PRCCharacterTwoPrimeReciprocalExcludesPrimeIdentity_iff_two_prime_reciprocal_forces
plain-language theorem explainer
Under local prime orientation, two normal forms of the orbit-2 reciprocal branch are equivalent: excluding any identity-oriented prime axis is the same as forcing every prime axis to be reciprocal. Native-cost uniqueness arguments cite this to switch freely between the contrapositive blocker and the positive transport form. The proof is a two-constructor Iff packaging the two one-direction lemmas.
Claim. Let $\chi$ be a map on rational orbits that is locally prime-oriented: each prime axis is sent either to itself or to its reciprocal. Then the following are equivalent: (i) if the distinguished orbit-$2$ prime axis is reciprocal-oriented, no native prime axis is identity-oriented; (ii) if the orbit-$2$ prime axis is reciprocal-oriented, every native prime axis is reciprocal-oriented.
background
In the Primitive Recognition Calculus, characters act on RatioOrbit displays (signed numerator over a nonzero orbit denominator). Prime axes are the distinguished directions associated to prime orbits; reciprocal orientation means the character sends an axis to its multiplicative inverse orbit.
Local prime orientation is the algebraic content of matching J-costs on a single prime direction: for every prime $p$, $\chi$ sends the $p$-axis either to itself or to its reciprocal. The two props compared here are branch normal forms for the distinguished orbit-$2$ axis. The exclusion form is the contrapositive blocker: reciprocal orientation at $2$ forbids any identity-oriented native prime. The forces form is the positive transport statement: reciprocal at $2$ implies reciprocal on every native prime.
Under local orientation alone, identity and reciprocal are the only options on each axis, so "no identity" and "all reciprocal" coincide once the orbit-$2$ hypothesis is fixed. The module develops these as interchangeable certificates toward native cost uniqueness.
proof idea
Term-mode Iff constructor. The forward direction applies PRCCharacterTwoPrimeReciprocalForcesPrimeReciprocal_of_local_excludes_prime_identity at the given local-orientation hypothesis: exclusion plus local orientation yields the forces form. The reverse direction is the unconditional lemma PRCCharacterTwoPrimeReciprocalExcludesPrimeIdentity_of_two_prime_reciprocal_forces, which turns universal reciprocal orientation into the negation of any identity orientation. No further case analysis is needed at this wrapper.
why it matters
Native cost uniqueness in Recognition Science needs a clean obstruction to mixed prime-axis orientations (identity on some primes, reciprocal on others). That mixed pattern would break trace coherence and allow non-unique J-cost realizations. This equivalence lets the uniqueness pipeline treat the atomic two-specific mixed-witness blocker and the positive reciprocal-branch transport form as the same certificate under local orientation.
Downstream it feeds prc_native_cost_uniqueness_blocker_certificate, which assembles the proved factorization targets and refutations that close the native-cost uniqueness blocker. In the broader forcing chain this sits under T5 J-uniqueness: characters realizing the same cost must share orientation data, and the orbit-$2$ branch is the calibrated witness that propagates reciprocal orientation across all primes. The result is bookkeeping rather than new physics, but it removes a dual-form friction point in the uniqueness certificate.
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