IndisputableMonolith.Verification.ODECoshUniqueCert
Verification certificate module for uniqueness of the cosh profile solving the T5 cost ODE. It packages the claim that the only admissible solution of the second-order ODE tied to the Recognition Composition Law is J(x)=cosh(log x)-1 (equivalently (x+x^{-1})/2-1). Auditors of the T5 forcing step cite it. The argument is assembled from FunctionalEquation helpers rather than a fresh analytic development.
claimCertificate that the unique $C^2$ solution of the T5 cost ODE on $\mathbb{R}_{>0}$ (subject to the normalization and convexity constraints of the Recognition Composition Law) is $J(x)=\cosh(\log x)-1=\frac{x+x^{-1}}{2}-1$.
background
Recognition Science forces the cost functional through the T5 step of the Unified Forcing Chain: the unique admissible $J$ is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. That uniqueness is usually obtained from the Recognition Composition Law
$$J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$$
together with regularity and normalization. An equivalent route treats $J$ as the solution of a second-order ODE obtained by differentiating the functional equation and imposing the fixed-point and convexity conditions that select the cosh profile.
The upstream module Cost.FunctionalEquation supplies the algebraic lemmas for that T5 uniqueness argument. The present verification module sits one layer above those lemmas and records a machine-checkable certificate that the ODE route yields the same unique cosh solution.
proof idea
This is a verification/certificate module, not a primary derivation. It imports Mathlib analysis primitives and the FunctionalEquation helper suite, then assembles a named certificate object (ODECoshUniqueCert) asserting uniqueness of the cosh solution to the T5 cost ODE. The logical work is delegated to the FunctionalEquation lemmas; the certificate packages hypotheses, the ODE statement, and the uniqueness conclusion into a single audit artifact for the verification layer.
why it matters in Recognition Science
T5 J-uniqueness is a landmark of the forcing chain: once $J$ is forced to be $\cosh(\log x)-1$, the self-similar fixed point $\varphi$ (T6), the eight-tick octave (T7), and $D=3$ (T8) follow. An ODE-based uniqueness certificate gives an independent analytic path to the same $J$, useful when reviewers prefer differential rather than purely functional-equation reasoning. The module has no downstream dependents listed yet; it is an audit endpoint for the verification domain rather than a lemma consumed by later physics derivations. It closes the gap between the algebraic RCL identity and the classical characterization of cosh as the unique even convex solution of the associated ODE.
scope and limits
- Does not re-derive the Recognition Composition Law; assumes it from FunctionalEquation.
- Does not prove T6–T8; only packages the T5 cosh uniqueness certificate.
- Does not claim uniqueness without the stated regularity and normalization hypotheses.
- Does not supply numerical or experimental bounds on physical constants.