PRCCharacterPrimeIdentityRespectsCanonicalAddTrace_of_trace_connected
plain-language theorem explainer
If a ratio-orbit character transports identity orientation along any finite δ-trace link between prime axes, then it transports identity through the canonical common extension orbitPositionTrace(p+r). Anyone proving native-cost uniqueness or prime-calibration forcing cites this reduction. The proof is a one-line specialization: every pair of prime axes is already known to be trace-connected via the sum orbit.
Claim. Let $\chi$ be a map on rational orbits. Suppose that whenever two prime axes $p$ and $r$ lie in the same finite $\delta$-trace component and $\chi$ fixes the identity orientation of $p$, then $\chi$ also fixes the identity orientation of $r$. Then $\chi$ transports identity orientation through the canonical common extension: if $\mathrm{orbitPositionTrace}(p)$ and $\mathrm{orbitPositionTrace}(r)$ both extend into $\mathrm{orbitPositionTrace}(p+r)$ and $\chi$ fixes the identity of $p$, then $\chi$ fixes the identity of $r$.
background
In the Primitive Recognition Calculus, ratio orbits package a signed numerator over a nonzero distinction-nat denominator. Characters $\chi$ act on these orbits; the native-cost uniqueness program asks which characters can arise from a cost functional built from the J-cost (the unique solution of the Recognition Composition Law).
Two related transport properties appear. Trace-connected transport says identity orientation moves along any finite $\delta$-trace witness linking two prime axes. Canonical-add-trace transport specializes the witness to the concrete common extension $\mathrm{orbitPositionTrace}(p+r)$, removing arbitrary choice of intermediate trace.
Upstream, every pair of prime axes is already known to be trace-connected: PRCPrimeAxisTraceConnected_proved supplies the witness $\mathrm{orbitPositionTrace}(p+r)$ together with the two extension facts. That theorem is what collapses the general connectedness hypothesis onto the canonical-add form.
proof idea
Term-mode specialization. Introduce primes $p,r$, the two extension hypotheses for $\mathrm{orbitPositionTrace}(p+r)$, and the identity-orientation hypothesis on $p$. Discharge the goal by applying the assumed trace-connected transport property at $(p,r)$, feeding it the universal connectedness witness PRCPrimeAxisTraceConnected_proved p hp r hr (which is exactly the sum-orbit common extension) together with the identity hypothesis. No further algebraic work.
why it matters
This is one direction of the equivalence between the two transport formulations (..._iff_trace_connected). Downstream, prime-calibration forcing uses it to upgrade a general trace-transport target into the canonical-add-trace target: once calibration forces identity transport along any connected pair, the canonical sum-orbit form follows immediately. The same lemma is wired into the native-cost uniqueness blocker certificate, which packages the remaining open factorization targets for the uniqueness program. In the broader RS forcing chain this sits under native J-cost uniqueness (T5), ensuring characters that respect the discrete δ-trace geometry cannot flip prime-axis identity orientation when axes merge by addition.
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