IndisputableMonolith.Verification.JlogStrictConvexCert
Certification module that packages the strict convexity of Jlog(t) = cosh t - 1 on the reals for the verification layer. Anyone citing the T5 J-uniqueness argument or the Recognition Composition Law needs this convexity fact on record. The module is a thin verification wrapper over the Cost.Convexity development rather than a fresh analytic proof.
claimThe module certifies that $J_{\log}(t) := \cosh t - 1$ is strictly convex on $\mathbb{R}$, and records the companion fact that $J(x) := \frac{1}{2}(x + x^{-1}) - 1$ is strictly convex on $\mathbb{R}_{>0}$.
background
Recognition Science forces a unique cost functional $J$ via the Recognition Composition Law and a short list of regularity axioms. In logarithmic coordinates the cost becomes $J_{\log}(t) = \cosh t - 1$, which is the form used in the T5 uniqueness step of the forcing chain.
The upstream module Cost.Convexity already proves both statements: $J_{\log}$ is strictly convex on $\mathbb{R}$, and the multiplicative form $J_{\mathrm{cost}}(x) = \frac12(x+x^{-1})-1$ is strictly convex on the positive reals. Those proofs are foundational for T5.
This verification module sits one layer above that development. It imports the convexity results and exposes them under a certificate-style name so downstream verification and audit tooling can point at a single, stable entry point.
proof idea
No independent analytic argument lives here. The module imports IndisputableMonolith.Cost.Convexity and re-exports or packages the already-proved strict convexity of $J_{\log}(t)=\cosh t-1$ (and the companion $J_{\mathrm{cost}}$ statement) as a verification certificate. Structure is therefore a thin wrapper over the upstream convexity theorems rather than a tactic-mode derivation.
why it matters in Recognition Science
Strict convexity of $J_{\log}$ is the analytic engine behind T5 J-uniqueness in the UnifiedForcingChain: once $J$ is forced to be (a multiple of) $\cosh(\log x)-1$, the rest of the forcing chain can pin $\varphi$, the eight-tick octave, and $D=3$. Packaging the fact in Verification keeps the audit surface clean: verifiers cite this certificate instead of reaching into Cost internals.
The module has no downstream edges recorded yet; its role is infrastructural, feeding any uniqueness, minimality, or defect-distance argument that needs $J_{\log}''>0$ on the nose. It closes no open sorry; it freezes a proved landmark for the verification graph.
scope and limits
- Does not reprove strict convexity; defers entirely to Cost.Convexity.
- Does not establish uniqueness of J; only supplies the convexity ingredient for T5.
- Does not address discrete or lattice restrictions of Jlog.
- Does not derive numerical constants, phi, or dimensional claims.