PRCTwoThreeCompositeLocalOrientationForTwoAdicAxisTwistTarget_of_no_failure_character
plain-language theorem explainer
If no ratio character witnesses a 2·3 composite-local orientation failure under two-adic axis twist, then every such character is forced to pick a canonical local orientation at the mixed composite 2·3. Native-cost uniqueness and fork-certificate authors cite this to discharge the positive branch from a negated countermodel. The proof is a one-line mpr application of the packaged equivalence.
Claim. If there is no ratio character $\chi$ that is a PRC ratio character, carries a two-adic axis twist, and fails $2\cdot 3$ composite-local orientation, then every ratio character that carries a two-adic axis twist satisfies $2\cdot 3$ composite-local orientation.
background
In the Primitive Recognition Calculus, ratio characters are maps on ratio orbits that preserve the multiplicative structure used to build the native cost. The two-adic axis twist is the branch where the character sends the orbit of $2$ to its reciprocal branch rather than the identity. The mixed composite $2\cdot 3$ is the first place where a two-adic twist can interact with a distinct native prime.
The positive target asserts that every ratio character with two-adic axis twist still chooses one of the two canonical local orientations at that composite. The failure character is the constructive countermodel surface: existence of a character that twists the two-adic axis and fails that local orientation. These two propositions are definitionally dual (universal positive form versus existential witness form).
The surrounding module develops native-cost uniqueness from doubled-trace and d'Alembert constraints on the J-cost side of the Recognition Composition Law, so this local orientation statement is a discrete branch blocker inside that uniqueness argument.
proof idea
One-line term proof. Apply the reverse direction (.mpr) of the already-proved biconditional equating the positive $2\cdot 3$ composite-local orientation target with the negation of the failure-character witness. The hypothesis is exactly that negation, so the target follows immediately.
why it matters
This lemma is the positive-branch half of the exact $2\cdot 3$ two-adic fork. Downstream, prcTwoThreeCompositeLocalForkCertificate packages the constructive branch (failure / ratio twist / calibrated twist) and the positive branch (local orientation as negation of each equivalent witness) under one name so certificate wiring need not repeat the equivalence chain.
That fork certificate feeds the conditional universal-foundation certificate, which assembles kernel, real-complete ordered field, and trace-logic pieces into a single PRC foundation bundle. In the broader Recognition forcing chain the statement is local bookkeeping on the two-adic mixed composite, not a T5–T8 landmark, but it keeps the native-cost uniqueness path free of an unoriented $2\cdot 3$ countermodel.
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