PRCPrimeCalibrationForcesNonunitIdentityComparableTraceTarget_refuted
plain-language theorem explainer
Prime calibration does not force identity orientation to respect comparability of finite δ-orbit traces on nonunit directions. Anyone auditing the native-cost uniqueness blocker chain or the PRC universal-foundation certificate cites this refutation. The proof is a short reduction: the target is equivalent to prime-floor successor transport, already refuted upstream.
Claim. It is false that every ratio-orbit character $\chi$ that is prime-direction calibrated must have identity orientation respecting comparability of finite $\delta$-orbit traces on nonunit directions. Equivalently, the trace-order sharpening of nonunit identity-branch transport under prime calibration fails.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ assigns to each ratio orbit another orbit and is meant to encode admissible cost data. Prime-direction calibration restricts how $\chi$ behaves on prime generators. The target proposition asserts that any such calibrated character must make the identity branch respect comparability of finite $\delta$-orbit traces on nonunit directions: a trace-order sharpening of nonunit identity-branch transport.
That target is definitionally a universal quantification over characters: character plus prime calibration implies the comparable-trace identity condition. An in-module equivalence identifies it with the prime-floor successor transport target (forcing transport along floor-successor steps on primes).
The local module develops native-cost uniqueness for PRC: which calibration hypotheses actually pin down the cost functional versus which over-strong transport claims fail. This declaration sits in the refutation spine that clears false uniqueness routes.
proof idea
Assume the comparable-trace identity target. Apply the forward direction of the recorded equivalence with prime-floor successor transport to obtain that weaker (but still false) transport target. Discharge by the already-proved refutation of prime-floor successor transport, which itself reduces further to the failure of prime-identity witness globalization on nonunits. Pure modus tollens along the iff chain; no new analytic work.
why it matters
This closes one concrete false route in native-cost uniqueness: prime calibration alone does not force identity orientation to respect nonunit comparable traces. Downstream, the nonunit identity-branch transport target and the nonunit orbit-orientation local comparable-trace target are both refuted by reducing to this lemma. The native-cost uniqueness blocker certificate aggregates such proved and refuted targets; the universal-foundation conditional certificate consumes that blocker spine. In the broader RS forcing picture this is housekeeping inside PRC cost uniqueness, not a T5–T8 landmark, but it prevents an over-strong calibration claim from being treated as a uniqueness theorem.
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