IndisputableMonolith.Verification.ConvexityCert
Verification certificate module for the strict convexity of the Recognition cost. It packages the Cost.Convexity results that Jlog(t)=cosh t-1 is strictly convex on the reals and Jcost(x)=½(x+x⁻¹)-1 is strictly convex on the positive reals. Anyone citing T5 J-uniqueness or the Recognition Composition Law would pull this cert. The module is a thin verification shell over the imported convexity proofs.
claimCertificate that $J_{\log}(t)=\cosh t-1$ is strictly convex on $\mathbb{R}$ and $J(x)=\frac{1}{2}(x+x^{-1})-1$ 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. Two equivalent presentations appear throughout the stack: the multiplicative form $J(x)=\frac12(x+x^{-1})-1$ on positive reals, and the additive form $J_{\log}(t)=\cosh t-1$ after the change of variables $x=e^t$.
Strict convexity of both presentations is the analytic engine behind uniqueness (forcing-chain step T5). The imported module Cost.Convexity already proves both statements; this Verification module exists to expose those facts as a named certificate consumable by downstream checkers and audit pages.
Local setting is the Verification domain: thin, import-only shells that re-export foundational Cost lemmas without adding new mathematics.
proof idea
No independent proof work lives here. The module imports Mathlib and IndisputableMonolith.Cost.Convexity, then surfaces the already-proved strict-convexity theorems for $J_{\log}$ and $Jcost$ as a verification certificate. Argument structure is therefore a one-hop re-export of the upstream convexity module.
why it matters in Recognition Science
T5 (J-uniqueness) in the UnifiedForcingChain rests on strict convexity of $J$: once convexity and the composition identity are in hand, the only admissible cost is $J(x)=\frac12(x+x^{-1})-1$ (equivalently $\cosh(\log x)-1$). This certificate makes that convexity hypothesis auditably available to verification consumers without forcing them to open the Cost layer.
Downstream used_by edges are empty at present; the module is a leaf cert intended for external checkers and for any future uniqueness or minimality arguments that need a named convexity token. It closes the verification side of the Cost.Convexity foundation rather than advancing a new forcing step.
scope and limits
- Does not reprove convexity; only certifies the Cost.Convexity results.
- Does not establish J-uniqueness (T5) or the Recognition Composition Law.
- Does not address discrete or lattice convexity, only the real/positive-real continuum statements.
- Does not derive numerical constants, mass formulae, or dimensional forcing.