PRCPrimeCalibrationForcesNoMixedPrimeWitnessesTarget_refuted
plain-language theorem explainer
Prime-direction calibration of a ratio character does not force the existential no-mixed-prime-witnesses property. Anyone tracking the PRC native-cost uniqueness blocker chain cites this as one more failed forcing target. The proof is a one-line transport: the target is equivalent to coherent prime orientation, already refuted upstream.
Claim. It is not the case that every ratio-orbit character $\chi$ that is prime-direction calibrated has no mixed prime witnesses. Equivalently, the universal claim "prime calibration forces the existential no-mixed-prime-witnesses property" is false.
background
In the Primitive Recognition Calculus, ratio characters $\chi$ act on ratio orbits and encode multiplicative structure used to build native costs. Prime-direction calibration is a local orientation constraint on how $\chi$ treats prime generators. A mixed prime witness is a pair of prime-level data in which one side is identity-oriented and the other is reciprocal-oriented; the no-mixed-witnesses target asserts that calibration alone forbids any such pair once an identity-oriented prime witness exists.
The target formalized here is the existential witness form of "no mixed prime orientation": for every ratio character, prime-direction calibration should imply absence of mixed prime witnesses. An equivalent coherent-orientation packaging of the same claim is already available in-module, together with a proved biconditional linking the two packages.
The surrounding module studies which calibration hypotheses actually force uniqueness of the native cost (via doubled-trace and d'Alembert structure). Several candidate forcing targets fail; this declaration records one of those failures.
proof idea
Assume the no-mixed-prime-witnesses target. Apply the in-module biconditional to rewrite it as the coherent prime-orientation target. Discharge the assumption by the already-proved refutation of that coherent target. The argument is pure propositional transport: intro, mp of the iff, then the upstream refutation.
why it matters
Native-cost uniqueness in PRC needs a clean account of which prime-calibration slogans survive. This refutation closes the existential no-mixed-witnesses slogan and immediately feeds sibling refutations (no-mixed orientation, nonunit no-mixed witnesses, and identity-witness globalization). Those negative results are collected into the native-cost uniqueness blocker certificate and surface in the universal-foundation conditional certificate.
In the broader Recognition forcing picture, cost uniqueness ultimately rests on the J-cost and the Recognition Composition Law rather than on ad-hoc prime-orientation constraints. Recording that prime calibration does not force no-mixed witnesses keeps the uniqueness route honest: failed local forcing targets are marked rather than silently assumed.
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