IndisputableMonolith.Verification.JcostStrictCert
Verification module packaging a strictness certificate for the J-cost functional used throughout Recognition Science. It records that J vanishes only at the identity and is positive elsewhere, tying the algebraic form forced by T5 to a machine-checkable Prop. Downstream uniqueness and forcing arguments cite this certificate rather than re-proving positivity. The module is thin: it imports Cost and exposes one named certificate object.
claimA verification certificate asserting that the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) satisfies $J(x)\ge 0$ for all $x>0$, with equality if and only if $x=1$.
background
Recognition Science derives physics from a single functional equation whose cost kernel is the J-function. Under the forcing chain, T5 fixes $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. The Recognition Composition Law then reads $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$.
The Cost module supplies the raw definition and basic identities. Strictness (non-negativity with unique zero at the multiplicative identity) is the analytic property that turns those identities into uniqueness and comparison lemmas: any competing cost that matches J on a generating set must coincide everywhere once the zero set is a singleton.
This Verification submodule isolates that strictness claim as a named certificate so later forcing and mass-ladder arguments can depend on a single opaque Prop rather than unfolding the closed form repeatedly.
proof idea
Definition-and-certificate module, not a deep proof development. It imports Mathlib and IndisputableMonolith.Cost, then exposes a single certificate object (JcostStrictCert) whose body is the strict-positivity statement for J. Any underlying inequality is discharged by elementary calculus or AM-GM on the closed form already fixed in Cost; the module's job is packaging, not re-derivation.
why it matters in Recognition Science
Strictness of J is the hinge between the algebraic RCL and the T5 uniqueness step in the UnifiedForcingChain: without a unique zero, competing costs could match J on orbits yet differ off them. Packaging the fact here keeps Verification-layer dependents (calibration, defect comparisons, ladder gaps) from re-proving $J(x)=0\Leftrightarrow x=1$. No downstream edges are recorded in the current graph, so the certificate is a leaf ready for import by uniqueness and comparison theorems rather than an already-wired parent.
scope and limits
- Does not re-derive the closed form of J; that lives in Cost and T5.
- Does not prove the Recognition Composition Law or any forcing step T0–T8.
- Does not address discrete eight-tick or dimension-three claims.
- Does not supply numerical bounds on alpha, masses, or Berry thresholds.
- Does not claim convexity beyond non-negativity with unique zero.