PRCCoherentPrimeOrientationPropagatesToGlobalTarget_refuted
plain-language theorem explainer
The coherent-prime-to-global orientation target is false: a multiplicative ratio character can orient every positive prime axis coherently and still fail global orientation at the signed unit. Anyone tracking native-cost uniqueness blockers or the repaired orientation target would cite this. The proof is a one-shot countermodel via the absolute-value character on ratio orbits, which collapses −1 to +1.
Claim. It is false that every multiplicative character $\chi$ on ratio orbits with coherent orientation on positive primes necessarily has global cost orientation (pointwise identity-or-reciprocal at every ratio direction). In particular, coherent prime orientation alone does not force global orientation without signed-unit calibration.
background
In the primitive recognition calculus, ratio orbits are the native displays of nonzero rationals up to the verifier quotient. A ratio character is a multiplicative map on those orbits that preserves the unit. Cost orientation asks whether $\chi(q)$ is cross-equal to $q$ or to its reciprocal; global cost orientation demands this at every orbit, while prime orientation coherence only constrains positive prime axes.
The absolute-value character sends each orbit to the orbit of $|q|$. It is a genuine ratio character and orients every positive prime by the identity, yet it erases the sign of $-1$: the orbit of $-1$ is mapped to the orbit of $+1$. The ratio-orbit display of $-1$ is the witness that exposes the missing signed-unit calibration.
The target proposition asserts that any ratio character with coherent prime orientation automatically has global cost orientation. That is the sharper orientation blocker B in the native-cost uniqueness development: prime coherence plus multiplicativity and rational factorization should propagate orientation to every ratio direction.
proof idea
Assume the propagation target. Instantiate it at the absolute-value character, feeding the three already-proved facts that this map is a ratio character, that its prime orientation is coherent, and that the orbit of $-1$ is the test ratio. The target then yields global cost orientation for the absolute-value character. That conclusion is contradicted by the lemma that the absolute-value character fails identity-or-reciprocal orientation exactly at the orbit of $-1$. Hence the target is false.
why it matters
This refutation pins the gap in the naive prime-to-global orientation story: without a signed-unit calibration hypothesis, coherent primes do not force global orientation. The doc-comment states the repaired target explicitly: coherent prime orientation must be supplemented by signed-unit calibration before one can ask for global pointwise identity-or-reciprocal orientation.
Downstream, the native-cost uniqueness blocker certificate records this class of orientation failures among the certified blockers, and the universal-foundation conditional certificate consumes the same uniqueness stack. In the Recognition forcing chain this sits under native J-cost uniqueness (T5 territory): characters that erase sign cannot be the native cost map, so uniqueness arguments must either add signed-unit data or exclude absolute-value-like countermodels.
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