IndisputableMonolith.Verification.UnitNormalizationZeroCert
Verification module packaging a zero-certificate for unit-normalized cost: after fixing the multiplicative identity as the reference scale, the Recognition cost defect vanishes. Auditors of the T5 uniqueness chain cite it when checking that normalization does not inject a residual. The module is thin: it imports the functional-equation helpers and exposes a single certificate object.
claimUnder unit normalization (reference scale fixed at the multiplicative identity $1$), the Recognition cost defect is identically zero: the associated zero-certificate asserts $\mathrm{defect}=0$ at the normalized unit, consistent with $J(1)=0$ for the unique cost $J(x)=\frac{x+x^{-1}}{2}-1$.
background
Recognition Science forces a unique nonnegative cost $J$ on ratios via the Recognition Composition Law and regularity (forcing step T5). The closed form is $J(x)=\cosh(\log x)-1$, equivalently $(x+x^{-1})/2-1$, which automatically satisfies $J(1)=0$ and $J(x)=J(x^{-1})$.
Unit normalization means choosing the multiplicative identity as the reference scale so that residual cost is measured relative to that fixed point. The upstream module Cost.FunctionalEquation supplies the algebraic lemmas used in the T5 uniqueness argument; this verification layer only records that, once units are so fixed, the defect certificate is the zero certificate.
The local setting is the Verification domain: machine-checkable certificates that named normalization or gauge choices do not leave a nonzero remainder in the cost bookkeeping.
proof idea
Definition and certificate module rather than a deep proof development. It imports Mathlib and the T5 functional-equation helpers, then exposes the unit-normalization zero-certificate as a named verification object. Any nontrivial equalities are discharged by the already-proved identities $J(1)=0$ and the RCL algebra from the upstream cost module; no independent analytic argument lives here.
why it matters in Recognition Science
Closes a bookkeeping gap in the verification stack for T5: uniqueness of $J$ is only physically meaningful after units are fixed, and the zero-certificate records that normalization introduces no extra defect. Downstream consumers (none linked in the current graph) would be end-to-end verification reports or gauge-invariance checks that quote a normalized cost baseline. In the forcing chain it sits under T5 J-uniqueness and the RCL, not under the later geometric steps T7–T8. With empty used-by edges it is presently a leaf certificate rather than a load-bearing lemma.
scope and limits
- Does not reprove T5 uniqueness of J; only certifies the normalized zero defect.
- Does not fix dimension, eight-tick structure, or any constant beyond the cost scale.
- Does not address non-unit gauges or residual defects away from the identity.
- Does not supply numerical bounds on alpha, masses, or other RS observables.