PRCTwoThreeCompositeLocalOrientationForTwoAdicAxisTwistTarget_of_prime_identity_branch_uniformity
plain-language theorem explainer
If prime calibration forces identity-branch uniformity on native prime axes, then every ratio character that twists the two-adic axis still picks a canonical local orientation at the first mixed composite 2·3. Foundation workers closing the two-adic branch blocker cite this bridge into the composite-local orientation target. The proof is a two-step term composition: uniformity kills two-adic axis-twist characters, and their absence discharges the target vacuously.
Claim. Assume every prime-direction-calibrated ratio character is identity-branch-uniform on native prime axes. Then every ratio character carrying a two-adic axis twist still chooses one of the two canonical local orientations at the first mixed composite $2\cdot 3$.
background
In the Primitive Recognition Calculus native-cost uniqueness module, ratio characters are maps on ratio orbits that encode multiplicative branch data for the cost. A two-adic axis twist means the character flips the native 2-axis off the identity branch. The positive $2\cdot 3$ composite-local orientation target says any such twisted character must still select one of the two canonical local orientations at the first mixed composite $2\cdot 3$; it is the constructive positive form of the current two-adic branch blocker.
The hypothesis is the trace-free branch-uniformity target: prime calibration should force every identity-oriented native prime axis to put all native prime axes on the identity branch. Upstream, that uniformity already implies there is no ratio character with a two-adic axis twist at all. A separate one-line lemma then turns absence of such twisted characters into the composite-local orientation target by vacuous truth.
proof idea
Pure term composition of two in-module lemmas. First apply the absurdity theorem: prime identity-branch uniformity yields $\neg$ (existence of a two-adic axis-twist ratio character). Feed that negation into the discharge lemma that assumes no such twisted character and returns the $2\cdot 3$ composite-local orientation target. That discharge lemma is itself vacuous: any purported twisted character would contradict the negation, so the universal claim holds. No extra case analysis or arithmetic.
why it matters
Closes one conditional edge on the two-adic branch blocker inside native-cost uniqueness: under prime identity-branch uniformity, the positive $2\cdot 3$ composite-local orientation obligation is secured. Downstream it is consumed by the universal foundation conditional certificate, which packages kernel, real-complete ordered field, and trace-logic certificates into a single foundation bundle. In the broader Recognition forcing chain this sits under native J-cost uniqueness (T5 landmark): ruling out rogue two-adic twists is part of forcing the cost to the unique $J(x)=(x+x^{-1})/2-1$ shape. The remaining open load is discharging the uniformity hypothesis itself rather than this bridge.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.