lin-0036

1.5 Structural laws governing the obstruction

One axiom does not remove the need for hypotheses. Axiom 1.1 supplies the universal carrier of a declared failure and characterizes its vanishing. Tangent closure, presentation descent, interaction structure, and repair transport do not follow from representability alone. The following laws register that additional structure explicitly.

Corollary 1.7 Base compositionality

An admissible model \(D\) satisfies its declared compositional structure exactly when its universal obstruction is trivial:

\[ D\models \mathbb S \quad \Longleftrightarrow \quad \operatorname {Fact}_{\mathbb S}(D)\neq \varnothing \quad \Longleftrightarrow \quad \mathcal O_{\mathbb S}(D)\cong 0_{\mathbb S,D}. \]

Proof ▼

The first equivalence is the definition of a model satisfying the learning sketch. The second is the triviality clause of Axiom 1.1.

Structural law 1.8 Tangent admissibility

If \(D\) is an admissible model of a tangent learning sketch, then \(TD\) is again admissible, and its action agrees with the tangent structure declared by the sketch.

Structural law 1.9 Presentation descent

Suppose the implementation is expressed in a redundant presentation \(\pi :P\to M\). Let \(p_P:TP\to P\) and \(p_M:TM\to M\) be the tangent projections. A proposed vector field \(V_P:P\to TP\), satisfying \(p_P\circ V_P=1_P\), represents a vector field on the model object only if it descends to some \(V_M:M\to TM\), with \(p_M\circ V_M=1_M\), such that

\[ T\pi \circ V_P=V_M\circ \pi \]

or if a registered gauge fixing or equivariant horizontal lift supplies an equivalent descent certificate.

Corollary 1.10 Tangent compositionality

Under tangent admissibility, Axiom 1.1 applies to \(TD\). Define the infinitesimal obstruction by

\[ \operatorname {INC}(D):=\mathcal O_{\mathbb S}(TD). \]

Then infinitesimal compositionality is the same declared universal problem applied after tangent lift:

\[ \operatorname {INC}(D)\cong 0_{\mathbb S,TD} \quad \Longleftrightarrow \quad \operatorname {Fact}_{\mathbb S}(TD)\neq \varnothing . \]

Proof ▼

Apply the triviality clause of Axiom 1.1 to the admissible realization \(TD\).

Structural law 1.11 Sketch-specific interaction

Every operation in a declared interaction signature \(\Omega _D\) determines a typed closure, factorization, or profile problem on \(\mathcal A_D\). Failure of that declared problem is admissible interaction data. No single operation, including the Lie bracket, is mandatory for every sketch.

Structural law 1.12 Conditional obstruction transport

A transport from base failure to an interaction-level obstruction,

\[ \mathfrak t_{D,\mathbb S}: \mathcal O_{\mathbb S}(D) \rightsquigarrow \mathcal O_{\Omega ,\mathbb S}(TD), \]

is additional sketch-relative data. It must be natural under admissible model transformations, descend through presentations, and preserve the semantics of the factorization problem. Its existence does not establish the empirical value of a scalarized tangent signal.

Structural law 1.13 Functorial repair

If \(\alpha :D\to D'\) is an admissible repair, then \(T\alpha :TD\to TD'\) is an admissible tangent repair. A repair advertised as compositionality-preserving must preserve the relevant factorization, and a repair advertised as local-to-global must commute with restriction up to the declared overlap coherence.

Proposition 1.14 Trivial obstruction forces zero scalarization

Suppose \(\mathcal E_{\mathbb S,D}\) has zero morphisms and \(0_{\mathbb S,D}\) is a zero object. If \(\mathcal O_{\mathbb S}(D)\cong 0_{\mathbb S,D}\), then every registered scalar observer

\[ \Phi _{\mathbb S}: \mathcal O_{\mathbb S}(D)\longrightarrow \mathbb R_{\geq 0} \]

is the zero morphism.

Proof ▼

An object isomorphic to a zero object is initial. Hence there is exactly one morphism from \(\mathcal O_{\mathbb S}(D)\) to the scalar codomain. The zero morphism is one such morphism, so it is equal to \(\Phi _{\mathbb S}\).

The converse—that a zero scalar observer forces the obstruction to be trivial—requires the observer to reflect triviality and is therefore an additional identifiability condition.

Condition 1.15 Observation specificity

For an observation profile \(\Psi _{\mathbb S}(D,e)\), identification of a repair requires

\[ \Psi _{\mathbb S}(D,e)=\Psi _{\mathbb S}(D,e') \quad \Longrightarrow \quad e\simeq e'. \]

Without this condition, the observations identify only an equivalence class or moduli space of repairs.

Admission contract

The registered repair language and structural laws make a repair judgment well typed. They do not make the repair beneficial. Admission requires evidence independent of the proposal signal and may assign zero weight to any base, tangent, or interaction observer that fails validation.