PRCPrimeCalibrationForcesPrimeIdentityCanonicalAddTraceTarget_iff_trace_transport
plain-language theorem explainer
Equivalence of two prime-calibration targets for identity orientation on ratio characters: the canonical additive finite orbit-trace form and the native prime-axis trace-transport form. Cited by anyone collapsing native-cost uniqueness blockers or packaging the PRC foundation certificate. Proof is a pure biconditional package of the two already-proved one-way implications.
Claim. The assertion that every prime-direction-calibrated ratio character has identity orientation respecting canonical additive finite $\delta$-orbit traces is equivalent to the assertion that every such character has identity orientation invariant along native prime-axis trace connections.
background
In the Primitive Recognition Calculus, ratio characters $\chi$ act on ratio orbits and encode orientation data used to build native cost. Prime-direction calibration is the constraint that $\chi$ aligns with the preferred prime-axis orientation. Two residual targets ask whether that calibration already forces identity orientation to travel along finite $\delta$-traces.
The smaller target requires identity orientation to be invariant under native prime-axis trace connectivity (the connectivity of that graph is already established upstream). The sharper target requires the same for the concrete common finite extension given by the orbit-position trace of $p+r$. Both are universal statements over calibrated ratio characters.
This module sits in the native-cost uniqueness development: the goal is to force the cost functional uniquely from character and trace data, so that competing orientations cannot survive prime calibration.
proof idea
Term-mode biconditional. The forward direction applies the lemma that canonical-add-trace respect implies trace-connected respect; the reverse applies the lemma that trace-connected respect implies canonical-add-trace respect. Each of those lemmas is a one-line intro-and-specialize over a calibrated character. No new analytic or combinatorial work occurs here: the two Props are identified by packaging the mutual implications.
why it matters
Collapses two named residual targets into one logical obligation inside native-cost uniqueness. Downstream, the native-cost uniqueness blocker certificate and the conditional universal-foundation certificate both sit on this uniqueness spine; equating the targets prevents double-counting and lets either formulation discharge the other. In the broader RS forcing picture this is bookkeeping on the cost side of the foundation (before J-uniqueness and the $\phi$ fixed point), not a new physical constant claim. It closes a scaffolding fork: prove one transport statement and the other comes free.
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