PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget_of_common_trace_extension
plain-language theorem explainer
Prime calibration that forces identity orientation to respect a common finite δ-trace extension also forces identity orientation to propagate across prime axes. Anyone closing native-cost uniqueness or the universal-foundation certificate cites this implication. The argument is a pointwise one-line application of the character-level common-extension-to-coherence lemma.
Claim. Assume that every ratio character $\chi$ that is prime-direction calibrated respects a common finite $\delta$-trace extension under identity orientation. Then every such $\chi$ is identity-trace-coherent across prime axes: identity orientation on a native prime axis propagates to all native prime axes.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits encoding how multiplicative structure is read. Prime-direction calibration pins the orientation of native prime axes. Identity-trace coherence asks that once identity orientation is fixed on one native prime axis, it propagates across the other prime axes.
The sharper common-trace-extension target strengthens that demand: identity orientation must respect an explicitly witnessed common finite $\delta$-trace extension (canonically via the orbit-position trace of a sum of primes). The module develops native-cost uniqueness by forcing characters that match the native cost to be the identity branch under these calibration hypotheses.
Upstream, the character-level lemma states that any single character that already respects the common finite $\delta$-trace extension is identity-trace-coherent. The present declaration lifts that implication from one character to the quantified calibration targets.
proof idea
One-line wrapper. Introduce a ratio character $\chi$ together with the ratio-character and prime-direction-calibration hypotheses. Apply the assumed common-trace-extension target at $\chi$ to obtain that $\chi$ respects the common finite $\delta$-trace extension. Feed that witness into the character-level lemma PRCCharacterPrimeIdentityTraceCoherent_of_common_trace_extension, which returns identity-trace coherence for $\chi$.
why it matters
This is one direction of the equivalence between the exact remaining trace-coherence target and the sharper common-trace-extension target. That equivalence is recorded immediately downstream and is the clean interface used when packaging native-cost uniqueness.
It feeds prc_native_cost_uniqueness_blocker_certificate, which assembles the proved factorization and refutation legs of the uniqueness blocker, and appears in the conditional universal-foundation certificate in UniversalFoundation. In the Recognition forcing chain this sits inside the foundation layer that pins the native cost before J-uniqueness (T5) and the self-similar fixed point $\phi$ (T6) are consumed by later physics.
No open scaffold remains on this arrow: both the character lemma and the target implication are proved.
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