PRCPrimeCalibrationForcesMixedNonunitIdentityWitnessReflectsPrimeWitnessTarget_proved
plain-language theorem explainer
Under prime-direction calibration, every PRC ratio character has mixed non-unit identity witnesses that reflect prime witnesses. Native-cost uniqueness and the mixed-witnesses split target cite this identity half. The proof feeds two already-proved calibration consequences (orbit-product display compatibility and local prime orientation) into a local reflection lemma.
Claim. For every map $\chi$ on ratio orbits that is a PRC ratio character and is prime-direction calibrated, the mixed non-unit identity witness of $\chi$ reflects the corresponding prime witness.
background
In Primitive Recognition Calculus, a ratio character is a structure-preserving map $\chi$ on ratio orbits. Native cost is recovered from such characters via doubled-trace and d'Alembert-type identities; uniqueness arguments therefore force characters to obey rigid calibration and reflection laws.
Prime-direction calibration fixes how $\chi$ acts on prime directions. The target proved here is the identity half of mixed-context reflection: in mixed non-unit settings, identity-side witnesses must mirror the prime witness of $\chi$. The reciprocal half is a sibling target.
Upstream, a local lemma already derives the reflection property from two structural hypotheses: orbit-product display compatibility and local prime orientation. Separate proved targets show that prime calibration forces both of those hypotheses.
proof idea
Term-mode discharge of the $\forall$-target. Introduce the character $\chi$, the ratio-character hypothesis, and prime-direction calibration. Apply the local reflection lemma, supplying (i) orbit-product display compatibility from the corresponding proved calibration target and (ii) local prime orientation from its proved calibration target. No further casework.
why it matters
Closes the identity half of mixed non-unit witness reflection under prime calibration. The immediate parent packages this with the reciprocal half into the mixed-witnesses split target. That split feeds the native-cost uniqueness blocker certificate, which records factorization and signed-admissible refutation obligations for uniqueness of the PRC native cost. The same proved target is listed in the universal-foundation conditional certificate chain. In the broader RS forcing picture this is bookkeeping inside cost uniqueness (toward J-uniqueness / T5), not a new physical constant law.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.