PRCPrimeCalibrationForcesPrimeIdentityCommonTraceExtensionTarget_of_comparable_trace
plain-language theorem explainer
If prime calibration already forces identity orientation to respect comparable finite δ-orbit traces, then it also forces respect for explicitly witnessed common finite δ-trace extensions. Native-cost uniqueness and prime-calibration propagation cite this implication when collapsing two orientation targets. The proof is a pointwise intro that hands the comparable-trace hypothesis to the character-level transport lemma.
Claim. Assume that every ratio-orbit character $\chi$ that is prime-direction calibrated has identity orientation respecting comparability of finite $\delta$-orbit traces. Then every such $\chi$ has identity orientation respecting an explicitly witnessed common finite $\delta$-trace extension.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits used to build a native cost. Prime-direction calibration constrains how $\chi$ acts on prime orbit directions. Two sharper orientation targets ask that, once calibrated, the identity orientation of $\chi$ transport correctly along finite $\delta$-traces: one via mere comparability of traces, and one via an explicitly witnessed common finite extension.
The comparable-trace target is the hypothesis here: for every ratio character that is prime-direction calibrated, identity orientation respects comparability of finite $\delta$-orbit traces. The common-trace-extension target is the conclusion: the same characters must respect an explicitly witnessed common finite $\delta$-trace extension.
Upstream, the character-level lemma already shows that respecting comparable traces implies respecting common extensions, by instantiating comparability at orbitPositionTrace. This declaration lifts that implication from a single character to the universal calibration targets.
proof idea
Term-mode proof with a short intro. Fix a ratio character $\chi$ together with the ratio-character and prime-direction-calibration hypotheses. Apply the assumed comparable-trace target at $\chi$ to obtain that identity orientation respects comparable traces. Feed that fact into the character-level lemma PRCCharacterPrimeIdentityRespectsCommonTraceExtension_of_comparable_trace, which converts comparable-trace respect into common-trace-extension respect. No further case analysis.
why it matters
Native cost uniqueness in PRC needs prime calibration to pin identity orientation tightly enough that cost characters cannot drift under finite $\delta$-trace transport. This lemma collapses the common-extension target onto the comparable-trace target, so the uniqueness pipeline only has to discharge the latter.
Downstream it feeds PRCPrimeCalibrationPropagationTarget_of_sharpened_orientation, which assembles local orientation targets into global prime-calibration propagation, and appears in the dependency cone of prc_native_cost_uniqueness_blocker_certificate. In the broader Recognition forcing chain this is bookkeeping inside the J-cost uniqueness story (T5), not a new physical constant, but it removes a redundant orientation obligation before cost factorization is certified.
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