PRCPrimeCalibrationForcesNonunitNoMixedWitnessesTarget_of_split
plain-language theorem explainer
If prime calibration both forbids mixed prime witnesses and lets prime-witness control reach nonunit composites, then it forbids mixed nonunit witnesses outright. Anyone assembling the native-cost uniqueness blocker or the split/combined equivalence cites this direction. The proof is a three-line term that feeds the first conjunct into the second.
Claim. Assume the split no-mixed-witness package: (i) every prime-direction-calibrated ratio character has no mixed prime witnesses, and (ii) under that calibration, prime-witness control extends to arbitrary nonunit witnesses. Then every prime-direction-calibrated ratio character $\chi$ satisfies the nonunit no-mixed-witnesses property: it cannot host both an identity-oriented and a reciprocal-oriented nonunit witness.
background
In the Primitive Recognition Calculus native-cost uniqueness module, ratio characters $\chi : \mathrm{RatioOrbit}\to\mathrm{RatioOrbit}$ are the algebraic stand-ins for candidate cost laws on multiplicative orbits. Prime-direction calibration orients $\chi$ on primes so that local witness data have a preferred identity or reciprocal side.
The combined target asserts that, once $\chi$ is a ratio character and prime-direction calibrated, it cannot carry both an identity-oriented nonunit witness and a reciprocal-oriented nonunit witness. The split package factors this into two obligations: first rule out mixed witnesses already at primes, then prove that prime-level control globalizes to all nonunit composites.
This lemma is the easy direction of that factorization: the conjunction of the two split targets implies the single combined target. The converse direction and the bridge from prime-only no-mixing live in sibling declarations in the same file.
proof idea
Term-mode, three lines. Introduce a ratio character $\chi$ together with the ratio-character and prime-calibration hypotheses. Project the split hypothesis to its two conjuncts. Apply the second conjunct (prime witnesses control nonunit witnesses) to $\chi$, feeding it the conclusion of the first conjunct (no mixed prime witnesses) at the same $\chi$. The resulting nonunit no-mixed-witnesses fact is exactly the combined target.
why it matters
Native-cost uniqueness in PRC needs a clean no-mixed-witness blocker so that calibrated characters cannot mix identity and reciprocal orientations on nonunits. This direction lets the development work with the split package and still discharge the combined target used by certificates.
It is one half of the iff between the combined and split targets, and it is the final step in the bridge that upgrades prime-only no-mixing to full nonunit no-mixing. Downstream it appears in prc_native_cost_uniqueness_blocker_certificate and in the conditional universal-foundation certificate, tying the witness-globalization step into the broader PRC foundation stack that supports J-uniqueness and the forcing chain.
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