IndisputableMonolith.Cost.ContDiffReduction
ContDiffReduction differentiates the d'Alembert equation under C^2 assumptions to obtain an ODE solved by hyperbolic cosines. Researchers verifying the T5 J-uniqueness step cite these lemmas to link the Recognition Composition Law to the explicit form J(x) = cosh(log x) - 1. The module consists of a chain of differentiation lemmas that reduce the functional equation at fixed points such as y = 1.
claimIf $J : (0,∞) → ℝ$ is twice continuously differentiable and satisfies the d'Alembert equation $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ for all $x,y > 0$, then the first derivative in the second variable yields an ODE whose unique solution is $J(x) = cosh(log x) - 1$.
background
This module sits in the Cost domain and imports from FunctionalEquation, whose doc-comment states it supplies lemmas for the T5 cost uniqueness proof. It works with the Recognition Composition Law (RCL) written in d'Alembert form and assumes contDiffTwo to justify differentiation. Key objects include the first and second derivatives of the equation, the reduction to an ODE, the cosh solution, and the uniqueness statement under these smoothness hypotheses.
proof idea
The module chains lemmas that first extract differentiability from contDiffTwo, compute the first derivative of the d'Alembert equation, evaluate the second derivative at zero, reduce the system to an ODE, solve explicitly to the cosh form, and close with uniqueness. Each step applies the functional equation after differentiation at specific points.
why it matters in Recognition Science
The module supplies the smoothness reduction needed for the T5 J-uniqueness step in the forcing chain, feeding the parent result in UnifiedForcingChain. It converts the abstract RCL into the concrete J form J(x) = (x + x^{-1})/2 - 1 once C^2 is granted, closing one link between the composition law and the explicit solution used downstream.
scope and limits
- Does not derive the Recognition Composition Law from axioms.
- Does not treat solutions lacking C^2 regularity.
- Does not compute numerical constants such as phi or alpha.
- Does not address spatial dimension or the eight-tick octave.
depends on (1)
declarations in this module (10)
-
lemma
contDiffTwo_differentiable -
lemma
contDiffTwo_differentiable_deriv -
lemma
hasDerivAt_deriv_of_contDiffTwo -
theorem
dAlembert_first_deriv_of_contDiff -
theorem
dAlembert_second_deriv_at_zero_of_contDiff -
theorem
dAlembert_to_ODE_of_contDiff -
theorem
dAlembert_to_ODE_hypothesis_of_contDiff -
theorem
composition_law_forces_reciprocity -
theorem
dAlembert_cosh_solution_of_contDiff -
theorem
washburn_uniqueness_of_contDiff