IndisputableMonolith.Verification.JcostMinimumCert
Verification module certifying that the Recognition J-cost attains a unique global minimum of zero at the identity scale. Auditors of the T5 J-uniqueness step and of nonnegativity lemmas cite it. The argument is a short analytic check on the closed form J(x)=(x+x^{-1})/2-1 together with elementary calculus or AM-GM.
claimThe cost $J(x)=(x+x^{-1})/2-1$ on $x>0$ satisfies $J(x)\ge 0$ with equality if and only if $x=1$. Equivalently $J(x)=\cosh(\log x)-1$, so the unique minimizer is the identity scale.
background
Recognition Science forces a unique cost functional $J$ on positive reals via the Recognition Composition Law and regularity. The closed form fixed by T5 is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. This $J$ is the yardstick for defect, mass rungs, and the forcing chain.
The ambient Cost module supplies the definition of $J$ and basic algebraic identities. The present module sits in Verification: it packages a machine-checkable certificate that $J$ is nonnegative and vanishes only at $x=1$, the identity recognition scale.
That minimum is the reference point for defect distance and for every later comparison of scales on the $\varphi$-ladder.
proof idea
Definition-plus-certificate module, not a long development. It imports the Cost definition of $J$, then records a short proof that $J(x)\ge 0$ with equality precisely at $x=1$, typically by rewriting $J(x)=(\sqrt{x}-1/\sqrt{x})^2/2$ or by calculus on $\cosh$. Downstream consumers treat the certificate as a black-box nonnegativity fact.
why it matters in Recognition Science
T5 J-uniqueness is only useful if the forced $J$ is a genuine cost: nonnegative, zero only at identity. This module locks that analytic fact into the Verification layer so later theorems (defect comparisons, mass formulas, forcing-chain steps that quote $J\ge 0$) need not re-prove the minimum. No downstream edges are recorded yet in the graph; the intended consumers are Cost-based inequality lemmas and any certificate that quotes the unique zero of $J$.
scope and limits
- Does not re-derive T5 uniqueness of J from the composition law.
- Does not treat discrete or lattice versions of the cost.
- Does not prove higher-order convexity or Hessian bounds beyond the global minimum.
- Does not connect the minimum to phi-ladder mass formulas or alpha bounds.