PRCPrimeCalibrationForcesPrimeIdentityCommonTraceExtensionTarget_of_trace_coherence
plain-language theorem explainer
Prime calibration that forces identity-orientation trace coherence also forces the sharper common finite δ-trace extension property on every calibrated ratio character. Native-cost uniqueness and universal-foundation certificates cite this implication when collapsing two residual targets. The proof is a pointwise one-line application of the character-level coherence-to-extension lemma.
Claim. Assume that every ratio character $\chi$ that is prime-direction calibrated is identity-orientation trace-coherent. Then every such $\chi$ also respects an explicitly witnessed common finite $\delta$-trace extension (the sharper trace-transport target).
background
In the Primitive Recognition Calculus, a ratio character is a map on ratio orbits that encodes orientation data for the native cost. Prime-direction calibration pins how that character behaves on native prime axes. Two residual targets remain after calibration: identity-orientation trace coherence (identity orientation propagates across prime axes), and a sharper common-trace-extension form (identity orientation respects an explicitly witnessed finite $\delta$-trace extension).
The module packages both as universal statements over calibrated characters. Upstream, the character-level lemma already shows that trace coherence of a single character implies that character respects a common finite $\delta$-trace extension, via a comparable-trace intermediate. The present declaration lifts that implication from one character to the quantified calibration targets.
proof idea
Term-mode, essentially a one-line wrapper. Introduce a ratio character $\chi$ together with the ratio-character and prime-direction-calibration hypotheses. Apply the ambient trace-coherence target at $(\chi,h\chi,h\mathrm{prime})$ to obtain character-level identity-orientation trace coherence. Feed that into the upstream lemma that converts character-level trace coherence into respect for a common finite $\delta$-trace extension. Discharge.
why it matters
This is the forward half of the equivalence between the residual trace-coherence target and the sharper common-trace-extension target; the sibling converse closes the iff. That equivalence lets the native-cost uniqueness blocker certificate treat the two residual obligations as interchangeable when assembling the uniqueness package. The same target also feeds the conditional universal-foundation certificate, which packages kernel, real-complete ordered field, and trace-logic obligations. In the Recognition forcing chain this sits inside the foundation layer that pins the native cost before J-uniqueness (T5) and the $\phi$ fixed point (T6) are used downstream.
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