IndisputableMonolith.Verification.JcostCoshFormCert
Verification module packaging a certificate that the Recognition cost equals its hyperbolic-cosine closed form. Auditors of T5 J-uniqueness cite it to confirm J(x) = cosh(log x) - 1 matches the algebraic expression (x + x^{-1})/2 - 1. The module is thin: it imports the functional-equation helpers and exposes a named cert object rather than a long proof development.
claimCertificate that the Recognition cost satisfies $J(x) = \cosh(\log x) - 1$, equivalently $J(x) = \frac{x + x^{-1}}{2} - 1$, on the positive reals as forced by the T5 uniqueness argument.
background
Recognition Science forces a unique cost functional $J$ on $\mathbb{R}_{>0}$ via the Recognition Composition Law and regularity. Landmark T5 states that uniqueness yields the closed form $J(x) = (x + x^{-1})/2 - 1$, which is identical to $\cosh(\log x) - 1$.
The upstream module Cost.FunctionalEquation supplies the lemmas used in that T5 uniqueness proof. This verification layer does not re-derive the functional equation; it packages the cosh identification as an auditable certificate object for the mirror.
Local setting is the Verification domain: small, import-light modules that pin named identities so downstream forcing-chain and constants work can cite a single cert rather than inline algebra.
proof idea
Definition and certificate module, not a multi-step tactic development. It imports Mathlib and the T5 functional-equation helpers, then exposes the sibling certificate JcostCoshFormCert asserting the algebraic/cosh agreement for $J$. Any equality work is delegated to those upstream lemmas; the module itself is a thin packaging layer.
why it matters in Recognition Science
T5 J-uniqueness is a landmark in the forcing chain (T0–T8): once $J$ is fixed as $\cosh(\log x)-1$, later steps can treat the cost as canonical. This module sits in Verification so that the cosh form is a named, checkable cert rather than a scattered rewrite. No downstream edges are recorded in the graph yet; the intended consumers are T5-facing uniqueness reports and any constant or mass-ladder arguments that quote the closed form of $J$. It does not itself prove dimension, eight-tick structure, or $\varphi$-fixed-point results (T6–T8).
scope and limits
- Does not prove T5 uniqueness from the composition law; only certifies the cosh closed form.
- Does not derive the Recognition Composition Law or force $\varphi$, eight-tick period, or $D=3$.
- Does not supply numerical bounds on $\alpha$ or mass-ladder rungs.
- Does not record downstream use sites in the current dependency graph.