PRCPrimeCalibrationForcesPrimeIdentityWitnessExcludesReciprocalTarget_of_no_mixed_prime_orientation
plain-language theorem explainer
Prime-direction calibration that already rules out mixed identity/reciprocal choices on distinct prime axes also forces the one-sided witness rule: once any identity-oriented native prime witness exists, no reciprocal-oriented prime witness is allowed. Anyone assembling native-cost uniqueness or the prime-calibration target lattice cites this implication. The proof is a one-line lift of the character-level no-mix lemma under the calibration hypotheses.
Claim. Assume that every ratio character $\chi$ that is prime-direction calibrated has no mixed prime orientation (identity and reciprocal orientations cannot be chosen independently on different native prime axes). Then every such calibrated $\chi$ also satisfies the one-sided witness exclusion: if there exists an identity-oriented native prime witness, there is no reciprocal-oriented native prime witness.
background
In the Primitive Recognition Calculus native-cost uniqueness module, ratio characters $\chi : \mathrm{RatioOrbit}\to\mathrm{RatioOrbit}$ encode how multiplicative ratio data is read. Prime-direction calibration fixes, on each native prime axis, whether the character acts by the identity or by reciprocal orientation. Two global constraints are then compared.
The no-mixed target asserts that calibration cannot leave independent mixed identity/reciprocal choices on different prime axes: once $\chi$ is a ratio character and prime-direction calibrated, it has no mixed prime orientation. The one-sided witness-exclusion target is weaker in appearance: if an identity-oriented prime witness exists, no reciprocal-oriented prime witness is allowed.
At the single-character level the implication is already proved: any character with no mixed prime orientation automatically satisfies identity-witness exclusion of reciprocal witnesses (by feeding the two witnesses into the no-mix predicate). The present declaration lifts that character fact to the quantified calibration targets.
proof idea
One-line wrapper. Introduce a ratio character $\chi$ together with the ratio-character and prime-direction-calibration hypotheses. Apply the no-mixed target hypothesis to obtain no mixed prime orientation for $\chi$. Feed that into the upstream character lemma PRCCharacterPrimeIdentityWitnessExcludesReciprocal_of_no_mixed_prime_orientation, which is itself a short rintro discharging the two existential witnesses against the no-mix predicate. The resulting property is exactly the body of the identity-witness-excludes-reciprocal target.
why it matters
This arrow is one half of the equivalence between the no-mixed prime-orientation target and the one-sided prime identity-witness exclusion target; the converse is the sibling implication, and together they form the iff used when collapsing the prime-calibration target lattice. Downstream it feeds the native-cost uniqueness blocker certificate and, through that stack, the conditional universal-foundation certificate.
In Recognition Science terms the result tightens how prime axes may be oriented once cost calibration is imposed: mixed identity/reciprocal readings are not free parameters. That discipline sits under the native $J$-cost uniqueness program (T5 $J$-uniqueness and the Recognition Composition Law), where admissible characters must not smuggle reciprocal twins alongside identity witnesses. Closing these target equivalences removes a blocker on the path from prime calibration to a unique native cost.
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