IndisputableMonolith.Verification.JcostStrictPosCert
Verification module certifying that the Recognition Science cost functional J is strictly positive away from the identity. Anyone citing T5 J-uniqueness or the Recognition Composition Law needs this positivity fact as a standing inequality. The module packages a machine-checkable certificate over the Cost layer rather than re-deriving J from scratch.
claimFor the RS cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), the module certifies $J(x)>0$ whenever $x>0$ and $x\neq 1$, with $J(1)=0$.
background
Recognition Science fixes a unique cost functional $J$ on the positive reals by the forcing chain (T5): $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. This $J$ is the unique continuous solution (up to scale) compatible with the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$.
Strict positivity of $J$ off the fixed point $x=1$ is the elementary analytic fact that makes $J$ a genuine cost: zero only at perfect match, positive under mismatch. The Cost import supplies the definition and basic algebraic identities; this Verification module isolates the positivity certificate so downstream proofs can cite a single named fact rather than reopen the calculus of $J$.
The local setting is pure verification: no new physics constants, no ladder or octave structure, only the inequality that underwrites defect and distance constructions built from $J$.
proof idea
Definition-plus-certificate module over IndisputableMonolith.Cost. The argument reduces strict positivity of $J$ to the elementary inequality $(x-1)^2/(2x)>0$ for $x>0$, $x\neq 1$, or equivalently to $\cosh u>1$ for $u=\log x\neq 0$. No deep lemmas beyond the Cost definition of $J$ and standard Mathlib facts on squares, division, and hyperbolic cosine. Structure is a named certificate object (JcostStrictPosCert) rather than a long tactic script.
why it matters in Recognition Science
Strict positivity of $J$ is a standing hypothesis for any defect, distance, or variational argument in the monolith: without $J(x)>0$ off identity, cost-minimization and uniqueness claims collapse. The module sits in the Verification domain so that T5 J-uniqueness and RCL-based identities can quote a checked certificate instead of an inline inequality. No downstream used_by edges are recorded yet; the intended consumers are defect-distance lemmas and any proof that treats $J$ as a strict Lyapunov or mismatch measure. It does not itself advance T6--T8 (phi, eight-tick, $D=3$), but those steps inherit a well-behaved cost only if this positivity fact is locked down.
scope and limits
- Does not re-prove uniqueness of J (T5); assumes the Cost definition.
- Does not address discrete ladder, mass formula, or phi-rung structure.
- Does not certify positivity of derived defects beyond raw J.
- Does not fix units or couple to c, hbar, G, or alpha.
- Does not supply quantitative lower bounds other than strict inequality.