PRCPrimeCalibrationForcesNonunitOrbitOrientationCoherentTarget_iff_sharpened
plain-language theorem explainer
Equivalence of two formulations of nonunit-orbit orientation coherence under prime calibration: the global target (every nonunit direction lands on one identity/reciprocal branch) and the sharpened source (local nonunit orientation plus prime-floor successor transport). Uniqueness and certificate authors cite it to collapse the two Prop packages. Proof is the term-mode pair of the two already-proved one-way implications.
Claim. The assertion that every prime-direction-calibrated ratio character is nonunit-orbit orientation-coherent is equivalent to the conjunction of (i) local nonunit orientation under the same hypotheses and (ii) prime-floor successor transport under those hypotheses.
background
In the Primitive Recognition Calculus, ratio characters assign to each ratio orbit another orbit, subject to multiplicative and reciprocity constraints. Prime-direction calibration pins the character on prime generators. Nonunit-orbit orientation coherence then demands that every nonunit direction is consistently identity-oriented or consistently reciprocal-oriented, never mixed; once that holds, mixed product factors are ruled out by nonunit non-self-reciprocity.
The global target packages that demand as a single universal statement over characters. The sharpened target splits the same demand into two more local pieces: local nonunit orientation on each direction, and transport of orientation along prime-floor successors. The module develops native-cost uniqueness by forcing characters through these calibration and coherence gates toward the unique J-cost shape.
Upstream, one direction extracts the two local pieces from global coherence; the converse rebuilds global coherence from local orientation plus successor transport via the character-level coherence lemma.
proof idea
Term-mode Iff introduction. The forward arrow is the theorem that global nonunit-orbit orientation coherence implies the sharpened package (local orientation and prime-floor successor transport). The reverse arrow is the theorem that the sharpened package implies global coherence, by feeding the two conjuncts into the character-level lemma that local orientation plus successor transport yield full nonunit-orbit orientation coherence. No extra tactics or arithmetic.
why it matters
Closes the bookkeeping gap between the global coherence target and its sharpened local source inside native-cost uniqueness. Downstream, the sharpened-target refutation applies the reverse direction of this equivalence to push a counterexample from the global target onto the sharpened package. The native-cost uniqueness blocker certificate and the universal-foundation conditional certificate sit further up the same dependency cone, so the equivalence keeps those certificates free to quote either formulation. In the broader RS forcing picture this is infrastructure for uniqueness of the native cost (the J-cost of T5), not a new physical constant claim.
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