PRCPrimeCalibrationForcesTwoPrimeReciprocalExcludesPrimeIdentityTarget_of_witness

In the Primitive Recognition Calculus, a ratio character is a map on ratio orbits that encodes how multiplicative structure is read as cost data. Prime-direction calibration fixes, for each native prime axis, whether the character sits on the identity branch or the reciprocal branch. Orbit-$2$ is the distinguished two-axis; mixed configurations (orbit-$2$ reciprocal while some other prime stays identity) are the obstruction class under study.

Two target propositions package the same rigidity claim at different granularities. The witness target says: under calibration, if orbit-$2$ is reciprocal then no identity-oriented native prime witness exists. The full target says the same without the witness packaging: no native prime axis may remain on identity. The character-level lemma already converts a witness exclusion for a fixed $\chi$ into the full exclusion for that $\chi$ by unpacking the existential witness into a universal quantifier over primes.

This module develops uniqueness of the native cost functional (the PRC avatar of the $J$-cost from the forcing chain). The two-reciprocal exclusion targets are intermediate rigidity statements on the way to that uniqueness.

One-line pointwise lift. Introduce a ratio character $\chi$ together with the ratio-character and prime-calibration hypotheses. Apply the assumed witness-target hypothesis at $\chi$ to obtain the character-level witness exclusion. Pass that to the upstream lemma that turns witness exclusion into full exclusion for a fixed character: if reciprocal orbit-$2$ excludes every identity-oriented prime witness, then it excludes identity orientation of every native prime. The resulting term inhabits the full target.

Closes the witness-to-target direction of the equivalence between the two-reciprocal exclusion target and its mixed-witness packaging; the matching iff theorem quotes both directions. Downstream, the same hypothesis form is fed into the implication that prime-identity calibration forces the orbit-$2$ axis onto identity as well, tightening branch rigidity across all native primes.

Those steps sit inside the native-cost uniqueness blocker certificate and, one module up, the conditional universal-foundation certificate. In framework terms this is local scaffolding for $J$-uniqueness (forcing step T5): ruling out mixed reciprocal/identity prime orientations narrows the admissible characters that could produce a native cost other than the canonical $J$. The declaration itself does not finish uniqueness; it only normalizes the exclusion target so later certificates can cite a single Prop.