PRCPrimeCalibrationForcesCharacterCrossEqRespectTarget_of_reduced_signCanonical_unique
plain-language theorem explainer
Under reduced sign-canonical uniqueness of ratio orbits, every prime-direction-calibrated ratio character automatically respects cross-equivalence. Native-cost uniqueness proofs cite this bridge when discharging the product-display compatibility target. The argument is a two-step term composition through the canonical-normalization intermediate.
Claim. Assume that any two reduced, sign-canonical ratio orbits that lie in the same cross-multiplication class are equal as raw orbits. Then every map $\chi$ on ratio orbits that is a ratio character and is prime-direction calibrated respects cross-equivalence: cross-equivalent inputs yield cross-equivalent (equivalently, equal after normalization) character values.
background
In the Primitive Recognition Calculus, ratio data live as raw ratio orbits. Cross-equivalence identifies displays that represent the same multiplicative class under cross-multiplication. A ratio character $\chi$ is a structure-preserving map on those orbits; prime-direction calibration constrains how $\chi$ acts along prime generators.
Reduced sign-canonical uniqueness is the remaining raw-display number-theory blocker for canonical normalization: two reduced, sign-canonical displays in the same cross class must be the same orbit. Canonical normalization is the intermediate target that turns that uniqueness into a well-defined normal-form map on orbits.
The conclusion target says prime calibration forces $\chi$ to respect cross-equivalence. Without it, $\chi$ is not yet quotient-native on the cross-class, so product-display compatibility for the native cost cannot be read off the character.
proof idea
Pure term composition, no tactics. First apply the upstream lemma that reduced sign-canonical uniqueness implies the canonical-normalization target (normal forms of cross-equivalent orbits coincide). Feed that hypothesis into the sibling lemma that canonical normalization already forces every ratio character (under prime calibration) to respect cross-equivalence. The composite is exactly the desired implication.
why it matters
This is the last glue step before the unconditional statement that prime calibration forces characters to respect cross-equivalence. The immediate parent applies the proved uniqueness instance and obtains the target as a theorem; that proved form sits in the native-cost uniqueness blocker certificate chain, which packages factorization and no-mixing obligations for the PRC native cost.
In framework terms it clears a display-level obstruction so the recognition character can descend to the cross-class quotient, a prerequisite for matching the native cost to the unique $J$-cost forced by the Recognition Composition Law and the T5 uniqueness landmark. It does not itself derive $J$ or $\phi$; it only removes a ratio-orbit ambiguity that would otherwise block uniqueness of the native cost presentation.
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