IndisputableMonolith.Cost.FunctionalEquationAczel
Module Cost.FunctionalEquationAczel asserts that the d'Alembert continuous-implies-smooth hypothesis holds for every solution H as an immediate consequence of Aczel's classification. Researchers deriving RCL inevitability or cost algebra results cite it to eliminate regularity assumptions in the T5 chain. The module structure is a direct import and application of the Aczel theorem to the functional equation setting.
claimEvery continuous solution $H$ to the d'Alembert equation $H(t+u) + H(t-u) = 2 H(t) H(u)$ with $H(0) = 1$ satisfies the smoothness hypothesis (i.e., $H$ is $C^∞$).
background
This module belongs to the Cost domain and imports AczelTheorem, whose doc-comment states that every continuous solution of $H(t+u) + H(t-u) = 2·H(t)·H(u)$ with $H(0)=1$ is $C^∞$, with the complete classification being the constant 1 or $H(t)=cosh(λt)$. It also imports FunctionalEquation, which supplies lemmas for the T5 cost uniqueness proof. The local setting is the Recognition Science derivation of cost properties from the J-uniqueness step in the T0-T8 forcing chain.
proof idea
The module applies the upstream Aczel theorem directly to conclude that dAlembert_continuous_implies_smooth_hypothesis holds for every H. It is a one-line consequence of the Aczel classification of continuous solutions; no separate argument is developed inside the module.
why it matters in Recognition Science
This module supplies the required smoothness result to the downstream FullUnconditional module, which proves the strongest form of RCL inevitability with both F and P forced and no assumption on P. It also supports CostAlgebra. In the framework it closes the regularity gap after T5 J-uniqueness, permitting the phi fixed point, eight-tick octave, and D=3 to proceed without extra hypotheses on the cost function.
scope and limits
- Does not classify or analyze discontinuous solutions to the d'Alembert equation.
- Does not prove the Aczel theorem or its 1966 classification.
- Does not extend the result to other functional equations or multi-variable cases.
- Does not address existence questions beyond the listed smooth solutions.