PRCPrimeCalibrationForcesPrimeIdentityCanonicalAddTraceTarget_iff_common_trace_extension
plain-language theorem explainer
Equates two formulations of the prime-calibration identity-orientation target: transport through the canonical additive orbit-position trace versus respect for an explicitly witnessed common finite δ-trace extension. Anyone closing native-cost uniqueness or the universal-foundation certificate cites this bridge. The proof is a two-constructor packing of the already-proved one-way implications.
Claim. The assertion that every prime-direction-calibrated ratio character forces identity orientation to transport through the concrete finite common extension $\mathrm{orbitPositionTrace}(p+r)$ is equivalent to the assertion that every such character forces identity orientation to respect an explicitly witnessed common finite $\delta$-trace extension.
background
In the Primitive Recognition Calculus, ratio characters are maps $\chi$ on ratio orbits that encode orientation and scaling data for the native cost. Prime-direction calibration is the hypothesis that $\chi$ is aligned on prime generators; under that hypothesis one wants identity orientation to be forced along finite trace comparisons.
Two Prop-level targets package that demand. The common-trace-extension target asks that identity orientation respect an explicitly witnessed common finite $\delta$-orbit trace extension (the sharper form). The canonical-add-trace target asks the same for the concrete finite common extension given by $\mathrm{orbitPositionTrace}(p+r)$.
Both sit inside the native-cost uniqueness development, where doubled-trace and d'Alembert structure are used to pin the cost character. The module already has one-way bridges between the two targets; this declaration records that they are equivalent.
proof idea
Term-mode Iff packing. The forward direction applies the existing lemma that any character satisfying the canonical-add-trace target satisfies the common-trace-extension target (by reducing the witnessed-extension obligation to the canonical additive orbit-position trace). The reverse direction applies the dual lemma, which lifts a common-trace-extension witness to the canonical-add-trace obligation. No new character-level reasoning occurs here.
why it matters
Native-cost uniqueness needs a single, stable statement of "prime calibration forces identity orientation on traces." Having two equivalent targets lets later certificates choose the formulation that matches the available lemmas (canonical additive traces versus abstract common extensions) without changing the logical content.
Downstream, the native-cost uniqueness blocker certificate and the universal-foundation conditional certificate both sit on this uniqueness spine. The equivalence is bookkeeping rather than a new physical claim, but it closes a formulation gap on the path from ratio-character calibration to uniqueness of the native cost (the J-cost side of the forcing chain). It does not itself discharge either target.
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