PRCPrimeCalibrationForcesNonunitOrbitOrientationCoherentSharpenedTarget_of_nonunit_coherent
plain-language theorem explainer
If prime calibration already forces a single coherent identity/reciprocal orientation on every nonunit ratio-orbit direction, then the sharpened source of that coherence holds: local nonunit orientation together with prime-floor successor transport. Cited by anyone closing the coherent-vs-sharpened equivalence or the native-cost uniqueness blocker. Proof is a two-component pair built from the two one-way projections of the coherent target.
Claim. Assume that every ratio-orbit character that is prime-direction calibrated is nonunit-orbit orientation coherent (a single identity/reciprocal branch on all nonunit directions). Then both of the following hold: (i) prime calibration forces local nonunit orientation on each nonunit direction, and (ii) prime calibration forces prime-floor successor transport along identity-oriented nonunit orbits.
background
In the Primitive Recognition Calculus, ratio-orbit characters assign to each ratio orbit another orbit in a multiplicative way. Prime-direction calibration pins the character on prime generators. Nonunit orbit orientation coherence is the global demand that every nonunit direction sits on one common identity-versus-reciprocal branch; mixed product factors are then ruled out by nonunit non-self-reciprocity.
The coherent target packages that global demand as a Prop: for every character $\chi$, if $\chi$ is a ratio character and prime-direction calibrated, then $\chi$ is nonunit-orbit orientation coherent. The sharpened target is the conjunction of two weaker-looking sources: local nonunit orientation (each nonunit direction oriented by itself) and prime-floor successor transport (identity orientation propagates along the prime-floor successor).
The module develops native-cost uniqueness by forcing characters through calibration and orientation constraints toward the unique J-cost shape. Upstream, the coherent target already implies each conjunct separately via the local-orientation and successor-transport extraction lemmas.
proof idea
Term-mode pair construction. The sharpened target is definitionally the conjunction of the local-orientation target and the prime-floor successor-transport target. Apply the upstream lemma that extracts local nonunit orientation from the coherent hypothesis, then the upstream lemma that extracts prime-floor successor transport from the same hypothesis (via local orientation plus adjacent no-mix). Pack the two results as an And.intro-style pair. No further rewriting.
why it matters
This is one direction of the equivalence between the coherent orientation target and its sharpened source (local orientation plus prime-floor successor transport). The sibling iff theorem uses it as the forward map, so either formulation may be assumed when discharging orientation constraints.
Downstream it feeds the native-cost uniqueness blocker certificate path: once orientation coherence is interchangeable with the sharpened pair, calibration arguments can route through whichever form is easier to prove or refute. In the Recognition forcing picture this sits under native J-cost uniqueness (the T5 J-uniqueness landmark), where characters must collapse to the unique cost compatible with the Recognition Composition Law. It does not itself prove uniqueness; it only equates two packaging styles of the orientation hypothesis used in that campaign.
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