PRCPrimeCalibrationForcesPrimeIdentityComparableTraceTarget_refuted
plain-language theorem explainer
Prime-direction calibration of ratio characters does not force identity orientation to respect comparability of finite δ-orbit traces. Anyone tracking native-cost uniqueness blockers or the orbit-successor identity cascade will cite this. The proof is a one-line transport through an iff onto the already-refuted trace-coherence target.
Claim. It is false that every ratio character $\chi$ which is prime-direction calibrated has identity orientation that respects comparability of finite $\delta$-orbit traces. Equivalently, the sharper trace-order target (prime calibration forces identity to respect comparable finite orbit traces) fails.
background
In the Primitive Recognition Calculus, ratio characters $\chi$ assign orbit data on ratio orbits. Prime-direction calibration is a local orientation constraint on those characters at prime steps. The comparable-trace target asks that, once calibrated, identity orientation must respect order comparisons among finite $\delta$-orbit traces (doubled-trace style bookkeeping used for native cost).
A sibling target, trace coherence, packages a related global coherence demand on the same identity orientation. An in-module equivalence identifies the two targets: the comparable-trace formulation holds if and only if the trace-coherence formulation holds. Upstream, the trace-coherence target is already refuted by reduction to a branch-uniformity failure.
The local module develops native-cost uniqueness and systematically records which calibration-to-identity targets survive and which collapse.
proof idea
Assume the comparable-trace target. Apply the right-to-left direction of the in-module equivalence PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget_iff_comparable_trace to obtain the trace-coherence target. Discharge the assumption by the prior theorem that refutes trace coherence (itself a short reduction onto branch-uniformity refutation). Pure term-mode transport; no new analytic work.
why it matters
This closes one node in the PRC native-cost uniqueness blocker lattice: the sharper trace-order identity target is dead. Downstream, the orbit-successor identity target is refuted by reducing to this result, so one-step successor invariance of identity under prime calibration also fails. The native-cost uniqueness blocker certificate and the universal-foundation conditional certificate consume this family of refutations when assembling what remains open versus settled in the foundation layer.
In framework terms, the cascade shows that prime calibration alone does not pin identity orientation on orbit traces tightly enough for a uniqueness route through comparable finite traces. That forces later uniqueness arguments to use stronger hypotheses (zero-calibrated factorization paths that are proved, signed-admissible paths that are separately refuted) rather than this identity-from-prime-calibration shortcut.
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