PRCPrimeCalibratedSamePrimeMixedPairWitnessCharacter_absurd
plain-language theorem explainer
There is no ratio-orbit character that is prime-direction calibrated and still carries a same-prime mixed-pair witness. Anyone closing native-cost uniqueness or prime-calibration force lemmas cites this to kill the same-axis mixed branch at the calibrated level. The proof unpacks the existential and applies the uncalibrated same-prime mixed-pair absurdity.
Claim. There do not exist a map $\chi$ on ratio orbits that is a PRC ratio character, is prime-direction calibrated, and admits a same-prime mixed-pair witness (a mixed branch supported on a single native prime orbit).
background
In the Primitive Recognition Calculus native-cost uniqueness development, ratio characters $\chi$ assign orbit data used to rebuild cost from doubled-trace structure. Prime-direction calibration restricts $\chi$ so that prime axes line up with native prime orbits rather than arbitrary mixed directions.
A same-prime mixed-pair witness asserts that a mixed branch of the character lives on one native prime orbit (same-axis model). The sibling proposition packages existence of a character that is simultaneously a ratio character, prime-calibrated, and equipped with such a witness.
Upstream, PRCCharacterSamePrimeMixedPairWitnesses_absurd already shows that no ratio character (calibrated or not) can carry a same-prime mixed-pair witness: unpacking yields a prime $p$ and a reciprocal identification that collapses under substitution. The calibrated package is therefore only an existential wrapper around an already-impossible core.
proof idea
Term-mode reductio. Assume a witness of the calibrated same-prime mixed-pair package; destructure the existential to a character $\chi$ together with the three conjuncts (ratio character, prime-direction calibration, same-prime mixed-pair witnesses). Discard the first two hypotheses and feed the third into PRCCharacterSamePrimeMixedPairWitnesses_absurd, which already proves that no character admits same-prime mixed-pair witnesses. Contradiction closes the claim.
why it matters
Killing the calibrated same-axis mixed branch is the half of the mixed-pair dichotomy that is unconditionally false. Downstream iff lemmas then reduce prime-calibration force targets (no mixed-prime witnesses; prime-pair product cost consistency) to the distinct-prime mixed-pair side alone: once same-prime is absurd, the target holds exactly when there is no calibrated distinct-prime mixed-pair witness character.
Those reductions feed prc_universal_foundation_conditional_certificate in UniversalFoundation, which assembles kernel, real-complete ordered field, and trace-logic certificates into the conditional universal-foundation package. In the broader RS forcing picture this is bookkeeping inside native-cost uniqueness (the J-cost / RCL side of T5), not a new forcing step: it clears an impossible mixed model so calibration can force cost consistency on genuine prime pairs.
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