IndisputableMonolith.Cost.FunctionalEquationAczel
The module Cost.FunctionalEquationAczel supplies the link between the d'Alembert functional equation and Aczél's classification of its continuous solutions. Researchers deriving T5 cost uniqueness or the full RCL inevitability cite it to obtain C^∞ regularity from continuity alone. The module imports the Aczél theorem and the T5 functional-equation helpers; the central hypothesis follows at once from the imported classification.
claimEvery continuous solution $H$ of $H(t+u)+H(t-u)=2H(t)H(u)$ with $H(0)=1$ is $C^∞$.
background
The module sits inside the Cost domain. It imports AczelTheorem, whose doc-comment states that every continuous solution of the d'Alembert equation $H(t+u)+H(t-u)=2H(t)H(u)$ with $H(0)=1$ is $C^∞$, the only possibilities being the constant function 1 and $H(t)=cosh(λt)$. The second import, FunctionalEquation, supplies lemmas used in the T5 cost-uniqueness argument. The DOC_COMMENT records that the hypothesis dAlembert_continuous_implies_smooth_hypothesis therefore holds for every $H$ as a direct consequence of the Aczél axiom.
proof idea
The module imports AczelTheorem and FunctionalEquation. The required smoothness implication is obtained directly from the Aczél classification of continuous solutions; no further reduction is performed inside the module.
why it matters in Recognition Science
This module removes the regularity obstacle that would otherwise block the T5 forcing step. It is imported by CostAlgebra and by DAlembert.FullUnconditional, the latter proving the strongest form of RCL inevitability in which both F and P are forced with no assumption on P. The result therefore supports the J-uniqueness claim inside the UnifiedForcingChain.
scope and limits
- Does not prove the Aczél theorem from first principles.
- Does not classify discontinuous solutions.
- Does not derive the explicit value of λ.
- Does not address the phi-ladder or mass formula.