Keywords. Organizational cybernetics; homotopy type theory; ∞-topos; lenses; neural tangent kernel; jets; constrained updates.

The company-level application is a proposed synthesis. Its starting hypotheses are that employee and customer traces can both become productive training data, that feedback changes the behavior being measured, and that data integration can create incentives for consolidation. These are conditional empirical and institutional claims. The mathematical results below concern explicitly specified models; they do not establish that every actual AI company collects all available traces or that a single company must emerge.

Wiener’s control-and-communication perspective [35], Ashby’s requisite variety [1], and Beer’s organizational cybernetics [2] motivate the separation of operation, coordination, adaptation, and policy. Modern categorical cybernetics supplies compositional interfaces and controllers [4, 27]; categorical learning supplies parametrized maps, learners, and reverse differentiation [9, 7]. Higher groupoids add a precise account of witnessed equivalence and its coherence. They are not substitutes for economic evidence or for differential equations.

Two further strands connect architecture and institutions: categorical deep learning studies algebraic architecture constraints [13, 12], while compositional game theory models interacting decision-makers [14]. Hedges explicitly proposes categorical-cybernetic analysis of AI-mediated markets and supply chains [16]. That proposal is a research agenda, not an established empirical law of AI companies.

Central question. Which observations and interventions can a firm perform, which specifications must those interventions preserve, and how do learning, governance, and the environment respond to one another? A model is useful when it makes these questions executable or falsifiable, rather than merely relabeling a company as a category.

2.1 Foundational conventions and notation

A category has objects, arrows with specified source and target, identity arrows, and associative composition. A functor preserves these data. A natural transformation consists of arrows between two functors’ values which commute with every source-category arrow. “Small” means that the objects and arrows belong to the chosen set-sized universe. Cop reverses arrows, and Fun(C,D) denotes functors and their natural transformations. A groupoid is a category whose arrows are invertible. An endomorphism has equal source and target; an automorphism is an invertible endomorphism. A terminal object 1 has one arrow from each object, and an initial object 0 has one arrow to each object, with the corresponding mapping spaces contractible in higher categories. Products and limits are defined by their universal mapping properties.

For a precise higher model, a simplicial set is a functor Δop→Set, where Δ has finite ordered sets [n]={0,…,n} and order-preserving maps. The horn Λin​⊆Δn consists of all faces except the ith. An ∞-category can be modeled by a simplicial set filling every inner horn (0<i<n); a Kan complex fills all horns and models an ∞-groupoid. The ∞-category S of spaces means homotopy types, modeled by Kan complexes with weak homotopy equivalences inverted. A weak homotopy equivalence induces a bijection of components and isomorphisms of all homotopy groups. The group πk​(X,x) consists of based homotopy classes of maps from the k-sphere to (X,x), for k≥1. All higher limits and pullbacks below are homotopy-coherent limits [25]. The core of a category retains its objects and equivalences.

In the internal type language, a universe U classifies the chosen small types. A dependent family B:A→U assigns a type to each a:A. Its dependent sum ∑a:A​B(a) contains pairs (a,b); its dependent product ∏a:A​B(a) contains sections assigning such a b to every a. A witness is an inhabitant of the stated type. A type X is contractible when there exist c:X and paths c=x for every x:X. For f:A→B, its fiber at b is ∑a:A​IdB​(f(a),b). An equivalence is a map with contractible fibers; Equiv(A,B) is the type of these maps and witnesses. A homotopy from f to g is a section of ∏a:A​IdB​(f(a),g(a)). Intensional equality uses such identity types, distinct from the definitional equality used in evaluation rules. The type Prop consists of mere propositions, defined below; Bool=1+1 has two alternatives. Decidability of a proposition P means a witness of P+(P→0), not just the formation of its type [32].

A presheaf of spaces is a functor Cop→S. A site is a small category with a Grothendieck topology: specified covering sieves, where a sieve is a collection of arrows to an object closed under precomposition. The axioms require the maximal sieve to cover, stability under pullback, and transitivity of covering. A sheaf has compatible local sections which glue uniquely, or up to coherent homotopy in the space-valued case. Descent data include sections on a cover and agreements on all iterated overlaps. The Čech nerve lists these overlaps; its limit is the space of such coherent data. An ∞-topos is an accessible left-exact localization of a presheaf ∞-category. Here a localization has a fully faithful right adjoint; left exact means preserving finite limits, and accessible means preserving κ-filtered colimits for some regular cardinal κ. A diagram is κ-filtered when every subdiagram with fewer than κ arrows has a compatible cocone. A geometric morphism f:E→F is an adjoint pair f∗:F⇄E:f∗​ with f∗ left exact [25, 22].

An open system exchanges inputs and outputs across a chosen boundary. A controller chooses inputs from observations and possibly memory; feedback feeds outputs into subsequent inputs. Governance specifies who may change controllers, objectives, interfaces, and admissibility rules. Resources are stocks or capacities consumed or replenished by operations. These terms acquire explicit maps, state variables, and authorization relations in the constructions below.

2.2 The identity tower

Work in intensional dependent type theory, with a chosen univalent universe when univalence is used. For x,y:A, an identity witness has type p:IdA​(x,y); for p,q:IdA​(x,y) there is a further type IdIdA​(x,y)​(p,q), and the process iterates. Reflexivity and identity elimination induce inverses and composition together with higher coherence. Identity elimination says that a family P(x,y,p) indexed by p:IdA​(x,y) has a section for all (x,y,p) once its reflexive instances P(x,x,reflx​) have sections, with the corresponding reflexive computation rule. It is the dependent substitution rule used for equality witnesses. This identity tower admits a weak ω-groupoid structure [24, 33]. It is stronger information than a binary relation of indistinguishability.

The groupoid model demonstrates that uniqueness of identity proofs is not a general consequence of intensional type theory [17]. Martin-Löf’s foundational presentation [26] and Voevodsky’s homotopy λ-calculus notes [34] provide foundational background. Grothendieck’s homotopical perspective [15] connects suitable models of higher groupoids to homotopy types.

Its universal property is Map(τ≤n​A,B)≃Map(A,B) for every n-type B; Map denotes the mapping space. The (−2)-types are the contractible types. Calling a proposition “yes or no” does not make it a decidable Boolean: P∨¬P is an additional assertion in constructive logic. Likewise, “discrete” in this note means homotopy 0-truncated; it need not mean discrete as a topological or smooth space. These distinctions matter when software must produce an actual certificate.

2.3 A topos of organizational contexts

This is an actual ∞-topos, obtained by accessible left-exact sheafification [25]. The selected site is a presentation of the semantics. Neither the bare organization nor an arbitrary category of neural networks automatically forms a topos. A 0-truncated part supports ordinary sheaf logic and the Mitchell–Bénabou language [22]; suitable universes and dependent type-theoretic structure support the higher identity language [32]. An arbitrary such model does not automatically supply every desired computational rule for a proof assistant.

A state in context U is a generalized element U→S. A global state is 1→S. A point of a topos, a geometric morphism Set→E in the ordinary setting, is a different notion. Local inhabitation, a global section, and an executable construction must not be conflated: a semantic witness becomes an effective procedure only through an appropriate constructive interpretation or implementation.

For differentiation, choose smooth parameter spaces and derivative operations. A smooth manifold is a Hausdorff, second-countable space covered by Euclidean charts with smooth transition maps; smooth maps are smooth in those charts. A concrete option is to replace the context site by a product with a site of Cartesian spaces and smooth maps, with the corresponding product topology, and to use smooth finite-dimensional charts for parameter objects. Probability is extra structure too. A standard Borel space is measurably isomorphic to the Borel space of a complete separable metric space. A Markov kernel K:X⇝Y assigns a probability measure K(x,−) to each x, with x↦K(x,B) measurable for every measurable B. Composition is (L∘K)(x,B)=∫Y​L(y,B)K(x,dy). A Markov category packages such composition in a symmetric monoidal category, with compatible commutative copy maps X→X⊗X and delete maps X→1, the latter natural and the unit terminal [11]. The tensor denotes parallel composition; copy need not be natural for stochastic maps. Neither calculus nor probability follows merely from the word “topos.”

3.1 States and admissible transitions

Let the ambient state object be

where H describes human roles and capabilities, D data and provenance, Θ models and parameters, O optimizer/runtime memory, C controllers, R resources, and G governance. Products may be replaced by dependent sums when, for example, an optimizer’s state depends on the architecture. Specify a predicate V:S0​→Prop and define

Thus an inhabitant of S contains both a proposed state and its validity witness. The predicate may include schema equations, authorization, resource bounds, deployment tests, and declared contractual conditions. Their concrete meaning must be supplied; a proof of the chosen predicate is not proof of every desirable social outcome.

This formalizes constrained database updates and speculative changes. A valid rollback concerns the modeled state: deletion of already transmitted information, or reversal of an irreversible physical act, requires additional operations and cannot be assumed from this proposition.

For an ordinary categorical database schema B, an instance is a functor D:B→Set [31]. Functoriality enforces the schema’s declared path equations. Authorization, statistical validity, and cross-table conditions beyond those equations must be specified separately. For a schema map u:B→B′, restriction Δu​ is precomposition by u; when they exist, its left and right adjoints Σu​ and Πu​ are the left and right Kan extensions. An adjunction is a natural equivalence between the corresponding mapping spaces. These migration constructions are not automatic proofs of the additional conditions.

Independent local approvals are not enough. Agreement on shared data, interfaces, and higher coherence is part of the premise. For example, two resource requests may each fit a budget locally and exceed it jointly; they do not form a valid global resource certificate.

3.2 Interfaces, lenses, and feedback

A deterministic open module consists of a state object S, an output map o:S→Y, and an update u:S×X→S. Closing an environment/controller map c:Y→X gives

More generally, c has its own state and the environment contributes inputs; the full product state must then be retained. In a stochastic interpretation, updates are Markov kernels and composition uses their integral composition law.

For interfaces (A+,A−) and (B+,B−), a simple lens is a pair

The second component carries responses, requested changes, or cotangent signals back through the interface. For L:A→B and M:B→C, composition is

This is the elementary bidirectional structure; dependent lenses permit response types to depend on the current output. Reverse differentiation sends a smooth map f:A→B to forward evaluation and cotangent pullback (a,bˉ)↦DfaT​bˉ in Euclidean coordinates. The chain rule makes this construction compositional. An optimizer closes the parameter port, as in categorical learning [7]. The tangent space at a smooth point a consists of derivations v:Ca∞​→R on smooth germs, linear over R and satisfying v(fg)=f(a)v(g)+g(a)v(f); these are infinitesimal directions. Its dual, the cotangent space, contains linear functionals on those directions. A parameter port is the part of the interface carrying parameters and their requested changes. To close it is to supply an update law for those changes.

Parametric maps and architecture constraints. A layer with parameters P is a map f:P×A→B. Composing it with g:Q×B→C gives a map with parameters Q×P, (q,p,a)↦g(q,f(p,a)). A reparametrization ρ:P→P′ relates f to f′:P′×A→B when f=f′∘(ρ×idA​). Such maps form the elementary structure behind the bicategorical Para construction; invertible reparametrizations give only its equivalence part. A 2-category has objects, 1-arrows, and 2-arrows between 1-arrows, with compatible vertical and horizontal composition; a bicategory weakens associative and unit laws to coherent invertible 2-arrows. An optic represents bidirectional behavior through a residual object M, with maps A+→M×B+ and M×B−→A−. Representatives are identified by the relations induced by changing residual objects through compatible maps. Thus residual state is retained for the backward response [12].

Algebraic architecture constraints can also be expressed through monads. In an ordinary category, a monad consists of an endofunctor T and natural transformations η:Id→T and μ:T2→T satisfying

An algebra a:TA→A satisfies a∘ηA​=idA​ and a∘μA​=a∘T(a). These equations constrain how composed operations are implemented. Categorical deep learning lifts this algebraic viewpoint to appropriate 2-categories of parametric maps, relating model constraints to architecture implementations [13]. The lift and the chosen ambient category are substantive parts of the construction. Architecture algebra does not by itself specify loss, optimizer memory, or a firm’s governance objective.

The temporal operations in (2) are generally irreversible. They belong to a category of transitions or systems. Only equivalences belong to its maximal higher groupoid. In an (∞,1)-category higher morphisms are invertible; its ordinary 1-morphisms need not be. Reflexivity, symmetry, and transitivity of a relation alone do not provide the entire identity tower.

An executable trace γ=(a1​,…,ak​) with specified state semantics denotes the composite [[γ]]=uak​​∘⋯∘ua1​​ when the types match. This makes a repeatable operation precise. An untyped log or a trace with missing causal context does not canonically denote an endomorphism. Recursive improvement can be modeled by an endomorphism of a type of certified agents; iteration additionally requires preserved admissibility and realizable operations, not merely a self-description.

Univalence asserts that the canonical map

is an equivalence [32]. It permits transport of type-dependent properties and structures along a supplied equivalence. The identity structure includes automorphisms; univalence does not reduce all equivalence witnesses to one proof. Nor is it derivable just from a verbal appeal to Leibniz’s principle. The structuralist interpretation motivates a chosen foundation; it does not establish a uniqueness theorem for all possible foundations.

For firms or learners, equivalence must be defined on structured systems. An appropriate object includes (S,o,u,V,resources,optimizer); its equivalences preserve the declared interfaces and these chosen structures. Equivalence of underlying state types alone transports no unspecified costs or control policies. A chosen universe may forget them entirely.

This gives a concrete requirement for a higher groupoid of learner presentations: equivalences of optimizer-equipped systems must preserve the relevant geometry. Gauge symmetries such as hidden-unit permutations may be organized into an action groupoid, and, with descent, a quotient stack [Θ/G]. Its stabilizers record presentation automorphisms. A quotient is not automatically a smooth manifold, and a kernel descends only when the required invariance holds. Here a group action is a map G×Θ→Θ preserving group multiplication and identity; its action groupoid has an arrow θg​gθ for each g∈G. The stabilizer of θ is {g:gθ=θ}. A stack is a groupoid-valued sheaf satisfying coherent descent, and the quotient stack is the sheafified action groupoid. Gauge symmetry here means a reparametrization preserving the declared learner structure. A moduli object classifies such structures together with their equivalences; it retains automorphisms rather than only their orbit set.

The point ∗ is the one-point homotopy type. The Eilenberg–Mac Lane space K(Z,k), for k≥1, is connected, has πk​=Z, and has all other positive-degree homotopy groups zero. Behavioral equivalence means equality of the declared observable input-output behavior; additional internal state is retained only when it is part of that declared structure. Thus a unique specified operation class need not have a unique implementation. Also, τ≤n​ is not in general a left-exact localization of an ∞-topos. In spaces,

so 0-truncation fails to preserve this homotopy pullback. Sheafification and homotopy truncation must therefore play distinct roles.

5.1 Finite models and continuous-time optimization

The notation Cr means derivatives through order r exist and are continuous; C∞ means this for every finite r. Write DF for the Jacobian, DkF for the k-linear derivative, ∇L for the gradient in the declared inner product, and ∇2L for its Hessian. A dot denotes a time derivative. A real symmetric matrix M is positive semidefinite, written M⪰0, when vTMv≥0 for every v; M⪰μI means M−μI⪰0. A Gram matrix has entries given by inner products of chosen vectors and is positive semidefinite. Neural tangent kernels below are matrix-valued Gram kernels; their meaning differs from a probabilistic Markov kernel. A loss is a scalar objective measuring the chosen discrepancy. A preconditioner maps its parameter covectors to update directions.

Fix a dataset and concatenate a differentiable network’s training outputs into F:Θ⊆Rp→Rq. For vector-valued outputs q includes both sample and output indices. Let ℓ:Rq→R be differentiable, set J(θ)=DF(θ) and g=∇ℓ(F(θ)), and prescribe

Here p is the number of parameter coordinates, q the number of stacked output coordinates, and N≥1 the sample count when a sample-normalized loss is used. A neural network means the chosen parametrized map fθ​; F stacks its outputs on that sample. The transpose is denoted T, I is the identity matrix of the required size, and ∥⋅∥ is the Euclidean norm or its induced multilinear operator norm. The matrix M⪰0 is a chosen preconditioner and v is an additional parameter drift. In geometric language M maps covectors to tangent vectors; the transpose uses the displayed Euclidean coordinates. Assume sufficient local regularity for the trajectory under consideration to exist. The following is a direct chain-rule calculation, not an infinite-width approximation.

For M=I, v=0, this is the empirical neural tangent kernel (NTK). In sample notation, including the output block structure,

Jacot, Gabriel, and Hongler established the foundational NTK viewpoint and a constant-kernel infinite-width regime under their hypotheses [19]. Linearized wide-network results [23] and tensor-program calculations for broad architectural classes [36] extend its scope. They do not imply that an arbitrary finite transformer or other deployed network has a constant kernel. Feature learning can change J and K; lazy training is a particular scaling regime [5] in which the parametrized map remains close to its linearization at initialization over the specified training interval.

5.2 Kernel evolution and the tangent hierarchy

For F, M sufficiently differentiable and no explicit time dependence of F,

The evolution already involves second parameter derivatives, and derivatives of the optimizer geometry. For the special scalar-output Euclidean flow with L=(2N)−1∑i​(fi​−yi​)2, define recursively

The labels yi​ are fixed scalar targets, fi​=fθ​(xi​), and I in H(m)(I) is an ordered m-tuple of sample indices; this local use is distinct from the identity matrix notation. Whenever the derivatives exist, the chain rule yields the exact identities

This is the recursive structure of the neural tangent hierarchy (NTH) [18]. These higher tensors are not, in general, positive semidefinite two-input kernels or fully symmetric tensors. Freezing a finite level is a closure approximation; the identities themselves do not bound its error. Huang and Yau obtain controlled truncations under specified smoothness, width, initialization, and data assumptions. Their approximation theorem is not asserted here for every modern architecture or optimizer.

6.1 Germs, the infinite jet tower, and holonomic sections

Fix an open parameter domain Θ⊆Rp; the same definitions apply in manifold charts. For scalar objectives write L=C∞(Θ,R). The standard geometric constructions of jets and their coordinate transformations are developed in [30] and [21]. The factorization results below are elementary consequences of explicitly retaining zeroth-order values and runtime state.

An inverse limit here is a compatible sequence under these forgetful maps; the germ direct limit identifies representatives agreeing after restriction. In Taylor coordinates the natural map Gθ​→Jθ∞​ sends a germ to its full derivative sequence. Its kernel is the ideal of flat germs, whose derivatives of every order vanish at θ. It can identify distinct smooth germs. In one variable, h(x)=e−1/x2 for x>0 and h(x)=0 for x≤0 has a nonzero germ at zero but zero infinite jet there. Every derivative of h on x>0 is an inverse-power polynomial times the exponential and tends to zero at the origin, which proves flatness. An analytic germ is represented by a locally convergent power series; for analytic germs the Taylor map is injective. Smooth germs, formal infinite jets, and analytic germs therefore require distinct types.

The assignment U↦C∞(U,R) is a sheaf: local smooth functions agreeing on overlaps glue to one smooth function. Germs are its stalks, namely the direct limits over neighborhoods. Prolongation and zeroth-order projection commute with restrictions on the corresponding holonomic jet sheaf. This gives the local-to-global organization of objective data inside smooth contextual semantics. An algorithm querying outside an open set U is not an operation determined solely by the objective restricted to U.

6.2 Jet algebra and its limitations

In a chosen parameter chart, the order-r parameter jet of a Cr map F is its Taylor polynomial modulo terms of degree r+1:

At a fixed point it records derivatives through order r, not the entire map. Under coordinate changes these records transform by the chain rule. They can be organized into jet bundles, whose fibers collect the jets at each base point, and compositional differential structures [3, 30, 21]. Here α=(α1​,…,αp​)∈Np is a multiindex, ∣α∣=∑j​αj​, α!=∏j​αj​!, and δα=∏j​δjαj​​. For a one-variable jet over a characteristic-zero algebra A, normalized Taylor coefficients obey

Equivalently, with unnormalized derivatives, (ab)(k)=∑j=0k​(jk​)a(j)b(k−j). Thus multiplication is a convolution, not pointwise multiplication of derivative lists. The product order shown also works for noncommutative coefficients when the variable is central and D is a derivation; arbitrary reordering is not permitted. An algebra here is an associative algebra over a field; characteristic zero ensures the displayed integer coefficients have their usual nonzero values.

The infinite jet, or formal power series, admits the coefficient shift without this finite-order loss. Truncated polynomial representatives can be differentiated after choosing a representative, but that is a chosen approximation to the germ’s derivative. Regular jets do not automatically include Laurent series with a pole at the base point. Shared differential formulas are not an equivalence of all polynomial, Laurent, smooth, and formal theories.

6.3 A universal jet representation of optimization algorithms

Code inspection and representation-dependent costs are explicit auxiliary data when they affect the algorithm. A randomized program may equivalently use a deterministic step with its seed in Ξ. The oracle is the interface answering objective or derivative queries; a black-box algorithm knows the objective through those answers rather than through a closed formula.

This is the precise sense in which every specified objective-based optimization algorithm is an instance of the global jet construction. It includes derivative-free methods because the objective’s values are its zeroth-order component. The theorem is a representation result: infinite derivatives can be available without each algorithm actually computing or using them. It also holds with global order-r holonomic sections for any r≥0; increasing the order enriches the accessible differential operations, rather than being necessary to recover a function already retained at order zero. Extending an optimizer to nonholonomic formal sections is a separate modeling choice and is not determined by this theorem.

Composition permits search, differentiation, filtering, acceptance, resource checks, and release gates to use one typed objective representation while retaining their separate runtime states.

Here germ-local means dependence only on agreement on a neighborhood; pointwise-jet determined means dependence only on the derivative data at the base point. These are distinct conditions. For example, choose a smooth function β supported in (1/2,3/2) with β(1)=1. The objectives L0​(x)=x2 and L1​(x)=x2−2β(x) have the same germ at zero. Searching the candidates {0,1} returns zero for L0​ and one for L1​. This legitimate value-based search cannot factor through the current germ or current pointwise jet. It does factor through the global holonomic jet section. Likewise the flat-germ example in Definition 6.1 explains why the infinite pointwise jet can forget information even in a germ-local problem.

Orders, queries, and optimizer state. Gradient methods read first-order jets of L at their query points. Newton methods read second-order jets. A line search selects a step length along a declared direction by querying candidate values; it can be expressed through zeroth-order evaluations of the global jet section. Finite differences use several such evaluations to approximate derivatives. Quasi-Newton methods update an approximate Hessian or inverse Hessian from successive gradient and displacement pairs, so their memory belongs to Ξ. Momentum and Adam combine first-order queries with retained memory. Population and randomized search methods retain several candidate points and query values at each. A trust-region method optimizes a local Taylor model under a bound ∥Δθ∥≤ρ with ρ>0; higher-order tensor methods use higher multilinear derivatives. Constraints and acceptance tests remain part of the update interface.

For a composite loss L=ℓ∘F, algorithms that use the network’s representation, such as Gauss–Newton, retain F and ℓ as auxiliary data, or use their jet sections jointly. Equality of the scalar loss alone need not identify this representation. Minibatch methods likewise specify the sampled objective family and the batch-selection state. Chart changes act on jets by the chain rule, and coordinate-invariant training still requires transport of the optimizer geometry as in Proposition 4.2.

For nonsmooth or discrete objectives, classical infinite derivative jets may not exist. The universal value interface remains: a function L:Θ→R is its global zeroth-order section. Subgradients, generalized derivatives, or surrogate derivatives can be additional declared oracle data. Consequently every objective-based algorithm has a global value/jet interface in its appropriate regularity setting. This does not identify arbitrary black-box search with a rule using only one smooth germ’s derivative sequence. The distinction is also computational: a runtime can expose finite jet queries on demand, without storing an infinite array or supplying an oracle for uncomputable objectives. A subgradient of a convex function L at θ is a vector v with L(y)≥L(θ)+v⋅(y−θ) for every y in its domain. Convexity means L(tx+(1−t)y)≤tL(x)+(1−t)L(y) for 0≤t≤1 on a domain containing each such line segment. Other generalized derivatives mean explicitly chosen set-valued replacements for classical derivatives; a surrogate derivative is an explicitly supplied replacement used by the algorithm, for example from a smooth approximation.

6.4 Second and higher-order optimizers

For L=ℓ∘F, the parameter Hessian is

Newton-type updates use this second jet. A damped positive-definite approximation can supply M=(H+λI)−1 in Theorem 5.1, where H is the declared symmetric Hessian approximation and damping λ is chosen so that H+λI is positive definite. Gauss–Newton retains the first term of (11); a raw indefinite inverse Hessian need not produce descent. Higher Taylor models, trust-region methods, and tensor methods can use larger jets, but a jet by itself is not an optimization algorithm: objective, step selection, constraints, and acceptance must also be chosen.

For an actual finite parameter step Δθ, Taylor’s theorem gives

with ∥Rr+1​∥≤B∥Δθ∥r+1/(r+1)! when the (r+1)st derivative is bounded by B along the connecting segment. This quantifies the step-size limitation of a finite jet description.

6.5 An explicitly defined input-jet tangent kernel

The phrase “jet tangent kernel” does not specify a single universal object. For a precise version, take continuous inputs x∈Rd, vector outputs, and multiindices ∣α∣≤r. Define jet observables Zα,θ​(x)=∂xα​fθ​(x). Assume the mixed derivatives needed below exist and commute. Stack the chosen sampled observables into Z(θ) and set

For M independent of input, differentiating the ordinary output block kernel gives KM;αβjet,r​=∂xα​∂x′β​KM​(x,x′).

This is appropriate for derivative matching and Sobolev objectives; Sobolev training is an established example [8]. A sampled Sobolev objective, for example, is 21​∑i,α​wα​∥Zα,θ​(xi​)−ti,α​∥2, with nonnegative weights wα​ and prescribed derivative targets ti,α​. The term means that function values and chosen derivatives both enter the loss. It is distinct from the parameter-jet tower in (10) and the NTH in (9). One may also construct weighted Gram matrices from higher parameter derivatives, but without an optimizer law such matrices do not determine training.