PRCPrimeCalibrationForcesNoMixedPrimeOrientationTarget_refuted
plain-language theorem explainer
Prime-direction calibration of a ratio character does not force a coherent non-mixed identity/reciprocal orientation across prime axes; the corresponding no-mixing target is false. Native-cost uniqueness work cites this as a blocker clearance. The proof is a short transport through the witnesses-form equivalence onto an already-refuted sibling.
Claim. It is false that every ratio-orbit map $\chi$ which is a ratio character and is prime-direction calibrated necessarily has no mixed prime orientation. In other words, prime cost calibration does not forbid independent mixed identity/reciprocal choices on different prime axes.
background
In the Primitive Recognition Calculus, native cost uniqueness is organized around ratio characters $\chi:\mathrm{RatioOrbit}\to\mathrm{RatioOrbit}$ and calibration hypotheses that pin how $\chi$ acts on prime axes. Prime-direction calibration constrains the local choice of identity versus reciprocal orientation per prime. The no-mixed-orientation target asserts that any such calibrated character must already be globally coherent: it cannot mix identity on some primes with reciprocal on others.
A parallel existential formulation speaks in terms of prime witnesses rather than the global orientation predicate. Those two target propositions are equivalent by a proved biconditional in this module. The witnesses form has already been refuted, so the orientation form is only a transport away from the same negative result.
Local setting is the native-cost uniqueness ledger: a catalogue of calibration targets that either hold or fail, used to certify which uniqueness routes are blocked.
proof idea
Assume the orientation target. Apply the right-to-left direction of the biconditional equating the no-mixed-witnesses target with the no-mixed-orientation target, obtaining the witnesses target. Discharge by the already-proved refutation of the witnesses target (itself reduced earlier to the coherent-orientation refutation). Pure logical transport; no new analytic content.
why it matters
This closes one named blocker in the native-cost uniqueness ledger: the hope that prime calibration alone forbids mixed identity/reciprocal choices is dead. Downstream, the identity-witness-excludes-reciprocal target and the reciprocal-witness-globalizes target are both refuted by reducing to this theorem via their own equivalences to the no-mixed-orientation target. The aggregate blocker certificate for native cost uniqueness records the pattern of proved versus refuted calibration targets; this entry supplies one of the negative limbs. In the broader Recognition forcing chain the point is narrow but sharp: J-uniqueness and $\phi$-fixed-point structure still need stronger hypotheses than bare prime-direction calibration if mixed prime orientations are to be ruled out.
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