IndisputableMonolith.Verification.CostUniquenessCert
Verification certificate module packaging the T5 cost-uniqueness theorem for the Recognition audit surface. It re-exports that any admissible cost on the positive reals equals the canonical J-cost. Physicists citing the forcing chain use it to confirm the uniqueness claim is wired into the verified stack. The module is a thin import-and-certify shell over CostUniqueness; no new mathematics is proved here.
claimCertificate that any cost $F:\mathbb{R}_{+}\to\mathbb{R}$ obeying symmetry $F(x)=F(x^{-1})$, unit normalization $F(1)=0$, strict convexity, and calibration equals the canonical cost $J(x)=(x+x^{-1})/2-1$ on $\mathbb{R}_{+}$.
background
Recognition Science forces the cost functional used throughout the framework by a short list of analytic axioms. The upstream module CostUniqueness states the main T5 result: any $F$ on the positive reals that is symmetric under $x\mapsto x^{-1}$, normalized at the unit, strictly convex, and calibrated must coincide with $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$).
That uniqueness sits inside the forcing chain (T0–T8): once $J$ is fixed, the self-similar fixed point $\varphi$, the eight-tick octave, and $D=3$ follow. The present module lives in the Verification domain and exists only to surface that uniqueness theorem as an auditable certificate, importing Mathlib and CostUniqueness and exposing a named cert object for the mirror pages.
proof idea
No independent proof lives in this module. It is a verification shell: import CostUniqueness, bind the consolidated T5 uniqueness theorem (symmetry, unit normalization, strict convexity, calibration imply $F=J$ on $\mathbb{R}_{+}$) to a certificate declaration, and stop. All analytic work (convexity, calibration identities, functional equation) remains upstream.
why it matters in Recognition Science
T5 J-uniqueness is a landmark of the forcing chain: without a unique cost, the Recognition Composition Law and the derived constants ($c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^{5}/\pi$, the $\alpha^{-1}$ band) lose their rigid anchor. This module does not feed further theorems by edge count; its role is audit packaging so the verified stack can point at a single cert that $J$ is forced. Downstream consumers of the forcing chain and of any mass or coupling derivation that quotes $J$ rely on that uniqueness being present and sorry-free in the certificate layer.
scope and limits
- Does not reprove T5; only certifies the upstream CostUniqueness theorem.
- Does not weaken or alter the axiom list (symmetry, unit norm, strict convexity, calibration).
- Does not address discrete or non-positive domains beyond $\mathbb{R}_{+}$.
- Does not derive $\varphi$, eight-tick structure, or $D=3$; those are later chain steps.
- Does not supply numerical bounds on $\alpha$ or mass-ladder claims.