PRCPrimeCalibrationForcesNonunitOrbitProductLocalOrientationTarget_of_identity_comparable_trace
plain-language theorem explainer
Under the hypothesis that prime calibration forces identity orientation to respect comparable finite δ-orbit traces on nonunit directions, prime calibration also forces product-factor local orientation to propagate through composite orbits. Native-cost uniqueness and universal-foundation certificates cite this bridge. The proof is a short intro-exact that feeds already-proved display compatibility and the no-mixed-orientation side of an iff into the product-propagation lemma.
Claim. Assume that every prime-direction-calibrated ratio character has identity orientation on nonunit directions that respects comparability of finite $\delta$-orbit traces. Then every such character also satisfies product-factor local-orientation propagation: prime-axis orientation is carried through composite orbit positions by the product decomposition.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits that encodes branch choice (identity versus reciprocal) along each direction. Prime-direction calibration pins orientation on prime axes. The native cost is recovered from doubled traces of such characters; uniqueness arguments therefore need orientation control off the prime axes as well.
Two intermediate targets appear here. The identity-comparable-trace target asks that, on nonunit directions, identity orientation respect order-comparability of finite $\delta$-orbit traces. The product local-orientation target asks that orientation on a composite orbit equal the product of orientations on its factors, so prime-axis data propagates.
Upstream, display compatibility (native equality of product orbit directions with ratio products of factor directions) is already proved from cross-equation respect. Separately, no-mixed-orientation on products is equivalent to the identity-comparable-trace target. The propagation lemma then says: display-compatible plus no-mixed implies product local-orientation propagation.
proof idea
Term-style after a single intro of the character, the ratio-character hypothesis, and prime calibration. Apply the propagation lemma PRCCharacterOrbitProductLocalOrientationPropagates_of_display_compatible_nomix, which needs three inputs: the character hypothesis, display compatibility, and no-mixed orientation.
Display compatibility is discharged by the already-proved theorem that prime calibration forces orbit-product display compatibility (itself reduced from cross-equation respect). No-mixed orientation is obtained by right-to-left application of the iff equating the no-mixed-orientation target with the identity-comparable-trace target, then specializing the resulting target to $\chi$. No further algebra is needed.
why it matters
This is a conditional bridge inside native-cost uniqueness: it converts the trace-order sharpening of identity-branch transport into the product-factor orientation step required off prime axes. Downstream it feeds the sibling theorem that upgrades product-local orientation to full nonunit orbit local orientation, and both sit under the native-cost uniqueness blocker certificate (zero-calibrated factorization proved, signed admissible factorization refuted).
It also appears in the universal-foundation conditional certificate path, which packages kernel, real-complete ordered field, and trace-logic ingredients. In the broader Recognition forcing chain this is foundation scaffolding for J-cost uniqueness (T5) and the Recognition Composition Law, not yet a physical constant claim: orientation control on characters is what lets the native cost match the unique J-cost on the ratio group.
The remaining open load is the identity-comparable-trace hypothesis itself; once that is discharged or replaced, this implication becomes unconditional product-local orientation under prime calibration.
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