PRCPrimeCalibrationForcesTwoPrimeIdentityTraceConnectedTarget_iff_prime_identity_trace_transport
plain-language theorem explainer
The two-prime identity trace-connected target is logically equivalent to the smaller prime-identity trace-transport target. Anyone tracking native-cost uniqueness blockers or Pass-81 reciprocal-twist reductions cites this. The proof is a pure Iff packaging of the two already-proved one-way implications.
Claim. The assertion that every prime-direction-calibrated ratio character forces identity orientation at the orbit-$2$ prime axis to transport along any finite $\delta$-trace connection to a native prime axis is equivalent to the assertion that every such character forces identity orientation to be invariant along native prime-axis trace connections.
background
In the Primitive Recognition Calculus native-cost uniqueness development, ratio characters $\chi$ act on ratio orbits. Prime-direction calibration fixes orientation on native prime axes. Two related blocker targets ask whether that calibration forces identity orientation to survive finite $\delta$-trace transport.
The smaller target requires identity orientation to respect native prime-axis trace connections (structural connectivity of that graph is already proved). The two-prime target strengthens the starting point: identity at the orbit-$2$ prime axis must transport along any finite $\delta$-trace path to a native prime axis. Pass 81 isolates the latter as the same blocker as reciprocal trace transport, viewed through reciprocal twist.
Both targets are universal statements over ratio characters that are PRC ratio characters and prime-direction calibrated, concluding the corresponding identity-respects-trace-connected property.
proof idea
Term-mode Iff constructor. Left-to-right applies the already-proved implication from the two-prime target to the smaller transport target. Right-to-left applies the converse implication from the smaller transport target back to the two-prime target. No new arithmetic or character reasoning occurs here; the declaration only packages the two one-way lemmas into a single equivalence.
why it matters
Equating the two formulations lets the development refute either blocker by refuting the other. Immediately downstream, the two-prime target is refuted by transporting the already-known refutation of the smaller prime-identity transport target across this iff. That refutation feeds the native-cost uniqueness blocker certificate, which records which factorization and calibration targets are proved versus refuted. The same equivalence is consumed by the universal-foundation conditional certificate, tying the PRC native-cost uniqueness ledger into the broader foundation stack. In framework terms this sits inside the uniqueness path for the native cost (the J-cost lineage forced at T5), not a new physical constant derivation.
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