IndisputableMonolith.Cost.AczelClass
This module encodes the Aczél smoothness requirement for continuous solutions of the d'Alembert functional equation within the Recognition Science cost framework. It supplies the mathematical basis needed for T5 J-uniqueness by guaranteeing that such solutions are C^∞. The module declares the property without proofs; the concrete instance appears in the AczelProof submodule via integration bootstrap.
claimAny continuous solution \(H:\mathbb{R}\to\mathbb{R}\) of the d'Alembert equation \(H(t+u)+H(t-u)=2H(t)H(u)\) with \(H(0)=1\) is infinitely differentiable (i.e., \(C^\infty\)).
background
Recognition Science derives T5 J-uniqueness from the functional equation satisfied by (J(x)=(x+x^{-1})/2-1), which is equivalent to (\cosh(\log x)-1). The d'Alembert equation arises directly as the addition formula for this J-cost. The AczelClass module imports Mathlib to access ContDiff and states the smoothness requirement that classifies continuous solutions as the constant 1, (\cosh(\lambda t)), or (\cos(\lambda t)), all of which are C^∞ (Aczél 1966, Ch. 3).
proof idea
This is a definition module, no proofs. It states the Aczél smoothness requirement and points to the unconditional dAlembert_contDiff_top theorem in AczelTheorem for the concrete instance.
why it matters in Recognition Science
The module feeds the AczelProof submodule, which establishes that continuous solutions are real analytic (ContDiff ℝ ⊤) via integration bootstrap, and supplies lemmas for the FunctionalEquation module used in the T5 cost uniqueness proof. It closes the smoothness step required before the phi-ladder mass formula and the eight-tick octave can be applied.
scope and limits
- Does not prove the smoothness theorem itself.
- Does not classify the explicit solution forms.
- Does not derive the J-cost equation or connect to phi or G constants.
- Does not address discrete or non-continuous cases.