items worsening. Within this move family, choosing which groups to snap with codes fixed, the format change is a small additive cost that no subset of the snap avoids by a resolved margin; a reconstruction-optimal code reassignment is a large, avoidable one that the current-state gradient cannot see. The same distinction resolves a historical number. An earlier pipeline first re-ran Lloyd quantization on the classic arm’s codebook, changing 9.74M strong-plane assignments, and then applied the grid snap: perplexity 20.93 → 36.17 (+72.8%), against 21.10 (+0.84%) when the assignments are kept and only the codebook is snapped; the snap is 1.53% of the historical increase on the log scale (Appendix D). The cost attributed to the format was the cost of a solver move judged by its proxy at a stale center: preserve the learned assignments, project only the codebook, and evaluate any code reassignment as a move on the deployment functional.

6

From Structure to Construction

Sections 3 and 4 prescribe what a constructor must evaluate: each move at its own midpoint, and each set of moves with its pairwise terms under every declared functional. Two constructors. One-shot midpoint selection: one hooked backward at each candidate’s own midpoint, then the additive choice of the set optimizing one declared functional subject to the others not worsening. Recentered Pareto beam: keep non-dominated intermediate states, extend each by candidates that leave it non-dominated, judge candidates at exact endpoints and recenter after every extension (a low-rank variant that estimates β: Appendix C). The reference competitor for both is exact endpoint search at the same budget; a midpoint backward costs 1.5–3.6 endpoint forwards (Appendix G). Repairing the re-selection step. Table 3 applies both to the 26-batch re-selection bank of Section 5. The one-shot and direct states beat the solver’s output on both functionals with intervals excluding zero (one-shot by 0.057 [0.033, 0.084] NLL and 0.166 [0.106, 0.224] ARC option-KL), and the screened beam does so at the point estimates. All three beat the start on ARC-Challenge option-KL and none resolves held-out NLL against it. The one-shot state, at half the budget, is separated from neither search by a resolved interval, and the gradient-free endpoint search selects eight of its nine batches. The screened beam, which cut candidates to four per state by the current-state row before judging them at exact endpoints, loses to the direct search on ARC by 0.041 [0.022, 0.059] at the same budget; in its nine contexts midpoint pricing reproduces the exact local choice 9/9 where the current row does 7

Preprint. Under review.

Table 3: Repairing the 4B code re-selection step: constructors frozen on the fit split, one state selected on validation, read once on untouched test blocks; changes from the step’s input, paired 95% intervals. Accuracy: zero-shot label choice (ARC-Challenge on the 586 test items never used in selection, HellaSwag 10,042, MMLU 14,042). The solver’s output, all 26 batches, changes these by +0.7, −6.9 [−7.7, −6.1] and −0.8 [−1.4, −0.1] points. one-shot midpoint selection

current-screened beam

direct endpoint beam

selection cost batches selected ∆ held-out NLL ∆ ARC-C option-KL

26 midpoint bwd. (≈94 fwd.) 9 −0.007 [−0.022, +0.007] −0.211 [−0.280, −0.139]

199 fwd.-equivalents 6 −0.0005 [−0.0073, +0.0068] −0.186 [−0.251, −0.121]

199 endpoint fwd. 8 −0.0004 [−0.0070, +0.0078] −0.226 [−0.301, −0.155]

ARC-C acc. (pp) HellaSwag acc. (pp) MMLU acc. (pp)

+2.7 [+0.3, +5.1] +2.9 [+2.1, +3.7] +2.8 [+2.2, +3.4]

+2.1 [−0.2, +4.3] +0.8 [+0.1, +1.4] +1.6 [+1.1, +2.2]

+4.1 [+1.7, +6.7] +2.4 [+1.7, +3.2] +3.1 [+2.6, +3.8]

Table 4: Constructors on standard GPTQ, Llama-3.2-1B INT3: each of 112 projections keeps its GPTQ codes or takes a same-rate alternative (RTN or 0.1% damping). Three one-shot additive selections (minimize predicted ∆ARC-Easy option-KL subject to ∆NLL ≤ 0) differ only in the price of a single replacement; the exact-endpoint beam spends the midpoint pricing budget (352 endpoint forwards) on exact endpoints. Changes from the GPTQ model on 114 unread test blocks and 297 unread items, paired 95% intervals. Upper block: GPTQ without activation ordering (perplexity 24.67 → 21.94 for the midpoint state, → 20.07 for the beam). Lower block: the activation-ordered GPTQ solution, same rules and budget (all three one-shot prices: Appendix E). constructor

replacements

∆ held-out NLL

∆ ARC-E option-KL

one-shot, own-midpoint price one-shot, current-state price one-shot, exact-endpoint price exact-endpoint beam

61 86 73 2

−0.117 [−0.223, −0.020] +0.451 [+0.262, +0.659] −0.109 [−0.222, −0.003] −0.206 [−0.260, −0.156]

+0.005 [−0.019, +0.030] +0.011 [−0.014, +0.035] +0.044 [+0.017, +0.071] −0.029 [−0.041, −0.017]

one-shot, own-midpoint price exact-endpoint beam

37 2

+0.033 [+0.006, +0.062] −0.001 [−0.004, +0.001]

−0.000 [−0.016, +0.014] −0.003 [−0.008, +0.001]

5/9. Its midpoint rows (72/72 exact signs against 32/72 for the current row) never entered its decisions, so its repair is exact-endpoint judging from the state reached. On three zero-shot tasks never used in selection the one-shot state gains 2.7–2.9 accuracy points over the step’s input, where the solver’s own output loses 6.9 on HellaSwag. At this support one midpoint read per move is enough, and screening at the current state costs 0.04 of option-KL. Standard GPTQ, and where the additive read stops. A constructor chooses per projection among the GPTQ codes and two same-rate alternatives, RTN and the 0.1%damping codes (Table 4). Midpoint prices select 61 replacements whose state is 11% lower in unread test perplexity than the GPTQ model, with ARC option-KL where it was. The objective did not move for any price because the composition, not the reads, is wrong. On the calibration units the midpoint state’s ARC prediction misses by +0.464: the 61 single-move reads are off by 0.005 in total, their composition by +0.459. The same budget spent on exact endpoints from the state reached finds, in one beam expansion, a two-projection change that improves both functionals (perplexity −19%) and beats the 61-change state on both (−0.089 [−0.164, −0.009] NLL, −0.034 [−0.057, −0.011] ARC). The activation-ordered solution leaves less to repair and shows the same boundary (Table 4, lower block): the midpoint one-shot changes 37 projections and raises test NLL by 0.033 [0.006, 0.062] against a predicted −0.0001, while the two-projection state of the same beam improves both functionals at the point estimate and beats it on NLL by 0.034 [0.008, 0.062]. Re-pricing the 224 alternatives at their midpoints from the state reached after 16 selected moves shows why (Figure 1). The ARC price of 121 of 208 comparable moves changes sign (rank correlation −0.25) while NLL prices keep rank 0.94, and a selection that recenters after every 16 moves improves both functionals on its first round on the development units and then trades them (−0.046 ARC, +0.090 NLL against its start on fresh units). A move’s utility is contextual at the granularity of a few moves. The constructor that respects it judges at exact endpoints from the state reached, 8

Preprint. Under review.

NLL: 25/208 reversed, ρ = +0.94

ARC option-KL: 121/208 reversed, ρ = −0.25

price after 16 edits (nats)

price after 16 edits (nats)

10−1 10−2 10−3 0 −10

−3

−10−2 −10−1 −10−1−10−2−10−3 0 10−3 10−2 10−1

10−2 10−3 0 −10−3 −10−2

−10−2 −10−3

price at the state before (nats)

0

10−3

10−2

price at the state before (nats)

same sign at both states

sign reversed

Figure 1: The same 208 moves priced at their own midpoints from two states 16 moves apart on the Llama-3.2-1B standard-GPTQ bank (start of the recentered selection and the state after its first round; symmetric-log axes, shaded quadrants are sign reversals). NLL prices keep sign and rank; ARC option-KL prices reverse on 121 of 208. A move’s utility for the objective functional is a property of the state it is read from. Both pairs of consecutive states: Appendix E.

sparse where the solver’s damage is sparse (two projections at 1B, eight of 26 batches at 4B); one midpoint read per move suffices where the support is small enough for the additive read to hold.

7

Related Work

Reconstruction-based and loss-aware PTQ. AdaRound (Nagel et al., 2020), BRECQ (Li et al., 2021), GPTQ, AWQ and OmniQuant (Frantar et al., 2023; Lin et al., 2024; Shao et al., 2024) optimize layer- or block-local reconstruction objectives and produce the moves we study. GuidedQuant (Kim et al., 2025) and DiscQuant (Chee et al., 2025) bring end-loss gradients into the layer objective and into rounding, evaluated at the current state; layer-local objectives are documented ignoring cross-block interactions (Shabanovi et al., 2024) and selecting high-loss grid points (Li et al., 2026). Set functions over quantization configurations. The coverage model (Hill, 2026) treats layer-level mixed-precision damage as a set function that saturation renders additive up to a monotone transform, the class Section 4 excludes for same-rate code edits; CoopQ (Zhao et al., 2025) (Shapley layer interactions) and cross-layer error compensation (Noda, 2026) work at layer granularity. Task-aware quantization and multi-objective selection. TaCQ (Xiao et al., 2025) and TAQ (LeVi et al., 2025) allocate precision from task gradients and hidden-state statistics; MGDA and PCGrad (Sener & Koltun, 2018; Yu et al., 2020) reconcile conflicting gradients in training; the fit/validation/test protocol follows Berk et al. (2013); Wasserman & Roeder (2009). Scope and limitations. The exhaustive lattices have eight primitives each, on one 0.6B bank and three moves of one 4B model; move-level results cover two Qwen3 scales and Llama-3.2-1B and 3B; intervals are paired bootstrap or Student-t over calibration units; no constructor uses the pairwise term itself. 9

Preprint. Under review.

Reproducibility Statement Every quantity in this paper is a paired difference between two legal hard states, or a gradient response, measured on stated statistical units; the appendix gives the construction rule of each bank, the split and selection rule of each constructor, which was fixed before the evaluation units were read, and the unit counts behind each interval (Appendices B–E). Proofs of Propositions 1–3 are in Appendix A. Our GPTQ implementation matches the reference implementation to zero decoded-weight difference in both configurations used (Appendix B and E). The anonymous supplementary material contains the code, the frozen banks with their selection records, the per-unit measurements behind every table, and the scripts that regenerate Figures 1–3 from those measurements, each script recomputing the numbers it draws and stopping if they differ from the ones printed here.

AI Use Statement Generative AI coding assistants were used to implement the measurement and search code and to run and monitor the experiments. The resulting GPTQ kernel was verified against the reference implementation to zero decoded-weight difference, and every number in the manuscript was traced by the authors to its source record. We have reviewed all AI-assisted work and take responsibility for the final content of this work, including text, claims and artifacts produced with the aid of generative AI.

References Richard Berk, Lawrence Brown, Andreas Buja, Kai Zhang, and Linda Zhao. Valid post-selection inference. The Annals of Statistics, 41(2):802–837, 2013. doi: 10.1214/ 12-AOS1077. URL https://doi.org/10.1214/12-AOS1077. Jerry Chee, Arturs Backurs, Rainie Heck, Li Zhang, Janardhan Kulkarni, Thomas Rothvoss, and Sivakanth Gopi. DiscQuant: A quantization method for neural networks inspired by discrepancy theory. arXiv preprint arXiv:2501.06417, 2025. URL https://arxiv.org/ abs/2501.06417. Elias Frantar, Saleh Ashkboos, Torsten Hoefler, and Dan Alistarh. GPTQ: Accurate posttraining quantization for generative pre-trained transformers. In International Conference on Learning Representations, 2023. URL https://arxiv.org/abs/2210.17323. Joshua Hill. Saturation makes quantization error additive: A coverage model with a certificate. arXiv preprint arXiv:2607.12266, 2026. URL https://arxiv.org/abs/2607.12266. Jinuk Kim, Marwa El Halabi, Wonpyo Park, Clemens J. S. Schaefer, Deokjae Lee, Yeonhong Park, Jae W. Lee, and Hyun Oh Song. GuidedQuant: Large language model quantization via exploiting end loss guidance. In Proceedings of the 42nd International Conference on Machine Learning, 2025. URL https://arxiv.org/abs/2505.07004. Amit LeVi, Raz Lapid, Rom Himelstein, Chaim Baskin, Ravid Shwartz-Ziv, and Avi Mendelson. You had one job: Per-task quantization using LLMs’ hidden representations. arXiv preprint arXiv:2511.06516, 2025. URL https://arxiv.org/abs/2511.06516. Hanyang Li, Jianhao Ma, and Ying Cui. Understanding quantization-aware training: Gradients at quantized weights bias to the low-loss basin. arXiv preprint arXiv:2606.09012, 2026. URL https://arxiv.org/abs/2606.09012. Yuhang Li, Ruihao Gong, Xu Tan, Yang Yang, Peng Hu, Qi Zhang, Fengwei Yu, Wei Wang, and Shi Gu. BRECQ: Pushing the limit of post-training quantization by block reconstruction. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=POWv6hDd9XH. 10

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Ji Lin, Jiaming Tang, Haotian Tang, Shang Yang, Wei-Ming Chen, Wei-Chen Wang, Guangxuan Xiao, Xingyu Dang, Chuang Gan, and Song Han. AWQ: Activation-aware weight quantization for on-device LLM compression and acceleration. In Proceedings of Machine Learning and Systems, volume 6, 2024. URL https://proceedings.mlsys.org/paper_files/ paper/2024/hash/42a452cbafa9dd64e9ba4aa95cc1ef21-Abstract-Conference.html. Markus Nagel, Rana Ali Amjad, Mart van Baalen, Christos Louizos, and Tijmen Blankevoort. Up or down? adaptive rounding for post-training quantization. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 7197–7206. PMLR, 2020. URL https://proceedings.mlr.press/ v119/nagel20a.html. Ryo Noda. Cross-layer error compensation and finite-sample feature-statistics matching for extreme low-bit quantization of large language models. arXiv preprint arXiv:2607.14630, 2026. URL https://arxiv.org/abs/2607.14630. Ozan Sener and Vladlen Koltun. Multi-task learning as multi-objective optimization. In Advances in Neural Information Processing Systems, volume 31, 2018. URL https://proceedings.neurips.cc/paper_files/paper/2018/hash/ 432aca3a1e345e339f35a30c8f65edce-Abstract.html. Khasmamad Shabanovi, Lukas Wiest, Vladimir Golkov, Daniel Cremers, and Thomas Pfeil. Interactions across blocks in post-training quantization of large language models. arXiv preprint arXiv:2411.03934, 2024. URL https://arxiv.org/abs/2411.03934. Wenqi Shao, Mengzhao Chen, Zhaoyang Zhang, Peng Xu, Lirui Zhao, Zhiqian Li, Kaipeng Zhang, Peng Gao, Yu Qiao, and Ping Luo. OmniQuant: Omnidirectionally calibrated quantization for large language models. In International Conference on Learning Representations, 2024. URL https://proceedings.iclr.cc/paper_files/paper/2024/hash/ c6483c8a68083af3383f91ee0dc6db95-Abstract-Conference.html. Larry Wasserman and Kathryn Roeder. High-dimensional variable selection. The Annals of Statistics, 37(5A):2178–2201, 2009. doi: 10.1214/08-AOS646. URL https://doi.org/10. 1214/08-AOS646. Hanqi Xiao, Yi-Lin Sung, Elias Stengel-Eskin, and Mohit Bansal. Task-circuit quantization: Leveraging knowledge localization and interpretability for compression. In Conference on Language Modeling (COLM), 2025. URL https://arxiv.org/abs/2504.07389. Tianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. In Advances in Neural Information Processing Systems, volume 33, 2020. URL https://proceedings.neurips.cc/paper_ files/paper/2020/hash/3fe78a8acf5fda99de95303940a2420c-Abstract.html. Junchen Zhao, Ali Derakhshan, Jayden Kana Hyman, Junhao Dong, Sangeetha Abdu Jyothi, and Ian Harris. CoopQ: Cooperative game inspired layerwise mixed precision quantization for LLMs. arXiv preprint arXiv:2509.15455, 2025. URL https://arxiv.org/abs/2509. 15455.

A

Proofs

A.1

Midpoint identity (Proposition 1)

Proof. Let u = d/2. Second-order Taylor expansion with integral remainder about m gives F (m ± u) = F (m) ± ∇F (m)⊤ u + 12 u⊤ ∇2 F (m)u + R± with Z 1 Z 1 LH ∥u∥3 |R± | = (1−s) u⊤ [∇2 F (m±su)−∇2 F (m)]u ds ≤ (1−s) LH s∥u∥3 ds = . (5) 6 0 0 Subtracting, the even terms cancel: F (q1 )−F (q0 ) = 2∇F (m)⊤ u+(R+ −R− ) = d⊤ ∇F (m)+r with |r| ≤ 2LH ∥d/2∥3 /6 = LH ∥d∥3 /24. For the current-state expansion, Taylor’s theorem with Lagrange remainder gives F (q1 ) − F (q0 ) = d⊤ ∇F (q0 ) + 12 d⊤ ∇2 F (ξ)d for some ξ on the segment. 11

Preprint. Under review.

An endpoint-centered bound must control two paths from W to the endpoints and scales as LH 3 3 6 (∥a − d/2∥ + ∥a + d/2∥ ) with a = m − W ; its ratio to the midpoint bound is at least 1 3 8(∥a∥/∥d∥ − 2 )+ , which grows without limit as the move becomes small relative to the total quantization displacement. A.2

Coefficients under constant curvature (Proposition 2)

Proof. With ∇2 F ≡ H on the convex hull of the lattice, F (q0 +Dx)−F (q0 ) = x⊤ D⊤ ∇F (q0 )+ 1 ⊤ ⊤ ai = d⊤ = D⊤ HD. Since x2i = xi on {0, 1}n , i ∇F (q0 ) and B P 2 x D HDxPexactly. Write P P 1 ⊤ 1 1 i Bii xi + i<k Bik xi xk , hence f (x) = i (ai + 2 Bii )xi + i<k Bik xi xk and 2 x Bx = 2 R≥3 ≡ 0. Comparing with the Möbius expansion (2) gives si = ai + 12 Bii and βik = Bik = 1 ⊤ d⊤ + 12 d⊤ . At the i Hdk . The own-midpoint response is di ∇F (q0 + 2 di ) = ai P i Hdi = si P 1 1 ⊤ full-proposal midpoint m = q0 + 2 D1, vi = di ∇F (m) = ai + 2 k Bik = si + 12 k̸=i βik . P P P P Summing over a subset, vx = xi + 12 i k̸=i βik xi , and subtracting f (x) = i siP P P Pi i i P P P 1 1 i si xi + i<k βik xi xk = i si xi + 2 i k̸=i βik xi xk leaves 2 i k̸=i βik xi (1 − xk ) = 1 ⊤ 2 x β(1 − x). Without constant curvature, for a subset S with displacement dS and complement dS̄ , the fundamental theorem of calculus along the segment from the own midpoint mS = q0 + 12 dS to the proposal midpoint m = mS + 12 dS̄ gives Z 1 X
⊤ 2 1 t (6) vi − dS ∇F (mS ) = 2 d⊤ S ∇ F m − 2 dS̄ dS̄ dt, i∈S

0

which reduces to 21 x⊤ β(1 − x) when the Hessian is constant. A.3

Sequential midpoint pricing

Proposition 3 (Sequential midpoint P Ppricing). For a path qt =3qt−1 + dt with midpoints mt , F (qT ) − F (q0 ) = t d⊤ t ∇F (mt ) + t rt with |rt | ≤ LH,t ∥dt ∥ /24, where LH,t bounds the Hessian’s Lipschitz constant on segmentP t; under constant curvature every rt vanishes, and within a fixed binary chart q(S) = q + 0 i∈S di the exact marginal of adding move i to an P already selected set S is si + k∈S βki . (For categorical replacements the constructors of Section 6 use the pathwise identity with state-specific moves.) Proof. Apply Proposition 1 to each segment [qt−1 , qt ] and sum; the telescoping left-hand sides give F with P(qT ) − F (q0 ). Under constant curvature H⊤each rt is zero. P For⊤the marginal, 1 ⊤ qS =P q0 + k∈S dk and m = qS + 12 di , d⊤ i ∇F (m) = di ∇F (q0 ) + k∈S di Hdk + 2 di Hdi = si + k∈S βki by (3). A.4

Undeclared functionals

Let an algorithm choose x = ̸ 0 from the responses of the declared functionals alone. For any ⊤ undeclared functional pick a response row w+ with w+ x > 0 and set w− = −w+ ; the two worlds present identical declared inputs, hence identical output x, and opposite signs for the undeclared change. No function of the declared responses constrains that sign; a measured response for the undeclared functional does, as the D5 prediction in Section 4.3 illustrates.

B

The 97-Move Bank

Sources: 22 partial transitions in a 0.6B context WA , 23 in WC , and 26 dose transitions along each of two consecutive steps of a solver acting on the Qwen3-4B master (bank labels M A B, the level-refit step from the grid-legal constrained hard state to free levels with codes fixed, and M B C, the code re-selection step by exhaustive magnitude-ranked assignment with free levels). A dose transition is a proxy-ranked batch of groups from the full proposal; each 12

Preprint. Under review.

is a legal same-rate move. Predictor and exact passes ran as separate jobs with endpoint outcomes unknown at prediction time. Table 5: Sign and rank agreement with the exact functional difference on the 97 frozen moves. Predictor ⊤

Reconstruction proxy d D(m−W ) Current state d⊤ ∇F (q0 ) Own midpoint d⊤ ∇F (q0 + 21 d)

predictor (signed log, per-panel scale)

1.5 1.0

(a) reconstruction proxy d D(m W)

3

sign 54/97 = 0.089

2

Sign

Pearson

Spearman

54/97 68/97 96/97

0.269 0.830 0.9995

0.089 0.652 0.9994

(b) current state d

F(q0)

(c) candidate midpoint d

sign 68/97 = 0.652

2

sign 96/97 = 0.999 single error (both near zero)

0.5

1

1

0.0

0

0

1

1

0.5

0.6B WA 0.6B WC 4B A B 4B B C

1.0 1.5

2

1

0

exact F (signed log)

1

2 3

2

F(m)

2 2

1

0

exact F (signed log)

1

2

2

1

0

exact F (signed log)

1

2

Figure 2: Three predictors against the exact difference on the same 97 moves; green quadrants are sign agreement. (a) Nearly every proxy value is negative because a solver only proposes moves its objective likes, which leaves the proxy near chance at ordering them. (b) The current-state gradient correlates but misorients the 4B re-selection cluster. (c) The midpoint response tracks the exact difference across the full dynamic range. Signed-log axes, per-panel y transform; only sign and rank are compared.

Table 6: Per-source results against the exact held-out functional difference. Source

Moves

Midpoint

Current

Proxy

Pearson

Spearman

0.6B WA 0.6B WC 4B level refit 4B code re-selection

22 23 26 26

21/22 23/23 26/26 26/26

16/22 20/23 20/26 12/26

13/22 20/23 17/26 4/26

0.99955 0.99912 0.99994 0.99967

0.99774 0.99901 0.99795 0.99385

Total

97

96/97

68/97

54/97

0.99953

0.99938

The midpoint response was estimated by a central difference at the midpoint; the two step sizes (h = 1/8 and 1/16) agree in sign on all 97 moves. Edge sizes follow cumulative power-of-two boundaries plus the exact final eligible count of each source (WA up to 524,288 groups, WC 1,048,576, level refit 11,608,025, re-selection 8,388,608). The single sign error is WA move e10: 512 groups, proxy −1.44 × 10−7 , midpoint +4.58 × 10−5 , exact −5.12 × 10−5 . Median residual |midpoint − exact|/|exact| per source: 1.44%, 1.47%, 0.36%, 1.18%. Scoring time: 9,683 s for all 97 moves on one GPU. Descriptive sample cost: the midpoint estimate stabilized on fewer calibration blocks than direct endpoint differencing on 4 moves, the same number on 90, and more on 3. Llama-3.2-1B standard-GPTQ bank. Model unsloth/Llama-3.2-1B (16 layers, 112 quantized projections); asymmetric INT3, row-wise groups of 128, block 128, dynamic per-group min/max scale and zero point, upper-Cholesky conditional error feedback, true sequential calibration (each layer’s inputs from the already quantized preceding layers), 128 calibration sequences of 2048 WikiText-103 tokens, FP32, no activation ordering, no MSE clipping; damping 1% (the GPTQ model) and 0.1% (the alternative). Our kernel reproduces the reference implementation (Frantar et al., 2023) to zero maximum decoded-weight difference on a real 2048 × 2048 projection; generation takes about 166 s per model. Rate: 3-bit codes plus identical scale/zero metadata, 3.375 bits per quantized projection weight for every state compared. On the calibration units, RTN → GPTQ(1%): NLL 3.519 → 2.946 13

Preprint. Under review.

(−0.573 [−0.743, −0.350]), ARC-Easy option-KL 0.259 → 0.229 (−0.030 [−0.049, −0.011]); GPTQ(1%)→(0.1%): NLL +0.439 [+0.103, +0.807], ARC −0.021 [−0.038, −0.005]. Audit bank: 98 legal interventions frozen from the three states, 75 RTN→GPTQ(1%) and 23 GPTQ(1%)→(0.1%), layer, projection-family and dose-chain batches, each scored in its own left context; NLL predictors on 28 FIT blocks, endpoints on those blocks and on 28 held-out WikiText-103 test blocks; ARC on 592 label-free ARC-Easy development items. Two reconstruction scores per move from the GPTQ(1%) calibration covariance: group-local (GPTQ’s objective) and full contextual layer reconstruction including cross-group terms. Predictor comparison is Table 1; same-unit NLL mean absolute error 0.0338 (current), 0.0431 (Gauss–Newton), 0.00111 (midpoint); ARC 0.0085, 0.0097, 0.00086. Held-out NLL versus ARC exact directions: both improve 37, NLL improves and ARC worsens 34, NLL worsens and ARC improves 12, both worsen 15; on the 62 moves resolved on both, 23/25/6/8. By source: RTN→GPTQ 75 moves, midpoint 75/75 NLL (FIT), 74/75 ARC, 67/75 held-out NLL, conflicts 37/75 (27/50 resolved); GPTQ(1%)→(0.1%) 23 moves, 22/23, 22/23, 19/23, conflicts 9/23. NLL scoring 7,130 s on one GPU. Llama-3.2-1B one-shot selection on standard GPTQ. State space: each of the 112 projections takes its GPTQ(1%) codes, its RTN codes or its GPTQ(0.1%) codes with the matching scale/zero, 224 single replacements from the GPTQ(1%) model, chosen without any functional score. FIT: the 28 calibration blocks and 592 items; VALIDATION 124 blocks and 297 items; TEST 114 blocks and 297 items, blocks and items never read before this construction (the items are the remaining clean ARC-Easy development ids). Selector: minimize predicted ∆ARC option-KL subject to predicted ∆NLL≤ 0, one alternative per projection, no support limit, solved to zero gap; prices from the own-midpoint derivative of each replacement, from one current-state gradient row, or from each replacement’s exact FIT endpoint; candidates and the midpoint primary frozen before VALIDATION/TEST. Selected 61/86/73 replacements. Midpoint state: VALIDATION −0.0872 [−0.1340, −0.0409] NLL, +0.0003 [−0.0287, +0.0288] ARC; TEST −0.1172 [−0.2230, −0.0200] (perplexity 24.67 → 21.94, relative change CI [−20.0%, −2.0%]), +0.0053 [−0.0193, +0.0295]. Current state: TEST +0.4505 [+0.2621, +0.6589], +0.0108 [−0.0139, +0.0349]. Exact-singleton state: TEST −0.1085 [−0.2218, −0.0028], +0.0439 [+0.0169, +0.0714]. Paired midpoint minus exact-singleton: TEST ARC −0.0386 [−0.0535, −0.0236], NLL −0.0086 [−0.0546, +0.0390]; midpoint minus current: TEST NLL −0.5677 [−0.7298, −0.4206]. Predicted composition on FIT from base F0 = 0.2292: ∆ARC −0.782 (current), −0.452 (midpoint), −0.473 (exact singleton), each below −F0 ; KL non-negativity therefore bounds the exact singleton state’s FIT residual P R(S) = F (qS ) − F (q0 ) − g∈S [F (qg ) − F (q0 )] by R(S) ≥ 0.2437 (measured below: +0.524). Cost on FIT: one current row 43 s; 224 own-midpoint derivatives 5,274 s; 224 exact endpoints 3,352 s. Earlier Llama solver variant. A first bank on the same model used a frozen-scale −1 code selection whose column error feedback took A−1 ij /Aii from the full damped inverse rather than GPTQ’s Cholesky factor (the two differ at 3/48 codes on a controlled example), with the same 98-move design. Held-out NLL signs: proxy 59, current 71, Gauss–Newton 87, midpoint 89 (resolved 76: 47/59/70/74); ARC-Easy option-KL 43/88/86/95 (resolved 70: 32/68/66/70); conflicts 54/98 and 28/52 resolved. Whole models on held-out blocks: RTN 3.558 → selection 3.259 (−0.299 [−0.335, −0.262]; ARC +0.006 [−0.016, +0.028]); the 0.1%-damping variant 3.816 (+0.557 [+0.513, +0.597]; ARC +0.334 [+0.290, +0.379]). Runs of 7,874 s and 16,826 s on one RTX 5090.

C

Exhaustive Lattices

Both lattices: 8 primitives, 256 exact hard states, three functionals, 14 centers, four-fold deterministic subset cross-validation, exact state order by integer mask. Witness primitives are bank batches {4, 5, 6, 10, 14, 15, 17, 20}; stratified primitives {2, 3, 6, 10, 14, 17, 20, 22} chosen from group count, separation, edge identity and a fixed random salt. Peak allocated memory: D0 26.7 GB, ARC 13.0 GB, HellaSwag 18.5 GB. 14

Preprint. Under review.

Table 7: Stratified lattice, four-fold subset cross-validation: sign accuracy / RMSE of each set-function model. Model additive coverage full quadratic sparse k=8 sparse k=16 low-rank r=1 low-rank r=4

D0 NLL

ARC KL

Hella KL

100.0% / 0.00292 71.1% / 0.0161 100.0% / 0.00052 100.0% / 0.00071 100.0% / 0.00051 100.0% / 0.00151 100.0% / 0.00052

90.6% / 0.0766 55.9% / 0.1195 92.6% / 0.0301 94.1% / 0.0309 94.5% / 0.0296 93.8% / 0.0559 92.6% / 0.0314

88.3% / 0.0505 65.2% / 0.0684 93.4% / 0.0153 93.4% / 0.0180 93.4% / 0.0154 91.4% / 0.0307 94.1% / 0.0157

Table 8: Pareto-front recall / precision. Exact fronts: 11 states (witness), 30 states (stratified). Model additive coverage full quadratic sparse k=8 sparse k=16 low-rank r=1 low-rank r=4

witness

stratified

54.5% / 85.7% 0% / 0% 63.6% / 100% 72.7% / 100% 63.6% / 100% 63.6% / 100% 72.7% / 100%

36.7% / 91.7% 10.0% / 75.0% 86.7% / 86.7% 86.7% / 78.8% 83.3% / 89.3% 56.7% / 73.9% 90.0% / 87.1%

Exact common-descent states: 69/256 (witness), 48/256 (stratified). Under a raw max–min rule the stratified lattice’s additive, quadratic, sparse, low-rank and recentered constructors all select mask 159, batches {2, 3, 6, 10, 14, 22}, with exact deltas (−0.0268, −0.1698, −0.0697); on the witness lattice the additive model selects mask 63 (−0.0404, −0.4469, −0.1299) and the quadratic family mask 182 (−0.0756, −0.1384, −0.0773), both on the front. Sparse k=4 produced one false-safe state on the witness lattice (predicted three negatives, exact ARC KL +0.0026). Center-probe tomography with four random centers and rank-4 truncation: sign accuracy 78.5/84.4/90.2% (witness) and 79.3/94.1/93.0% (stratified); with all thirteen nonzero probes the stratified ARC and HellaSwag RMSE are 0.0775 and 0.0489 against 0.0301 and 0.0153 for the endpoint-fitted quadratic. Cross-functional pair coordinates (3 × 28): first two singular directions carry 99.89% (witness) and 99.90% (stratified) of the energy; with rank at most three this indicates a weak third direction rather than a shared subspace, and we treat the per-functional participation ranks in Table 2 as the load-bearing figures. Dividing βik by the decoded-weight norms ∥di ∥2 ∥dk ∥2 of the two moves leaves the participation ranks at 2.04–2.64 with the top four√spectral components carrying at least 99.67% of the energy in every cell (normalizing by Gi Gk group counts: 2.04–2.32); on the stratified ARC cell, however, 97.8% of the normalized squared energy is incident to one two-group move, a star pattern whose spectrum is concentrated by construction. The compression is a property of these lattices, not evidence of a shared latent subspace across moves. On the extrapolation split the full quadratic lowers the ARC RMSE from 0.0997 to 0.0470 and the HellaSwag RMSE from 0.0719 to 0.0245 relative to the additive model. Free-intercept affine control (same folds and splits): witness pooled recall 63.6% against 54.5% zero-intercept and 63.6% quadratic; stratified 30.0% against 36.7% and 86.7%; low-to-high extrapolation 80.0%/52.0% (witness/stratified) for both additive variants. Full-fit recall on the witness lattice is 72.7/81.8/63.6% (zero-intercept / affine / quadratic), so the witness front does not favour the quadratic under every metric. In the reverse direction (fit on support ≥ 5, evaluate on support ≤ 3) the quadratic family recovers 100% of the witness front against 50% for the additive model and 67–100% of the stratified front against 33%, with one to four false-safe states for some quadratic variants; high-to-low transport supports the structure without certifying it. On the witness lattice the low-to-high direction gives 80% recall for both the additive and quadratic models, with false-safe states 3 → 0. 15

Preprint. Under review.

Table 9: Set-function controls on the two lattices. Left: four-fold subset cross-validation of the optimal monotone single-index model h(a⊤ x) (isotonic link, Isotron direction updates, starts at the additive direction, the coordinate axes and 32 frozen random directions) against the quadratic family. Right: Hamming-stratified extrapolation on the stratified lattice, fitted on the 93 states of support ≤ 3 and evaluated on the 163 states of support 4–8. FS: false-safe states (predicted improving on all three functionals, exact worsening on at least one). 4-fold CV Pareto recall (FS) Support ≤ 3 → 4–8, stratified Model

witness

stratified

Pareto recall / precision (FS)

additive (zero intercept) 54.5% affine (free intercept) 63.6% monotone single-index 45.5% (8) full quadratic 63.6% sparse k=16 63.6% low-rank r=4 72.7%

36.7% 30.0% 30.0% (11) 86.7% 83.3% 90.0%

52.0% / 86.7% (1) 52.0% / — 32.0% / 80.0% (1) 92.0% / 100% (0) 92.0% / 88.5% (0) 80.0% / 87.0% (0)

Contextual reversals. For each lattice and functional, all 8 × 128 = 1,024 same-edit contrasts f (S∪{i}) − f (S) and 28 × 64 = 1,792 ordering contrasts f (S∪{i}) − f (S∪{j}) were tested with 10,000 paired bootstrap draws over the shared statistical units (token-weighted blocks for NLL, individual items for the option-KL functionals); the maximum standardized contrast error within each cell sets the simultaneous critical value (3.92–4.17 at α = 0.05/6 across the six cells), and a witness requires both contexts resolved with opposite signs. Resolved same-edit sign reversals out of 8 edits / ordering reversals out of 28 pairs: stratified D0 0/1, ARC P 6/14, HellaSwag 4/11; witness D0 0/6, ARC 5/14, HellaSwag 4/12. Under F (S) = h( i∈S ai ) with monotone h, neither reversal can occur, since the index increment of edit i is ai in every context and the index difference of S∪{i} against S∪{j} is ai − aj . Front recall under resampling, quality tolerance and fixed choice. Using the pooled four-fold predictions without refitting, 2,000 paired resamples of the statistical units (Monte Carlo error at most 1.1 points) give mean stratified recall 35.9% (additive), 33.3% (affine), 84.4% (full quadratic) and 87.8% (rank 4); witness 59.0/68.0/67.0/77.1%. Quality-tolerant coverage asks whether an exact endpoint on the predicted front is within 0, 1 or 2 median state standard errors of every exact-front endpoint on each functional: stratified 66.7/83.3% (additive), 70.0/86.7% (affine) and 100/100% for both quadratics at one and two standard errors; on the witness lattice every model reaches 100% at one. Fixed-choice utility: the state minimizing the worst standardized functional among those predicted to improve all three (standardization by the lattice’s exact-state standard deviations, fixed for all models and resamples; state 0 if none), scored by regret against the exact best feasible state. Stratified: additive and affine select mask 150 (regret 0.0716; on the exact front in 24.8% of resamples), full quadratic mask 21 (0.0396; 99.95%), rank 4 mask 149 (0.0855; 43.3%); all three selected states improve all three functionals in every resample. Witness: additive, affine and full quadratic select mask 55 (0.0267), rank 4 mask 54 (0.0184). The tolerances are descriptive and are not deployment-equivalence margins. Response operator against exact pairwise coefficients. With βik = f (ei +ek ) − f (ei ) − f (ek ) and the response operator fitted by 2[r(z) − r(0)] = Bz on the thirteen non-zero centers (free diagonal, off-diagonal entries compared, diagonal absorbed into the linear term), sign agreement is 92.9/64.3/78.6% (witness D0/ARC/Hella) and 67.9/82.1/85.7% (stratified), Spearman 0.67–0.94, normalized RMSE 0.51–0.81; 3.6–32.1% of the pairwise discrepancies fall within 1.96 paired-bootstrap standard errors (1,000 replicates, joint resampling of blocks and microbatches). The operator recovered from the eight unit centers alone mispredicts the response at the full and four random centers on 31/39/34 (witness) and 34/39/37 (stratified) of 40 components with paired 95% radii excluding zero, and the residual grows with the center’s distance from the base state. The gauge-aligned response quadratic nonetheless reproduces 97.7/92.6/92.2% and 100/94.1/94.5% of exact endpoint signs. 16

Preprint. Under review.

10−1 10−2 10

−3

10−4

D0 NLL, witness ARC KL, witness Hella KL, witness D0 NLL, stratified ARC KL, stratified Hella KL, stratified

10−5 10−6 10−7 1

2

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Walsh degree

7

8

(c) spectrum of the 8 × 8 pairwise matrix

(b) stratified lattice, ARC option-KL

100

additive, RMSE 0.077

0.6

full quadratic, RMSE 0.030 exact Pareto front (30)

0.4 0.2 0.0

|λi|/|λ1| of the fitted β

cross-validated prediction (nats)

fraction of non-constant energy

(a) energy by degree 100

10−1

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0.2

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exact Δ ARC-Easy option-KL (nats)

1

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3

4

5

6

7

8

eigenvalue index

Figure 3: Structure of the set functions on the two exhaustive 0.6B lattices. (a) Walsh energy by degree as a share of the non-constant energy, per functional (open markers: witness lattice; filled: stratified). (b) Stratified lattice, ARC-Easy option-KL: exact endpoint of each of the 256 states against its four-fold cross-validated prediction from the additive model and from the full quadratic; circles mark the 30 states of the exact three-functional Pareto front. (c) Eigenvalue magnitudes of each functional’s fitted 8 × 8 pairwise matrix, relative to the largest. Script scripts/fig interaction atlas.py recomputes every quantity from the frozen 256-state measurements and asserts the values of Table 2 before drawing.

Anchored low-rank search and its anchor-budget simulation. The anchored variant of the beam takes the backbone of B from antithetic probe pairs z ± = (1 ± u)/2, for which 2[r(z + ) − r(z − )] = Bu, fits the functional-specific residual on 4–12 exact pair endpoints chosen before their values are read, and solves the quadratic pseudo-Boolean problem with the same beam; it recovers 83% of the stratified and 55% of the witness front. On the stratified lattice, 315 fixed configurations (exact pair budgets 4, 8, 12; anchor selectors by response disagreement, leverage, manifest diversity, a hybrid and a hash control; unit-, alland random-center rank-4 backbones; affine, subspace and backbone-plus-exact-residual models) were evaluated with anchors chosen before any unqueried pair value was read. Quality point (12 disagreement anchors, all-center backbone, exact residual): Pareto recall 83.3%, precision 89.3%, 0 false-safe, 1 false-reject, emitted mask 156 = {6, 10, 14, 22} with exact deltas (−0.0256, −0.1927, −0.0843) and zero regret under the response-standardized utility; additive response-only search: recall 43.3%, emitted {2, 3, 6, 14, 22} with regret 0.080. Eight leverage anchors: recall 90.0%, precision 84.4%; four diversity anchors: 80.0% recall, no improvement of the emitted state. Witness lattice, same quality recipe: recall 54.5%, 4 false-safe, 8 false-reject. Forward-equivalent cost per functional including one confirmation: 106–155 against 37 for exact pair enumeration on eight primitives. An exact-endpoint oracle beam that reads endpoints during search finds 63%, 97% and 100% of the front at widths 4, 8 and 12 using 69, 95 and 104 unique endpoints; it is the active-query reference for any recentered method.

Recentered search backend (Llama-3.2-1B, INT3). A categorical state over 16 disjoint proxy-dose batches with three code alternatives each (RTN, the code selection at 1% and at 0.1% damping) supports add, remove and one-for-one replacement, exact materialization of any mixed state, and hooked backward rows that agree with the scalar current-state and own-midpoint derivatives to 10−9 –10−11 . On two FIT blocks the cheap screen (current row, proxy, structural diversity, deterministic random) ranks the exact best of 32 atomic moves first, and screening plus a top-4 exact judge costs 126 s against 410 s for enumerating all 32 endpoints (break-even at K = 26); a per-move Gauss–Newton screen signs 31/32 moves correctly yet ranks the exact best 29th. A closed-form legal scale refit on a 1024-group batch lowers the weight squared error by 9.06% and raises FIT NLL by 0.0018, so refit enters the search as a candidate judged at exact endpoints rather than as a default. Four own-midpoint anchor rows drift from the base row by 8.7 × 10−4 on average with 2 of 128 sign changes. 17

Preprint. Under review.

Manifest structural kernel. Features per primitive: group count, separation, occupancy ratio, strong Hamming, layer histogram, projection-family histogram, layer centroid and spread, proxy scale, own-singleton response. Models: nine-feature ridge and symmetric bilinear forms of rank one and two. The two lattices share five of eight primitives, so transfer is scored on the eighteen pair identities unseen in training. Witness→stratified best sign agreement 61.1/50.0/44.4% (D0/ARC/Hella), stratified→witness 77.8/55.6/66.7%, with Spearman between −0.19 and 0.70 and no model best in both directions on all three functionals; substituting the predicted β into the quadratic gives Pareto recall 23–33% or four to ten false-safe states.

level refit: 197/255 improve both 0.00 −0.02 −0.04 −0.06 −0.08

full (255)

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code re-selection: 2/255 improve both 0.20

full (255)

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non-baseline state

improves both functionals

Δ ARC-Challenge option-KL (10−3 nats)

The 4B Solver Lattices and the Grid-Snap Cells Δ ARC-Challenge option-KL (nats)

Δ ARC-Challenge option-KL (nats)

D

grid snap: 1/255 improve both 10 full (255)

8 6 4 2 0 0.0

0.5

1.0

1.5

Δ D2 NLL (10−3 nats)

base state

Figure 4: Exact attainable sets of the level-refit, code re-selection and grid snap lattices (Section 5): the 255 non-baseline states of each lattice in the (D2 NLL, ARC-Challenge optionKL) plane, states improving both functionals in green. On the re-selection lattice the full state (mask 255), the current-state additive selection (mask 66, unsafe), the midpoint additive selection (mask 72, safe) and the safe singleton (mask 64) are circled; the grid-snap lattice is drawn at a 103 finer ARC scale. Script scripts/fig bridge attainable.py asserts the common-descent counts and full-state deltas of Table 10 from the frozen measurements before drawing. Table 10: Three exhaustive 256-state lattices of eight manifest-selected dose batches on Qwen3-4B under D2 NLL and ARC-Challenge option-KL (Section 5). Common descent: states improving both (cert.: paired intervals excluding zero); D2 sign accuracy of the current-state and midpoint additive predictions over all 255 states; false-safe: predicted improving on both, exact worsening on one. Resolved-state counts for the grid snap are in the text below. Move

common descent

level refit 197/255 code re-selection 2/255 grid snap 1/255 (0 cert.)

D2 sign, all 255 false-safe current / midpoint current / midpoint 85.1 / 97.3% 2.75 / 99.6% 48.2 / 98.4%

40 / 1 33 / 1 19 / 1

The 4B lattices act on two states of Qwen3-4B. The level-refit and re-selection lattices act on the master of the grid-trained arm: its grid-legal constrained hard state (E2M1 magnitude pair × UE4M3 scale per group of 128, current assignment), the free-level state obtained by re-solving each group’s two magnitudes with codes fixed, and the re-selected-code state obtained by exhaustive magnitude-ranked assignment with free levels; the reconstruction objective decreases along this ladder. The grid-snap lattice acts on the classic free-codebook arm’s 1600-step endpoint (cell A below) and its plane-preserving snap onto the grid (Bplane ). The historical implementation of that projection re-ran Lloyd’s algorithm before snapping (Blegacy ); the two are separated below. 18

Preprint. Under review.

Table 11: Endpoint cells on Qwen3-4B (held-out WikiText perplexity), reproduced in the Exp. C′ recovery. A: free two-bit codebook recovered for 1600 steps. Bplane : A’s code planes kept, codebook snapped to the NVFP4 grid (code-plane reproduction error 0). Blegacy : Lloyd re-quantization of A (9,736,593 strong-plane assignments changed, maximal requantization error 1.0744) followed by the same snap. Cell

Construction

A Bplane Blegacy

free endpoint grid snap of A, codes fixed Lloyd re-assignment of A, then snap

PPL

vs. A

20.9293 21.1049 36.1654

— +0.84% +72.80%

On the log-perplexity scale Bplane accounts for 1.53% of the A→Blegacy damage. Grid-snap lattice. The bank covers the 28,385,280 groups whose codebook changes under the snap, split into 26 dose batches by positive proxy cost; the L8 selected batches {3, 6, 9, 13, 15, 17, 22, 25} before any functional was read. Full snap: D2 +0.00180859 (CI [−0.00083762, +0.00419934]), ARC option-KL +0.00978934 (CI [+0.00827162, +0.01157384], all 586 item deltas positive). Point-level common descent: one state (batches {6, 9}; D2 −3.7 × 10−8 , ARC −1.4 × 10−9 , both intervals spanning zero); point-level Pareto front {6, 18, 19, 22, 25, 40, 50, 52}. Degree-two energy 0.1165% (D2) and 0.0134% (ARC); degreethree-and-higher below 10−6 . Over all 255 states the current-state additive rule signs 48.2% (D2) and 75.3% (ARC) and the midpoint rule 98.4% and 98.0%, with 19 and 1 false-safe states; restricted to states whose paired 95% interval excludes zero (10,000 replicates) the counts are 103/103 and 224/228 (current) against 103/103 and 228/228 (midpoint). The exact-singleton additive rule signs 98.8%/99.2% with no false-safe state and abstains. Sums of the 28 pair coefficients: D2 −9.1 × 10−5 [−1.46 × 10−4 , −3.6 × 10−5 ], ARC −2.2 × 10−4 [−4.8 × 10−4 , +2.3 × 10−5 ]. Solver-step lattices. Both lattices use the same manifest-only selection rule on the 26 dose batches of each step (level-refit batches {1, 4, 8, 12, 16, 19, 22, 24}, re-selection batches {3, 7, 10, 12, 14, 18, 22, 23}); the 26 known batch outcomes were not consulted, and the selected batches are representative of them (refit: 3/8 selected and 6/18 omitted batches harm D2; re-selection: 7/8 and 15/18). Functionals: D2 odd NLL on 28 held-out blocks and ARC-Challenge option-KL on 586 label-free development items; 18 centers (zero, eight unit, full, four antithetic pairs). Exact fronts: refit 10 states, 197 common-descent (3 singletons, 194 compositions); re-selection 4 states {8, 64, 66, 72}, common-descent {64, 72} only. Full quadratic R2 : 0.999999/0.999978 (refit D2/ARC), 0.999342/0.997375 (re-selection); degreetwo energy 0.014%/0.235% and 2.413%/4.560%. Sums of the 28 pair coefficients: refit ARC +0.0104 [+0.0083, +0.0125] (harmful), the other three unresolved. Rank-4 truncation of the 18-center operator on the re-selection D2 falls to 53.9% sign agreement, consistent with the center dependence of Appendix C. Peak memory 75 GiB (refit) and 60 GiB (re-selection) on one RTX PRO 6000.

E

Selection Runs: Protocols and Results

All runs share one contract: banks frozen before functionals are measured, selection on a fit split, confirmation on a disjoint outer split read once, materialization of the exact-decodable state, own-midpoint rescoring on the artifact, and target endpoints read only after that. Intervals are paired Student-t or item/block bootstrap over calibration units. Composition does two different jobs. On the ARC 4B bank below no single primitive improved both declared functionals, yet one pair does at exact endpoints (Table 12), with ARC-Easy accuracy 491 → 505/594 against 501 and 488 for its components: the pairwise term expands what is reachable. On the 0.6B HellaSwag bank a single primitive already improves all three declared functionals and beats the selected pair on the target (−0.078 against −0.065 KL) while the pair beats it on both generic domains (Table 13): the pairwise 19

Preprint. Under review.

term rebalances slack between functionals. Every constructor re-scores its candidate at the candidate’s own midpoint before reading any target endpoint, the direct test of (4); on the 1.7B CommonsenseQA bank this reversed all three candidates that had passed common-center confirmation, so no state was emitted, and a WinoGrande bank gave the same outcome. ARC 4B (development). T = {D2 NLL, ARC-Easy option-KL}, 26 primitives. Of 16 pairs passing a two-means-and-two-halves screen, {5, 17} had the largest weakest standardized margin (3.23 against 1.36). Materialized state: 65,552 groups (16 from primitive 5, 65,536 from 17), all 252 projections decoding exactly. Own-midpoint rescoring: D2 −0.00457/ − 0.00458/ − 0.00458 (coarse/fine/Richardson, 22/28 units negative), target −0.04429/ − 0.04441/ − 0.04445 (53/74). Held-out: D2 odd NLL 2.99799 → 2.99672; ARCEasy 491 → 505/594 (17 rescues, 3 regressions, McNemar p = 0.00258); gold margin +0.282 [+0.234, +0.328]; top-1 margin +0.192; option CE −0.0553; teacher-KL −0.0426. Undeclared D5: predicted +0.0029088 from the response matrix, observed +0.0029598. The D2 outer upper bound was +0.00027; the exact component controls of Table 12 were run after the pair was frozen. Table 12: ARC 4B pair and its components at exact endpoints, paired 95% intervals. Negative is improvement. State

D2 NLL

ARC option-KL acc. /594

primitive 5 +0.00167 [+0.00027, +0.00313] −0.04951 [−0.06136, −0.03742] primitive 17 −0.00259 [−0.00533, −0.00019] +0.01672 [+0.00290, +0.03068] pair {5, 17} −0.00127 [−0.00421, +0.00129] −0.04264 [−0.05880, −0.02653]

501 488 505

TruthfulQA 4B (pre-registered). Three functionals, 26 primitives, minimum-cardinality selector; one fit-feasible singleton (batch 8, 128 groups). Validation: D2 −0.00284, D3 −0.00321, target −0.00729 with halves −0.00888 and −0.00539; the target’s standardized margin 1.66 fell short of the Bonferroni critical value 2.47 (UCB +0.00353), so no state was materialized and the 171 evaluation items were not scored. HellaSwag 0.6B (consensus protocol). 23 primitives, three functionals, five-fold consensus with a four-fold identity requirement. Support 1 reached 5/5 on {14} (pooled out-of-fold weakest margin 6.76); support 2 reached 4/5 on {4, 14} (7.78); no support from 3 to 8 reached four folds. Outer confirmation: −0.0292, −0.0404, −0.0395 with upper bounds −0.0230, −0.0339, −0.0236; own-midpoint rescoring negative on 27/28, 37/38 and 48/50 units. Table 13: HellaSwag components at exact endpoints (0.6B, 1,820 FINAL items for the target columns). All quantized states answer the same 424 items correctly; the teacher answers 631. State primitive 4 (8 groups) primitive 14 (8,192) pair (8,200)

D2 NLL

D3 NLL

target KL

gold margin

−0.00817 −0.01714 −0.02489

−0.00632 −0.03347 −0.03911

+0.01669 −0.07831 −0.06458

−0.02916 +0.10801 +0.07488

Pair intervals on FINAL: target KL [−0.0705, −0.0587], gold margin [+0.0610, +0.0887], option CE −0.0713 [−0.0825, −0.0599], top1–top2 gap −0.1730 [−0.1837, −0.1624]; endpoint interactions relative to component sums +0.00043, +0.00068, −0.00296, −0.00398. 4B re-selection bank: one-shot midpoint selection. Same bank, splits and functionals as the beam below. One hooked backward at the own midpoint of each of the 26 singleton batches on the FIT units (rows already measured by the beam’s anchor jobs were reused); the selector uses the i-th coordinate of the i-th row only, sums additively, and takes the set minimizing predicted ARC-Challenge option-KL subject to predicted ∆D2 ≤ 0, frozen before any endpoint was read: {0, 1, 5, 6, 15, 17, 21, 22, 24} (additive FIT prediction −0.00041, 20

Preprint. Under review.

−0.134). Exact changes from B: FIT −0.00139 / −0.1098, VALIDATION −0.00445 / −0.1175, TEST −0.00728 [−0.02175, +0.00663] / −0.2112 [−0.2798, −0.1389]. TEST paired against the solver’s output C: −0.0571 [−0.0837, −0.0327] / −0.1659 [−0.2239, −0.1057]; against the beam’s primary: −0.0068 [−0.0188, +0.0025] / −0.0256 [−0.0533, +0.0003] (10,000 paired draws; D2 over seven blocks, ARC over ten stored microbatch means of 154 items). Cost: 26 own-midpoint jobs, each 3.63 endpoint forwards at the median ratio measured on the beam’s 36 anchor jobs, against the beam’s 199 forward-equivalents. 4B re-selection bank: recentered exact-endpoint beam. Bank: the 26 dose batches of the re-selection step; start B (free levels, codes fixed); functionals D2 odd NLL and ARCChallenge option-KL; FIT 14 blocks and 18 microbatches of 16 items, VALIDATION 7 blocks and 144 items, TEST 7 blocks and 154 items, all disjoint; beam 4, add-only, slack 0 relative to B. Per state: cheap screen (current row, proxy, structural, deterministic random) to four candidates, four own-midpoint rows, four exact FIT endpoints; persistent Pareto beam with exact re-anchoring; stop when the frontier is exhausted (nine rounds). Five exact Pareto states on FIT; frozen four-state slate; validation policy fixed before reading (minimize ARC optionKL subject to ∆D2≤ 0) selected {6, 10, 15, 17, 21, 24}. TEST: D2 3.586104 → 3.585646 (−0.000458, paired 95% [−0.007320, +0.006771], 4/7 blocks negative); ARC option-KL 0.759318 → 0.573742 (−0.185575, [−0.250825, −0.121108], 10/10 microbatches negative). Replay audit: zeroing or negating all 36 midpoint rows changes none of the nine round decisions or the final frontier; the 36 anchor jobs (10.4 h) produced the 72/72 versus 32/72 diagnostic and nothing the decisions depended on. Median cost ratio of an own-midpoint job to an exact endpoint job: 3.63; total search budget 199 endpoint evaluations. Local pricing replay at the nine archived base states, same four measured candidates, same scales, feasibility and tie rules: own-midpoint pricing reproduces the exact local beam head 9/9 with 0/36 false-feasible candidate occurrences, the recentered current row 5/9 with 16/36; in the four disagreements (rounds 4, 6, 7, 9) exact and midpoint pricing keep the incumbent where the current row expands. 4B re-selection bank: direct endpoint control. Same bank, splits, beam semantics, slate rule and validation policy; the frozen budget of 199 FIT endpoint evaluations is spent on exact endpoints without any current-state or midpoint scoring. Selected state {0, 1, 5, 6, 15, 17, 21, 24}. Changes from B: FIT −0.00260 / −0.1166, VALIDATION −0.00643 / −0.0861, TEST −0.00037 [−0.00697, +0.00780] / −0.2262 [−0.3013, −0.1554]. TEST paired against C: −0.0502 [−0.0792, −0.0278] / −0.1810 [−0.2525, −0.1073]; against the current-screened beam’s primary: +0.0001 [−0.0040, +0.0039] / −0.0407 [−0.0585, −0.0217]; against the one-shot state: +0.0069 [−0.0004, +0.0166] / −0.0150 [−0.0429, +0.0109]. On VALIDATION the one-shot state had the lower ARC option-KL (direct minus one-shot +0.0314 [+0.0042, +0.0597]); the frozen selection was not revisited. Search wall-clock 64.7 min on one GPU with a persistent evaluator and cached decoded bank, not comparable to the historical run’s timing. Zero-shot accuracy of the frozen states. Protocol: all states frozen before any accuracy was read; ARC-Challenge test (1,172 items), HellaSwag validation (10,042), MMLU test (14,042); zero-shot restricted candidate-label likelihood, i.e. the options are shown and the model’s most likely label is its answer (for HellaSwag this is a label choice over the four shown endings, not the length-normalized ending completion of the evaluation harness); Qwen3-4B with its direct-answer chat template and thinking disabled, Llama with a plain prompt; per item the prediction, gold margin and restricted-option KL to the FP master are stored; differences use 10,000 paired item bootstrap draws and exact McNemar tests. Of the ARC-Challenge items, 586 had served as the 4B repair’s FIT, VALIDATION and TEST option-KL units; the main text reports the other 586. Qwen3-4B accuracy (ARC-C / HellaSwag / MMLU): FP 85.32/68.74/63.37; B 66.13/48.73/44.93; C 66.55/41.83/44.17; oneshot 68.77/51.64/47.73; direct 69.80/51.17/48.07; beam primary 68.43/49.50/46.52. Changes from B on all 1,172 ARC-C items: C +0.43 [−1.54, +2.39], one-shot +2.65 [+0.94, +4.35], direct +3.67 [+1.88, +5.46], beam +2.30 [+0.68, +3.92]; on the 586 unused items, C +0.68 [−1.88, +3.41] (McNemar p = 0.70), one-shot +2.73 [+0.34, +5.12] (p = 0.040), direct +4.10 [+1.71, +6.66] (p = 0.0015), beam +2.05 [−0.17, +4.27] (p = 0.088). Option-KL differences from B are negative with intervals excluding zero for all three repaired states on 21

Preprint. Under review.

all three tasks. Llama-3.2-1B: FP 36.18/27.30/36.24; standard GPTQ(1%) 23.98/25.80/26.27; midpoint one-shot 25.60/25.52/26.53, changes +1.62 [−1.79, +4.95], −0.28 [−1.49, +0.93], +0.26 [−0.77, +1.26]; option-KL changes −0.026, −0.073, −0.027, all resolved. Queue 4 h 56 min on one GPU. FIT decomposition of the Llama one-shot states. With p(S) the selector’s price sum, e(S) the sum of exact single-replacement endpoints from the common start and ∆(S) the exact endpoint of the composed state on the same FIT units, p − e is the single-move integration error and ∆ − e the composition residual. (NLL, ARC option-KL): midpoint state ∆ = (−0.106, +0.012), p = (−0.0003, −0.452), e = (−0.023, −0.447), composition (−0.083, +0.459), integration (+0.023, −0.005); current state ∆ = (+0.760, +0.002) (NLL paired CI [+0.408, +1.165]), p = (−0.802, −0.782), e = (+1.032, −0.370), composition (−0.272, +0.373), integration (−1.835, −0.412); exact-singleton state ∆ = (−0.066, +0.051), p = e = (−0.0005, −0.473), composition (−0.066, +0.524). Llama-3.2-1B matched exact-endpoint beam. Budget frozen before search as the one-shot midpoint pricing time divided by the archived mean exact-singleton time, ⌊5,274.0/14.96⌋ = 352 endpoint forwards, all charged (the 224 singleton answers were reused where the search requested them; 128 composed states were materialized and scored). A width-4 exact Pareto beam from the GPTQ(1%) model exhausted the budget after one expansion with a three-state slate of support two; the frozen validation rule selected {9 → RTN, 12 → GPTQ(0.1%)}. VALIDATION: NLL 3.0975 → 2.9314 (−0.1661 [−0.2118, −0.1240]), ARC 0.2347 → 0.1995 (−0.0352 [−0.0492, −0.0219]); TEST: 3.2056 → 2.9993 (−0.2063 [−0.2603, −0.1555]), 0.2110 → 0.1821 (−0.0289 [−0.0407, −0.0173]). Against whole RTN on TEST: −0.4929 / −0.0753; against whole GPTQ(0.1%): −0.5862 / −0.0130 [−0.0331, +0.0075]. Against the 61-change midpoint state, same units: VALIDATION −0.0789 [−0.1047, −0.0518] / −0.0355 [−0.0602, −0.0109], TEST −0.0891 [−0.1645, −0.0085] / −0.0342 [−0.0574, −0.0112]. This control was run after the shared VALIDATION/TEST had been read for the one-shot states; it was pre-registered and its rule did not depend on them. Llama-3.2-1B recentered selection on fresh units. Start: the 61-change midpoint state. Each round prices all 224 one-batch alternatives at their own midpoints from the current state on the development FIT units and selects at most 16 categorical changes minimizing summed ARC price subject to summed NLL price ≤ 0; the predicted state is advanced to without veto. Exact FIT deltas per round (NLL, ARC): (−0.0057, −0.0498), (+0.0095, −0.0013), (−0.0039, +0.0639) against predictions (−0.0002, −0.286), (−0.0018, −0.182), (−0.0015, −0.181). Price drift over the 208 moves that remain non-current at consecutive centers: NLL Spearman 0.937 and 0.923 with 25/208 and 25/208 sign flips; ARC Spearman −0.247 and −0.091 with 121/208 and 105/208. Fresh units: 56 VALIDATION and 56 TEST blocks of 2048 tokens from a newly downloaded C4 validation shard (zero block-hash overlap with any prior unit) and two unopened ARCEasy sets of 594 items each. Frozen validation rule (minimize ARC option-KL subject to point NLL no worse than the start; the direct state and whole models as controls only) selected round 2. TEST: GPTQ(1%) 3.9677 / 0.2353; RTN 4.1357 / 0.2515; GPTQ(0.1%) 4.4127 / 0.2153; direct primary 3.8064 / 0.2063; one-shot start 3.8067 / 0.2250; rounds 1–3 3.8371 / 0.2013, 3.8963 / 0.1791, 3.9453 / 0.2854. Round-2 primary against the start: +0.0896 [+0.0180, +0.1837] / −0.0460 [−0.0553, −0.0367]; against the direct primary +0.0899 [+0.0134, +0.1765] / −0.0272 [−0.0424, −0.0129]; against GPTQ(1%) −0.0714 [−0.1685, +0.0314] / −0.0562 [−0.0728, −0.0399]. The direct primary against the start on these fresh units: −0.0003 [−0.0812, +0.1073] / −0.0187 [−0.0364, −0.0003]. Pricing cost 15,745 s for three rounds. Figure 5 shows the price drift for both consecutive pairs of centers; the tidy table behind it is figures/data/recentered price drift.csv and the script scripts/fig price reversal.py, which re-derives the four sign-flip counts and rank correlations from the frozen price journals before drawing. Scale gate. A pre-registered rule capped FIT pricing at 24 h. Llama-3.1-8B fits the GPU (76.7 GB peak, producers 25 min) but its 448 replacements project to 40.2 h (26.2 midpoint, 22

Preprint. Under review.

NLL: 25/208 reversed, ρ = +0.94

price after round 1 (nats)

price after round 1 (nats)

10

ARC option-KL: 121/208 reversed, ρ = −0.25

−1

10−2 10−3 0 −10

−3

−10

−2

−10−1

10−2 10−3 0 −10−3 −10−2

−10−1−10−2−10−3 0 10−3 10−2 10−1

−10−2 −10−3

10−3

10−2

price at round 0 center (nats)

NLL: 25/208 reversed, ρ = +0.92

ARC option-KL: 105/208 reversed, ρ = −0.09 10−2

price after round 2 (nats)

10−1

price after round 2 (nats)

0

price at round 0 center (nats)

10−2 10−3 0 −10−3 −10−2 −10−1

10−3

0

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−10−2 −10−1−10−2−10−3 0 10−3 10−2 10−1

−10−2

price at round 1 center (nats)

−10−3

0

10−3

10−2

price at round 1 center (nats)

same sign at both states

sign reversed

Figure 5: Own-midpoint prices of the 208 comparable moves at consecutive centers of the recentered selection (top: start to round 1; bottom: round 1 to round 2). Symmetric-log axes; shaded quadrants are sign reversals. NLL: 25 and 25 reversals, Spearman 0.94 and 0.92; ARC option-KL: 121 and 105 reversals, Spearman −0.25 and −0.09. 14.0 exact singleton), so the scale runs used Llama-3.2-3B (28 layers, 196 projections, 392 replacements). Llama-3.2-3B without activation ordering: a configuration failure. The first 3B run used the 1B configuration (no activation ordering). Its GPTQ(1%) model is far worse than RTN (TEST NLL: RTN −3.586 [−4.593, −2.519] relative to it; ARC −0.173); the reference GPTQ implementation run with the same configuration reproduces it exactly (zero decoded-weight difference on all 196 projections, FIT NLL 7.053878 in both), so the failure belongs to the configuration on this model, not to our kernel. The run is kept as a record, not as a scale result. FIT predictors against exact single-replacement endpoints: midpoint 392/392 NLL (Spearman 0.9996) and 390/392 ARC (0.9993); current state 318/392 (0.835) and 365/392 (0.957). Selections: midpoint 130 batches, exact singleton 130 (4 differ), current 137 (11 and 13 differ). TEST against GPTQ(1%): midpoint −3.712 [−4.719, −2.644] / −0.041 [−0.127, +0.046], exact singleton −3.712 / −0.052, current −3.693 / −0.002; midpoint against current −0.0186 [−0.0219, −0.0151] / −0.0391 [−0.0592, −0.0194], against RTN −0.1255 [−0.1344, −0.1166] / +0.1322 [+0.0910, +0.1738] (28 NLL blocks, 297 ARC items per partition). Llama-3.2-3B with activation ordering. GPTQ with act order and static groups (groups defined on the original column order so that every state stays a contiguous g128 INT3 23

Preprint. Under review.

state; block 128, sequential, 1% and 0.1% damping), matching the reference implementation to zero decoded-weight difference. Whole models on the calibration units (NLL / ARC option-KL): RTN 2.5130 / 0.3948; GPTQ(1%) without act-order 7.0539 / 0.5424; act-order GPTQ(1%) 2.2808 / 0.3695; act-order GPTQ(0.1%) 2.2607 / 0.3915. Act-order GPTQ(1%) against RTN: NLL −0.2322 [−0.2712, −0.1700], ARC −0.0253 [−0.0542, +0.0034]. Bank: 196 projections, each keeping the act-order GPTQ(1%) codes or taking RTN or the act-order 0.1% codes, 392 replacements priced by the current-state row, the own midpoint and the exact singleton endpoint on the FIT units. Exact endpoints: 269/392 replacements raise NLL and 194/392 raise ARC option-KL (by source: RTN 152 and 100 of 196, 0.1% 117 and 94); the two directions conflict on 183/392 (NLL improves and ARC worsens 54, NLL worsens and ARC improves 129, both improve 69, both worsen 140), with Pearson 0.20 and Spearman 0.03 between the two coordinates. Predictors: midpoint 390/392 NLL (Spearman 0.9996, MAE 0.00037, largest error 0.032) and 391/392 ARC (0.9993, 0.00035); current state 153/392 (0.292, 0.0044, largest error 0.59) and 357/392 (0.931, 0.0040); both directions jointly 389/392 against 141, conflict classified on 389/392 against 164; by source the midpoint is 195/196 and 196/196 (RTN) and 195/196 on both (0.1%). The frozen one-shot selectors (minimize predicted ARC subject to predicted NLL ≤ 0) change 149 (current), 132 (midpoint) and 134 (exact singleton) batches; the midpoint and exact-singleton selections differ at 2 of 196 batches, the current selection at 24 and 22, and summed over exact singleton endpoints the current selection violates its NLL constraint (+0.210) before any composition residual. No composed state was materialized; these are selector-level results. Cost: current row 120 s; 392 midpoints 23,279 s (59.4 s each); 392 exact endpoints 15,497 s (39.5 s each), ratio 1.50.

Llama-3.2-1B with activation ordering. The same producer on Llama-3.2-1B gives a bank of 224 replacements of the act-order GPTQ(1%) codes by RTN or the act-order 0.1% codes. Exact endpoints on the FIT units: 184/224 raise NLL, 153/224 raise ARC option-KL, and the two directions conflict on 81/224; midpoint signs 223/224 (NLL) and 222/224 (ARC), current-state 50/224 and 179/224. Constructors with the rules, splits and 352-forward budget of the no-act-order bank: the one-shot selectors change 81 (current), 37 (midpoint) and 35 (exact singleton) projections with predicted FIT deltas (−0.460, −0.359), (−0.00005, −0.093) and (−0.0002, −0.088); their VALIDATION deltas against the base are (+0.424, +0.016), (+0.021, −0.001) and (+0.013, +0.014), TEST (+0.339, +0.036), (+0.033, −0.0005) and (+0.028, +0.016), the midpoint state’s TEST NLL increase resolved ([+0.006, +0.062]) and its ARC change not ([−0.016, +0.014]). The width-4 beam exhausted its budget after one expansion with four two-change FIT Pareto states; the frozen validation rule selected {0, 50} → RTN (layer-0 q-projection, layer-7 k-projection). Against the base: VALIDATION −0.0004 [−0.0032, +0.0014] / −0.0066 [−0.0115, −0.0017]; TEST −0.0014 [−0.0043, +0.0007] / −0.0032 [−0.0081, +0.0013]. Against the one-shot states on TEST: midpoint −0.0345 [−0.0624, −0.0082] / −0.0027 [−0.0167, +0.0115]; exact singleton −0.0295 [−0.0541, −0.0081] / −0.0193 [−0.0360, −0.0024]; current −0.341 [−0.429, −0.186] / −0.0396 [−0.0586, −0.0203]. As on the no-act-order bank, the control was frozen and run after the shared one-shot VALIDATION/TEST had been read.

1.7B CommonsenseQA and WinoGrande (abstentions). CommonsenseQA: 25 primitives, 11.0M physically changed proxy-improving groups; four stable fit candidates, three passing response-based outer confirmation; own-midpoint rescoring reversed the target on all three; FINAL unread. The full same-unit drift audit is Table 14. WinoGrande: original protocol returned no certified state; on the same outer units, candidates {17, 23} and {17, 22, 23} had target fit means −0.032 and −0.051 and outer full-center means +0.025 and +0.014 (fit-to-outer sample-transport differences +0.057 and +0.065, Welch intervals [−0.011, +0.125] and [−0.003, +0.132]), and outer medoid-center means +0.139 and +0.108 (full-to-medoid center-transport differences +0.114 and +0.094, paired intervals [+0.092, +0.137] and [+0.073, +0.115]); FINAL unread. 24

Preprint. Under review.

Table 14: All nine same-unit paired drift cells on the 1.7B bank (Section 4.2). Unit-level: support-1 drift positive on 32/32 blocks and 100/100 target microbatches; support-2 on 32/32 and 98/100; support-4 negative on 32/32 D0 and 29/32 D1 blocks, positive on 100/100 target microbatches. Candidate Functional Proposal center Own center support-1 support-1 support-1 support-2 support-2 support-2 support-4 support-4 support-4

F

D0 NLL D1 NLL target KL D0 NLL D1 NLL target KL D0 NLL D1 NLL target KL

−0.00209 −0.00179 −0.00219 −0.00186 −0.00134 −0.00262 −0.05442 −0.04372 −0.00355

+0.00249 +0.00311 +0.00376 +0.00297 +0.00376 +0.00262 −0.35554 −0.26543 +0.06962

Drift Simultaneous CI +0.00457 +0.00491 +0.00595 +0.00482 +0.00509 +0.00524 −0.30112 −0.22171 +0.07317

[+0.00359, +0.00556] [+0.00365, +0.00616] [+0.00548, +0.00642] [+0.00362, +0.00603] [+0.00360, +0.00658] [+0.00462, +0.00586] [−0.37392, −0.22832] [−0.31977, −0.12366] [+0.06600, +0.08033]

Response Structure Across Domains

A 12 × 52 response matrix over six domains under NLL and full-vocabulary teacher-KL is algebraically full rank with all twelve singular modes above a block-bootstrap noise floor; participation rank 2.71 with 75.9% of energy in two modes, 1.79 (NLL) and 1.69 (KL) within families; within-domain NLL–KL correlation 0.15–0.63. Held-out completion under a frozen four-fold split: nonnegative matched-family cone 78.5% sign accuracy, Spearman 0.744; source identity 66.0%; ten-anchor unconstrained fit 72.3% against 76.9% for the strongest single anchor. On eight moves frozen by generic quadrant before any ARC-Easy output was read, KL-family signatures called 7/8 gold-margin directions, the NLL family 3/8. F.1

Teacher-KL is conditional on the teacher

Stratifying option-KL effects by whether the teacher answers correctly inverts the sign on three of the measured ARC-Easy moves between teacher-correct and teacher-wrong strata. Improving KL to a teacher transports its decisions, including the wrong ones.

G

Compute

Timing on one RTX PRO 6000 (Qwen3-0.6B, 28 D2 blocks, 23-primitive WC bank), excluding the shared 8.1 s model load: response setup 0.41 s; 28 response forward+backward 10.13 s (0.362 s per block, 23 coordinates each); baseline plus 23 primitive forwards 25.4 s; sparse exact patch construction 93.9 s, one-time patch transfer 2.3 s, 23 apply+revert 0.09 s; CPU subset search over 880,969 subsets 51.4 s. Search-side totals: response 61.9 s, sparse exact 173.2 s (2.80×), full-artifact exact 279.5 s. Break-even bank size 9.6 primitives. Confirming B, E14 and the pair on exact endpoints: 3.19 s. Peak allocated memory: response 21.3 GB, sparse endpoint 10.3 GB, single primitive endpoint 4.9 GB. Lattice measurements: D0 26.7 GB, ARC 13.0 GB, HellaSwag 18.5 GB; 512 distinct exact masks and 84 functionalcenter sweeps across both lattices. Cost of one own-midpoint backward in exact-endpoint forwards, measured on the constructor banks: median 3.63 on the Qwen3-4B re-selection bank (Appendix E), 1.57 on the Llama-3.2-1B bank (5,274 s against 3,352 s for the same 224 replacements) and 1.50 on the act-order Llama-3.2-3B bank (59.4 s against 39.5 s per replacement), the range quoted in Section 6.

25

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