lin-0064

4.3 Infinitesimal non-compositionality

Definition 4.3 Infinitesimal non-compositionality

The first-order infinitesimal non-compositionality of \(D\) is

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

When iterated tangent structure is available,

\[ \operatorname {INC}^{(n)}(D) := \mathcal O_{\mathbb S}(T^nD). \]

The base and tangent obstructions answer related but distinct questions. The object \(\mathcal O_0(D)\) asks whether the realized diagram satisfies the declaration. The object \(\mathcal O_1(D)=\operatorname {INC}(D)\) asks whether admissible infinitesimal perturbations satisfy its tangent lift.

Proposition 4.4 Exact identities differentiate

Suppose two smooth composites \(p,q:X\to Y\) agree after restriction to an open subset \(U\subseteq X\). Then \(T(p|_U)=T(q|_U)\) on \(TU\). Consequently, an exactly commuting smooth learning diagram has trivial first-order route incompatibility along every admissible tangent direction based in that domain.

Proof ▼

Functoriality gives the tangent maps \(Tp\) and \(Tq\). Since the base maps agree on an open domain, their derivatives agree there, hence \(Tp=Tq\).

The converse does not say that a small sampled tangent residual proves an exact base identity. It may reflect limited probes, low support, a null direction, or an insensitive observation.