PRCPrimeCalibrationForcesNoMixedPrimeWitnessesTarget_iff_trace_coherence
plain-language theorem explainer
Prime-direction calibration of a ratio character forbids mixed prime witnesses if and only if it forces identity-orientation trace coherence across native prime axes. Native-cost uniqueness and the universal-foundation certificate cite this bridge. The proof is a two-step Iff.trans through the intermediate no-mixed-orientation target.
Claim. The following are equivalent: (i) every prime-direction-calibrated ratio character $\chi$ has no mixed prime witnesses; (ii) every such $\chi$ is prime-identity trace-coherent (identity orientation on one native prime axis propagates to all native prime axes).
background
In the Primitive Recognition Calculus, ratio characters $\chi$ act on ratio orbits and encode how recognition cost orients along multiplicative axes. Prime-direction calibration means the character has already fixed a consistent cost sign on each native prime axis. Two residual obstruction targets remain after that calibration.
The no-mixed-prime-witnesses target is the existential form: once an identity-oriented prime witness exists, no reciprocal-oriented prime witness may appear on another axis. The prime-identity trace-coherence target is the global propagation form: identity orientation on any native prime axis forces every native prime axis onto the identity branch.
An intermediate target (no mixed prime orientation) sits between them. Upstream, the module already proves witnesses $\leftrightarrow$ no-mixed-orientation and no-mixed-orientation $\leftrightarrow$ trace coherence; this declaration only composes those links.
proof idea
Pure term-mode Iff composition. Apply PRCPrimeCalibrationForcesNoMixedPrimeWitnessesTarget_iff_no_mixed_prime_orientation to rewrite the witnesses target as the no-mixed-orientation target, then PRCPrimeCalibrationForcesNoMixedPrimeOrientationTarget_iff_trace_coherence (itself a two-direction constructor pair) to rewrite that as prime-identity trace coherence. No new arithmetic or character analysis is introduced here.
why it matters
This is the clean bridge that lets the native-cost uniqueness story speak in either witness language or trace-coherence language. Downstream, prc_native_cost_uniqueness_blocker_certificate packages the uniqueness blockers, and prc_universal_foundation_conditional_certificate consumes the foundation stack (kernel, real complete ordered field, trace logic). Ruling out mixed independent prime inversions is exactly the "sharper orientation blocker" needed so that J-cost uniqueness (T5 in the forcing chain) can be stated without residual prime-axis sign flips. Without this equivalence, the certificate layer would have to carry two parallel formulations of the same obstruction.
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