PRCPrimeCalibrationForcesNonunitOrbitOrientationLocalBranchAgreementTarget_of_coherent
plain-language theorem explainer
Prime calibration that forces a single coherent orientation on all nonunit ratio-orbit directions already yields the positive normal form: local nonunit orientation plus two-branch agreement. Anyone closing the native-cost uniqueness or universal-foundation certificates cites this implication. The proof is a pair constructor that applies two short reductions, one per conjunct.
Claim. Assume that every ratio character $\chi$ that is prime-direction calibrated has a single coherent orientation on all nonunit orbits. Then the same calibration forces both (i) local nonunit orbit orientation and (ii) agreement of the two orientation branches.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits compatible with the multiplicative structure used to build the native cost. Nonunit orbits carry an orientation: identity-type versus reciprocal-type. Global coherence means every nonunit direction shares one orientation; mixed identity/reciprocal factors are then ruled out by nonunit non-self-reciprocity.
The coherent target packages the claim that prime-direction calibration forces that global coherence for every such $\chi$. The local-branch-agreement target is the positive normal form of the same blocker: the conjunction of a local-orientation target and a two-branch-agreement target. Upstream, local orientation is recovered from coherence by case-splitting the global identity-or-reciprocal alternative; branch agreement is likewise inherited from coherence.
proof idea
Term-mode pair construction for the conjunction. The first conjunct is discharged by introducing $\chi$, character and prime-calibration hypotheses, applying the coherent target, then calling PRCCharacterNonunitOrbitLocalOrientation_of_coherent. The second conjunct is the sibling one-liner PRCPrimeCalibrationForcesNonunitBranchAgreementTarget_of_coherent applied to the same coherent hypothesis. No further algebra.
why it matters
This is one direction of the equivalence between the coherent target and the local-branch-agreement normal form; the sibling converse closes the iff used as a rewrite interface inside native-cost uniqueness. Downstream it is also wired into prc_universal_foundation_conditional_certificate, so the foundation certificate can quote either packaging of the nonunit branch-coupling blocker. In the Recognition forcing chain this sits under native J-cost uniqueness (T5 landmark): coherent orientation on nonunit orbits is what keeps product factors from mixing and thereby pins the cost functional to the unique $J$. It does not itself discharge the coherent target; it only normalizes the packaging once coherence is assumed.
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