PRCPrimeCalibrationForcesPrimeFloorNoAdjacentMixedOrientationTarget_refuted
plain-language theorem explainer
Refutes the claim that every prime-direction-calibrated ratio character forbids adjacent mixed identity/reciprocal orientations on the prime floor. The two-adic axis-twist character is a counterexample: calibrated, yet reciprocal on the 2-orbit and identity on the 3-orbit. Cited by native-cost uniqueness blocker and universal-foundation certificates. Proof is a direct counterexample from the twist branch data.
Claim. It is false that every ratio-orbit map $\chi$ which is a PRC ratio character and is prime-direction calibrated necessarily satisfies the prime-floor no-adjacent-mixed-orientation condition. Equivalently: there exists a prime-direction-calibrated ratio character for which some pair of adjacent nonunit orbit directions carries opposite identity versus reciprocal orientations.
background
In the Primitive Recognition Calculus, ratio characters $\chi$ act on ratio-orbit data and may be constrained by prime-direction calibration: on each prime axis the character is forced to a fixed orientation class. Orientations of nonzero orbit directions are recorded by cross-equality either to the direction itself (identity) or to its reciprocal.
The target proposition asserts a universal implication: every calibrated ratio character has no adjacent mixed orientations on the prime floor (adjacent nonunit orbit directions cannot carry opposite identity/reciprocal orientations). That target is the second component of the prime-floor successor blocker for native-cost uniqueness.
Orbit positions live in the base-neutral DistinctionNat (zero and successor). The two-step orbit is the successor of one; its successor is the three-step orbit. Reciprocal and identity orientation predicates are stated for arbitrary nonzero orbits, not only primes, so composite positions can participate in trace transport.
proof idea
Assume the universal target and derive a contradiction from the two-adic axis-twist character.
- Obtain prime-direction calibration of that character from the two-adic axis-twist branch lemma.
- Instantiate the assumed target at this character (using that it is a ratio character) to get the no-adjacent-mixed-orientation property.
- From the twist branch, read off reciprocal orientation on the two-orbit and identity orientation on the three-orbit (successor of two), rewriting via the orbit-direction identity/reciprocal defs and the two/three prime-direction abbreviations.
- Feed the two-orbit (prime, nonunit) into the no-adjacent property; its second conjunct forbids the pair (reciprocal on two, identity on three). That pair is exactly what step 3 produced.
Thus the universal target is false.
why it matters
Native-cost uniqueness in PRC needs a clean account of which prime-floor constraints actually hold. This refutation kills one candidate universal constraint (prime calibration forcing no adjacent mixed orientations), so the blocker certificate can record a sharp negative rather than an open hypothesis.
Downstream it is consumed by prc_native_cost_uniqueness_blocker_certificate (the uniqueness-blocker package) and by the conditional universal-foundation certificate. In the broader Recognition stack this sits in foundation work that feeds the forcing chain toward J-uniqueness and the native cost, not yet at T5–T8 physics landmarks, but it prunes false rigidity assumptions on character orientation before cost uniqueness is sealed.
It closes a scaffolding-style target by counterexample rather than by proof of the positive claim.
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