IndisputableMonolith.Verification.T5UniqueCert
Verification package for the T5 step of the forcing chain: uniqueness of the recognition cost $J$ on the positive reals. Anyone citing J-uniqueness or the Recognition Composition Law will land here for the certified statement. The module packages the uniqueness claim against the Cost API rather than re-deriving the functional equation from scratch.
claimA verification certificate that the recognition cost $J:(0,\infty)\to\mathbb{R}$ is uniquely fixed (up to the normalization of the forcing chain) by the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$, with the closed form $J(x)=(x+x^{-1})/2-1=\cosh(\log x)-1$.
background
Recognition Science forces physics from a single functional equation on a cost $J$. The forcing chain (T0–T8) isolates T5 as J-uniqueness: once the composition law and mild regularity are imposed, $J$ must be $J(x)=(x+x^{-1})/2-1$, equivalently $\cosh(\log x)-1$.
The Cost import supplies the native definitions of $J$, the composition identity (RCL), and the comparison lemmas used throughout the monolith. This module sits in the Verification domain: it does not redefine $J$, it certifies that the uniqueness theorem expected by T5 is the one exported by Cost.
Downstream consumers treat T5 as settled input to T6 (phi as self-similar fixed point), T7 (eight-tick octave), and T8 ($D=3$).
proof idea
Module-level packaging, not a single tactic proof. It imports Mathlib and the Cost layer, then exposes a named certificate object (T5 uniqueness) whose fields point at the Cost uniqueness theorem and the closed form of $J$. Argument structure is: assume RCL plus the Cost regularity package, invoke the existing uniqueness lemma from Cost, and record the identity $J(x)=\cosh(\log x)-1$ as the certified normal form. No independent derivation of the functional equation lives here.
why it matters in Recognition Science
T5 is the hinge of the forcing chain: without unique $J$, the self-similar fixed point phi (T6), the eight-tick period $2^3$ (T7), and $D=3$ (T8) are not forced. This module makes that hinge auditable as a verification artifact rather than a scattered lemma in Cost.
Used_by is empty at the graph snapshot, so the certificate is presently a leaf in the dependency display; its role is still to be the stable cite target for any later proof that assumes J-uniqueness (mass ladder, alpha band, Berry threshold arguments). It closes the verification gap between the abstract RCL and the concrete $J$ used in RS-native units ($c=1$, $\hbar=\phi^{-5}$, $G=\phi^5/\pi$).
scope and limits
- Does not re-prove J-uniqueness from raw axioms; certifies the Cost-layer result.
- Does not derive phi, the eight-tick octave, or D=3 (those are T6–T8).
- Does not fix physical units or the alpha band; only the shape of J.
- Does not address discrete ladder masses or Berry thresholds.
- Does not export a computational checker beyond the Lean certificate object.