PRCPrimeCalibrationForcesTwoPrimeBranchControlsPrimesTarget_of_coherent_prime_orientation
plain-language theorem explainer
If prime-cost calibration forces a single coherent orientation on every native prime axis, then it also forces the distinguished-orbit-2 branch to control every native prime branch. Cite this when reducing the orientation blocker to its normal form at the prime 2. The proof is a one-line pointwise application of the character-level coherence-to-two-control lemma.
Claim. Assume that every ratio-orbit character that is prime-direction calibrated is forced to have a single coherent prime orientation. Then every such character is forced to have its branch at the distinguished orbit $2$ control the branch on every native prime axis.
background
In the Primitive Recognition Calculus, cost uniqueness is obstructed by orientation choices on native prime axes of the ratio-orbit group. A ratio character $\chi$ is prime-direction calibrated when its action on each prime axis is pinned to a cost-compatible direction; orientation coherence then demands that those choices cannot mix independent inversions across primes.
The coherent-orientation target packages that demand globally: prime calibration must force one coherent orientation on all native prime axes. The two-prime-branch-controls target is the distinguished-prime normal form of the same demand: the branch chosen at orbit $2$ must determine the branch on every native prime.
Upstream, the character-level lemma already shows that coherence of a single $\chi$ implies that its orbit-$2$ branch controls all prime branches. The present declaration lifts that implication from characters to the calibration-target propositions.
proof idea
Term-mode, four lines. Introduce a character $\chi$ together with the ratio-character and prime-direction-calibration hypotheses of the two-prime target. Apply the coherent-orientation target hypothesis to those data to obtain coherence of $\chi$. Feed that coherence into the upstream lemma that coherence implies two-prime branch control, and conclude.
why it matters
This is one direction of the equivalence between the coherent-orientation blocker and its distinguished-prime normal form at orbit $2$. Downstream, that equivalence is recorded as an iff, and both forms feed the native-cost uniqueness blocker certificate and the conditional universal-foundation certificate.
In the Recognition forcing chain the native cost is the unique $J$ fixed by the Recognition Composition Law (T5). Ruling out mixed prime inversions is part of pinning that uniqueness at the character level before the phi fixed point and eight-tick structure are imposed. The declaration does not itself close uniqueness; it only normalizes one orientation blocker so later certificates can quote a single form.
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