PRCPrimeCalibrationForcesPrimeIdentityBranchUniformityTarget_of_no_mixed_prime_orientation

In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits that encodes the multiplicative skeleton of a candidate native cost. Prime-direction calibration restricts how $\chi$ acts on the native prime axes (the orbit directions generated by primes). Two structural constraints matter here: local prime orientation (each prime axis is either identity- or reciprocal-oriented) and no mixed prime orientation (the identity/reciprocal choice cannot flip independently across different primes).

Branch uniformity is the global form of that discipline: once any native prime axis sits on the identity branch, every native prime axis must. The module packages these as calibration targets: propositions asserting that prime cost calibration forces the corresponding structural property for every calibrated ratio character.

Upstream, local prime orientation under calibration is already proved unconditionally. Separately, local orientation plus no-mixing implies branch uniformity for a fixed character. This declaration lifts that fixed-character fact to the target level under the no-mixing target hypothesis.

Term-mode proof by introducing a calibrated ratio character $\chi$ and discharging the branch-uniformity goal with one application of the fixed-character lemma: local orientation together with no mixed orientation yields identity-branch uniformity.

Local orientation is supplied by the already-proved local prime-orientation target applied to $\chi$. No-mixing is supplied by the assumed no-mixing target applied to the same $\chi$. No further case analysis or arithmetic is required.

This is one half of the equivalence between the no-mixing target and the prime identity branch-uniformity target; the sibling converse closes the iff. That equivalence collapses two seemingly different calibration obligations into one, which simplifies the native-cost uniqueness blocker certificate and the conditional universal-foundation certificate that consume these targets.

In the Recognition forcing picture, native cost uniqueness is the bridge from the Recognition Composition Law and J-uniqueness (T5) to a single admissible cost on ratio orbits. Forbidding mixed prime orientations is the arithmetic content that keeps the prime-axis skeleton coherent; branch uniformity is the form needed for trace-free transport along the prime-axis graph. Closing their equivalence removes a structural fork in the uniqueness chain.