PRCPrimeCalibratedTwoAdicAxisTwistCharacter_of_two_three_local_orientation_failure_character
plain-language theorem explainer
A ratio-orbit character that twists the two-adic axis and fails local orientation on the 2·3 composite yields a prime-calibrated two-adic axis-twist character. Native-cost uniqueness arguments cite this bridge when turning composite-local countermodels into calibrated valuation witnesses. The proof is a two-step term composition: drop the orientation-failure conjunct, then promote the remaining twist character to the calibrated form.
Claim. If there exists a ratio-orbit character $\chi$ that is a two-adic axis twist and fails the $2\cdot 3$ composite-local orientation condition, then there exists a ratio-orbit character that is prime-direction calibrated and a two-adic axis twist.
background
In the Primitive Recognition Calculus native-cost uniqueness development, ratio-orbit characters are maps $\chi$ on ratio orbits that encode multiplicative branch choices for the cost. A two-adic axis twist is the branch pattern that sends the orbit of $2$ to the reciprocal side. Prime-direction calibration strengthens that pattern by fixing how native prime directions are valued, giving a concrete calibrated model rather than a bare twist.
The hypothesis is the witness form of $2\cdot 3$ composite-local failure: some ratio character is a two-adic axis twist yet fails composite-local orientation on the $2\cdot 3$ direction. The module treats that failure surface as the constructive countermodel equivalent to the reduced two-adic ratio-character target.
The conclusion is the calibrated two-adic axis-twist existence statement. Upstream, the failure character already packages a ratio character with the twist conjunct; a sibling lemma strips the negated orientation conjunct to a pure twist-ratio character, and another lemma upgrades any such twist-ratio character to the prime-calibrated form.
proof idea
Term-mode composition of two in-module lemmas. First apply the extractor that turns a $2\cdot 3$ composite-local orientation failure character into a two-adic axis-twist ratio character (destructure the existential and keep the ratio-character and twist conjuncts, discarding the negated local-orientation conjunct). Feed that intermediate witness into the upgrade lemma that builds a prime-calibrated two-adic axis-twist character from any two-adic axis-twist ratio character. No extra algebraic work occurs at this site.
why it matters
This bridge sits on the native valuation route that refutes the current character-rigidity branch: calibrated two-adic axis-twist models are the concrete objects used to force contradictions against rigidity. Immediately downstream it feeds the mixed-character construction that packages prime-calibrated two-prime reciprocal identity with a non-two prime mixed branch, and it supplies the contrapositive absurdity lemma: absence of any calibrated two-adic axis twist rules out every $2\cdot 3$ composite-local orientation failure character.
It also appears in the conditional universal-foundation certificate assembly, so the local countermodel-to-calibration step is part of the PRC foundation packaging rather than a dead-end lemma. Within Recognition Science this is foundation-layer character bookkeeping on the path toward unique native cost (the J-cost uniqueness landmark T5), not a direct mass or coupling computation.
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