L EARNING IN I NFINITESIMAL N ON -C OMPOSITIONAL S KETCHES A P REPRINT Sridhar Mahadevan Adobe Research and University of Massachusetts, Amherst [email protected], [email protected]
July 17, 2026
arXiv:2607.15107v1 [cs.LG] 16 Jul 2026
A BSTRACT This paper develops a categorical framework – Learning in Infinitesimal Non-Compositional Sketches (L INCS ) – that reframes machine learning (ML) as the repair of non-compositionality: failures of diagrams to factor through their intended quotient sketches lifted to the tangent category setting. ML problems are specified as sketches (graphs with commutativity conditions D, limit cones L, and colimit cocones K), generalizing the usual scalarization of loss functions or vector space assumptions. Non-compositionality is defined purely as failure of a universal factorization problem, not as arithmetic error between the desired and actual predictions. The central object is Infinitesimal NonCompositionality (INC): given a learning sketch S = (S, D, L, K), whose underlying graph is S, and a model D : J → C, the base defect is the obstruction to factorization Obs(FactS (D)). The tangent lift applies the tangent functor T to obtain T D : J → C, and INC is the obstruction Obs(FactS (T D)) —asking whether infinitesimal perturbations preserve the compositionality constraints. The paper also introduces Tangent Learning Sketches, which are sketches equipped with CockettCruttwell tangent structure, ensuring that if D is admissible, so is T D. This allows the definition of L INCS categories where learning data includes the pair (FactS (D), FactS (T D)). Beyond tangent factorization, the framework admits sketch-specific interaction enrichments. Lie-bracket closure is the intrinsic antisymmetric specialization; when a connection is declared, the full second-order jet also contains connection-dependent symmetric acceleration. Parameterized realizations must descend through presentation redundancies, and computational tangent signals are admitted contingently rather than assigned mandatory weight. The paper defines the INC endofunctor TINC , which iterates the tangent lift, producing a tower D, T D, T 2 D, · · · of factorization problems. ML is thereby formulated as the search for a coalgebraic fixed point where successive tangent unfoldings stabilize (νTINC ). Using the Aczel–Mendler theorem, we prove existence of a final INC coalgebra whenever TINC admits a set-based class realization that creates its final carrier. Barr’s theorem yields a set-sized alternative for accessible realizations and a regular-cardinal bound on the final carrier. For complete metric realizations in which TINC is contractive, we also prove existence and uniqueness of the stabilized INC behavior, geometric convergence of the exact tower, and a finite-error bound for approximate unfoldings. A detailed experimental evaluation of L INCS is underway in a number of concrete ML settings, including deep learning, large language models, and reinforcement learning, and is described in companion papers. Keywords Tangent categories · sketches · infinitesimal learning · categorical machine learning · diagrammatic backpropagation · connections · Lie brackets · quotient geometry · coalgebras · Kan extensions · reinforcement learning
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Contents 1
Introduction
2
2
Related Work
3
3
Learning Sketches and Tangent Learning Sketches
5
4
Examples of Infinitesimal Non-Compositionality
7
5
Diagrammatic Backpropagation Revisited
10
5.1
10
From global to local non-compositionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Tangent Lift of Diagrammatic Backpropagation
11
7
L INCS Axioms
12
7.1
Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
7.2
Universal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
8
L INCS as a Coalgebra
15
9
L INCS as a Category
18
10 Interaction Profiles and Higher INC
19
11 Homotopical L INCS
20
12 Future Directions
21
13 Summary
23
1
Introduction
The dominant language of machine learning (ML) is optimization: choose a loss function and minimize it (Hastie et al., 2009; Murphy, 2012; Sutton and Barto, 2018). This language is powerful, but it hides a common structure shared across many learning objectives. A loss often scalarizes a failure of composition. A supervised loss scalarizes a failure of a predictor to compose input with target behavior. A Bellman residual scalarizes a failure of a value function to commute with the Bellman operator. A contrastive loss scalarizes failure of representation invariance across views. A sheaf loss scalarizes failure of local sections to glue. Diagrammatic Backpropagation (DB) made this viewpoint explicit: learning can be driven by failures of diagrammatic compositionality (Mahadevan, 2026c). A DB loss scalarizes failure of a diagram of computations to commute. The paradigmatic Transformer model used in large language models (LLMs) provides a suggestive example. After embedding, a length-m token sequence is represented as a point of a Euclidean array, and the content-only Transformer map F is permutation equivariant: for a permutation matrix P , the intended sketch contains the commuting condition F (P X) = P F (X). Thus “scramble the input, then apply the model” and “apply the model, then scramble the output” are two paths that should agree, and an equivariance loss scalarizes their failure to compose. This statement deliberately factors out positional encodings, causal masks, and other order-dependent structure, which break or refine the permutation symmetry in an implemented language model. From the L INCS viewpoint the scalar loss is only the base-level shadow: one can also ask whether this equivariance square remains coherent under perturbations of embeddings, attention maps, and parameters, and whether new obstructions appear under the iterated lifts T2 , T3 , . . .. This suggests that L INCS may expose structural properties of language models that are invisible to the ordinary scalar 2
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Learning compositionality problem
J
D
C
Infinitesimal compositionality problem
T
J
TD
C
tangent lift
qD
qD D̄ ?
J/∼D
TD ? J/∼D
Obs(FactS (D))
INC(D) = Obs(FactS (TD))
transport the defect
Figure 1: Every learning compositionality problem has a tangent lift. At left, a candidate model D may fail to factor through the quotient specified by the learning sketch S. Applying the tangent functor produces the corresponding factorization problem for TD without first scalarizing the base defect. The obstruction to this lifted factorization is infinitesimal non-compositionality, INC(D). objective. Section 4 answers this question explicitly: exact smooth equivariance propagates through every tangent order, whereas a nonzero higher obstruction localizes where approximate or deliberately broken equivariance fails to survive perturbation. Using this paradigmatic example as a theme, L INCS explores the result of formulating a wide range of ML problems, from deep learning to large language models and reinforcement learning, as a categorical compositionality problem lifted by the tangent functor. We can therefore ask not only whether a base factorization problem is solved, but whether its infinitesimal transport is coherent. The central thesis is that every learning compositionality problem has a tangent lift. Figure 1 depicts this passage at the level of the factorization problem itself. The lift defines a new learning signal – Infinitesimal Non-Compositionality (INC). The usual ML formulation asks: is the scalarized defect small? L INCS asks: is the defect small, and does its tangent factorization behave correctly? This paper develops L INCS as a mathematical framework and illustrates the framework in a number of practical settings, from diagrammatic backpropagation in deep learning (Mahadevan, 2026c), Lie-algebraic LoRA adapters (A LLORA) for fine-tuning Transformer models (Mahadevan, 2026a), Lie-algebroidal skill optimization methods (L ASKO) (Mahadevan, 2026g), and latent-confounded causal discovery using Lie-brackets (Mahadevan, 2026b). Contributions.
The main contributions of this paper include the following:
1. It defines Infinitesimal Non-Compositionality (INC) as the tangent lift of diagrammatic factorization failure. 2. It formulates obstruction localization, a functorial decomposition of global non-compositionality into compatible local learning signals, and shows that this decomposition is preserved by tangent lift. 3. It formulates L INCS objects, L INCS morphisms, and INC categories using tangent-category structure. 4. It introduces interaction-enriched INC: bracket closure is the intrinsic antisymmetric specialization, while connection-equipped sketches may also retain symmetric acceleration and full second-order jet data. 5. It distinguishes the categorical tangent lift from its parameterized realization by requiring optimizer fields to descend through presentation quotients or to be supplied with an equivariant horizontal lift. 6. It gives a coalgebraic view of L INCS , proves existence of final INC coalgebras under set-based class or accessible set semantics, bounds the set-sized carrier at a regular cardinal, and proves geometric convergence for contractive metric realizations. 7. It proposes functoriality and universal-completion principles for L INCS categories. 8. It relates L INCS to Kan-invariant learning, tangent categories, infinitesimal causality, Lie-bracket learning, and homotopical repair. 9. It sketches how GIRL, Differential KET, Diagrammatic Backpropagation, causal Lie-bracket learning, ALLORA, and LASKO instantiate the same core axioms with domain-specific interaction signatures and contingent computational admission.
2
Related Work
Sketches and categorical theories. The use of sketches to present structured theories goes back to Ehresmann’s work on sketches and algebraic structures (Ehresmann, 1968). Sketches were later developed as a flexible categorical language 3
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for theories and models, notably in the work of Barr and Wells (Barr and Wells, 1999), Makkai and Paré (Makkai and Paré, 1989), and Adámek and Rosický (Adámek and Rosický, 1994). Following the Makkai-Paré presentation, we regard a sketch as a quadruple (S, D, L, K), where S is an underlying graph, D is a class of commutativity conditions, L is a class of distinguished cones, and K is a class of distinguished cocones. L INCS uses sketches in this spirit: a learning problem is specified by a graph of formal computations together with the commutative, limiting, and colimiting constraints that a strict learning model would satisfy. Accessible categories. Accessible-category theory provides a natural size-controlled setting for this sketch semantics. Under standard smallness hypotheses, many categories of models presented by sketches are accessible or locally presentable; conversely, accessible categories admit presentation results in categorical model theory (Makkai and Paré, 1989; Adámek and Rosický, 1994). This connection is relevant to L INCS because accessibility supplies set-sized families of presentable models and closure under filtered colimits, offering a plausible technical setting for constructing categories of learning models and studying tangent or L INCS completions without uncontrolled size growth. Tangent categories. Rosický introduced abstract tangent functors, and Cockett and Cruttwell developed tangent categories as an axiomatic account of tangent bundle structure (Rosický, 1984; Cockett and Cruttwell, 2014). The subsequent theory of differential bundles, tangent fibrations, and connections provides a categorical setting in which vector fields, curvature, and transport can be studied without choosing coordinates (Cockett and Cruttwell, 2018, 2017). L INCS builds directly on this tangent-categorical substrate by treating the Cockett-Cruttwell structure itself as sketch-presentable data acting on learning sketches and their factorization problems. Weil-algebra semantics for tangent structure. Leung gives a functorial classification of the Cockett–Cruttwell axioms in terms of Weil algebras (Leung, 2017, Thm. 14.1). If M is a category, specifying tangent structure on M is equivalent, up to isomorphism, to specifying a strong monoidal functor F : (N-Weil1 , ⊗, N) −→ (End(M), ◦, 1M ) that preserves the foundational pullbacks and the equalizer encoding the universality of vertical lift. The dual-number algebra W = N[x]/(x2 ) is sent to the tangent functor T = F (W ), while the structural maps of W are sent to the projection, zero, addition, vertical lift, and canonical flip. Thus N-Weil1 acts as a classifying theory, or initial tangent structure, rather than merely providing examples of tangent functors. This result supplies the precise functorial semantics behind the tangent action used by L INCS; the additional L INCS data are the learning sketches, their admissible models, and the factorization obstructions transported by that action. Functorial semantics of Lie theory. MacAdam’s thesis reconnects two strands of Ehresmann’s work—sketch theory and many-object Lie theory—through tangent categories (MacAdam, 2022). It presents Lie algebroids by involution algebroids, a tangent-categorical sketch whose models in smooth manifolds form the category of Lie algebroids, and then identifies involution algebroids with suitably exact tangent functors out of the classifying category Weil1 . It also places Lie differentiation in a nerve–realization context induced by an infinitesimal groupoid object. This is directly relevant to L INCS: it supplies a rigorous precedent for both sketch-presented tangent structure and Lie-algebroidal semantics. The distinction is that MacAdam classifies infinitesimal geometric structure, whereas L INCS studies the obstruction to a learning model and its tangent lift satisfying a chosen sketch. Compositional learning. Several lines of work have used category theory to describe learning compositionally. Backpropagation has been formulated functorially (Fong et al., 2019), Universal Decision Learners formulate reinforcement learning and function approximation in terms of Kan-invariant extensions (Mahadevan, 2026h), and Kan Extension Transformers use Kan-extension structure to unify attention, diffusion-style completion, and predict-detach self-conditioning (Mahadevan, 2026f). Diagrammatic Backpropagation treats noncommuting diagrams as learning signals (Mahadevan, 2026c). L INCS extends this viewpoint by studying not only base non-compositionality, but also tangent obstructions, sketch-specific interaction profiles and closure, coalgebraic stabilization, and homotopical repair. Kan Extensions and Universal Decision Learners: Universal Decision Learners (UDL) formulate decision learning in terms of Kan-invariant extensions of local decision data (Mahadevan, 2026h). In this view, a learner extends partial information along a functor and seeks invariance under a universal extension. Bellman equations, policy evaluation, and function approximation become forms of local-to-global coherence. L INCS adds a tangent refinement: Kan invariance in the base category
⇝
Kan invariance in the tangent category.
If a learned extension is a left or right Kan extension in C, then L INCS asks whether its tangent lift satisfies the corresponding universal property in TC. This suggests a combined principle: learn by universal extension, then require infinitesimal coherence of the extension. In this sense, L INCS can be viewed as tangent Kan-invariant learning. 4
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3
Learning Sketches and Tangent Learning Sketches
We introduce the core notions of learning sketches and tangent learning sketches in this section. Let C be a category whose objects are states, representations, hypotheses, local models, sections, policies, or world-model fragments. Its morphisms are computations, transitions, encoders, decoders, interventions, update rules, or gluing maps. A diagram D : J → C expresses a desired compositional relationship. To motivate the sketch-based formalism, note that in a bare category, an arithmetic loss function cannot be encoded by an expression such as g ◦ f − f ◦ g. L INCS translates these numerical losses into a categorical notion of non-compositionality, without assuming additive, metric, Hilbert, or vector-space enrichment. We begin with a sketch-theoretic formulation. Let S = (S, D, L, K) be a sketch in the sense of Makkai and Paré: S is a graph, D is a class of commutativity conditions in S, L is a class of cones, and K is a class of cocones. Write J = Path(S) for the free category on S. The commutativity data D generate a congruence ∼D on the morphisms of J, and hence a quotient functor qD : J → J/∼D . The cone and cocone data specify the universal constraints a strict model is expected to realize. A diagram D : J → C satisfies the commutativity part of the sketch precisely when it factors through qD : D
J −→ C D = D̄qD . ↓ qD ↗D̄ J/∼D Thus non-compositionality is not an arithmetic difference. It is the failure of a universal factorization problem, enriched by the relevant cone and cocone constraints, to have a solution. Definition 1 (Learning sketch). A learning sketch is a quadruple S = (S, D, L, K), where S is a graph of formal learning operations, D is a class of path-equations expressing intended commutativity, L is a class of cones expressing limiting constraints, and K is a class of cocones expressing colimiting constraints. If J = Path(S), the equations in D generate a quotient qD : J → J/∼D . The passage from categories to sketches gains expressive power but weakens automatic transport. Spivak makes this tradeoff explicit for database schemas: specified limit and colimit cones can express constraints such as one table being the product of two others, but the resulting sketch formalism no longer inherits the full collection of built-in data-migration functors available for unconstrained categorical schemas (Spivak, 2014). The same issue matters for L INCS . A map of the underlying learning graphs need not preserve the designated cones, cocones, admissible models, or their repair spaces. Accordingly, transport between learning sketches is additional structure: a L INCS morphism or change of sketch must state which constraints and factorization problems it preserves. This is why the later functoriality results impose preservation of learning sketches and quotient factorization problems rather than deriving it from an arbitrary schema map. There is also a limitation at the level of presentation itself. Barr and Wells show that not every natural category of structured objects and structure-preserving morphisms is the category of Set-valued models of an ordinary sketch: for example, certain categories defined by preservation of subinitial objects, and groups with center-preserving homomorphisms, are not sketchable in this sense (Barr and Wells, 1992). They also indicate how higher-order sketches can recover some structures beyond this boundary. Thus the L INCS definitions are relative to a chosen sketch-presentable class of learning models and morphisms. When admissibility or preservation conditions quantify over structure that an ordinary sketch cannot present, a higher-order, enriched, or fibrational replacement may be required. The tangent action on factorization problems is already formulated in a way that permits such a refinement. Definition 2 (Model of a learning sketch). Given a category C, a candidate model of a learning sketch S in C is a functor D : J → C, where J = Path(S). It is strict, or compositional, when there exists a functor D̄ : J/∼D → C such that D = D̄qD , and the images of the distinguished cones and cocones satisfy the universal requirements specified by L and K. Definition 3 (Non-compositionality as factorization failure). The non-compositionality of a model D : J → C of a learning sketch is the factorization problem FactS (D) = {D̄ : J/∼D → C | D = D̄qD and D̄ realizes L, K}. The model is compositional exactly when FactS (D) is inhabited. When only the commutativity component is under discussion, we write this as FactqD (D). 5
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Definition 4 (Tangent sketch). A tangent sketch is a sketch T = (ST , DT , LT , KT ) whose graph contains formal operations for a tangent endofunctor T , projection, zero, addition, vertical lift, and canonical flip, and whose commutativity, cone, and cocone data present the Cockett-Cruttwell tangent-category axioms. A model of T in a category C is precisely a choice of tangent-category structure on C, up to the level of strictness encoded by the sketch. Definition 5 (Tangent learning sketch). A tangent learning sketch is a learning sketch S = (S, D, L, K) equipped with an action of a tangent sketch on its factorization problem. Concretely, if J = Path(S) and D : J → C is admissible in a model (C, T) of the tangent sketch, then TD : J → C is again admissible and the base factorization problem FactS (D) is transported to the tangent factorization problem FactS (TD). Thus a tangent learning sketch is not merely a learning sketch interpreted in a tangent category; it is a sketch whose intended learning constraints are stable under the Cockett-Cruttwell tangent structure. Remark 1 (Infinitesimal objects and representable tangent semantics). Leung’s classification identifies general tangent structure on C with a strong monoidal functor from N-Weil1 to End(C) preserving the designated foundational pullbacks and vertical-lift equalizer (Leung, 2017, Thm. 14.1). MacAdam gives a complementary representable realization of this functorial semantics and uses it in the semantics of Lie theory (MacAdam, 2022). Following Cockett and Cruttwell, an infinitesimal object in a symmetric monoidal category is an object D equipped with zero, augmentation, multiplication, and coaddition maps satisfying the stated (co)universality axioms. Equivalently, in the symmetric monoidal closed setting it determines a strict symmetric monoidal functor D(−) : Weil1 −→ C. It induces the representable tangent functor T = [D, −] on C and the dual tangent functor T = D ⊗ (−) on C op . More generally, tangent structure on C is equivalently expressed by a Weil1 sketch action Weil1 × C −→ C. Consequently, a representable L INCS model admits a concrete reading: T D probes the learning diagram D by the infinitesimal object D, and INC(D) asks whether this infinitesimal probe still realizes the factorization, limit, and colimit constraints of the learning sketch. MacAdam’s infinitesimal object is therefore not itself an INC obstruction; it represents the tangent structure with which the obstruction is formed. Remark 2 (Lie-algebroidal realization of a tangent learning sketch). When a tangent learning sketch is realized over smooth state or parameter spaces, it can admit a more structured, Lie-algebroidal interpretation. MacAdam makes this presentation precise: involution algebroids form a tangent-categorical sketch, and their category in smooth manifolds is isomorphic to the category of Lie algebroids (MacAdam, 2022). For each sketch object x, let Ax → Mx be a Lie algebroid over the realized object Mx = D(x), with anchor ρx : Ax −→ T Mx . The bundle Ax represents the admissible infinitesimal learning operations, whereas the anchor sends them to the actual tangent directions along which the realized model can change. A sketch arrow f : x → y is equipped, when defined, with an algebroid transport Af : Ax → Ay over D(f ) satisfying the anchor square Ax ↓ ρx T Mx
Af
−−→ T (Df )
−−−−→
Ay ↓ ρy
T (Df )ρx = ρy Af .
T My .
Thus the path equations of the learning sketch constrain not only the base maps D(f ) but also their induced transports of admissible infinitesimal operations. For two parallel sketch paths, compositionality requires the corresponding anchored transports to agree; tangent non-compositionality measures failure of this compatibility after passage through the anchors. The Lie-algebroid identity ρx ([s, t]Ax ) = [ρx (s), ρx (t)] then relates bracket closure of abstract learning operations to closure of their realized vector fields. This is an additional realization of a tangent learning sketch, not a claim that every tangent sketch canonically carries a Lie algebroid. MacAdam’s result provides the appropriate sketch semantics for this additional structure; the anchor and bracket conditions below specify how that structure acts on a particular learning diagram. It connects L INCS directly to LASKO, where skill edits are modeled as anchored sections and order-sensitive skill interactions are detected by their Lie brackets (MacAdam, 2022; Mahadevan, 2026g). 6
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A failed factorization may indicate a hidden variable, a missing morphism, a defective representation, an unmodeled constraint, a failure of gluing, or a missing causal direction. The learning problem is not merely to reduce error, but to repair structure so that a universal property becomes closer to being satisfied.
4
Examples of Infinitesimal Non-Compositionality
To make the L INCS formalization more concrete, we now list several concrete instances of the L INCS pattern. In each case, the base learning system has a compositionality defect, and the infinitesimal version asks whether that defect remains controlled after tangent lift. We spell out the first example in detail, since it is the running template for the later architectures. Transformer equivariance. The question posed in the introduction has a precise answer. Let Fθ : Rm×d → Rm×e be a smooth content-only Transformer and let a permutation matrix P act on the token axis. Define the base equivariance defect EP,θ (X) = Fθ (P X) − P Fθ (X). This is the additive scalarizable representative of the obstruction to the equivariance square. Its first tangent lift, evaluated on an input perturbation V , is DEP,θ (X)[V ] = DFθ (P X)[P V ] − P DFθ (X)[V ]. If parameters are also perturbed by θ̇, the full tangent defect adds ∂θ Fθ (P X)[θ̇] − P ∂θ Fθ (X)[θ̇]. Consequently, tangent INC tests whether the local sensitivity of the Transformer intertwines the permutation action, rather than merely whether the two finite outputs happen to agree. More generally, the nth input-level prolongation is represented by Dn Fθ (P X)[P V1 , . . . , P Vn ] − P Dn Fθ (X)[V1 , . . . , Vn ]. Hence, if Fθ (P X) = P Fθ (X) holds identically on a smooth invariant domain, differentiating the identity shows that every displayed tangent and higher-order obstruction vanishes. The iterated L INCS question is therefore not mysterious in the ideal equivariant model: exact equivariance is inherited by all tangent orders. It becomes informative for learned or approximate equivariance, restricted data support, finite-precision implementations, and architectures containing positional encodings, causal masks, routing, or other order-dependent components. In those cases the base defect measures output-level symmetry breaking, the first lift measures its sensitivity, and higher lifts distinguish curvature and interaction effects that a small base loss can conceal. Thus the INC tower provides a graded diagnostic of where and at what differential order a language model departs from its intended equivariance sketch. GIRL. Gradient Infinitesimal Reinforcement Learning (GIRL) is the L INCS lift of gradient temporal-difference learning. Its base learning problem is policy evaluation in a Markov reward process: a value representation should be compatible with the Bellman update, and the tangent learning problem asks whether infinitesimal perturbations of states, values, parameters, or rewards preserve that compatibility. Let X denote the state object of an MRP, P π : X → X the transition morphism under policy π, Vθ : X → R a parameterized value representation, and Bπ : R → R,
Bπ (v) = r + γv,
the scalar Bellman update induced by the reward and discount. The GIRL base square is X ↓ Vθ R
Pπ
−−−→ B
π −−− →
X ↓ Vθ R.
As a learning sketch, GIRL is the quadruple SGIRL = (SGIRL , DGIRL , LGIRL , KGIRL ). The graph SGIRL has two formal objects x and v, standing for states and values, and generating arrows p : x → x,
u : x → v, 7
b : v → v.
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Under a model Dθ : Path(SGIRL ) → C, these are interpreted as Dθ (x) = X ,
Dθ (v) = R,
Dθ (p) = P π ,
Dθ (u) = Vθ ,
Dθ (b) = Bπ .
The commutativity component is the single Bellman equation DGIRL = { u ◦ p ∼ b ◦ u }. Equivalently, the quotient qD : Path(SGIRL ) → Path(SGIRL )/∼D identifies the two paths p
u
u
x− →x− → v,
b
x− →v→ − v.
The limiting component LGIRL records the admissible feature/value representation structure, for example the finitedimensional parameter or feature cone through which Vθ is required to factor in linear TD. The colimiting component KGIRL records the sampling or empirical aggregation structure used to form the TD estimating equations, such as the finite-sample cocone that aggregates transition observations into the stochastic approximation objective. In the minimal Bellman sketch these two classes may be empty; in GTD and GTD-MP they carry the extra representation and empirical averaging structure needed by the saddle objective. The base non-compositionality is therefore not the arithmetic Bellman residual itself. It is the obstruction Obs FactSGIRL (Dθ ) to realizing the Bellman sketch. A scalar TD, GTD, or GTD-MP loss is obtained only after applying a scalarization to this obstruction. The corresponding tangent learning sketch equips SGIRL with the Cockett-Cruttwell tangent action. Its model sends Dθ to TDθ : Path(SGIRL ) → C, whose interpreted arrows are TP π , TVθ , and TBπ . The tangent Bellman square is TX ↓ TVθ TR
TP π
−−−−→ TB
π −−−− →
TX ↓ TVθ TR.
The GIRL infinitesimal non-compositionality is INCGIRL (Dθ ) = Obs FactSGIRL (TDθ ) . After scalarization, this yields the tangent Bellman error developed in the companion GIRL paper (Mahadevan, 2026d). At the categorical level considered here, GIRL augments the base Bellman factorization problem with an infinitesimal Bellman-closure term; the concrete LINCS-GTD-MP saddle formulation is introduced in that companion work. Problem class
Representative obstruction or loss
GIRL (Mahadevan, 2026d) DB (Mahadevan, 2026c) BRIDGE/SKFM/IC (Mahadevan, 2026b) ALLORA (Mahadevan, 2026a) LASKO (Mahadevan, 2026g)
Bellman and tangent Bellman loss compositional backpropagation and tangent diagram loss causal Lie-bracket closure loss LoRA matrix commutator and quotient interaction profiles Markdown-category skill-composition loss
A detailed experimental evaluation of L INCS across these problem classes—GIRL, DB, BRIDGE/SKFM/IC, ALLORA, and LASKO—is currently underway and will be reported in future work. GIKET. Kan Extension Transformers learn by extending local neighborhoods into global representations (Mahadevan, 2026f). The base defect is a failure of neighborhood extension or Kan invariance; the INC lift asks whether the extension varies coherently under infinitesimal perturbations of tokens, embeddings, or neighborhoods. Differential DB. Diagrammatic Backpropagation repairs noncommuting computational diagrams, and is the canonical example from which L INCS grows. A useful concrete instance is Sudoku solving. A candidate Sudoku grid determines many local views: its rows, columns, blocks, and individual cells. A correct solution is not merely a vector of predicted digits; it is a diagram whose overlapping local views agree and whose rows, columns, and blocks satisfy the Sudoku constraints. 8
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Let X be a formal object for a candidate grid, Ri the i-th row object, Kj the j-th column object, Bb the b-th block object, and Cij the cell object at position (i, j). The Sudoku DB learning sketch is SDB = (SDB , DDB , LDB , KDB ). The graph SDB contains the view maps ρi : X → Ri ,
κj : X → Kj ,
βb : X → Bb ,
rij : Ri → Cij ,
kij : Kj → Cij ,
bij : Bb → Cij ,
and the cell-projection maps
where b is the block containing cell (i, j). The commutativity component DDB says that every overlapping view gives the same cell: rij ρi ∼ kij κj ∼ bij βb for all (i, j). The limiting component LDB records that a grid is assembled from compatible local cell assignments, for example as a product-like cone of cell predictions. The colimiting component KDB records gluing: row, column, and block views are identified along their shared cells. Additional Sudoku validity maps can be included in SDB , such as maps Ri , Kj , Bb → Perm9 , whose sketch constraints require each row, column, and block to realize the digit set exactly once. A neural Sudoku solver is then a candidate model Dθ : Path(SDB ) → C whose objects may be soft digit distributions, logits, constraint states, or learned local representations. The DB obstruction Obs FactSDB (Dθ ) measures failure of the predicted local views to glue into a globally consistent Sudoku solution. A scalar DB loss is obtained by measuring these overlap and constraint obstructions. The tangent learning sketch for Differential DB applies the tangent structure to the same Sudoku factorization problem: TDθ : Path(SDB ) → C. Its INC term is INCDB (Dθ ) = Obs FactSDB (TDθ ) . This asks whether infinitesimal changes in logits, givens, local cell beliefs, or constraint messages preserve row-columnblock compatibility. In a crossword puzzle, the analogous sketch replaces Sudoku rows, columns, and blocks by across clues, down clues, and shared letter cells. The Berkeley Crossword Solver uses neural candidate generation followed by loopy belief propagation and local search to enforce such overlap constraints among potential answers (Wallace et al., 2022). In L INCS terms, DB measures failure of clue-wise word assignments to agree on overlaps, while Differential DB measures whether that agreement is stable under infinitesimal changes in clue embeddings or letter beliefs. ALLORA. ALLORA trains low-rank neural adapters with Lie-algebraic commutator penalties so independently useful adapters compose more predictably (Mahadevan, 2026a). Its base defect is adapter order-sensitivity; its reflective L INCS version studies quotient-projectable adapter-flow interactions, including both bracket obstructions and connection-dependent symmetric acceleration in the representation bundle. The low-rank presentation ∆ = BA has a GL(r) redundancy, (B, A) 7→ (BQ, Q−1 A). Consequently, an optimizer field in factor coordinates is not automatically a vector field on the quotient of effective updates. A valid tangent realization must either fix a balanced gauge or provide an equivariant horizontal lift through the presentation map π(B, A) = BA. Once this descent datum is fixed, two task/commutator directional interactions may be retained separately. The Lie bracket records their antisymmetric difference; a declared connection also defines their symmetric acceleration. Which component is useful for predicting adapter composition is a sketch- and realization-specific question, not a categorical preference for antisymmetry. LASKO. LASKO models agentic skill optimization over controlled Lie algebroids, where edits to prompts, schemas, tools, validators, and workflow artifacts are treated as anchored sections whose brackets reveal order-sensitive repair interactions (Mahadevan, 2026g). Its L INCS version studies vector-bundle bracket defects for latent knowledge and skill-update manifolds. 9
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5
Diagrammatic Backpropagation Revisited
Diagrammatic Backpropagation can be expressed as the repair of factorization failures (Mahadevan, 2026c). Let Dθ : J → C be a learned diagram depending on parameters θ, obtained from the path category of a learning sketch S = (S, D, L, K), and let q = qD : J → J/∼D encode the intended commutativity constraints. DB attempts to adjust θ so that Dθ factors through q, possibly together with the limit and colimit constraints specified by L and K. In enriched settings, this factorization problem is often scalarized by a loss: LDB (θ) = Φ(Factq (Dθ )), where Φ is an application-specific measure of factorization failure, such as a norm, energy, likelihood, divergence, Bregman distance, or empirical risk. Such scalarizations are important computationally, but they are not primitive in the categorical definition. DB therefore replaces pointwise error with compositional error. The learning signal is no longer only “prediction minus target”; it is a failure of a structured diagram to satisfy a universal compositionality condition. Remark 3. Ordinary losses fit this pattern after scalarization. Supervised learning uses a diagram in which an input is mapped by a model and compared with a target map. Autoencoding compares an input with the reconstruction composite. Bellman residual minimization compares a value function with its Bellman transform. Contrastive learning compares two representation paths induced by augmentations. Sheaf learning compares restrictions along overlaps. 5.1
From global to local non-compositionality
Diagrammatic Backpropagation can also be read as a localization principle. A global learning diagram is covered by smaller computational diagrams, and its global failure of compositionality is restricted to failures attached to those local pieces. This suggests a categorical generalization of the role played by the chain rule in ordinary backpropagation. Let {ia : Ja → J}a∈A be a family of subsketch inclusions whose images cover the objects, generating arrows, and specified constraints of a learning sketch S. Write Sa for the induced local sketch and Da = Dia for the restriction of a model D : J → C. Definition 6 (Obstruction localization). An obstruction localization for the cover {Ja }a∈A is a natural assignment ΛD : Obs FactS (D) −→ Comp({Obs(FactSa (Da ))}a∈A ) , where Comp denotes families of local obstruction data satisfying the required compatibility conditions on overlaps Ja ×J Jb . The localization is complete when a compatible family of inhabited local factorization problems glues to an inhabited global factorization problem. The compatibility term is essential. Vanishing of each isolated local defect need not imply global compositionality unless the chosen local repairs agree on their shared objects, paths, cones, and cocones. Thus obstruction localization is a descent statement for factorization problems, rather than an additive decomposition of a numerical error. Proposition 1 (Tangent stability of obstruction localization). Suppose restriction along each ia preserves admissible models and factorization problems, the obstruction assignment is natural under these restrictions, and tangent lift commutes with restriction: (TD)ia ∼ = T(Dia ). Then every obstruction localization ΛD has a tangent localization ΛTD : INC(D) −→ Comp({INC(Da )}a∈A ) . If the base localization is complete and tangent lift preserves the overlap descent data, the tangent localization is complete as well. ∼ TDa . Naturality of the factorization and obProof. Restricting the tangent model to Ja gives (TD)ia = struction assignments therefore sends the global tangent obstruction Obs(FactS (TD)) to the compatible family {Obs(FactSa (TDa ))}a∈A , which is precisely the displayed family of local INC objects. Preservation of overlap descent transports the local-to-global gluing property to the tangent level. For a feedforward network, the indexing category is a chain of layer maps and the terminal supervised error is a scalarization of a global factorization obstruction. Reverse-mode differentiation applies the chain rule to transport the sensitivity of that scalarization through the chain, producing a local parameter-update signal at each layer. In this sense, ordinary backpropagation is the scalarized, additive instance of obstruction localization. L INCS retains the same global-to-local architecture before scalarization: it localizes base obstructions, tangent obstructions, and, when defined, interaction-profile or higher-order obstructions. This opens the possibility of “backpropagating” structured obstruction objects even when subtraction, norms, or scalar losses are unavailable. 10
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6
Tangent Lift of Diagrammatic Backpropagation
We now assume that C carries tangent structure. Tangent categories axiomatize the behavior of tangent bundles in a categorical setting (Cockett and Cruttwell, 2014). A tangent category has an endofunctor T:C→C together with structure maps such as projection, zero, addition, vertical lift, and canonical flip, satisfying axioms abstracting the tangent bundle of a smooth manifold. In this paper, tangent learning sketches are the bridge between these two levels: they specify ordinary learning constraints and require those constraints to remain meaningful after applying the tangent functor. Thus the tangent functor lifts not only objects and morphisms in C, but also the sketch-theoretic factorization problems that define compositional learning. The tangent-category axioms also support a notion of vector field and, in the standard tangent-categorical setting, a Lie bracket of vector fields. This is important for L INCS because infinitesimal coherence is not exhausted by the existence of FactS (TD). A learning diagram may generate tangent directions that are individually admissible, while their bracket lies outside the proper admissible distribution represented by the current architecture or repair language. In that case the learned infinitesimal dynamics are not closed relative to that distribution. A connection, when additionally declared, exposes symmetric as well as antisymmetric second-order interaction. Definition 7 (Tangent model). Let D : J → C be a model of a learning sketch S = (S, D, L, K), with J = Path(S). Its tangent model is TD : J → C, obtained by applying the tangent functor objectwise and morphismwise: (TD)(j) = T(Dj),
(TD)(u) = T(Du).
Remark 4 (Weil-algebra indexing of tangent models). Leung’s classifier makes the iteration in this definition explicit (Leung, 2017, Sec. 14). For the dual-number object W = N[x]/(x2 ) and the classifying functor F : N-Weil1 → End(C), one has T = F (W ). Strong monoidality identifies the n-fold iterate with Tn ∼ = F (nW ), where nW is the n-fold coproduct of W , whereas the fiber-product power T (n) used in the tangent axioms is represented by F (W n ). Consequently INC(n) (D) may be read as the obstruction to the learning factorization problem after the Weil prolongation indexed by nW . The distinction between nW and W n separates iterated or mixed infinitesimal directions from the pullback powers required for additive tangent bundles. The base factorization problem asks whether computations compose: FactS (D) ̸= ∅. The tangent factorization problem asks whether infinitesimal perturbations compose: FactS (TD) ̸= ∅. Remark 5 (Tangent structure and quotient sketches). The Cockett-Cruttwell axioms require the tangent functor to preserve the finite limits used in the tangent structure, together with compatibility of the projection, zero, addition, vertical lift, and canonical flip. The quotient qD : J → J/∼D , however, is induced by the commutativity component of a sketch and may involve colimit-like data in the category of indexing shapes. The cone and cocone components L, K add further universal constraints. L INCS therefore does not assume that T literally preserves the quotient construction or all sketch data as colimits. Instead, it assumes that the factorization problem induced by the sketch is stable under tangent lift: whenever D : J → C is admissible, TD is again an admissible model of the same learning sketch, and FactS (TD) is the transported tangent factorization problem. Condition 1 (Tangent-stable factorization fibration). The factorization fibration p : Fact(C) → Learn(C) is tangentstable when there is a lift of the tangent functor to factorization problems, T
Fact(C) −−Fact −→ ↓p T Learn(C) − →
Fact(C) ↓p Learn(C),
so that the fiber over D is sent to the fiber over TD. This is the fibrational compatibility condition replacing any blanket assumption that T preserves all quotient sketches as colimits. 11
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Definition 8 (Scalarized L INCS objective). When scalarizations are available, a L INCS objective associated with a diagram Dθ is LLIN CS (θ) = Lbase Obs(Factq (Dθ )) + λLtan INC(Dθ ) , where λ = 0 is permitted. More generally, scalarized L INCS may include higher-order tangent, bracket, connection, curvature, or jet terms. These are candidate signals supplied by the categorical realization; their nonzero computational admission is an additional validation decision rather than an axiom of tangent compositionality. Thus L INCS is not a new optimizer. It is a rule for lifting learning compositionality problems into tangent structure.
7
L INCS Axioms
In this section, we propose an axiomatic structure for L INCS . First, we state the underlying theoretical object more formally. Definition 9 (Tangent compositionality problem). Let D : J → C be a model of a learning sketch S = (S, D, L, K), with commutativity quotient q = qD : J → J/∼D . The tangent compositionality problem of D is FactS (TD), or Factq (TD) when only the commutativity quotient is under discussion. It asks whether the tangent-lifted model TD satisfies the same sketch-theoretic factorization problem as D. Definition 10 (Infinitesimal Non-Compositionality). The infinitesimal non-compositionality of D is the obstruction to inhabiting its tangent compositionality problem: INC(D) := Obs FactS (TD) . When higher-order tangent structure is available, the n-th order INC is INC(n) (D) := Obs FactS (Tn D) . Thus INC has a factorization form. A learning diagram may additionally determine vector fields, tangent sections, a connection, or other prolonged operations. These data refine tangent INC, but they do not replace its factorization definition. Definition 11 (Interaction signature). An interaction signature ΩD for an admissible model D is a declared family of operations on an admissible tangent distribution AD ⊆ VectD . It may contain the Lie bracket, covariant directional derivatives for a specified connection, curvature or torsion, canonical-flip comparisons, or higher jet operations. The signature is part of the learning sketch or its realization; no single second-order operation is required for every sketch. Definition 12 (Connection-aware second-order profile). Suppose the realization of D carries a connection ∇ and generates admissible fields X, Y . Its ordered second-order profile is 2 J∇,D (X, Y ) = ∇X Y, ∇Y X . The associated antisymmetric and symmetric components are A∇ (X, Y ) =
1 ∇X Y − ∇Y X , 2
S∇ (X, Y ) =
1 ∇X Y + ∇Y X . 2
For a torsion-free connection, 2A∇ (X, Y ) = [X, Y ]. In contrast, S∇ depends on the declared connection and is not primitive tangent category structure. Definition 13 (Bracket-closure INC). Let VectD denote the class of tangent directions, vector fields, or tangent sections generated by an admissible learning model D, and let AD ⊆ VectD be the directions declared admissible for the current architecture or repair language. The bracket-closure INC of D is the closure problem Cl[,] (AD ) :
{ [X, Y ] | X, Y ∈ AD } ⊆ AD .
The model is bracket-closed relative to AD when this inclusion holds. Failure of the inclusion is a second-order infinitesimal non-compositionality. Taking AD to be the entire tangent bundle would make this test vacuous for genuine smooth vector fields, whose bracket is automatically tangent; structural nonclosure is meaningful relative to a proper admissible distribution such as a fixed adapter placement, shared factor subspace, or repair vocabulary. 12
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This definition avoids subtracting morphisms in the base category. The primitive obstruction is not a residual g ◦f −f ◦g. It is the failure of tangent directions generated by non-compositional learning data to be involutive under the Lie bracket supplied by the tangent structure. Bracket nonclosure is one interaction obstruction. A connection-aware realization 2 may instead or additionally observe S∇ , A∇ , or the full ordered profile J∇ without asserting that antisymmetry is the unique carrier of useful tangent information. If Factq (D) is scalarized by a loss, then Obs(Factq (TD)) may induce a gradient-like or Jacobian-like regularizer. But L INCS is not merely gradient regularization. The object being lifted is not necessarily a scalar objective; it is the universal compositionality problem whose obstruction produced the loss. Remark 6. Backpropagation differentiates a scalar loss. L INCS lifts the categorical factorization problem whose scalarization produced the loss. This distinction matters because the tangent lift can preserve structure that scalarization discards. 7.1
Axioms
We now propose an axiomatization of L INCS categories. Definition 14 (L INCS object). A L INCS object is a pair (X, SX ), where X ∈ Ob(C) and SX is a specified class of learning sketches whose models land in, are indexed by, or are interpreted around X. Informally, X is a state, representation, policy, local model, or world-model object together with the compositionality problems it is expected to satisfy. Definition 15 (L INCS morphism). A L INCS morphism f : (X, SX ) → (Y, SY ) is a morphism f : X → Y in C together with a transport of learning sketches and their models from SX to SY , such that tangent lift commutes with this transport. Concretely, if D : J → C is a transported model, then transporting TD agrees with tangent-lifting the transported model. Definition 16 (INC category). An INC category is a tangent category (C, T) equipped with: 1. a class SkC of tangent learning sketches S = (S, D, L, K); 2. for each S, with J = Path(S), a class of admissible models D : J → C; 3. for each sketch S, the associated base factorization problem FactS (D), including the quotient induced by D and the cone/cocone constraints induced by L, K; 4. for each admissible model D, the tangent model TD : J → C and tangent factorization problem FactS (TD); 5. when higher interaction data are assigned to learning models, an interaction signature ΩD , an admissible distribution AD , and the corresponding closure or profile problem; bracket closure Cl[,] (AD ) is the intrinsic antisymmetric specialization. Axiom 1 (Base compositionality). For every admissible model D : J → C of a learning sketch S, D is compositional if and only if FactS (D) is inhabited. Axiom 2 (Tangent admissibility). If D : J → C is an admissible model of a tangent learning sketch S, then TD : J → C is also an admissible model of S, and the assignment D 7→ TD agrees with the tangent action specified by the sketch. Axiom 3 (Presentation descent). Suppose a computational realization is expressed in a parameter presentation π : P → M with redundant representatives. An optimizer or repair field VP : P → T P is admissible as a field on the model object M only when it is π-projectable: there is a field VM : M → T M such that T π ◦ VP = VM ◦ π. Alternatively, the realization must declare a gauge fixing or an equivariant horizontal lift h satisfying T π◦h(VM ) = VM . Tangent obstruction signals that fail this descent condition are presentation artifacts and are not INC data on M . Axiom 4 (Tangent compositionality). For every admissible model D : J → C of a learning sketch S, infinitesimal compositionality is the factorization property FactS (TD) ̸= ∅. Thus tangent compositionality is defined by the same universal property as base compositionality, but applied after the tangent functor. INC is the obstruction Obs(FactS (TD)) to this property. 13
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Axiom 5 (Sketch-specific interaction). For every admissible model D equipped with an interaction signature ΩD and admissible tangent distribution AD , each operation declared by ΩD determines a closure, factorization, or profile problem. Failure of a declared closure or factorization property is an admissible INC signal. When [, ] ∈ ΩD , this includes the relative bracket-closure problem Cl[,] (AD ) :
[X, Y ] ∈ AD
for all X, Y ∈ AD .
2 When a connection belongs to the realization, J∇ , S∇ , and A∇ are admissible profile data.
The sketch does not categorically mandate that the antisymmetric projection be primary. Axiom 6 (Conditional non-compositionality transport). For a model and sketch carrying an interaction signature, a declared transport is a natural assignment tD,S : Obs FactS (D) ⇝ ObsΩ,S (TD).
The dashed arrow emphasizes that transport is additional sketch-relative data, not an equality between base failure and bracket closure. An admissible transport must be natural under model transformations, descend through any parameter presentation, and preserve the semantics of the factorization problem being repaired. Bracket transport is recovered when ΩD = {[, ]}; its existence alone does not imply that its scalarization is informative for a particular repair target. Axiom 7 (Functoriality of repair). If a transformation of models α : D → D′ is an admissible repair, then Tα : TD → TD′ is an admissible infinitesimal repair. Moreover, repair preserves factorization: if D′ factors through q, then repaired models factor through q whenever the repair is declared compositionality-preserving. For a cover by local subsketches, a repair declared descent-compatible must also commute with restriction up to the specified coherence, resU ◦RG ∼ = RU ◦ resU , and satisfy the corresponding cocycle conditions on overlaps. Functorial repair, descent-compatible repair, and empirically useful repair are distinct notions. Axiom 8 (Optional scalarization). An INC category may be equipped with scalarization maps Φq : {obstructions to factorization problems over q} → R≥0 , but such maps are additional computational structure, not part of the bare categorical definition. When present, Φq should vanish on inhabited factorization problems: Factq (D) ̸= ∅ ⇒ Φq Obs(Factq (D)) = 0. Condition 2 (Observation specificity and identifiability). A computational realization may retain a profile-valued observation ΨS (D, e) before scalarization, containing local, overlap, counterfactual, or downstream effects of a candidate repair e. Structural identification from this profile requires the separate condition ΨS (D, e) = ΨS (D, e′ )
=⇒
e ≃ e′ .
When this implication fails, the sketch identifies only an equivalence class or moduli space of repairs. Defect reduction alone does not imply recovery of a privileged architecture. Definition 17 (Validated scalarized realization). A validated scalarized L INCS realization consists of profile or obstruction scalarizations together with an admission rule, fitted without test leakage, that may assign zero weight to any auxiliary tangent signal. A typical base-first objective is X Lval gk (V)λk Φk (INCk (D)), D = Φq Obs(Factq (D)) + k∈ΩD
where V is validation data, gk (V) ∈ {0, 1} or R≥0 , and sparse solutions with gk = 0 are admissible. For a connectionaware two-field realization the learned coefficients may be written (µs , µa ) for symmetric and antisymmetric profiles. If these profiles are inserted directly into a vector update rather than a scalar loss, signed regularized coefficients and the appropriate step-size scaling are allowed. b K . A stochastic Condition 3 (Stochastic consistency). Suppose an interaction profile is estimated from minibatches by Ψ L INCS realization is consistent when its aggregated estimator converges to the profile of the expected admissible field, b K −→ ΨE Ψ in the declared probabilistic or metric sense. A finite-sample certificate may take the form 2 b K − ΨE ≤ C + b2K , E Ψ bK → 0. K Same-minibatch and independently crossed directional terms may have different variance because their covariance differs, even when they converge to the same expected-field profile. This estimator convergence is distinct from the contractivity of the coalgebraic INC endofunctor. 14
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Definition 18 (L INCS category). A L INCS category is an INC category together with a rule that, for every admissible model D : J → C, regards the pair Factq (D), Factq (TD) as the basic learning datum. When an interaction signature is present, the datum is refined to Factq (D), Factq (TD), ObsΩ,D (TD) , with Cl[,] (AD ) one possible component. If optional scalarizations are present, this datum may be converted into an objective X LD = Φq Obs(Factq (D)) + λΦTq INC(D) + µk Φkq INCk (D) . k∈ΩD
The bare category supplies candidate interaction data; a validated scalarized realization determines which coefficients µk are nonzero. Proposition 2 (Functoriality of INC data). Let F : C → D be a tangent functor between tangent categories, so that F TC ∼ = TD F . Suppose F preserves learning sketches, admissible models, and quotient factorization problems. Then F maps INC data in C to INC data in D: F Obs(Factq (TC D)) 7−→ Obs FactF q (TD (F D)) . In particular, tangent functors preserving the learning structure preserve tangent compositionality problems and their obstructions. If interaction signatures, connections, admissible distributions, or presentation quotients are included, preservation of those additional data is a separate hypothesis; it does not follow from tangent functoriality alone. Proof. Since F preserves the learning sketch and quotient, the base factorization problem Factq (D) is sent to the corresponding factorization problem FactF q (F D). Since F is tangent, F (TC D) ∼ = TD (F D). Therefore the image of the tangent factorization problem Factq (TC D) is the tangent factorization problem FactF q (TD (F D)). Applying the obstruction assignment gives the claimed map on INC data. 7.2
Universal Properties
The universal property question is: what is the minimal tangent learning structure generated by a category of base learning diagrams? Definition 19 (L INCS completion, informal). Let C be a category equipped with learning sketches and admissible models. A L INCS completion of C is a tangent category LINCS(C) together with an embedding i : C → LINCS(C) such that base factorization problems in C acquire tangent factorization problems in LINCS(C), and any sketch/modelpreserving functor from C into a L INCS category factors through i. Conjecture 1 (Free L INCS completion). For a suitable class of sketch-equipped categories C, there exists a free L INCS category LINCS(C) satisfying the universal property: HomLINCSCat (LINCS(C), D) ∼ = HomSkCat (C, U D), where U forgets tangent factorization structure. This conjecture is the L INCS analogue of free tangent or differential completions. It says that L INCS is not merely an added regularizer; it is a universal completion that freely adds infinitesimal factorization structure to a base learning category. For interaction-enriched L INCS, the universal property is relative to the declared signature Ω: a bracket-closed completion, a connection-equipped jet completion, and a higher-operation completion are distinct specializations rather than one mandatory Lie-algebraic completion.
8
L INCS as a Coalgebra
There is a second universal perspective on L INCS , related to the standard coalgebraic treatment of state-based systems and final semantics (Rutten, 2000). A tangent category contains an endofunctor T : C → C. An endofunctor does not by itself make every object into a coalgebra, but it determines a category of coalgebras once one chooses structure maps γ : X → TX. 15
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In a tangent category, vector fields are the basic examples: a vector field on X is a section v : X → TX of the tangent projection pX : TX → X. Thus vector fields can be read as T -coalgebras compatible with tangent structure. This gives a coalgebraic reading of INC. A learning model D : J → C has a tangent tower D,
TD,
T2 D,
...
and a corresponding tower of factorization problems Factq (D),
Factq (TD),
Factq (T2 D),
....
The sequence resembles a system repeatedly unfolding its own infinitesimal behavior. Each stage asks whether the previous compositionality problem remains coherent after one more application of the tangent functor. Definition 20 (INC endofunctor). The INC endofunctor is the operation TINC : INC(C) → INC(C) defined on admissible tangent factorization problems by TINC Factq (Tn D) = Factq (Tn+1 D), whenever these iterated tangent models are admissible. Equivalently, after applying the obstruction assignment, TINC Obs(Factq (Tn D)) = Obs(Factq (Tn+1 D)). Definition 21 (L INCS coalgebra). A L INCS coalgebra is an INC object A ∈ INC(C) equipped with a structure map γ : A → TINC A that transports base INC data to its next tangent lift. In concrete learning models, γ may be induced by a vector field, update rule, policy perturbation, local flow, or infinitesimal repair map. Definition 22 (Coalgebraic L INCS fixed point). A coalgebraic L INCS fixed point is a L INCS coalgebra (A, γ) for which the comparison A −→ TINC A is stable in the chosen categorical sense: isomorphism, equivalence, bisimulation, or convergence under an applicationspecific scalarization. At such a point, applying the tangent lift produces no essentially new INC data. For practical learning systems, the most useful notion of stability is likely not literal isomorphism, but enriched convergence: the INC tower stabilizes when the obstruction objects at successive levels are related by an approximate bisimulation and a scalar Lyapunov functional decreases to a fixed tolerance. For a neural network, reaching νTINC means that further tangent unfolding does not reveal new out-of-distribution directions, declared interaction defects, or higher-order factorization obstructions not already represented by the learned tangent model. This formulation says that L INCS can be studied as the stable behavior of iterated infinitesimal non-compositionality. The base category supplies learning diagrams; the tangent functor unfolds their infinitesimal defects; the sketch-specific interaction signature tests whether generated directions remain coherent and internally expressible; and the coalgebraic fixed point is the limit at which these unfoldings become self-consistent. Definition 23 (Set-based class realization). A set-based class realization of the INC endofunctor consists of a fully faithful semantics U : INC(C) −→ Class and a set-based endofunctor F : Class → Class together with a natural isomorphism ∼ F U. U TINC = Here set-based means that, for every class A and every x ∈ F (A), there are a set A0 ⊆ A and x0 ∈ F (A0 ) such that x = F (iA0 ,A )(x0 ), where iA0 ,A : A0 ,→ A is inclusion. We say that the realization creates the final carrier when the carrier and structure map of the final F -coalgebra lie in the essential image of U , and coalgebra morphisms between realized objects are reflected by U . Theorem 1 (Existence of a final INC coalgebra). Let TINC admit a set-based class realization. Then the realizing endofunctor F has a final coalgebra in Class. If the realization creates the final carrier, then TINC has a final coalgebra in INC(C). 16
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Proof. The first statement is the Final Coalgebra Theorem of Aczel and Mendler: every set-based endofunctor on the category of classes has a final coalgebra (Aczel and Mendler, 1989). Let (Z, ζ : Z → F Z) be this final coalgebra. When the realization creates the final carrier, choose A ∈ INC(C) with U A ∼ = Z. The natural isomorphism U TINC ∼ = FU transports ζ to a TINC -coalgebra structure on A. For any other realized coalgebra, finality of (Z, ζ) gives a unique coalgebra morphism into Z; fullness and reflection lift it uniquely to A. Hence A is final in INC(C). Definition 24 (Accessible set realization). An accessible set realization of TINC consists of a fully faithful functor V : INC(C) −→ Set, an accessible endofunctor G : Set → Set, and a natural isomorphism V TINC ∼ = GV . As above, the realization creates the final carrier when the final G-coalgebra lies in the essential image of V and coalgebra morphisms between realized objects are reflected by V . Theorem 2 (Set-sized final INC coalgebra). If TINC admits an accessible set realization, then its realizing endofunctor G has a final coalgebra in Set. If the realization creates the final carrier, this coalgebra lifts to a final TINC -coalgebra in INC(C). More quantitatively, let κ > ℵ0 be regular and suppose that 2λ ≤ κ for every λ < κ. If G is κ-accessible and |A| < κ
|GA| ≤ κ,
=⇒
then the final G-coalgebra has cardinality at most κ. Proof. Barr proves that the forgetful functor from coalgebras of any accessible endofunctor on Set has a right adjoint, and therefore that its coalgebra category has a terminal object (Barr, 1993). His regular-cardinal refinement gives the stated bound of κ on the terminal carrier under the displayed hypotheses. The lifting from Set to INC(C) is the same fully faithful transfer used in the preceding theorem. Conjecture 2 (L INCS –final-coalgebra comparison). Under the hypotheses of either preceding existence theorem, the universal L INCS completion is equivalent to the resulting final INC coalgebra: LINCS(C) ≃ νTINC , with the equivalence compatible with the embedding of base learning models and their iterated tangent factorization problems. The Aczel–Mendler theorem discharges existence once the INC endofunctor has a set-based class realization. Barr’s theorem avoids proper classes whenever TINC instead has an accessible realization on sets, and its regular-cardinal form bounds the size of the final carrier. What remains conjectural is the comparison: that the final coalgebra so obtained also satisfies the free-completion universal property proposed for LINCS(C). Independently, convergence of the concrete INC tower can be proved under explicit contractivity hypotheses by metric coinduction, which establishes properties of a limit from invariance under a contractive approximation step (Kozen and Ruozzi, 2009). Definition 25 (Metric realization of INC). A metric realization of an INC fiber is a complete metric space (MD , dD ) of obstruction states associated with an admissible model D, together with a map FD : MD → MD realizing the action of TINC . It is ρ-contractive when there is a constant 0 ≤ ρ < 1 such that dD (FD x, FD y) ≤ ρdD (x, y)
for all x, y ∈ MD .
The realization is bisimulation-conservative when distance zero identifies precisely the INC states that are bisimilar in the obstruction fiber. Theorem 3 (Contractive coalgebraic stabilization). Let (MD , dD , FD ) be a ρ-contractive metric realization of an INC fiber, and let x0 ∈ MD represent the base obstruction Obs(Factq (D)). Define the realized INC tower by xn+1 = FD xn . Then: 1. there is a unique fixed point x∗ = FD x∗ ; 2. the tower converges geometrically, with dD (xn , x∗ ) ≤
ρn dD (x1 , x0 ); 1−ρ
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3. if an approximate tower satisfies dD (e xn+1 , FD x en ) ≤ ϵ at every stage, then dD (e xn , x∗ ) ≤ ρn dD (e x0 , x∗ ) +
1 − ρn ϵ, 1−ρ
and hence lim supn dD (e xn , x∗ ) ≤ ϵ/(1 − ρ). If, in addition, the realization is bisimulation-conservative, then the exact tower stabilizes at a unique INC behavior up to bisimulation. If this behavior lifts to a final TINC -coalgebra in the obstruction fibration and the universal L INCS embedding is dense and preserved by TINC , then it realizes the comparison LINCS(C) ≃ νTINC . Proof. This is the contractive fixed-point construction underlying metric coinduction (Kozen and Ruozzi, 2009), specialized to the realized INC tower. For m > n, contractivity gives dD (xk+1 , xk ) ≤ ρk dD (x1 , x0 ). The triangle inequality therefore yields dD (xm , xn ) ≤
m−1 X
ρk dD (x1 , x0 ) ≤
k=n
ρn dD (x1 , x0 ). 1−ρ
Thus (xn ) is Cauchy. Completeness supplies a limit x∗ , and the Lipschitz continuity of FD gives FD x∗ = limn FD xn = limn xn+1 = x∗ . If y∗ is another fixed point, then dD (x∗ , y∗ ) = dD (FD x∗ , FD y∗ ) ≤ ρdD (x∗ , y∗ ), so dD (x∗ , y∗ ) = 0, proving uniqueness. Letting m → ∞ in the Cauchy estimate proves the geometric bound. For the approximate tower, dD (e xn+1 , x∗ ) ≤ dD (e xn+1 , FD x en ) + dD (FD x en , FD x∗ ) ≤ ϵ + ρdD (e xn , x∗ ). Induction solves this recurrence and gives the stated finite-n and asymptotic bounds. Bisimulation-conservativity turns uniqueness at distance zero into uniqueness up to bisimulation. Under the final lifting and density hypotheses, the lifted fixed behavior has the universal property of νTINC , while density and preservation extend the comparison from presentable learning models to their L INCS completion. This proves the final conditional statement. The scalar certificate used in applications fits this theorem when the obstruction energy X En = Φ Obs(Factq (Tn D)) + µk,n Φkn + ηΦhn k∈ΩD
is a Lyapunov upper bound for dD (xn , x∗ ). In finite-sample learning, an inequality En+1 ≤ ρEn + ϵ then gives En ≤ ρn E0 +
1 − ρn ϵ, 1−ρ
which is the scalar counterpart of approximate coalgebraic stabilization. The theorem therefore proves the convergent fixed-point part of the conjecture. Establishing finality without the lifting hypotheses, and proving that the resulting final coalgebra satisfies the free-completion universal property, remain open.
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L INCS as a Category
We now collect the preceding ideas into an explicit categorical picture. Definition 26 (Learning diagram category). A learning diagram category is a category Learn(C) whose objects are admissible models D : J → C of learning sketches, and whose morphisms are model transformations preserving the intended learning semantics. Definition 27 (Factorization fibration). A factorization fibration over Learn(C) is a functor p : Fact(C) → Learn(C) whose fiber over D : J → C is the factorization problem Factq (D) associated with its learning sketch. 18
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Definition 28 (INC fibration). The INC fibration is the tangent-lifted factorization fibration pT : INC(C) → Learn(TC), whose fiber over TD is the tangent factorization problem Factq (TD). Definition 29 (Interaction-profile fibration). When admissible learning models carry signatures ΩD , the interactionprofile fibration is a functor pΩ : IntINC(C) → Learn(TC), whose fiber over TD is the declared closure, factorization, or profile problem ObsΩ,D (TD). Connection-equipped 2 fibers include the ordered profile J∇ and its symmetric and antisymmetric projections. Definition 30 (Bracket-closure fibration). When [, ] ∈ ΩD , the bracket-closure specialization is a functor p[,] : BrINC(C) → Learn(TC), whose fiber over TD is the closure problem Cl[,] (AD ). Definition 31 (Conditional interaction transport). An interaction transport is a functor or, when transport is only partially defined, a fibered correspondence over learning diagrams BΩ : Fact(C) ⇝ IntINC(C) such that the fiber over a base model D relates the base factorization problem Factq (D) to the declared interaction profile generated by its tangent lift. It is admissible only under the naturality, presentation descent, and sketch-fidelity requirements of conditional non-compositionality transport. Taking ΩD = {[, ]} recovers a bracket-transport functor into BrINC(C). A L INCS category can then be seen as a category of learning diagrams equipped with both a factorization fibration and its tangent lift: Learn(C) ← Fact(C), Learn(TC) ← INC(C). With interaction transport, the picture refines to B
Ω Fact(C) −−− → IntINC(C) → Learn(TC).
Coalgebraically, the same category carries an unfolding operation TINC : INC(C) → INC(C), so that L INCS learning can also be viewed as constructing coalgebras A → TINC A whose iterated tangent unfoldings stabilize. Learning is a section-selection or repair problem: choose morphisms, parameters, coalgebra maps, or extensions that make base and tangent factorization problems closer to being inhabited, make the induced directions coherent under the sketch-specific interaction signature, and make the INC tower closer to a fixed point. A computational realization may still assign zero admission weight to an available interaction signal.
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Interaction Profiles and Higher INC
The factorization definition of INC captures first-order infinitesimal coherence: does the tangent-lifted diagram satisfy the same universal factorization property as the base diagram? Second-order structure is richer than one distinguished projection. Lie brackets test order-sensitive noncommutativity and involutivity. A connection-equipped realization also retains the ordered derivatives ∇X Y and ∇Y X, whose symmetric sum measures joint acceleration relative to that connection. In a tangent category, vector fields and their Lie brackets can be defined from tangent structure. Thus, once a learning model D determines an admissible distribution AD , L INCS can ask whether AD is closed under bracket. This is the tangent analogue of asking whether a distribution is involutive. Infinitesimal Causality models interventions as tangent directions and studies their closure and noncommutativity (Mahadevan, 2026e); BRIDGE/SKFM uses Lie-bracket geometry to expose latent confounded causal structure (Mahadevan, 2026b). These applications justify bracket closure as an important specialization, but not as the unique second-order observation for every learning sketch. 19
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Definition 32 (Sketch-relative interaction transport). Let D : J → C be an admissible model with base factorization problem FactS (D) and interaction signature ΩD . An interaction transport for D is a natural, presentation-descending assignment tD,S : Obs FactS (D) ⇝ ObsΩ,S (TD). It maps base non-compositionality to a candidate sketch-specific interaction profile. It does not assert that every component of that profile identifies the base repair or has positive predictive value after scalarization. Definition 33 (Lie-bracket INC). The Lie-bracket INC of D relative to AD is the obstruction INC[,] (D) = { [X, Y ] | X, Y ∈ AD , [X, Y ] ∈ / AD }. The model is bracket-compositional relative to AD when INC[,] (D) is empty. Definition 34 (Connection-profile INC). For a connection-equipped realization, connection-profile INC is the obstruction profile obtained from 2 J∇,D (X, Y ) = ∇X Y, ∇Y X , together with any sketch-declared factorization, closure, or observation maps. Its symmetric and antisymmetric projections may be scalarized separately. Neither a nonzero symmetric acceleration nor a nonzero bracket is by itself a defect: defect status is relative to the constraints and admissible repair distribution declared by the sketch. The distinction matters computationally. The bracket is invariant under coordinate change for projectable fields, whereas the symmetric projection requires the declared connection. Conversely, order-sensitive torque need not be the mechanism that predicts future failure: accumulated joint acceleration may carry more information in a particular sketch. A validated realization therefore admits the two components separately, for example with sparse coefficients (µs , µa ), and permits either coefficient to vanish. This gives a hierarchy: base factorization problems → tangent factorization problems → interaction profiles → relative closure, curvature, and torsion → higher jets. Higher-order L INCS studies this hierarchy as successive, signature-relative refinements of compositionality.
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Homotopical L INCS
Strict INC measures failure of strict commutativity. In many learning systems, however, exact equality of diagrams is too rigid. A representation, policy, world model, or reasoning graph may be correct only up to a controlled deformation. This suggests a homotopical refinement of L INCS . Strict repair asks for an exact factorization: strict repair:
D = D̄q.
Homotopy repair asks for a coherent deformation from the learned diagram to a factored diagram: homotopy repair:
D ≃ D̄q.
Thus DB minimizes strict diagram defects, L INCS minimizes tangent-lifted defects, and homotopical L INCS asks whether those tangent defects are null-homotopic, deformable, or obstructed. Definition 35 (Homotopical INC). In a homotopical, model-categorical, or ∞-categorical enrichment of C, the homotopical infinitesimal non-compositionality of D is INCh (D) = Obs Factq (TD) up to homotopy . It vanishes when the tangent compositionality problem is inhabited up to coherent homotopy. Interaction profiles and homotopy describe complementary layers of path dependence. The bracket is the local, infinitesimal shadow of noncommuting flows; connection-dependent symmetric acceleration records joint local bending; homotopy is the global coherence class of finite deformations. This yields an obstruction-theoretic ladder: defect → tangent defect → interaction or closure defect → homotopy class or cohomological obstruction. Layer
Object
Defect
Base Tangent Interaction Homotopy
category tangent category fields with signature/connection model category or ∞-category
non-commuting diagram infinitesimal non-composition relative non-closure or jet defect deformation obstruction
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12
Future Directions
The results above isolate several concrete problems whose resolution would turn L INCS from a foundations framework into a mature categorical theory of learning. Identifying the L INCS completion. The central open problem is the L INCS –final-coalgebra comparison. The Aczel–Mendler and Barr theorems establish existence of final INC coalgebras under class-based or accessible set-based realizations, while the contractive stabilization theorem controls a metric realization of the same unfolding. What remains is to construct the canonical comparison LINCS(C) −→ νTINC and give conditions under which it is fully faithful and essentially surjective. A promising route is to show that admissible learning sketches form an accessible category, that TINC preserves the relevant filtered colimits, and that the presentable objects generate both sides. Such a result would connect the free-completion and final-semantics viewpoints rather than treating them as parallel descriptions. Intrinsic criteria for stabilization. The contractive theorem assumes a complete metric realization and a constant ρ < 1. An important next step is to derive these hypotheses from the learning sketch and tangent structure themselves. This requires identifying when scalarizations of factorization, interaction-profile, closure, and homotopy obstructions define a complete behavioral metric and when tangent lift is contractive or eventually contractive in that metric. Stochastic learning further calls for versions of metric coinduction with martingale noise, biased tangent estimators, and data-dependent contraction factors. These results would turn stabilization from an assumed analytic property into a verifiable certificate for a learning system. Higher-order obstruction theory. First-order INC tests tangent factorization, whereas interaction profiles detect relations among generated infinitesimal directions. These should not be identified: tangent factorization is the primary first-order condition, while Lie-bracket nonclosure is a derived, antisymmetric second-order obstruction. With a declared connection, the full ordered pair (∇X Y, ∇Y X) and its symmetric acceleration must be retained before selecting a projection. A higher theory should construct prolonged learning sketches S(n) whose models are Tn D equipped not only with the original factorization constraints, but also with the canonical flips, vertical lifts, and coherence maps of iterated tangent structure. The corresponding hierarchy would take the form INC(n) (D) = Obs(FactS(n) (Tn D)) ,
n ≥ 0.
2
At level T , the canonical flip and any declared connection should organize both symmetric and antisymmetric infinitesimal interactions, including acceleration, bracket, torsion, and curvature profiles. At level T3 , the relevant coherence conditions should include Jacobiator- and Bianchi-type obstructions; at still higher levels, jets and higher or L∞ -style brackets may provide the appropriate language. In a homotopical or ∞-categorical setting, one should then determine when these local higher defects integrate to global deformation obstructions and when they vanish up to coherent homotopy. Such a prolongation theory would organize higher INC signals structurally, rather than as an unstructured collection of repeated derivative penalties. Reflective diagrammatic backpropagation. The coalgebraic tower also suggests a “meta” form of DB in which the learning mechanism becomes part of the diagram being tested. Let B denote a DB procedure that sends a global obstruction to a compatible family of local obstruction signals and then to parameter or architectural repairs. A reflective DB sketch would include B, its localization maps, and its repair maps as morphisms. Its commutativity conditions would ask, for example, whether localizing an obstruction and then repairing its local pieces agrees with repairing the global diagram and then restricting the result. Thus the relevant question is not anthropomorphic self-awareness, but a precise structural one: is the backpropagation mechanism itself compositional with respect to the learning problem it is intended to repair? Ordinary DB detects non-compositionality in a target computation. First-order reflective L INCS-DB would detect instability in how DB transports and repairs that obstruction, while T2 , T3 , . . . would test interactions among repair directions and expose defects that appear only after the learner changes itself. A nonzero meta-obstruction could then motivate an architectural operation—adding a computational path, module, state variable, skip connection, or compatibility constraint—rather than only a parameter update. This is analogous to BRIDGE/SKFM: those methods use bracket nonclosure to reveal latent causal directions, whereas reflective L INCS-DB may use a sketch-specific interaction profile to reveal latent structure in the learning architecture itself. Developing the corresponding meta-sketch, its admissible architectural edits, and a convergent coalgebraic repair procedure is a natural subject for a separate theoretical and experimental paper. 21
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There is also a more speculative stabilization question. The tower D, TD, T2 D, . . . is not by itself a spectrum, so repeated tangent lift does not automatically place L INCS in stable homotopy theory. Nevertheless, if a homotopical realization of prolonged sketches supplies compatible structure maps and a regime in which suspension and looping become inverse, then stabilized INC obstructions might define spectrum-valued invariants. This would replace the question of whether a defect vanishes at one finite tangent order by the stable question of which obstruction class persists under all further prolongations. Determining whether natural L INCS models, including Transformer equivariance sketches, admit such a stabilization is an open problem. Topological L INCS through the nerve. There is a complementary route from categorical L INCS to topology. For a fixed learning sketch and diagram D, let RINC (D) denote a category whose objects are admissible base and tangent repairs and whose morphisms are structure-preserving transformations between them. Its nerve N RINC (D)n = Fun([n], RINC (D)) is a simplicial set: vertices are repairs, edges are transformations between repairs, and higher simplices encode coherent chains of transformations. The geometric realization |N RINC (D)| therefore turns the repair category into a classifying space. This construction parallels the use of nerves and classifying spaces to study equivalence classes of categorical causal models in Universal Causality and its higher algebraic K-theoretic extension (Mahadevan, 2023, 2025). MacAdam’s Weil nerve suggests an infinitesimal refinement of this construction that should be distinguished from the ordinary simplicial nerve above (MacAdam, 2022, Ch. 4). For an involution algebroid A in a tangent category C, the Weil nerve is a fully faithful embedding NWeil : Inv(C) −→ [Weil1 , C] that records A as a transverse-limit-preserving tangent functor V 7→ A(V ); Segal-like exactness conditions characterize which such functors arise from involution algebroids. Accordingly, if the admissible infinitesimal repairs of D assemble into an involution algebroid AINC (D) in a category-valued tangent semantics, its Weil nerve should produce repair categories RVINC (D) := NWeil (AINC (D))(V ), V ∈ Weil1 . Taking the ordinary nerve and geometric realization objectwise then gives a Weil-indexed family of spaces V 7−→ N RVINC (D) . The monoidal unit records the base repair topology, the dual-number generator records first-order repair directions, and its Weil composites organize higher and mixed prolongations. Thus tangent information is retained before passing to homotopy type, rather than being forgotten by a single classifying space construction. Chapter 5 of MacAdam’s thesis supplies a complementary integration step. In the category W of Weil spaces, an infinitesimal groupoid ∂ : Weilop 1 → Gpd(W) induces an infinitesimal nerve N∂ with a left adjoint Lie realization | − |∂ , constructed by left Kan extension. The resulting tangent adjunction | − |∂ : Inv(W) ⇄ Gpd(W) : N∂ preserves products and base spaces (MacAdam, 2022, Ch. 5). For topological L INCS, the realization |AINC (D)|∂ is therefore a natural candidate for a groupoid of finite repairs integrating the infinitesimal repair algebroid. The adjunction unit AINC (D) → N∂ |AINC (D)|∂ tests whether the infinitesimal repair semantics is recovered after integration. Future work should construct a comparison between the classifying space of this realized groupoid and |N RINC (D)|. Failure of the unit, or of that comparison, to be an equivalence would then define an integration obstruction: a local infinitesimal repair calculus that does not assemble into the proposed global topology of finite repairs. Such a topological L INCS would distinguish more than the existence or scalar cost of a repair. Connected components could classify inequivalent repair regimes, loops could record nontrivial cycles of transformations, and higher homotopy or homology groups could detect coherent higher obstructions. Applying this construction to every prolonged category (n) RINC (D) would produce a tower of spaces associated with D, TD, T2 D, . . .. A central question is whether the tangent lift induces natural maps between these nerves and, under additional delooping or group-completion hypotheses, whether their stabilized homotopy type supplies the spectrum-valued INC invariants envisioned above. Functorial constructions across learning domains. The examples in this paper suggest that Bellman consistency, Kan extension, diagrammatic backpropagation, sheaf gluing, and causal intervention can all be viewed as instances of the same tangent factorization pattern. A stronger result would specify functors between these model categories and prove that they preserve INC, declared interaction transport, and coalgebraic stabilization. Such transfer theorems would clarify which guarantees are genuinely domain independent and which depend on additive, probabilistic, or smooth enrichment. 22
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Computational semantics and empirical validation. A practical implementation should compile a learning sketch into its base and tangent factorization problems, verify presentation descent, generate profile-valued observations before scalarization, and expose certificates for declared interaction closure and successive INC stabilization. Empirical work can then test a precise claim: tangent coherence should improve robustness to perturbations that are invisible to the base compositionality loss. Appropriate evaluations should therefore report base error, tangent error, gauge or presentation invariance, one-sided and symmetric/antisymmetric controls, stability under stochastic aggregation, held-out admission, and out-of-distribution consistency, rather than treating INC as an undifferentiated regularizer. The problem classes summarized in Section 4 provide initial domains for such tests, but the longer-term objective is a reusable categorical scientific-computing interface in which sketches, tangent lifts, and obstruction objects are first-class program structures.
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Summary
L INCS begins with a simple observation: every learning compositionality problem has a tangent lift. But the consequences are broad. If losses are scalarizations of obstructions to factorization, then tangent losses are scalarizations of obstructions to tangent factorization. They reveal how errors move, interact, commute, fail to close, deform, and unfold under repeated tangent lift. Parameterized realizations must first descend through their presentation redundancies, and connection-dependent interaction profiles must declare the geometry that defines them. Diagrammatic Backpropagation introduced diagrammatic non-compositionalities as learning signals. L INCS introduces infinitesimal non-compositionalities as candidate learning signals. Its coalgebraic form suggests a further refinement: learning is the search for stable tangent unfoldings of non-compositionality. The categorical framework supplies base, tangent, and sketch-specific interaction data; a validated computational realization decides contingently which auxiliary signals to admit. The resulting framework suggests a tangent-categorical reformulation of machine learning: learn not only the function, policy, representation, or world model, but the presentation-invariant infinitesimal geometry and coalgebraic self-consistency of the factorization problems by which it is learned.
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