IndisputableMonolith.Verification.T5UniqueCert
Verification package for the T5 step of the forcing chain: uniqueness of the recognition cost J on the positive reals. A working physicist checking the RS derivation would cite it when auditing that J(x)=(x+x^{-1})/2-1 is the only admissible cost. The module packages certificate-style statements imported from the Cost layer rather than re-proving the functional equation from scratch.
claimCertificate module asserting uniqueness of the recognition cost $J:(0,\infty)\to\mathbb{R}$ with $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), as required by step T5 of the unified forcing chain.
background
Recognition Science derives physics from a single functional equation on a cost $J$. The forcing chain (T0–T8) isolates successive constraints; T5 is J-uniqueness: the only admissible cost is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$.
The Recognition Composition Law (RCL) $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ is the algebraic skeleton. The Cost import supplies the concrete $J$ and related identities used by verification certificates.
This module sits in the Verification domain: it does not redefine $J$, but exposes a named certificate object that downstream audits can point at when claiming T5 is discharged.
proof idea
Definition and certificate module, not a standalone proof development. It imports Mathlib and the Cost layer, then exposes a certificate-shaped object (and a thin predicate) that records T5 uniqueness for the verification graph. Argumentative content lives upstream in Cost and the unified forcing chain; here the structure is packaging and naming for audit.
why it matters in Recognition Science
T5 is a landmark in the forcing chain: once $J$ is unique, T6 can force $\phi$ as the self-similar fixed point, then T7 the eight-tick octave and T8 spatial dimension $D=3$. A machine-checkable uniqueness certificate is what makes the later chain steps non-optional rather than ansatz.
No downstream used_by edges are recorded on this page, so the module is an audit endpoint: it exists so verification reports can cite a single T5 uniqueness certificate rather than scattering Cost lemmas. It closes the verification gap between the analytic form of $J$ and the claim that no other cost satisfies the RS axioms.
scope and limits
- Does not re-prove J-uniqueness from the RCL; defers to Cost and the forcing chain.
- Does not derive T6–T8 (phi, eight-tick, D=3) inside this module.
- Does not fix numerical constants (c, hbar, G, alpha) or mass-ladder rungs.
- Does not supply used_by consumers; it is a verification endpoint, not a lemma library.