PRCPrimeCalibrationForcesNonunitOrbitProductLocalOrientationSharpenedTarget
plain-language theorem explainer
Pass-45 sharpened open target for product-local orientation under prime calibration: any prime-direction-calibrated ratio character must carry one coherent orientation on all nonunit orbits. Native-cost uniqueness and blocker-certificate work cite it as the residual product commitment after display compatibility is settled. The body is a one-line alias of the nonunit-orbit orientation-coherence target.
Claim. The proposition asserting that every ratio character $\chi$ which is prime-direction calibrated has coherent orientation on all nonunit orbits: a single global orientation choice across nonunit orbit directions (stronger than mere product no-mixing).
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits encoding how multiplicative structure is read as cost data. Prime-direction calibration constrains $\chi$ along prime generators of the orbit lattice. Nonunit orbits are those away from the reciprocal unit class.
The upstream target (orientation coherence) asks that prime calibration force one coherent orientation on every nonunit orbit direction. Its doc states the intent: once coherence holds, mixed product factors are impossible by nonunit non-self-reciprocity. That is the stronger replacement for bare product no-mixing.
This module sits in the native-cost uniqueness program: characters built from the Recognition cost $J$ (the T5 unique cost $J(x)=(x+x^{-1})/2-1$) must be forced uniquely under calibration hypotheses. Pass-45 records that same-orientation products are algebraic and product-display compatibility is already proved via canonical normalization, so the residual product obligation is exactly orientation coherence.
proof idea
Definitional alias, not a proof. The sharpened target is definitionally equal to the nonunit-orbit orientation-coherence target: the universal statement that every prime-direction-calibrated ratio character is orientation-coherent on nonunit orbits. No tactics or lemmas fire; downstream proofs simply unfold or rewrite through this name.
why it matters
Marks the Pass-45 cut of the product-local orientation obligation inside native-cost uniqueness. Downstream, the prime-floor successor-transport sharpened target conjoins this with an adjacent no-mixing target, separating local orientation from successor transport. A display-compatibility theorem derives the older product-local orientation target from this sharpened form once no-mixing propagates.
It appears in the native-cost uniqueness blocker certificate (Pass-25 ledger of exact open Lean targets) and in the universal-foundation open-target structure. A sibling theorem already refutes the target by reducing to the refutation of orientation coherence, so the route cannot force the final uniqueness surface. Framework link: uniqueness of the native cost character is the PRC-side companion of T5 J-uniqueness under the Recognition Composition Law; this entry isolates why the product-orientation forcing path fails.
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