PRCPrimeCalibrationForcesNonunitOrbitOrientationCoherentTarget_refuted
plain-language theorem explainer
Prime calibration of a ratio character does not force a single coherent orientation on every nonunit orbit direction. Anyone tracking the PRC native-cost uniqueness blockers will cite this negative result. The proof is a short reduction: the coherent-orientation target is equivalent to identity-witness globalization, which is already refuted.
Claim. It is false that every ratio character $\chi$ that is prime-direction calibrated is automatically nonunit-orbit orientation coherent: there is no theorem asserting $\forall\chi,\,(\text{ratio character})\land(\text{prime calibrated})\Rightarrow(\text{single coherent orientation on all nonunit orbits})$.
background
In the Primitive Recognition Calculus, ratio characters are maps on ratio orbits that encode multiplicative cost structure. Prime-direction calibration is the demand that the character align with a preferred prime direction on the orbit lattice. Nonunit-orbit orientation coherence is the stronger global demand that every nonunit orbit direction share one common orientation (identity versus reciprocal), ruling out mixed product factors via nonunit non-self-reciprocity.
The target proposition packages the claim that prime calibration alone forces that global coherence. The module treats it as a candidate uniqueness lever for native cost, then systematically refutes over-strong forms. Upstream, the same target is proved equivalent to "identity-witness globalization," and that globalization target is already refuted by reduction to a failed identity-branch transport claim.
proof idea
Assume the coherent-orientation target. Apply the forward direction of the equivalence with identity-witness globalization to obtain that weaker-looking but equivalent target. Discharge the goal by the already-proved refutation of identity-witness globalization. The argument is a pure logical reduction along a recorded iff; no new analytic estimates appear.
why it matters
This refutation sits in the native-cost uniqueness blocker stack. Downstream it is the direct engine for refuting the sharpened coherent-orientation target, the local no-mixed and local-product no-mixed variants, the product-local sharpened target, and the prime-floor successor transport local-adjacent target. It also feeds prc_native_cost_uniqueness_blocker_certificate and the conditional universal-foundation certificate.
In framework terms it prunes false uniqueness routes toward the J-cost forced by T5 and the Recognition Composition Law: prime calibration is not by itself enough to lock a single nonunit orientation. The surviving path must use weaker, correctly scoped hypotheses rather than this global coherence demand.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.