PRCPrimeCalibrationForcesPrimeIdentityComparableTraceTarget_iff_prime_floor_successor_transport
plain-language theorem explainer
Prime calibration of ratio characters forces identity orientation to respect comparable finite δ-orbit traces if and only if it forces successor transport above the self-reciprocal unit floor. Native-cost uniqueness and universal-foundation certificates cite this bridge between the sharper trace-order target and the corrected successor target. The proof is a pure Iff.trans of two already-proved intermediate equivalences.
Claim. The following are equivalent for the primitive recognition calculus: (i) every prime-direction-calibrated ratio character has identity orientation that respects comparability of finite $\delta$-orbit traces; (ii) every such character has identity successor transport on orbits lying above the self-reciprocal unit floor (not additive transport out of the unit orbit itself).
background
In the primitive recognition calculus, a ratio character $\chi$ is a map on ratio orbits that encodes orientation data for the native cost. Prime-direction calibration restricts $\chi$ on prime axes. Two sharper forcing targets are compared here.
The first target says that, under prime calibration, identity orientation must respect comparability of finite $\delta$-orbit traces: if two finite traces are comparable, the character cannot flip identity inconsistently. The second is the corrected successor target after the reciprocal-character check: calibration should force successor transport above the self-reciprocal unit floor, rather than additive transport leaving the unit orbit.
Upstream, the prime identity-comparable target is already equivalent to a nonunit identity-comparable intermediate, and that intermediate is equivalent to the prime-floor successor-transport target. This declaration closes the remaining link in that chain.
proof idea
Term-mode one-liner: compose two prior equivalences by Iff.trans.
First apply the equivalence of the prime identity-comparable trace target with the nonunit identity-comparable intermediate. Then apply the equivalence of that nonunit intermediate with the prime-floor successor-transport target. No new case analysis; the result is pure transitivity of $\leftrightarrow$.
why it matters
Native-cost uniqueness in Recognition Science needs a clean dictionary among blocker targets: trace-order identity, nonunit identity, and successor transport above the unit floor. This iff is that dictionary entry for the prime-calibrated case.
It is consumed by the native-cost uniqueness blocker certificate and by the conditional universal-foundation certificate. Those packages assemble kernel, ordered-field, and trace-logic facts into a single foundation claim. Aligning the sharper trace-order target with the corrected successor target removes an ambiguity that would otherwise leave the uniqueness blocker ill-posed.
In the broader forcing chain this sits inside the primitive recognition calculus that underwrites J-uniqueness and the native cost before T5–T8 landmarks are invoked; it does not itself force $\phi$ or $D=3$, but it keeps the cost-character side coherent for those steps.
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