PRCTwoThreeCompositeLocalOrientationForTwoAdicAxisTwistTarget_of_no_non_two_composite_cost_defect_character
plain-language theorem explainer
Absence of a prime-calibrated mixed-prime composite J-cost defect character forces every two-adic axis-twist ratio character to pick a canonical local orientation at the first 2·3 composite. Foundation workers closing the two-adic branch blocker and the conditional universal-foundation certificate cite this. The proof is a short term chain: no-defect implies no failure character, then the target↔no-failure biconditional.
Claim. If there is no ratio character $\chi$ that is prime-direction calibrated and carries a two-prime reciprocal-identity non-two composite cost defect, then every ratio character that carries the two-adic axis twist still admits one of the two canonical local orientations at the mixed composite $2\cdot 3$.
background
In the Primitive Recognition Calculus native-cost uniqueness development, ratio characters are maps on ratio orbits obeying the PRC character axioms. The two-adic axis-twist branch is the residual obstruction after pure two-power calibration: characters that twist along the 2-adic axis must still be constrained at the first mixed composite $2\cdot 3$.
The positive target asserts that every such axis-twist character chooses one of the two canonical local orientations at that composite. Its failure character is the constructive countermodel surface. Separately, the calibrated non-two composite cost-defect character is the Pass-95-style blocker rewritten so the defect is visible in composite $J$-cost (the RS cost $J(x)=(x+x^{-1})/2-1$).
Upstream, the target is definitionally equivalent to the negation of the failure character, and the failure character is absurd once the calibrated composite cost-defect model is denied.
proof idea
Term-mode one-liner. From the hypothesis that the calibrated two-prime reciprocal-identity non-two composite cost-defect character does not exist, apply the absurdity lemma to conclude there is no $2\cdot 3$ composite-local orientation failure character. Feed that into the reverse direction of the biconditional equating the positive two-adic axis-twist orientation target with the negation of the failure character.
why it matters
This discharges the positive $2\cdot 3$ composite-local orientation target under the natural no-defect hypothesis, converting the Pass-95 cost-defect blocker into a usable orientation constraint on the two-adic branch. Downstream it is consumed by the conditional universal-foundation certificate in UniversalFoundation, which packages kernel, real-complete ordered field, and trace-logic certificates into one PRC foundation bundle.
In the broader RS forcing picture this sits inside native-cost uniqueness for ratio characters: ruling out composite $J$-cost defects at mixed primes is part of forcing the unique cost functional (T5 $J$-uniqueness) and keeping the discrete recognition calculus aligned with the Recognition Composition Law before dimension and octave forcing (T7–T8).
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