scripts.cost_unit_axiom_audit
Audit script for the cost-unit axiom stack that underwrites T5 J-uniqueness. It wires FunctionalEquation, GaugeOrbitClassification, UnitFromMinimality, and MonotoneMultiplicativePower so a referee can check that the unit of cost is forced by minimality rather than postulated. Cite it when verifying that the Recognition Composition Law plus gauge orbits pin a unique cost gauge. Structure is import-and-check scaffolding, not a standalone theorem.
claimModule assembling the cost-unit axiom audit: uniqueness of the cost functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) under the Recognition Composition Law, gauge-orbit classification, unit-from-minimality, and monotone multiplicative powers.
background
Recognition Science forces the cost functional through the T5 step of the unified forcing chain: $J$ is the unique symmetric cost compatible with the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$. In RS-native units the same $J$ sets the dimensionless yardstick against which masses and couplings are read on the $\varphi$-ladder.
The imported Cost modules supply the pieces of that uniqueness argument. FunctionalEquation holds the algebraic identities used in the T5 proof. GaugeOrbitClassification organizes residual gauge freedom on cost orbits. UnitFromMinimality derives the cost unit from a minimality principle rather than an external normalization. MonotoneMultiplicativePower controls how costs scale under multiplicative powers, closing monotonicity gaps in the uniqueness argument.
This script lives under scripts/ and is an audit surface: it re-exports and exercises those four modules so the axiom footprint of the cost unit can be inspected in one place.
proof idea
No standalone proof body. The module is an import-and-audit harness: it pulls FunctionalEquation (T5 helper lemmas), GaugeOrbitClassification, UnitFromMinimality, and MonotoneMultiplicativePower, then exposes a thin entry point (enters) so CI or a human auditor can confirm the cost-unit axiom stack typechecks and has the expected dependency cone. Argument structure is compositional: uniqueness of $J$ is inherited from the imported Cost lemmas, not re-proved here.
why it matters in Recognition Science
T5 J-uniqueness is a landmark of the forcing chain: once $J(x)=(x+x^{-1})/2-1$ is forced, $\varphi$ (T6), the eight-tick octave (T7), and $D=3$ (T8) sit on a fixed cost gauge. An explicit audit of the unit-from-minimality and gauge-orbit hypotheses makes the non-postulated character of that gauge checkable.
Downstream consumers are the Cost and Foundation developments that quote T5; this module itself has no used_by edges because it is a scripts-level auditor rather than a library theorem. Its value is refereeing: it packages the axiom surface a skeptic must accept before granting cost uniqueness, and it flags any sorry or hypothesis interface that would weaken the T5 claim.
scope and limits
- Does not prove T5 J-uniqueness; only audits imported Cost modules.
- Does not derive numerical constants ($c$, $\hbar$, $G$, $\alpha$).
- Does not discharge hypothesis interfaces inside UnitFromMinimality or GaugeOrbitClassification.
- Does not address mass-ladder rungs or Berry-threshold phenomenology.
- Does not export reusable theorems; it is a scripts-level check surface.