scripts.cost_unit_axiom_audit
Audit script for the cost-unit axiom package that underwrites T5 J-uniqueness. It pulls the functional-equation helpers, gauge-orbit classification, monotone multiplicative powers, and unit-from-minimality lemmas into one place so a referee can check that the unit of the recognition cost is forced rather than assumed. Anyone citing the T5 uniqueness chain or the Recognition Composition Law normalization would land here. Structure is import-and-assemble: no new theorems, only a consolidated axiom surface.
claimCollect and expose the axiom surface that forces the recognition cost unit: the unique (up to gauge) cost $J$ satisfying the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$, with $J(x)=(x+x^{-1})/2-1$, unit fixed by minimality and monotone multiplicative scaling.
background
Recognition Science derives the cost functional from a single composition law (RCL). The T5 step of the forcing chain asserts that $J(x)=\cosh(\log x)-1$ is the unique solution (up to the gauge orbit) once regularity and normalization are fixed. The imported FunctionalEquation module supplies the algebraic lemmas for that uniqueness argument.
Unit choice is not free: UnitFromMinimality pins the additive zero and scale by a minimality condition, while MonotoneMultiplicativePower controls how costs behave under $x\mapsto x^p$. GaugeOrbitClassification quotients out residual reparameterizations so that "the" unit is a true invariant rather than a gauge artifact.
This script sits outside the monolith proper. It is an audit harness: it re-imports those four Cost modules so the axiom and lemma surface that justifies the cost unit can be inspected in one compilation unit.
proof idea
Definition and import module, not a proof module. It wires four Cost dependencies (functional-equation helpers for T5, gauge-orbit classification, monotone multiplicative powers, unit-from-minimality) and exposes a thin audit entrypoint (enters). No local theorems are proved; the argument is the upstream T5 uniqueness and unit-forcing chain, re-exported for axiom review.
why it matters in Recognition Science
T5 J-uniqueness is a landmark of the forcing chain (T0–T8): once $J$ and its unit are forced, $\phi$, the eight-tick octave, and $D=3$ can be derived downstream. An explicit cost-unit axiom audit matters because every later constant (including $\hbar=\phi^{-5}$ and the $\alpha^{-1}$ band) inherits its normalization from that unit.
No downstream monolith theorems list this script as a dependency (used_by is empty); it is tooling for maintainers and referees, not a link in the proved chain. Its value is hygiene: confirming that the unit is discharged by minimality and gauge classification rather than smuggled in as an extra axiom.
scope and limits
- Does not prove T5 J-uniqueness; only re-imports the Cost lemmas that do.
- Does not derive $\phi$, eight-tick structure, or $D=3$.
- Does not fix physical units ($c$, $\hbar$, $G$); only the dimensionless cost unit.
- Does not add new axioms; it audits the existing unit-from-minimality surface.
- Does not appear in the proved downstream forcing chain (zero used_by edges).