PRCPrimeCalibrationForcesNonunitOrbitProductLocalOrientationTarget_of_display_compatible_nomix
plain-language theorem explainer
Prime-direction calibration of a ratio character, together with nonunit orbit orientation coherence, forces product-local orientation propagation on composite orbits. Native-cost uniqueness arguments cite this to close the product-factor step of the orientation chain. The proof is a short term reduction: proved display compatibility plus coherence-implies-no-mixing feed the existing propagation lemma.
Claim. Assume the sharpened prime-calibration commitment: every prime-direction-calibrated ratio character $\chi$ has nonunit orbit orientation coherence. Then every such $\chi$ also satisfies product-local orientation propagation: prime-axis orientation is carried through composite orbit positions via product factors.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits that encodes how recognition cost orients multiplicative structure. Prime-direction calibration pins $\chi$ on prime axes. Product-local orientation propagation is the step that extends that pinning from primes to composite orbit positions by factoring products.
The sharpened target (Pass-45) isolates the remaining product commitment as nonunit orientation coherence: same-orientation products are already algebraic, and product-display compatibility is already proved via canonical normalization. Coherence is what remains, and it is known to imply product no-mixing (no mixed-orientation factor pairs).
Upstream, display compatibility under prime calibration is a proved target, and a separate lemma converts nonunit coherence into the no-mixed-orientation hypothesis. A third lemma states that display compatibility plus no-mixing already yield product-local orientation propagation for any ratio character.
proof idea
Term-mode reduction, not a new calculation. Fix a ratio character $\chi$ that is prime-direction calibrated. Apply the propagation lemma that needs three inputs: the character hypothesis, product-display compatibility, and product no-mixed orientation.
Display compatibility is supplied by the already-proved prime-calibration display-compatibility target at $(\chi)$. No-mixing is supplied by applying the coherence-to-no-mixing lemma to the sharpened hypothesis at $(\chi)$. The propagation lemma then returns product-local orientation propagation, which is exactly the unsharpened target.
why it matters
This declaration discharges the product-local orientation target from the Pass-45 sharpened coherence commitment, so the uniqueness pipeline can treat product-factor orientation as settled once nonunit coherence is in hand. It feeds prc_native_cost_uniqueness_blocker_certificate, which packages the native-cost uniqueness blocker certificate (zero-calibrated factorization proved; signed admissible factorization refuted).
In the broader Recognition Science foundation, native cost uniqueness is the PRC route toward the unique $J$-cost of the forcing chain (T5: $J(x)=(x+x^{-1})/2-1$), before $\phi$ and the eight-tick structure. Closing product orientation under prime calibration is one of the algebraic gates between character axioms and that uniqueness claim. The remaining open load sits on establishing or assuming the sharpened coherence hypothesis itself, not on this implication.
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