3.13 What the differential account does not claim

. Leung’s Weil theory provides algebraic semantics for infinitesimal probes. Involution algebroids provide the established one-direction bridge, while classical double Lie algebroids organize two compatible classes of smooth variation. Involution algebroids replace Jacobi at the internal level by a Yang–Baxter-style flip law, from which Jacobi is recovered on sections. The DIAL bridge remains to be constructed. None of these structures supplies domain meaning, a novelty criterion, or a unique finite extension. A tangent or double-infinitesimal geometry can formalize local possibility without proving that every historical creative jump decomposes into infinitesimal steps.

Four limitations remain visible. First, tangent structure is relative to a chosen category and interpretation. A missing object or observer may be invisible to every current probe. Second, the proposed split into assimilatory and proto-accommodative directions is semantic data, not a canonical decomposition supplied by a double vector bundle. Third, local differential evidence usually underdetermines the global package change. Fourth, preservation of tangent structure is only one admission criterion: a scientifically creative theory must also survive substantive formal, simulator-relative, or empirical tests.

These limitations are productive. They identify the division of labor used in the rest of the book: Weil probes expose local alternatives, DIAL repair geometry diagnoses their interaction, registered controls test competing explanations, frontier models propose, a finite package comparison records the candidate global change, and independent procedures admit or reject it. When the presentation changes, sketch surgery records that component. Chapter 4 next supplies an anchored geometry for the skills that conduct this exploration.