PRCPrimeCalibrationForcesPrimeIdentityTraceTransportTarget_of_two_prime_identity_trace_connected
plain-language theorem explainer
Assuming prime calibration forces identity at the orbit-2 prime axis to transport along any finite δ-trace link to a native prime axis, the same calibration forces identity orientation to stay invariant along native prime-axis trace connections. Cited by the two-target equivalence, the native-cost uniqueness blocker certificate, and the universal-foundation conditional certificate. Proof routes through the reciprocal-branch exclusion ladder, then combines two-prime identity transport with the forced two-prime identity map at character level.
Claim. If every ratio character that is prime-direction calibrated has identity orientation at the orbit-$2$ prime axis transported along any finite $\delta$-trace connection to a native prime axis, then every such character has identity orientation invariant along native prime-axis trace connections.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits encoding orientation (identity versus reciprocal) along prime axes. Prime-direction calibration pins how $\chi$ sits on native prime axes. A $\delta$-trace connection is a finite chain of elementary trace steps linking two axes in the prime-axis trace graph; structural connectivity of that graph is already established upstream.
Two target propositions sit in this module. The two-prime identity trace-connected target asks that, under prime calibration, identity at the orbit-$2$ axis transport along any finite $\delta$-trace link to a native prime axis. The smaller prime-identity trace-transport target asks only that identity orientation be invariant along native prime-axis trace connections. Pass 81 isolates the two-prime form as the same blocker as reciprocal trace transport, seen through reciprocal twist.
Upstream character-level glue already shows: if a fixed $\chi$ respects two-prime identity along trace connections and identity at any calibrated prime forces identity at orbit $2$, then $\chi$ respects prime-identity along native prime-axis trace connections.
proof idea
Term-mode chain on the hypothesis $h_{\mathrm{two}}$. First twist $h_{\mathrm{two}}$ into the two-prime reciprocal trace-connected target, then lift that to the reciprocal-forces-prime-reciprocal target, then to the two-reciprocal-excludes-prime-identity target, and finally to the one-sided target that prime calibration forces identity at orbit $2$ from identity at any calibrated prime.
Specialize both $h_{\mathrm{two}}$ and that forced-two target at a calibrated character $\chi$. Finish by applying the character-level lemma that two-prime identity trace respect plus forced two-prime identity yields full prime-identity trace respect. No new analytic work: pure target-to-target transport through the reciprocal exclusion ladder.
why it matters
Closes one direction of the equivalence between the two-prime identity trace-connected target and the smaller prime-identity trace-transport target, so the two blockers may be used interchangeably. That equivalence and this implication feed the native-cost uniqueness blocker certificate, which packages factorization and signed-admissible refutation facts for the uniqueness program.
Downstream it also appears in the universal-foundation conditional certificate (kernel, real complete ordered field, and trace-logic bundle). In the Recognition forcing chain this sits inside native $J$-cost uniqueness work that underwrites T5-style cost uniqueness: only the calibrated identity orientation may survive along prime-axis traces, blocking reciprocal contamination of the native cost character.
No open scaffold remains on this arrow; the claim is fully proved and only the broader uniqueness certificate still aggregates sibling blockers.
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