PRCPrimeCalibrationForcesTwoPrimeReciprocalExcludesPrimeIdentityWitnessTarget_of_identity_forces_two
plain-language theorem explainer
If prime calibration forces identity on the orbit-2 axis whenever any calibrated prime axis is identity-oriented, then a reciprocal orientation on orbit 2 excludes any identity-oriented native prime witness. Character-rigidity and native-cost uniqueness arguments cite this bridge. The proof is a two-step term composition through the non-witness exclusion target.
Claim. Assume that every prime-direction-calibrated ratio character which is identity-oriented on some calibrated prime axis is identity-oriented on the orbit-$2$ prime axis. Then every such character that is reciprocal-oriented on the orbit-$2$ prime axis admits no identity-oriented native prime witness.
background
In the Primitive Recognition Calculus, ratio characters $\chi$ act on ratio orbits and encode signed orientation data along prime axes. Prime-direction calibration restricts how $\chi$ may assign identity versus reciprocal orientation on native primes. The orbit-$2$ axis is distinguished: it is the first even prime and anchors mixed-orientation rigidity arguments used later for native cost uniqueness.
The hypothesis target says calibration forces a one-sided implication: identity anywhere on a calibrated prime axis forces identity at orbit $2$. The conclusion target is the mixed-witness exclusion: if orbit $2$ sits on the reciprocal branch, no native prime may remain identity-oriented as a witness. Both targets quantify over ratio characters that are prime-direction calibrated.
Upstream, the non-witness exclusion target is already known to follow from the identity-forces-two hypothesis, and the witness form follows from the non-witness form by a separate lifting lemma.
proof idea
Pure term composition, no tactics. First apply the upstream lemma that turns the identity-forces-two target into the non-witness two-prime-reciprocal exclusion target. Feed that result into the second upstream lemma, which lifts non-witness exclusion to the mixed-witness exclusion target. The composite is exactly the desired implication.
why it matters
This is one direction of the local equivalence between the one-sided distinguished-axis target and the two-specific mixed-witness exclusion target. That equivalence packages two formulations of the same rigidity demand used in native-cost uniqueness bookkeeping.
Downstream it appears in the iff linking those targets, and it contributes (via the uniqueness blocker path) to the native-cost uniqueness blocker certificate and the conditional universal-foundation certificate. In the broader RS forcing picture this is scaffolding for J-cost uniqueness (T5): ruling out mixed prime-axis orientations that would admit non-native cost characters.
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