sec-tangent-uocl

7.5 Tangent UOCL: learning how a world can vary

Ordinary UOCL identifies a category relative to base queries. Tangent UOCL (TUOCL) takes its hypotheses in \(\mathbf{TanCat}\), or in a declared category of models carrying tangent structure. Its transcript may therefore contain both base evidence and tangent evidence: observations of infinitesimal probes, their projections, sums, lifts, and iterated interchange.

Definition 7.7 Tangent UOCL instance

A tangent UOCL instance is a UOCL instance whose hypothesis fibration

\[ p_{\mathrm{tan}}:\int H_{\mathrm{tan}} \longrightarrow \mathbf{Pres} \]

has tangent-category hypotheses, whose comparison maps are strong or explicitly lax tangent morphisms, and whose query doctrine decomposes as \(\mathcal Q^{\mathrm{base}}\cup \mathcal Q^{\mathrm{tan}}\). The tangent queries test the structure maps \((T,p,0,+,\ell ,c)\) and the coherence laws on which later infinitesimal calculations depend.

For a comparison \(F:\mathcal C\to \mathcal D\), tangent compatibility includes a coherent comparison

\[ \tau _F:F T_{\mathcal C}\Longrightarrow T_{\mathcal D}F, \]

invertible in the strong case, satisfying the projection, zero, addition, lift, and flip axioms. Thus tangent assimilation preserves not only a learned composite but also the admissible variations of that composite. Tangent accommodation repairs \(T\) or one of its coherence maps when new infinitesimal evidence cannot be absorbed by the current tangent realization.

Proposition 7.8 Base evidence cannot identify tangent structure

Suppose two tangent hypotheses \(\mathbb C_1^{\mathrm{tan}}\) and \(\mathbb C_2^{\mathrm{tan}}\) have equivalent underlying categories but are not equivalent in \(\mathbf{TanCat}\). Any presentation and query doctrine that factors through \(U_{\mathrm{tan}}\) leaves them observationally indistinguishable. Hence categorical identification of the base does not imply tangent identification.

Proof ▼

Every admitted observation depends only on the common image under \(U_{\mathrm{tan}}\), so corresponding base queries have equivalent answers. Their disagreement lies entirely in structure forgotten by that functor and cannot be separated without a tangent-sensitive probe.

The proposition is the tangent analogue of the finite-transcript obstruction. It explains why differentiating a learned model after the fact is not the same as learning the world’s infinitesimal geometry: automatic differentiation supplies a tangent mechanism for a presentation, but does not establish that the mechanism is identified or invariant under an equivalent presentation.