PRCCharacterPrimeWitnessesControlNonunitWitnesses_iff_mixed_reflects
plain-language theorem explainer
For a ratio-orbit character χ, the composite bridge “prime no-mixing controls all nonunit witnesses” is equivalent to its reflection “mixed nonunit witnesses force mixed prime-axis witnesses.” Anyone tracking uniqueness of the native PRC cost via character constraints will cite this. The proof is a pure Iff pairing of the two already-proved one-way implications.
Claim. Let $\chi$ be a map on rational orbits. Then the statement “if $\chi$ has no mixed prime witnesses, then it has no mixed nonunit witnesses” is equivalent to “if $\chi$ has mixed nonunit witnesses (identity and reciprocal directions), then it already has mixed prime-axis witnesses.”
background
In the Primitive Recognition Calculus, characters act on RatioOrbit (K4.7: integer numerator over a nonzero orbit denominator). Cost uniqueness arguments isolate prime-axis witnesses first, then must pass from prime no-mixing to arbitrary nonunit no-mixing.
PRCCharacterPrimeWitnessesControlNonunitWitnesses is exactly that composite bridge: no mixed primes implies no mixed nonunits. Its reflection form PRCCharacterMixedNonunitWitnessesReflectPrimeWitnesses says the converse direction product propagation does not give: existence of mixed nonunit witnesses (both identity and reciprocal orbit directions) forces existence of mixed prime-axis witnesses.
This module packages native-cost uniqueness hypotheses around doubled-trace and d’Alembert structure for such characters; the present declaration only equates the two bridge formulations.
proof idea
Term-mode Iff constructor. The forward direction is PRCCharacterMixedNonunitWitnessesReflectPrimeWitnesses_of_prime_control (assume prime-control, then contrapose a mixed-nonunit hypothesis against absence of mixed primes). The reverse is PRCCharacterPrimeWitnessesControlNonunitWitnesses_of_mixed_reflects (feed a mixed-nonunit witness into the reflection hypothesis and apply prime no-mixing). No further algebra.
why it matters
After primes are isolated, native-cost uniqueness still needs a bridge from prime no-mixing to full nonunit no-mixing. Recording that bridge as equivalent to its reflection lets later arguments choose the more convenient polarity (direct control vs. witness reflection) without re-proving either side.
No downstream users are wired yet (used_by empty), so this is infrastructure inside PRCNativeCostUniqueness: it closes the logical gap between the two named composite-bridge Props that sibling character/trace lemmas already mention. It does not itself force J-uniqueness (T5) or the RCL; it only equates two witness-control formulations used on the way to native cost uniqueness for PRC characters.
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