PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget_iff_common_trace_extension
plain-language theorem explainer
Equivalence of two remaining prime-calibration targets in the primitive recognition calculus: identity-orientation trace coherence across prime axes is the same as identity orientation respecting a common finite δ-trace extension. Anyone collapsing or refuting native-cost uniqueness blockers cites this bridge. The proof is pure bidirectional packaging of the two one-way transport lemmas.
Claim. The assertion that every prime-direction-calibrated ratio character has identity orientation trace-coherent across prime axes is equivalent to the assertion that every such character has identity orientation respecting a common finite $\delta$-trace extension.
background
In the primitive recognition calculus, a ratio character $\chi$ is a map on ratio orbits obeying the multiplicative character laws used to build native cost. Prime-direction calibration means $\chi$ is already fixed on the native prime axes in the preferred orientation. Two residual targets remain after that calibration.
The trace-coherence target asks that identity orientation propagate across all prime axes: once calibrated, $\chi$ must be identity-trace-coherent on those axes. The common-trace-extension target is the sharper transport form: identity orientation must respect an explicitly witnessed common finite $\delta$-trace extension (the concrete finite common extension along orbit-position traces).
Upstream, each direction is already proved separately: coherence implies the common-extension property by applying the pointwise transport lemma on each calibrated character, and conversely common extension implies coherence by the matching reverse transport lemma.
proof idea
Term-mode Iff constructor. The forward arrow is the already-proved implication from the trace-coherence target to the common-trace-extension target; the reverse arrow is the already-proved implication the other way. No new algebra: the theorem only packages those two one-way lemmas into a single $\leftrightarrow$.
why it matters
This bridge lets the native-cost uniqueness development treat the two residual prime-calibration targets as interchangeable. Downstream, the common-trace-extension target is refuted by reducing through this equivalence to the already-refuted trace-coherence target. The same equivalence feeds the native-cost uniqueness blocker certificate and, one module up, the conditional universal-foundation certificate.
In the Recognition forcing picture this sits inside the foundation layer that pins the native cost before J-uniqueness (T5) and the $\phi$ fixed point (T6) are invoked: clearing or certifying these prime-axis identity-transport targets is part of showing the cost functional is forced rather than chosen.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.