dAlembert_cosh_solution_of_contDiff
plain-language theorem explainer
Any C² function Hf from reals to reals that satisfies the d'Alembert equation, normalizes to Hf(0)=1, and has second derivative 1 at the origin must equal cosh. Researchers closing the T5 uniqueness step in Recognition Science cite the result to remove an explicit regularity seam. The proof first converts the functional equation to the ODE Hf''=Hf under the ContDiff assumption, records evenness and zero first derivative at the origin, then applies the cosh initial-value uniqueness lemma.
Claim. Let $Hf : ℝ → ℝ$. If $Hf(0)=1$, $Hf(t+u)+Hf(t-u)=2 Hf(t) Hf(u)$ for all real $t,u$, $Hf$ is twice continuously differentiable, and the second derivative of $Hf$ at 0 equals 1, then $Hf(t)=cosh(t)$ for all real $t$.
background
The ContDiff Reduction module for T5 shows that a C² solution to the d'Alembert equation already satisfies the ODE H''=H. Here Hf corresponds to the shifted cost H=J+1, where J(x)=(x+x^{-1})/2-1 from the T5 uniqueness statement; the upstream CostAlgebra.H records that the Recognition Composition Law becomes the d'Alembert equation under this shift. The module documentation states that normalization, composition, and calibration plus C² regularity force the canonical reciprocal cost without assuming reciprocity.
proof idea
The term proof first calls dAlembert_to_ODE_of_contDiff to obtain the ODE deriv(deriv Hf)t = Hf t. It then applies dAlembert_even to establish evenness of Hf, contDiffTwo_differentiable to extract differentiability at 0, and even_deriv_at_zero to obtain deriv Hf 0 = 0. The final step invokes ode_cosh_uniqueness_contdiff on the resulting initial-value problem.
why it matters
The theorem supplies the cosh solution on the C² surface and is invoked directly by washburn_uniqueness_of_contDiff, which states that normalization, the composition law, calibration, and C² regularity of H=G+1 already force the canonical reciprocal cost. It advances the sharpened T5 surface described in the module doc, closing part of the regularity seam in the T0-to-T8 forcing chain. The result ties to the J-uniqueness landmark where the self-similar fixed point yields the cosh form.
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