PRCPrimeCalibrationForcesPrimeIdentityBranchUniformityTarget_not_of_ratio_character_axis_twist
plain-language theorem explainer
If a ratio character carrying two-adic axis twist exists, the prime-calibration identity-branch-uniformity target is false. Native-cost uniqueness work that tests calibration-to-uniformity implications cites this. The proof is a one-line contrappositive application of the dual absurdity lemma.
Claim. Assume there exists a map $\chi$ on ratio orbits that is a ratio character and realizes two-adic axis twist. Then the following fails: every ratio character that is prime-direction calibrated is automatically prime-identity-branch uniform.
background
In the Primitive Recognition Calculus native-cost uniqueness development, cost candidates are probed through ratio characters: maps $\chi$ on ratio orbits obeying the character axioms (multiplicativity and related orbit laws). Prime-direction calibration asks that native prime axes sit in a preferred orientation relative to the identity branch. The branch-uniformity target then asserts that any such calibrated character forces every native prime axis onto the identity branch.
Separately, two-adic axis twist is an uncalibrated construction target: a ratio character that flips or twists the two-adic axis relative to the identity orientation. The module records that once this branch behavior is carried by a ratio character, prime-calibration fields become automatic in related passes, so the twist is a sharp countermodel probe.
An upstream lemma already shows the dual: if the uniformity target holds for all calibrated characters, then no two-adic axis-twist ratio character can exist. The present statement is the logical flip used when a concrete twist character is on hand.
proof idea
Term-mode proof by contradiction. Assume the uniformity target. Feed that hypothesis into the upstream absurdity lemma PRCTwoAdicAxisTwistRatioCharacter_absurd_of_prime_identity_branch_uniformity, which concludes there is no two-adic axis-twist ratio character. That directly contradicts the given twist hypothesis. No further case analysis or character arithmetic is needed.
why it matters
This is the bridge from a constructed two-adic twist character to an outright refutation of the trace-free branch-uniformity target. The immediate parent is the unconditional refutation theorem, which instantiates this lemma at the constructed twist witness and concludes the uniformity target is false.
In the Recognition Science forcing picture, native cost uniqueness is meant to pin the J-cost (the unique solution of the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$) and rule out exotic character branches. Refuting over-strong uniformity targets clarifies which calibration hypotheses are admissible and which overclaim identity-branch collapse for all prime axes. The result is local to the PRC character interface; it does not itself force T5 J-uniqueness, but it prunes a false lemma shape on the path toward native-cost rigidity.
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