IndisputableMonolith.Cost.ContDiffReduction
Under C^2 regularity, the Recognition Composition Law in d'Alembert form reduces to a second-order ODE whose positive solutions are forced to the J-cost shape cosh(log x)-1. Cost-uniqueness arguments cite this when the continuous case has already been upgraded to twice continuous differentiability. The module differentiates in the second variable, evaluates at the identity, and matches the ODE to the known cosh family.
claimIf $F:\mathbb{R}_{>0}\to\mathbb{R}$ is $C^2$ and satisfies the d'Alembert composition law $F(xy)+F(x/y)=2F(x)F(y)+2F(x)+2F(y)$, then (under the T5 normalizations) $F(x)=J(x)=\frac{x+x^{-1}}{2}-1=\cosh(\log x)-1$. Intermediate steps give the first and second derivatives of the law and the ODE reduction at the identity.
background
Recognition Science forces the cost functional through T5: the unique normalized solution of the Recognition Composition Law (RCL) is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. The parent module Cost.FunctionalEquation supplies algebraic helpers for that uniqueness proof.
This module treats the regularly differentiable route. One assumes a $C^2$ candidate $F$ on the positive reals obeying the same d'Alembert identity as RCL. Differentiability lets one pass from the functional equation to pointwise derivative identities, then to an ordinary differential equation at the multiplicative identity.
Sibling lemmas package the calculus facts (a $C^2$ map is differentiable with continuous derivative; the derivative itself has a derivative) and the successive differentiations of the composition law in the second variable.
proof idea
Structure is a calculus reduction ladder, not a single theorem. First, standard $C^2$ facts give differentiability of $F$ and of $F'$. Next, differentiate the d'Alembert identity once in the second slot to obtain a first-derivative form of the law; evaluate at the identity to pin $F'(1)$. Differentiate again to read off $F''(1)$ and produce a linear second-order ODE for $F$ along rays. Solve that ODE under the T5 boundary data to recover the cosh (equivalently $J$) profile. A parallel track records the same ODE as a named hypothesis interface, and a final lemma packages the full implication: composition law plus $C^2$ forces $J$-cost.
why it matters in Recognition Science
T5 in the Unified Forcing Chain demands uniqueness of the cost $J$. Measurable or continuous routes still need a bridge into a rigid analytic class; this module is that bridge for the $C^2$ case. Downstream consumers are the T5 uniqueness assembly and any argument that already has reciprocity plus smoothness and only needs the ODE-to-cosh step. The terminal sibling law_of_logic_forces_jcost_of_contDiff is the headline export: RCL under continuous twice-differentiability yields $J$. Without this reduction, the forcing chain would stall between the algebraic composition identity and the explicit closed form used for the phi-ladder and the alpha band.
scope and limits
- Does not treat merely continuous or measurable solutions without a C^2 upgrade.
- Does not derive the composition law itself; that is an input hypothesis.
- Does not fix discrete or eight-tick structure (T7); only the real-variable cost shape.
- Does not by itself close full T5 uniqueness outside the ContDiff route.
- Does not address dimensional forcing (T8) or the phi fixed-point step (T6).
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
law_of_logic_forces_jcost_of_contDiff