PRCPrimeCalibrationForcesCharacterCrossEqRespectTarget_proved
plain-language theorem explainer
Prime direction calibration on a ratio-orbit character forces that character to respect ratio cross-equivalence, so it descends to a quotient-native map. Anyone proving product-display compatibility or native cost uniqueness cites this. The proof is a one-line discharge: reduced sign-canonical ratio uniqueness implies the target via an intermediate reduction lemma.
Claim. Every map $\chi$ from ratio orbits to ratio orbits that is a PRC ratio character and is prime-direction calibrated necessarily respects ratio cross-equivalence: equivalent ratio presentations are sent to equivalent images, so $\chi$ is well-defined on the quotient.
background
In the Primitive Recognition Calculus, costs and characters live on ratio orbits rather than raw positive reals. A PRC ratio character is a structure-preserving map $\chi$ on those orbits. Prime-direction calibration pins how $\chi$ treats the preferred orientation of prime factors (identity versus reciprocal), tying the character to the native calibration anchor used throughout RS measurement.
Cross-equivalence identifies ratio presentations that differ only by the ledger's reciprocal automorphism (swap source/target and invert the ratio) or by other presentation noise that must not affect native cost. Until $\chi$ respects that relation, it is only a raw orbit map, not a quotient-native character.
The module's uniqueness program isolates this as a sharper intermediate target: prime calibration alone should force cross-equivalence respect, which then feeds product-display compatibility (no mixed identity/reciprocal factor orientations under native multiplication).
proof idea
One-line term wrapper. Apply the reduction lemma that any proof of reduced sign-canonical ratio uniqueness yields the cross-equivalence respect target, supplying the already-proved uniqueness theorem PRCReducedSignCanonicalRatioUniqueTarget_proved.
That reduction itself routes through a normalize-ratio canonical intermediate: uniqueness of the reduced sign-canonical representative implies a canonical normalization of ratios, which forces calibrated characters to identify cross-equivalent inputs.
why it matters
This closes the cross-equivalence respect target inside native cost uniqueness. Downstream, PRCPrimeCalibrationForcesOrbitProductDisplayCompatibilityTarget_proved applies it directly: product-display compatibility is obtained from cross-equivalence respect, ruling out mixed identity/reciprocal orientations under native multiplication.
It also sits on the path into the native-cost uniqueness blocker certificate, which packages zero-calibrated factorization targets for the broader uniqueness audit. In framework terms, the step keeps characters aligned with the reciprocal automorphism and the identity (zero-cost) event that anchor J-cost geometry, so later uniqueness of the native cost (the T5 J-shape lineage) is not blocked by presentation artifacts.
No open scaffold remains on this node: the claim is fully proved.
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