lin-0265

22.4 Higher interaction signatures

First-order INC asks whether the tangent lift of a declared factorization is inhabited. It does not exhaust infinitesimal structure. Generated directions can interact through brackets, connections, curvature, torsion, or higher jets. The application must declare which of these operations has meaning.

Definition 22.2 Interaction transport

Let \(D:J\to \mathcal C\) be an admissible model with factorization problem \(\operatorname {Fact}_{\mathbb S}(D)\) and interaction signature \(\Omega _D\). An interaction transport is a natural, presentation-descending assignment

\[ \mathfrak t_{D,\mathbb S}: \operatorname {Obs}\! \left(\operatorname {Fact}_{\mathbb S}(D)\right) \rightsquigarrow \operatorname {Obs}_{\Omega ,\mathbb S}(TD). \]

It maps a base obstruction to a candidate interaction profile. It does not assert that every component identifies a repair or has predictive value after scalarization.

For admissible fields \(X,Y\), bracket INC records relative failure of closure:

\[ \operatorname {INC}_{[,]}(D) = \{ [X,Y]\mid X,Y\in \mathcal A_D,\ [X,Y]\notin \mathcal A_D\} . \]

This antisymmetric operation underlies the causal screen used by BRIDGE and SKFM [ Mahadevan , 2026b ] . Its semantics do not transfer automatically to adapters, policies, or skills.

With a declared connection \(\nabla \), the ordered profile

\[ J^2_{\nabla ,D}(X,Y) = \bigl(\nabla _XY,\nabla _YX\bigr) \]

retains both the antisymmetric component associated with the bracket and the symmetric component associated with joint acceleration. Either component may be irrelevant in a particular sketch. Sparse empirical admission should be allowed to set its coefficient to zero.

Level	Declared object	Characteristic question	Possible witness
Base	learning diagram \(D\)	Do the required finite routes and universal properties hold?	parallel-path or factorization obstruction
Tangent	lifted diagram \(TD\)	Does first-order behavior preserve the declaration?	tangent route incompatibility
Interaction	fields with signature \(\Omega _D\)	Do generated directions close and interact as declared?	bracket, acceleration, torsion, or curvature profile
Higher	prolonged sketch \(\mathbb S^{(n)}\)	Are iterated tangent coherences realized?	flip, jet, Jacobiator, or Bianchi-type obstruction
Homotopical	enriched factorization problem	Can the failure be coherently deformed away?	homotopy or cohomological obstruction class

Table 22.5 A hierarchy of increasingly structured compositionality tests.

Weil algebras provide an indexing language for these prolongations [ Leung , 2017a , 2017c ] . If \(W\) is the first-order dual-number object and

\[ F:\mathsf{Weil}_1\longrightarrow \operatorname {End}(\mathcal C) \]

classifies the tangent structure, then \(T=F(W)\). Iterated and mixed Weil probes distinguish repeated directional differentiation from the fiber powers used in the tangent axioms. A mature higher LINCS theory should construct prolonged sketches

\[ \mathbb S^{(n)} \quad \text{with}\quad \operatorname {INC}^{(n)}(D) = \operatorname {Obs}\! \left( \operatorname {Fact}_{\mathbb S^{(n)}}(T^nD) \right), \]

including canonical flips, vertical lifts, and any declared connection data.