PRCPrimeCalibrationForcesNonunitNoMixedWitnessesSplitTarget_refuted
plain-language theorem explainer
The split form of the prime-calibration no-mixed-witness blocker is false: one cannot both rule out mixed prime witnesses and push prime-witness control to nonunit composites under the stated calibration. Native-cost uniqueness and universal-foundation certificate work cite this closure. The proof is a short transport through the equivalence with the already-refuted unsplit target.
Claim. It is not the case that both of the following hold under prime calibration: (i) there are no mixed prime witnesses, and (ii) control of prime witnesses extends to nonunit composite witnesses. Equivalently, the conjunction that defines the split no-mixed-witness target is false.
background
In the Primitive Recognition Calculus native-cost uniqueness development, cost characters are constrained by prime calibration. A recurring obstruction is whether calibration forces a global "no mixed witnesses" regime: first that mixed prime witnesses are impossible, then that prime-level witness control reaches nonunit composites.
The split target packages exactly that two-step program as a conjunction. An equivalence theorem identifies the split package with the unsplit no-mixed-witness target. The unsplit target has already been refuted by reducing it to the failure of the no-mixed-prime-witnesses half.
Local setting is the PRC native-cost uniqueness module: blocker propositions are stated as named targets, then either proved or refuted, and the outcomes are assembled into certificates for uniqueness and for the conditional universal foundation.
proof idea
One-line transport. Assume the split target. Apply the reverse direction of the equivalence between the unsplit no-mixed-witness target and its split form to obtain the unsplit target. Discharge by the already-proved refutation of the unsplit target (itself a reduction to the refuted no-mixed-prime-witnesses half). No new arithmetic or witness construction appears here.
why it matters
Closes the split packaging of the no-mixed-witness blocker so certificate assembly can treat split and unsplit forms uniformly as refuted. Downstream, prc_native_cost_uniqueness_blocker_certificate records which factorization and witness targets are proved versus refuted; this refutation is part of that ledger. The same status feeds prc_universal_foundation_conditional_certificate in the universal-foundation layer.
In the broader Recognition chain this is bookkeeping inside native-cost uniqueness (J-cost / character uniqueness under calibration), not a forcing-chain landmark (T5–T8). It sharpens what prime calibration does not buy: mixed-witness globalization fails, so uniqueness routes must avoid that blocker path.
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