PRCPrimeCalibrationForcesNoMixedNonunitOrbitOrientationTarget_refuted
plain-language theorem explainer
Prime calibration of ratio-orbit characters does not force global nonunit orientation coherence: an identity-oriented nonunit direction can coexist with a reciprocal-oriented one. Cited by anyone auditing native-cost uniqueness blockers or the universal foundation certificate. Proof is a short transfer: the target is equivalent to a product-form orientation claim already refuted upstream.
Claim. It is false that every ratio-orbit character $\chi$ which is prime-direction calibrated necessarily has no mixed nonunit orbit orientation. Equivalently: prime calibration does not forbid an identity-oriented nonunit direction from coexisting with a reciprocal-oriented nonunit direction.
background
In the primitive recognition calculus, a ratio-orbit character is a map on ratio orbits satisfying the character axioms used to build native cost. Prime-direction calibration restricts how such a character treats prime directions. Nonunit orbit orientation records, for each nonunit direction, whether the character is identity-oriented or reciprocal-oriented.
The target proposition asserts a branch-coupling law: under prime calibration, no identity-oriented nonunit direction may coexist with any reciprocal-oriented nonunit direction. Together with local nonunit orientation, that would be global nonunit orientation coherence.
An equivalent product-form target (no mixed orientation on orbit products) is already known to fail, and an iff links the two formulations. The present result is the corresponding negative statement for the nonunit-orbit form.
proof idea
Assume the nonunit-orbit target. Apply the reverse direction of the equivalence between the orbit-product no-mixed target and the no-mixed-nonunit target, obtaining the product-form claim. Discharge by the already-proved refutation of that product-form target (itself reduced to failure of nonunit identity-branch transport). Pure logical transfer; no new analytic work.
why it matters
This closes one branch-coupling blocker in the native-cost uniqueness ledger: prime calibration alone does not enforce global nonunit orientation coherence. It is consumed by the next refutation in the chain (nonunit identity witness excludes reciprocal) and sits among the negative facts aggregated into the native-cost uniqueness blocker certificate. That certificate, with the kernel and trace-logic packages, feeds the conditional universal foundation certificate. In the Recognition forcing picture this is bookkeeping on the cost side of T5-style J-uniqueness, not a new forcing step: it records that a hoped-for calibration implication is false, so uniqueness arguments must route elsewhere.
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