PRCPrimeCalibrationForcesOrbitProductNoMixedOrientationTarget_refuted
plain-language theorem explainer
Prime calibration of ratio characters does not force product no-mixed orientation under native orbit multiplication. Researchers tracking native-cost uniqueness blockers cite this refutation. The proof is pure logical transport: the product target is equivalent to the already-refuted nonunit identity branch-transport target, so the latter's negation yields the former's.
Claim. It is false that every prime-direction-calibrated ratio character $\chi$ on ratio orbits is free of mixed identity/reciprocal factor orientations under native orbit multiplication. Equivalently, prime calibration alone does not force the product no-mixing property.
background
In the Primitive Recognition Calculus, a ratio character is a map on ratio orbits compatible with the native cost structure. Prime-direction calibration constrains how the character sits on prime generators. The product no-mixing target asserts that any such calibrated character keeps a single coherent orientation (identity versus reciprocal) across factors under native multiplication, so mixed identity/reciprocal factor orientations are ruled out.
An equivalent formulation is the nonunit identity branch-transport target: prime calibration should force identity-branch transport on nonunit orbits. That transport claim is already refuted upstream. The present module records a cascade of such target refutations that feed the native-cost uniqueness blocker certificate.
proof idea
One-line logical transport. Assume the product no-mixing target. Apply the forward half of the equivalence
PRCPrimeCalibrationForcesOrbitProductNoMixedOrientationTarget ↔ PRCPrimeCalibrationForcesNonunitIdentityBranchTransportTarget
to obtain the nonunit identity branch-transport target. Invoke the already-proved refutation of that branch-transport target to reach a contradiction. No new analytic content is introduced.
why it matters
Closes one node in the refutation cascade for hoped-for forcing properties of prime calibration. Immediately downstream, the no-mixed-nonunit-orbit-orientation target is refuted by reducing to this result via another equivalence. The cascade contributes to prc_native_cost_uniqueness_blocker_certificate, which packages proved and refuted factorization targets constraining how far prime calibration can pin down the native cost.
In Recognition Science foundation work, these blockers show that uniqueness of the native cost (the J-cost side of the forcing chain, T5 J-uniqueness and the Recognition Composition Law) cannot be obtained from prime calibration alone without further structure.
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