ora-0100
7.5.1 Differential presentations of tangent worlds
7.5.1 Differential presentations of tangent worlds
TUOCL need not assume that its tangent hypotheses are primitive. Let \(\mathbf{DiffCat}_{\mathrm{adm}}\) be a declared class of differential categories satisfying the hypotheses needed for the coalgebra construction. Write
schematically for the assignment sending a differential category to its coEilenberg–Moore category of coalgebras with the induced tangent structure [ Cockett et al. , 2020 ] . A differential-realized TUOCL instance factors its tangent hypotheses through this assignment. Its tangent queries observe \(\mathsf{Coalg}_{!}(\mathbb X)\); stronger differential queries may also test the modality \(!\), the deriving transformation \(d\), and their coherence.
Suppose admissible differential hypotheses \(\mathbb X_1\) and \(\mathbb X_2\) are inequivalent as differential categories, while \(\mathsf{Coalg}_{!}(\mathbb X_1)\) and \(\mathsf{Coalg}_{!}(\mathbb X_2)\) are equivalent for every admitted tangent query. Any transcript and query doctrine factoring through \(\mathsf{Coalg}_{!}\) leaves the two differential realizations observationally indistinguishable.
Every admitted answer is computed after applying \(\mathsf{Coalg}_{!}\). By hypothesis the resulting tangent models agree on the entire tangent query doctrine, so no such answer can separate their preimages. A separating observation must inspect differential-presentation structure not retained by the realized tangent world.
The result is deliberately conditional: it does not assert that such pairs exist for every chosen doctrine. It identifies the exact obstruction whenever the coalgebra realization fails to be query-faithful. Accordingly, learning admits a three-rung ladder:
Each step requires new probes and a new persistence claim. DIAL needs the middle rung to type infinitesimal defects. A learned differentiation engine or resource semantics may require the third; ordinary predictive success guarantees neither.
Let a TUOCL execution have strong tangent comparison maps whose underlying base comparisons are conservative on \(\mathcal Q^{\mathrm{base}}_0\), and whose tangent comparison cells are conservative on \(\mathcal Q^{\mathrm{tan}}_0\). If both classes are closed under reindexing and composition, then all settled base and tangent answers persist along every finite continuation.
Apply Proposition 7.3 to the base comparisons. Strong tangent coherence identifies the tangent comparison for a composite with the composite of the reindexed tangent cells. Closure then makes every settled tangent observer invert that composite as well.
This yields a sharper convergence ladder. A learner may stabilize behaviorally on base queries while its inferred tangent structure continues to change. Tangent stabilization requires eventual constancy of both the base hypothesis and its tangent structure up to tangent equivalence.
A UOCL hypothesis is DIAL-ready for a decision declaration when it is decision-ready, its relevant world and mechanism objects carry identified tangent structure, evidence and action maps admit coherent tangent lifts, tangent observers separate the repair-relevant defects, and unresolved tangent comparisons are represented as explicit obligations rather than silently treated as derivatives.
DIAL readiness therefore does not require a differential-category realization. When such a realization is claimed, however, its modality and deriving transformation become additional learned warrants rather than automatic consequences of the tangent axioms.
This criterion gives the fourth book a precise relation to Infinitesimal Creativity. DIAL specifies how infinitesimal defects can guide localized creative repair. TUOCL explains how an agent might acquire, test, preserve, and accommodate the tangent structure that makes those defects meaningful.