IndisputableMonolith.Foundation.GeneralizedDAlembert.SmoothnessTop
The SmoothnessTop module supplies the result that a function continuously differentiable to every finite order is infinitely differentiable. It is cited in the Generalized D'Alembert development to replace a strict polynomial-degree bound with a continuity-based regularity condition. The argument is a direct transfer of the standard intersection-of-C^n classes result from real analysis. This step supports Move 3 by weakening the regularity hypothesis on the route-independence combiner P.
claimIf a function is continuously differentiable of every finite order, then it is infinitely differentiable.
background
The module belongs to the Foundation layer and is imported by GeneralizedDAlembert.lean. That parent file carries out Move 3: discharging the polynomial regularity requirement on the route-independence combiner P by means of continuity. The Translation Theorem had required P to be a polynomial of total degree at most two, yet the quartic-log counterexample shows that some regularity is necessary while the degree bound is stronger than needed.
proof idea
This is a definition module whose single sibling declaration transfers the standard real-analysis fact that the intersection of all finite differentiability classes equals the infinite class. No local tactics or reductions are performed; the result is obtained by direct appeal to the ambient analysis library.
why it matters in Recognition Science
The module feeds the parent GeneralizedDAlembert module and thereby completes Move 3 of the regularity discharge. It relaxes the original degree-at-most-two polynomial hypothesis of the Translation Theorem to a continuity condition, consistent with the quartic-log counterexample. The result therefore sits inside the broader effort to derive physical structure from the Recognition Composition Law and the forcing chain without extraneous algebraic restrictions.
scope and limits
- Does not prove the Translation Theorem.
- Does not construct or refute the quartic-log counterexample.
- Does not derive any Recognition Science constants or dimensions.
- Does not address the J-cost or phi-ladder structures.