PRCPrimeCalibrationForcesNonunitOrbitProductLocalOrientationTarget_refuted
plain-language theorem explainer
Prime calibration of a ratio character does not force product-factor local orientation to propagate through composite orbit positions. Anyone tracking native-cost uniqueness blockers or the conditional universal-foundation certificate will cite this negative result. The proof is a one-step reduction: the product target implies a weaker nonunit-orbit orientation target already refuted by a two-adic axis-twist counterexample.
Claim. It is not the case that every ratio character $\chi$ that is prime-direction calibrated necessarily has product-factor local orientation propagation: there exist ratio characters that are prime-calibrated yet fail to carry prime-axis orientation through composite orbit positions via the product rule.
background
In the Primitive Recognition Calculus, ratio characters are maps $\chi$ on ratio orbits that encode directional cost data compatible with the native $J$-cost structure. Prime-direction calibration asks that $\chi$ align with a preferred orientation on prime axes. Product local orientation propagation is the sharper demand that this prime-axis orientation be transported to composite orbit positions by a product-of-factors rule.
The module studies which calibration hypotheses actually force uniqueness of the native cost. The product-propagation target was proposed as a "sharper source of nonunit local orientation": if prime calibration forced product propagation, one would obtain nonunit orientation on composite orbits and tighten the uniqueness argument.
Upstream, the product target is known to imply a weaker nonunit-orbit local-orientation target. That weaker target is already refuted: the two-adic axis-twist character is prime-direction calibrated yet fails nonunit local orientation on some orbits.
proof idea
Assume the product local-orientation target. Apply the implication lemma that any character satisfying product propagation (under prime calibration) also satisfies the weaker nonunit-orbit local-orientation target. Feed that derived target into the already-proved refutation of the weaker target, which is witnessed by the two-adic axis-twist character. Contradiction. The whole argument is a three-line reduction; no new counterexample is constructed here.
why it matters
This closes a false lead in the native-cost uniqueness program: the product-factor propagation step cannot be forced from prime calibration alone, so it cannot serve as the missing bridge to nonunit orientation on composite orbits. Downstream it feeds the native-cost uniqueness blocker certificate, which records which candidate forcing routes are open and which are dead, and appears in the conditional universal-foundation certificate that packages kernel, ordered-field, and trace-logic status.
In the broader Recognition Science chain this sits inside foundation work that supports T5 $J$-uniqueness and the Recognition Composition Law route to a unique cost. Refuting over-strong calibration claims keeps the uniqueness argument honest: only routes that survive counterexamples (such as two-adic twists) remain candidates for forcing the native $J$.
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