1 Introduction↩︎

The dominant language of machine learning (ML) is optimization: choose a loss function and minimize it [1]–[3]. 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 [4]. 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 \(F(PX)=P F(X)\). 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 Lincs 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 \(\mathsf{T}^2,\mathsf{T}^3,\ldots\). This suggests that Lincs may expose structural properties of language models that are invisible to the ordinary scalar objective. Section 4.0.0.1 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, Lincs 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? Lincs asks: is the defect small, and does its tangent factorization behave correctly? This paper develops Lincs as a mathematical framework and illustrates the framework in a number of practical settings, from diagrammatic backpropagation in deep learning [4], Lie-algebraic LoRA adapters (Allora) for fine-tuning Transformer models [5], Lie-algebroidal skill optimization methods (Lasko) [6], and latent-confounded causal discovery using Lie-brackets [7].

1.0.0.1 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 Lincs objects, Lincs 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 Lincs , 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 Lincs categories.

8. It relates Lincs 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↩︎

2.0.0.1 Sketches and categorical theories.

The use of sketches to present structured theories goes back to Ehresmann’s work on sketches and algebraic structures [8]. Sketches were later developed as a flexible categorical language for theories and models, notably in the work of Barr and Wells [9], Makkai and Paré [10], and Adámek and Rosický [11]. Following the Makkai-Paré presentation, we regard a sketch as a quadruple \((S,\mathcal{D},\mathcal{L},\mathcal{K})\), where \(S\) is an underlying graph, \(\mathcal{D}\) is a class of commutativity conditions, \(\mathcal{L}\) is a class of distinguished cones, and \(\mathcal{K}\) is a class of distinguished cocones. Lincs 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.

2.0.0.2 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 [10], [11]. This connection is relevant to Lincs 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 Lincs completions without uncontrolled size growth.

2.0.0.3 Tangent categories.

Rosický introduced abstract tangent functors, and Cockett and Cruttwell developed tangent categories as an axiomatic account of tangent bundle structure [12], [13]. 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 [14], [15]. Lincs 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.

2.0.0.4 Weil-algebra semantics for tangent structure.

Leung gives a functorial classification of the Cockett–Cruttwell axioms in terms of Weil algebras [16]. If \(\mathcal{M}\) is a category, specifying tangent structure on \(\mathcal{M}\) is equivalent, up to isomorphism, to specifying a strong monoidal functor \[F:(\mathbb{N}\text{-}\mathsf{Weil}_1,\otimes,\mathbb{N}) \longrightarrow (\operatorname{End}(\mathcal{M}),\circ,1_{\mathcal{M}})\] that preserves the foundational pullbacks and the equalizer encoding the universality of vertical lift. The dual-number algebra \(W=\mathbb{N}[x]/(x^2)\) 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 \(\mathbb{N}\text{-}\mathsf{Weil}_1\) 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 Lincs; the additional Lincs data are the learning sketches, their admissible models, and the factorization obstructions transported by that action.

2.0.0.5 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 [17]. 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 \(\mathsf{Weil}_1\). It also places Lie differentiation in a nerve–realization context induced by an infinitesimal groupoid object. This is directly relevant to Lincs: 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 Lincs studies the obstruction to a learning model and its tangent lift satisfying a chosen sketch.

2.0.0.6 Compositional learning.

Several lines of work have used category theory to describe learning compositionally. Backpropagation has been formulated functorially [18], Universal Decision Learners formulate reinforcement learning and function approximation in terms of Kan-invariant extensions [19], and Kan Extension Transformers use Kan-extension structure to unify attention, diffusion-style completion, and predict-detach self-conditioning [20]. Diagrammatic Backpropagation treats noncommuting diagrams as learning signals [4]. Lincs 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.

2.0.0.7 Kan Extensions and Universal Decision Learners:

Universal Decision Learners (UDL) formulate decision learning in terms of Kan-invariant extensions of local decision data [19]. 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. Lincs adds a tangent refinement: \[\text{Kan invariance in the base category} \quad\leadsto\quad \text{Kan invariance in the tangent category}.\] If a learned extension is a left or right Kan extension in \(\mathcal{C}\), then Lincs asks whether its tangent lift satisfies the corresponding universal property in \(\mathsf{T}\mathcal{C}\). This suggests a combined principle: learn by universal extension, then require infinitesimal coherence of the extension. In this sense, Lincs can be viewed as tangent Kan-invariant learning.

3 Learning Sketches and Tangent Learning Sketches↩︎

We introduce the core notions of learning sketches and tangent learning sketches in this section. Let \(\mathcal{C}\) 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 \(D:J\to\mathcal{C}\) 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 \(g\circ f-f\circ g\). Lincs 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 \[\mathbb{S}=(S,\mathcal{D},\mathcal{L},\mathcal{K})\] be a sketch in the sense of Makkai and Paré: \(S\) is a graph, \(\mathcal{D}\) is a class of commutativity conditions in \(S\), \(\mathcal{L}\) is a class of cones, and \(\mathcal{K}\) is a class of cocones. Write \(J=\operatorname{Path}(S)\) for the free category on \(S\). The commutativity data \(\mathcal{D}\) generate a congruence \(\sim_{\mathcal{D}}\) on the morphisms of \(J\), and hence a quotient functor \[q_{\mathcal{D}}:J\to J/{\sim_{\mathcal{D}}}.\] The cone and cocone data specify the universal constraints a strict model is expected to realize. A diagram \(D:J\to\mathcal{C}\) satisfies the commutativity part of the sketch precisely when it factors through \(q_{\mathcal{D}}\): \[\begin{array}{ccc} J & \xrightarrow{D} & \mathcal{C}\\ \downarrow q_{\mathcal{D}} & \nearrow_{\bar D}\\ J/{\sim_{\mathcal{D}}} \end{array} \qquad D=\bar D q_{\mathcal{D}}.\] 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.

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 [21]. The same issue matters for Lincs . 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 Lincs 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 [22]. They also indicate how higher-order sketches can recover some structures beyond this boundary. Thus the Lincs 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.

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 Lincs formalization more concrete, we now list several concrete instances of the Lincs 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.

4.0.0.1 Transformer equivariance.

The question posed in the introduction has a precise answer. Let \(F_\theta:\mathbb{R}^{m\times d}\to\mathbb{R}^{m\times e}\) be a smooth content-only Transformer and let a permutation matrix \(P\) act on the token axis. Define the base equivariance defect \[E_{P,\theta}(X)=F_\theta(PX)-PF_\theta(X).\] This is the additive scalarizable representative of the obstruction to the equivariance square. Its first tangent lift, evaluated on an input perturbation \(V\), is \[D E_{P,\theta}(X)[V] =D F_\theta(PX)[PV]-P D F_\theta(X)[V].\] If parameters are also perturbed by \(\dot{\theta}\), the full tangent defect adds \[\partial_\theta F_\theta(PX)[\dot{\theta}] -P\partial_\theta F_\theta(X)[\dot{\theta}].\] 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 \(n\)th input-level prolongation is represented by \[D^nF_\theta(PX)[PV_1,\ldots,PV_n] -P D^nF_\theta(X)[V_1,\ldots,V_n].\] Hence, if \(F_\theta(PX)=PF_\theta(X)\) holds identically on a smooth invariant domain, differentiating the identity shows that every displayed tangent and higher-order obstruction vanishes. The iterated Lincs 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.

4.0.0.2 GIRL.

Gradient Infinitesimal Reinforcement Learning (GIRL) is the Lincs 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 \(\mathcal{X}\) denote the state object of an MRP, \(P^\pi:\mathcal{X}\to\mathcal{X}\) the transition morphism under policy \(\pi\), \(V_\theta:\mathcal{X}\to\mathbb{R}\) a parameterized value representation, and \[B_\pi:\mathbb{R}\to\mathbb{R}, \qquad B_\pi(v)=r+\gamma v,\] the scalar Bellman update induced by the reward and discount. The GIRL base square is \[\begin{array}{ccc} \mathcal{X} & \xrightarrow{\;P^\pi\;} & \mathcal{X}\\ \downarrow V_\theta && \downarrow V_\theta\\ \mathbb{R}& \xrightarrow{\;B_\pi\;} & \mathbb{R}. \end{array}\]

As a learning sketch, GIRL is the quadruple \[\mathbb{S}_{\mathrm{GIRL}} = (S_{\mathrm{GIRL}}, \mathcal{D}_{\mathrm{GIRL}}, \mathcal{L}_{\mathrm{GIRL}}, \mathcal{K}_{\mathrm{GIRL}}).\] The graph \(S_{\mathrm{GIRL}}\) has two formal objects \(x\) and \(v\), standing for states and values, and generating arrows \[p:x\to x, \qquad u:x\to v, \qquad b:v\to v.\] Under a model \(D_\theta:\operatorname{Path}(S_{\mathrm{GIRL}})\to\mathcal{C}\), these are interpreted as \[D_\theta(x)=\mathcal{X},\quad D_\theta(v)=\mathbb{R},\quad D_\theta(p)=P^\pi,\quad D_\theta(u)=V_\theta,\quad D_\theta(b)=B_\pi.\] The commutativity component is the single Bellman equation \[\mathcal{D}_{\mathrm{GIRL}} = \{\,u\circ p \sim b\circ u\,\}.\] Equivalently, the quotient \(q_{\mathcal{D}}:\operatorname{Path}(S_{\mathrm{GIRL}})\to \operatorname{Path}(S_{\mathrm{GIRL}})/{\sim_{\mathcal{D}}}\) identifies the two paths \[x\xrightarrow{p}x\xrightarrow{u}v, \qquad x\xrightarrow{u}v\xrightarrow{b}v.\] The limiting component \(\mathcal{L}_{\mathrm{GIRL}}\) records the admissible feature/value representation structure, for example the finite-dimensional parameter or feature cone through which \(V_\theta\) is required to factor in linear TD. The colimiting component \(\mathcal{K}_{\mathrm{GIRL}}\) 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 \[\operatorname{Obs}\big(\operatorname{Fact}_{\mathbb{S}_{\mathrm{GIRL}}}(D_\theta)\big)\] 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 \(\mathbb{S}_{\mathrm{GIRL}}\) with the Cockett-Cruttwell tangent action. Its model sends \(D_\theta\) to \[\mathsf{T}D_\theta:\operatorname{Path}(S_{\mathrm{GIRL}})\to\mathcal{C},\] whose interpreted arrows are \(\mathsf{T}P^\pi\), \(\mathsf{T}V_\theta\), and \(\mathsf{T}B_\pi\). The tangent Bellman square is \[\begin{array}{ccc} \mathsf{T}\mathcal{X} & \xrightarrow{\;\mathsf{T}P^\pi\;} & \mathsf{T}\mathcal{X}\\ \downarrow \mathsf{T}V_\theta && \downarrow \mathsf{T}V_\theta\\ \mathsf{T}\mathbb{R}& \xrightarrow{\;\mathsf{T}B_\pi\;} & \mathsf{T}\mathbb{R}. \end{array}\] The GIRL infinitesimal non-compositionality is \[\mathsf{INC}_{\mathrm{GIRL}}(D_\theta) = \operatorname{Obs}\big(\operatorname{Fact}_{\mathbb{S}_{\mathrm{GIRL}}}(\mathsf{T}D_\theta)\big).\] After scalarization, this yields the tangent Bellman error developed in the companion GIRL paper [23]. 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.

A detailed experimental evaluation of Lincs across these problem classes—GIRL, DB, BRIDGE/SKFM/IC, ALLORA, and LASKO—is currently underway and will be reported in future work.

4.0.0.3 GIKET.

Kan Extension Transformers learn by extending local neighborhoods into global representations [20]. 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.

4.0.0.4 Differential DB.

Diagrammatic Backpropagation repairs noncommuting computational diagrams, and is the canonical example from which Lincs 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.

Let \(X\) be a formal object for a candidate grid, \(R_i\) the \(i\)-th row object, \(K_j\) the \(j\)-th column object, \(B_b\) the \(b\)-th block object, and \(C_{ij}\) the cell object at position \((i,j)\). The Sudoku DB learning sketch is \[\mathbb{S}_{\mathrm{DB}} = (S_{\mathrm{DB}}, \mathcal{D}_{\mathrm{DB}}, \mathcal{L}_{\mathrm{DB}}, \mathcal{K}_{\mathrm{DB}}).\] The graph \(S_{\mathrm{DB}}\) contains the view maps \[\rho_i:X\to R_i,\qquad \kappa_j:X\to K_j,\qquad \beta_b:X\to B_b,\] and the cell-projection maps \[r_{ij}:R_i\to C_{ij},\qquad k_{ij}:K_j\to C_{ij},\qquad b_{ij}:B_b\to C_{ij},\] where \(b\) is the block containing cell \((i,j)\). The commutativity component \(\mathcal{D}_{\mathrm{DB}}\) says that every overlapping view gives the same cell: \[r_{ij}\rho_i \sim k_{ij}\kappa_j \sim b_{ij}\beta_b \qquad \text{for all }(i,j).\] The limiting component \(\mathcal{L}_{\mathrm{DB}}\) records that a grid is assembled from compatible local cell assignments, for example as a product-like cone of cell predictions. The colimiting component \(\mathcal{K}_{\mathrm{DB}}\) records gluing: row, column, and block views are identified along their shared cells. Additional Sudoku validity maps can be included in \(S_{\mathrm{DB}}\), such as maps \(R_i,K_j,B_b\to \mathsf{Perm}_9\), 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 \[D_\theta:\operatorname{Path}(S_{\mathrm{DB}})\to\mathcal{C}\] whose objects may be soft digit distributions, logits, constraint states, or learned local representations. The DB obstruction \[\operatorname{Obs}\big(\operatorname{Fact}_{\mathbb{S}_{\mathrm{DB}}}(D_\theta)\big)\] 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: \[\mathsf{T}D_\theta:\operatorname{Path}(S_{\mathrm{DB}})\to\mathcal{C}.\] Its INC term is \[\mathsf{INC}_{\mathrm{DB}}(D_\theta) = \operatorname{Obs}\big(\operatorname{Fact}_{\mathbb{S}_{\mathrm{DB}}}(\mathsf{T}D_\theta)\big).\] This asks whether infinitesimal changes in logits, givens, local cell beliefs, or constraint messages preserve row-column-block 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 [24]. In Lincs 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.

4.0.0.5 ALLORA.

ALLORA trains low-rank neural adapters with Lie-algebraic commutator penalties so independently useful adapters compose more predictably [5]. Its base defect is adapter order-sensitivity; its reflective Lincs version studies quotient-projectable adapter-flow interactions, including both bracket obstructions and connection-dependent symmetric acceleration in the representation bundle.

The low-rank presentation \(\Delta=BA\) has a \(\mathrm{GL}(r)\) redundancy, \((B,A)\mapsto(BQ,Q^{-1}A)\). 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 \(\pi(B,A)=BA\). 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.

4.0.0.6 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 [6]. Its Lincs version studies vector-bundle bracket defects for latent knowledge and skill-update manifolds.

5 Diagrammatic Backpropagation Revisited↩︎

Diagrammatic Backpropagation can be expressed as the repair of factorization failures [4]. Let \(D_\theta:J\to\mathcal{C}\) be a learned diagram depending on parameters \(\theta\), obtained from the path category of a learning sketch \(\mathbb{S}=(S,\mathcal{D},\mathcal{L},\mathcal{K})\), and let \(q=q_{\mathcal{D}}:J\to J/{\sim_{\mathcal{D}}}\) encode the intended commutativity constraints. DB attempts to adjust \(\theta\) so that \(D_\theta\) factors through \(q\), possibly together with the limit and colimit constraints specified by \(\mathcal{L}\) and \(\mathcal{K}\).

In enriched settings, this factorization problem is often scalarized by a loss: \[\mathcal{L}_{DB}(\theta)=\Phi(\operatorname{Fact}_q(D_\theta)),\] where \(\Phi\) 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.

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 \(\{i_a:J_a\to J\}_{a\in A}\) be a family of subsketch inclusions whose images cover the objects, generating arrows, and specified constraints of a learning sketch \(\mathbb{S}\). Write \(\mathbb{S}_a\) for the induced local sketch and \(D_a=D i_a\) for the restriction of a model \(D:J\to\mathcal{C}\).

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.

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. Lincs 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.

6 Tangent Lift of Diagrammatic Backpropagation↩︎

We now assume that \(\mathcal{C}\) carries tangent structure. Tangent categories axiomatize the behavior of tangent bundles in a categorical setting [13]. A tangent category has an endofunctor \[\mathsf{T}:\mathcal{C}\to\mathcal{C}\] 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 \(\mathcal{C}\), 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 Lincs because infinitesimal coherence is not exhausted by the existence of \(\operatorname{Fact}_{\mathbb{S}}(\mathsf{T}D)\). 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.

The base factorization problem asks whether computations compose: \[\operatorname{Fact}_{\mathbb{S}}(D)\neq\varnothing.\] The tangent factorization problem asks whether infinitesimal perturbations compose: \[\operatorname{Fact}_{\mathbb{S}}(\mathsf{T}D)\neq\varnothing.\]

Thus Lincs is not a new optimizer. It is a rule for lifting learning compositionality problems into tangent structure.

7 Lincs Axioms↩︎

In this section, we propose an axiomatic structure for Lincs . First, we state the underlying theoretical object more formally.

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.

This definition avoids subtracting morphisms in the base category. The primitive obstruction is not a residual \(g\circ f-f\circ g\). 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 may instead or additionally observe \(S_\nabla\), \(A_\nabla\), or the full ordered profile \(J^2_\nabla\) without asserting that antisymmetry is the unique carrier of useful tangent information.

If \(\operatorname{Fact}_q(D)\) is scalarized by a loss, then \(\operatorname{Obs}(\operatorname{Fact}_q(\mathsf{T}D))\) may induce a gradient-like or Jacobian-like regularizer. But Lincs 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.

7.1 Axioms↩︎

We now propose an axiomatization of Lincs categories.

7.2 Universal Properties↩︎

The universal property question is: what is the minimal tangent learning structure generated by a category of base learning diagrams?

This conjecture is the Lincs analogue of free tangent or differential completions. It says that Lincs 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 Lincs, the universal property is relative to the declared signature \(\Omega\): 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 Lincs as a Coalgebra↩︎

There is a second universal perspective on Lincs , related to the standard coalgebraic treatment of state-based systems and final semantics [25]. A tangent category contains an endofunctor \[\mathsf{T}:\mathcal{C}\to\mathcal{C}.\] An endofunctor does not by itself make every object into a coalgebra, but it determines a category of coalgebras once one chooses structure maps \[\gamma:X\to\mathsf{T}X.\] In a tangent category, vector fields are the basic examples: a vector field on \(X\) is a section \(v:X\to\mathsf{T}X\) of the tangent projection \(p_X:\mathsf{T}X\to X\). Thus vector fields can be read as \(T\)-coalgebras compatible with tangent structure.

This gives a coalgebraic reading of INC. A learning model \(D:J\to\mathcal{C}\) has a tangent tower \[D,\quad \mathsf{T}D,\quad \mathsf{T}^2D,\quad \ldots\] and a corresponding tower of factorization problems \[\operatorname{Fact}_q(D),\quad \operatorname{Fact}_q(\mathsf{T}D),\quad \operatorname{Fact}_q(\mathsf{T}^2D),\quad \ldots .\] 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.

Equivalently, after applying the obstruction assignment, \[\mathsf{T}_{\mathsf{INC}}\big(\operatorname{Obs}(\operatorname{Fact}_q(\mathsf{T}^nD))\big) = \operatorname{Obs}(\operatorname{Fact}_q(\mathsf{T}^{n+1}D)).\]

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 \(\nu\mathsf{T}_{\mathsf{INC}}\) 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 Lincs 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.

The Aczel–Mendler theorem discharges existence once the INC endofunctor has a set-based class realization. Barr’s theorem avoids proper classes whenever \(\mathsf{T}_{\mathsf{INC}}\) 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 \(\mathsf{LINCS}(\mathcal{C})\). 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 [28].

The scalar certificate used in applications fits this theorem when the obstruction energy \[E_n = \Phi\big(\operatorname{Obs}(\operatorname{Fact}_q(\mathsf{T}^nD))\big) + \sum_{k\in\Omega_D}\mu_{k,n}\Phi^k_n + \eta\Phi^{\mathrm h}_n\] is a Lyapunov upper bound for \(d_D(x_n,x_\ast)\). In finite-sample learning, an inequality \(E_{n+1}\leq\rho E_n+\epsilon\) then gives \[E_n\leq\rho^nE_0+\frac{1-\rho^n}{1-\rho}\epsilon,\] 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.

9 Lincs as a Category↩︎

We now collect the preceding ideas into an explicit categorical picture.

A Lincs category can then be seen as a category of learning diagrams equipped with both a factorization fibration and its tangent lift: \[\mathsf{Learn}(\mathcal{C}) \leftarrow \mathsf{Fact}(\mathcal{C}), \qquad \mathsf{Learn}(\mathsf{T}\mathcal{C}) \leftarrow \mathsf{INC}(\mathcal{C}).\] With interaction transport, the picture refines to \[\mathsf{Fact}(\mathcal{C}) \xrightarrow{\;B_\Omega\;} \mathsf{IntINC}(\mathcal{C}) \to \mathsf{Learn}(\mathsf{T}\mathcal{C}).\] Coalgebraically, the same category carries an unfolding operation \[\mathsf{T}_{\mathsf{INC}}:\mathsf{INC}(\mathcal{C})\to\mathsf{INC}(\mathcal{C}),\] so that Lincs learning can also be viewed as constructing coalgebras \[A\to\mathsf{T}_{\mathsf{INC}}A\] 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.

10 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 \(\nabla_XY\) and \(\nabla_YX\), 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 \(\mathcal{A}_D\), Lincs can ask whether \(\mathcal{A}_D\) 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 [29]; BRIDGE/SKFM uses Lie-bracket geometry to expose latent confounded causal structure [7]. These applications justify bracket closure as an important specialization, but not as the unique second-order observation for every learning 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 \((\mu_s,\mu_a)\), and permits either coefficient to vanish.

This gives a hierarchy: \[\begin{align} &\text{base factorization problems} \to \text{tangent factorization problems} \to \text{interaction profiles}\\ &\to \text{relative closure, curvature, and torsion} \to \text{higher jets}. \end{align}\] Higher-order Lincs studies this hierarchy as successive, signature-relative refinements of compositionality.

11 Homotopical Lincs↩︎

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 Lincs .

Strict repair asks for an exact factorization: \[\text{strict repair:}\qquad D=\bar Dq.\] Homotopy repair asks for a coherent deformation from the learned diagram to a factored diagram: \[\text{homotopy repair:}\qquad D\simeq \bar Dq.\] Thus DB minimizes strict diagram defects, Lincs minimizes tangent-lifted defects, and homotopical Lincs asks whether those tangent defects are null-homotopic, deformable, or obstructed.

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: \[\text{defect} \to \text{tangent defect} \to \text{interaction or closure defect} \to \text{homotopy class or cohomological obstruction}.\]

12 Future Directions↩︎

The results above isolate several concrete problems whose resolution would turn Lincs from a foundations framework into a mature categorical theory of learning.

12.0.0.1 Identifying the Lincs completion.

The central open problem is the Lincs –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 \[\mathsf{LINCS}(\mathcal{C})\longrightarrow\nu\mathsf{T}_{\mathsf{INC}}\] 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 \(\mathsf{T}_{\mathsf{INC}}\) 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.

12.0.0.2 Intrinsic criteria for stabilization.

The contractive theorem assumes a complete metric realization and a constant \(\rho<1\). 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.

12.0.0.3 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 \((\nabla_XY,\nabla_YX)\) and its symmetric acceleration must be retained before selecting a projection. A higher theory should construct prolonged learning sketches \(\mathbb{S}^{(n)}\) whose models are \(\mathsf{T}^nD\) 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 \[\mathsf{INC}^{(n)}(D) = \operatorname{Obs}\!\left(\operatorname{Fact}_{\mathbb{S}^{(n)}}(\mathsf{T}^nD)\right), \qquad n\geq 0.\] At level \(\mathsf{T}^2\), the canonical flip and any declared connection should organize both symmetric and antisymmetric infinitesimal interactions, including acceleration, bracket, torsion, and curvature profiles. At level \(\mathsf{T}^3\), the relevant coherence conditions should include Jacobiator- and Bianchi-type obstructions; at still higher levels, jets and higher or \(L_\infty\)-style brackets may provide the appropriate language. In a homotopical or \(\infty\)-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.

12.0.0.4 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 \(\mathcal{B}\) 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 \(\mathcal{B}\), 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 Lincs-DB would detect instability in how DB transports and repairs that obstruction, while \(\mathsf{T}^2,\mathsf{T}^3,\ldots\) 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 Lincs-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.

There is also a more speculative stabilization question. The tower \(D,\mathsf{T}D,\mathsf{T}^2D,\ldots\) is not by itself a spectrum, so repeated tangent lift does not automatically place Lincs 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 Lincs models, including Transformer equivariance sketches, admit such a stabilization is an open problem.

12.0.0.5 Topological Lincs through the nerve.

There is a complementary route from categorical Lincs to topology. For a fixed learning sketch and diagram \(D\), let \(\mathcal{R}_{\mathsf{INC}}(D)\) denote a category whose objects are admissible base and tangent repairs and whose morphisms are structure-preserving transformations between them. Its nerve \[N\mathcal{R}_{\mathsf{INC}}(D)_n = \operatorname{Fun}([n],\mathcal{R}_{\mathsf{INC}}(D))\] is a simplicial set: vertices are repairs, edges are transformations between repairs, and higher simplices encode coherent chains of transformations. The geometric realization \(|N\mathcal{R}_{\mathsf{INC}}(D)|\) 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 [30], [31].

MacAdam’s Weil nerve suggests an infinitesimal refinement of this construction that should be distinguished from the ordinary simplicial nerve above [17]. For an involution algebroid \(A\) in a tangent category \(\mathcal{C}\), the Weil nerve is a fully faithful embedding \[N_{\mathsf{Weil}}: \operatorname{Inv}(\mathcal{C})\longrightarrow[\mathsf{Weil}_1,\mathcal{C}]\] that records \(A\) as a transverse-limit-preserving tangent functor \(V\mapsto A(V)\); 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 \(A_{\mathsf{INC}}(D)\) in a category-valued tangent semantics, its Weil nerve should produce repair categories \[\mathcal{R}_{\mathsf{INC}}^{V}(D) :=N_{\mathsf{Weil}}(A_{\mathsf{INC}}(D))(V), \qquad V\in\mathsf{Weil}_1.\] Taking the ordinary nerve and geometric realization objectwise then gives a Weil-indexed family of spaces \[V\longmapsto\left|N\mathcal{R}_{\mathsf{INC}}^{V}(D)\right|.\] 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 \(\mathcal{W}\) of Weil spaces, an infinitesimal groupoid \(\partial:\mathsf{Weil}_1^{\mathrm{op}}\to\operatorname{Gpd}(\mathcal{W})\) induces an infinitesimal nerve \(N_{\partial}\) with a left adjoint Lie realization \(|-|_{\partial}\), constructed by left Kan extension. The resulting tangent adjunction \[|-|_{\partial}: \operatorname{Inv}(\mathcal{W}) \; \rightleftarrows \; \operatorname{Gpd}(\mathcal{W}) :N_{\partial}\] preserves products and base spaces [17]. For topological Lincs, the realization \(|A_{\mathsf{INC}}(D)|_{\partial}\) is therefore a natural candidate for a groupoid of finite repairs integrating the infinitesimal repair algebroid. The adjunction unit \(A_{\mathsf{INC}}(D)\to N_{\partial}|A_{\mathsf{INC}}(D)|_{\partial}\) 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 \(|N\mathcal{R}_{\mathsf{INC}}(D)|\). 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 Lincs 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 \(\mathcal{R}_{\mathsf{INC}}^{(n)}(D)\) would produce a tower of spaces associated with \(D,\mathsf{T}D,\mathsf{T}^2D,\ldots\). 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.

12.0.0.6 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.

12.0.0.7 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.

13 Summary↩︎

Lincs 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. Lincs 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.

References↩︎