IndisputableMonolith.Verification.ReciprocalSymmetryEvenCert
Verification certificate that the Recognition cost is even under reciprocal inversion: the unique T5 cost satisfies J(x)=J(x^{-1}) for all positive x. Auditors of the T5 uniqueness chain cite this module when checking that reciprocal symmetry is packaged as a stand-alone cert rather than left implicit in the functional equation. The module imports the FunctionalEquation helpers and exposes a single certificate object.
claimCertificate that the T5 cost $J$ is reciprocal-even: $J(x)=J(x^{-1})$ for all $x>0$, equivalently $J(e^{t})=J(e^{-t})$ (evenness in the logarithmic coordinate).
background
Recognition Science forces a unique nonnegative cost $J$ on the positive reals by the Recognition Composition Law and regularity (T5). The closed form is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. A basic symmetry of this $J$ is reciprocal evenness: replacing $x$ by $x^{-1}$ leaves $J$ unchanged, so the cost depends only on the unordered pair ${x,x^{-1}}$.
The upstream module Cost.FunctionalEquation supplies the algebraic lemmas used in the T5 uniqueness argument. The present verification module sits in the Verification domain and packages the reciprocal-even property as an explicit certificate object rather than leaving it as an inline lemma inside the forcing chain.
proof idea
Definition and certificate module, not a multi-step derivation. It imports Mathlib and the FunctionalEquation helpers, then exposes the certificate ReciprocalSymmetryEvenCert asserting evenness of $J$ under $x\mapsto x^{-1}$. Any nontrivial content is a thin wrapper around the closed form or the functional-equation identities already proved upstream for T5.
why it matters in Recognition Science
T5 uniqueness is a landmark of the forcing chain: once $J$ is fixed, $\phi$, the eight-tick octave, and $D=3$ follow (T6–T8). Reciprocal evenness is part of the structural fingerprint of that $J$ (and of any cost obeying the RCL in log-symmetric form). Packaging it as a Verification certificate lets downstream audits and export surfaces cite a named object instead of re-deriving $J(x)=J(x^{-1})$ ad hoc. No downstream consumers are wired in the current graph; the module is a leaf cert for the T5 cost package.
scope and limits
- Does not re-prove T5 uniqueness of J; only certifies reciprocal evenness.
- Does not derive the closed form of J from the RCL inside this module.
- Does not address odd parts, higher multipoles, or costs outside the T5 class.
- Does not feed a named parent theorem in the current dependency graph (used_by empty).