PRCPrimeCalibratedTwoAdicAxisTwistCharacter_of_ratio_character_axis_twist
plain-language theorem explainer
Any ratio character that implements two-adic axis-twist branch behavior is automatically prime-direction calibrated, so the uncalibrated twist target upgrades to the calibrated model. Native-cost uniqueness and character-rigidity arguments cite this to collapse the two existence statements. The proof unpacks the existential and applies the automatic calibration lemma on the twist hypothesis.
Claim. If there exists a ratio-orbit map $\chi$ that is a ratio character and satisfies the two-adic axis-twist branch condition, then there exists a ratio character that is both prime-direction calibrated and two-adic axis-twisting. Equivalently, the uncalibrated two-adic axis-twist construction target implies the prime-calibrated two-adic axis-twist model.
background
In the Primitive Recognition Calculus native-cost uniqueness development, cost candidates are probed through ratio characters: maps $\chi$ on ratio orbits that preserve the multiplicative structure used to read off doubled-trace and $J$-cost data. Two special branch conditions appear. The two-adic axis twist asks that $\chi$ flip or reorient the distinguished two-orbit axis in a controlled way. Prime-direction calibration asks that, on every prime orbit, $\chi$ act as the identity (or the calibrated native direction).
The uncalibrated target is the existence of a ratio character carrying only the two-adic axis twist. The calibrated target strengthens this by also requiring prime-direction calibration. Pass 115 records that the calibration field is not independent once the twist is carried by a genuine ratio character.
The upstream lemma states exactly that: any character satisfying the two-adic axis-twist condition is automatically prime-direction calibrated (proved by case split on whether the prime is the two-orbit).
proof idea
Term-mode packing after a single destructuring. Unpack the hypothesis as an existential triple $(\chi, h_\chi, h_{\mathrm{branch}})$: a ratio character together with the two-adic axis-twist witness. Feed $h_{\mathrm{branch}}$ to the upstream lemma that two-adic axis twist implies prime-direction calibration, obtaining the missing calibration field. Repack $(\chi, h_\chi, \mathrm{calibration}, h_{\mathrm{branch}})$ as a witness for the calibrated existence statement. No further arithmetic is required.
why it matters
This is the upgrade half of the equivalence between the uncalibrated and prime-calibrated two-adic axis-twist targets; the matching iff theorem is assembled from this arrow and its converse. Downstream, it turns any construction of an uncalibrated twist (including the two-three local orientation-failure character) into a calibrated countermodel, which is the native valuation route to refuting the current character-rigidity branch.
Several force-target negation lemmas (prime-identity, prime-pair product cost consistency, two-prime mixed composite cost consistency) route through the uncalibrated hypothesis and rely on this upgrade, or on the absurdity form that no calibrated twist implies no uncalibrated twist. The universal foundation conditional certificate also depends on the same bridge. In the broader Recognition forcing chain this sits inside native $J$-cost uniqueness (T5-adjacent character rigidity), not yet at $\varphi$ or $D=3$.
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