IndisputableMonolith.Foundation.GeneralizedDAlembert.SmoothnessTop
The SmoothnessTop module supplies the infinite differentiability result needed to relax the polynomial-degree bound on the route-independence combiner in the Translation Theorem. Researchers refining regularity assumptions inside the Generalized D'Alembert framework would cite it when replacing a degree-at-most-two hypothesis with a weaker smoothness condition. The module consists of a single lifting lemma that converts finite-order differentiability into the top-level case.
claimIf a map between manifolds is $C^k$ for every finite $k$, then it is $C^\infty$.
background
The module belongs to the Foundation layer and is imported by GeneralizedDAlembert. That parent file describes Move 3 as the step that discharges polynomial regularity using continuity. The Translation Theorem currently requires the route-independence combiner P to be a polynomial of total degree at most two; the quartic-log counterexample shows that some regularity is necessary but that the degree bound is stronger than needed.
The module imports only Mathlib and contains no new Recognition Science definitions. It supplies the top-level smoothness object that lets the continuity argument proceed without the original polynomial restriction.
proof idea
The module contains one lemma that lifts differentiability of all finite orders to infinite differentiability. The argument applies the standard Mathlib result that a function smooth at every finite order is smooth at the top level.
why it matters in Recognition Science
The module feeds the GeneralizedDAlembert file and completes Move 3 by discharging the polynomial regularity requirement via continuity. It thereby supports weakening the degree-at-most-two hypothesis on the combiner P inside the Translation Theorem, directly addressing the limitation shown by the quartic-log counterexample.
scope and limits
- Does not prove the Translation Theorem itself.
- Does not construct or analyze the quartic-log counterexample.
- Does not quantify the minimal regularity needed beyond infinite differentiability.
- Does not connect to the phi-ladder, J-cost, or other Recognition Science constants.