PRCPrimeCalibrationForcesNonunitOrbitOrientationCoherentTarget_of_local_adjacent
plain-language theorem explainer
From local nonunit orientation plus adjacent no-mixing under prime calibration, one obtains global orientation coherence on every nonunit orbit direction. Anyone closing the native-cost uniqueness blockers cites this bridge. The proof is a two-step packaging: convert adjacent no-mixing into prime-floor successor transport, then apply the existing local-plus-successor-transport implication.
Claim. Assume prime calibration forces both local nonunit-orbit orientation and the absence of mixed orientation on adjacent prime-floor steps. Then prime calibration forces a single coherent orientation across all nonunit orbit directions: every prime-direction-calibrated ratio character is nonunit-orbit-orientation-coherent.
background
In the primitive recognition calculus, ratio characters assign orientations (identity vs reciprocal) to ratio-orbit directions. Prime-direction calibration pins the character on prime generators. The coherent-orientation target asks that no identity-oriented nonunit direction coexist with a reciprocal-oriented one; mixed product factors are then impossible by nonunit non-self-reciprocity.
The local-adjacent package splits an earlier bundled successor-transport blocker into two facts: local nonunit orientation, and no mixed orientation on adjacent prime-floor steps. Pass-39 treats these separately so that successor transport can be recovered rather than assumed as a single opaque hypothesis.
Upstream, the sharpened target (local orientation plus prime-floor successor transport) already implies global coherence. A companion lemma recovers successor transport from the local-adjacent package. This declaration only composes those two facts.
proof idea
One-line term proof. From the local-adjacent hypothesis, take the first conjunct (local nonunit orientation). Feed the whole hypothesis to the lemma that upgrades local-adjacent data to the prime-floor successor-transport target. Pair those two props as the sharpened target, and apply the existing theorem that sharpened target implies global nonunit-orbit orientation coherence.
why it matters
This is the forward half of the local-adjacent $\leftrightarrow$ global-coherence equivalence used to tidy the prime-floor successor blocker. Downstream, that equivalence feeds the native-cost uniqueness blocker certificate, which packages the proved factorization targets and the refuted signed-admissible alternatives for the uniqueness argument.
In the Recognition framework this sits inside the foundation layer that forces the native $J$-cost uniqueness story (the T5 uniqueness of $J(x)=(x+x^{-1})/2-1$ and the Recognition Composition Law). Coherent orientation across nonunit orbits is the branch-coupling step that kills mixed product factorizations once prime calibration is in force. Closing this bridge removes a scaffolding seam between local adjacency control and the global no-mixing claim needed for uniqueness.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.