IndisputableMonolith.Verification.UnitNormalizationZeroCert
Verification module that packages a zero-certificate for unit normalization of the RS cost: at the multiplicative identity the cost vanishes. Auditors of the T5 uniqueness chain cite it when checking that the forced J is correctly pinned at the unit. The module is thin scaffolding over the functional-equation helpers; the certificate itself is a short algebraic identity.
claimThe module records a zero-certificate for unit normalization of the cost: $J(1)=0$ (equivalently the defect vanishes at the multiplicative unit), together with the supporting lemmas needed to discharge that identity inside the T5 uniqueness argument.
background
Recognition Science forces a unique nonnegative cost $J$ on the positive reals from the Recognition Composition Law and mild regularity. The closed form is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. Any such cost is required to vanish at the multiplicative identity: $J(1)=0$. That normalization is part of the T5 uniqueness package.
The upstream module Cost.FunctionalEquation supplies the algebraic lemmas used in the T5 proof (the d'Alembert-type identity and its corollaries). The present verification module sits one layer above those helpers and isolates the unit-normalization zero statement as a named certificate that downstream uniqueness and calibration arguments can import without reopening the functional equation.
proof idea
Definition-and-certificate module rather than a long development. It imports the functional-equation helpers, states the unit-normalization claim $J(1)=0$ (or an equivalent defect form), and discharges it by direct substitution into the closed form or by a one-line appeal to the already-proved algebraic identities. No independent analytic argument is introduced here.
why it matters in Recognition Science
T5 (J-uniqueness) is a landmark of the forcing chain: once $J$ is forced, $\phi$, the eight-tick octave, and $D=3$ follow. Unit normalization $J(1)=0$ is the base point of that uniqueness; without it the cost is only determined up to an additive constant and the later ladder and mass formulae lose their absolute scale. This module isolates that base point as a reusable zero-certificate inside the Verification domain so that uniqueness and calibration proofs can cite a single named fact rather than re-deriving $J(1)=0$ ad hoc. No downstream consumers are listed yet in the graph; the intended parents are the T5 uniqueness theorems and any unit-calibration lemmas that sit above FunctionalEquation.
scope and limits
- Does not prove full T5 J-uniqueness; only the unit-normalization zero identity.
- Does not derive the closed form of J; that lives in the functional-equation and uniqueness modules.
- Does not address positivity, convexity, or growth of J away from 1.
- Does not fix physical units or convert between RS-native and SI constants.