PRCPrimeCalibrationForcesMixedNonunitReciprocalWitnessReflectsPrimeWitnessTarget_proved
plain-language theorem explainer
Under prime-direction calibration, every ratio character reflects mixed nonunit reciprocal witnesses onto prime witnesses. Cost-uniqueness and universal-foundation certificates cite this as the reciprocal half of the mixed-context reflection package. The proof is a short term application of the local reflection lemma, feeding proved orbit-product compatibility and local prime orientation.
Claim. For every map $\chi$ on ratio orbits that is a ratio character and is prime-direction calibrated, the mixed nonunit reciprocal-witness reflection property holds: reciprocal mixed nonunit witnesses are reflected onto prime witnesses.
background
In the Primitive Recognition Calculus native-cost uniqueness module, ratio characters are structure-preserving maps $\chi$ on ratio orbits. Prime-direction calibration forces the character's action on prime directions to match a fixed orientation convention used throughout the uniqueness argument.
The target proved here is the reciprocal half of the mixed-context reflection package: in mixed nonunit contexts, reciprocal witnesses must reflect onto prime witnesses once calibration is assumed. The companion identity half is proved separately and later paired into a split target.
Upstream, local reflection already holds once orbit-product display compatibility and local prime orientation are available. Calibration is known to force both of those intermediate targets, so the global reciprocal reflection target reduces to assembling those facts.
proof idea
Term-mode proof. Introduce the character $\chi$, the ratio-character hypothesis, and prime-direction calibration. Apply the local lemma that reciprocal mixed nonunit reflection follows from orbit-product display compatibility plus local prime orientation. Discharge those two hypotheses by the already-proved calibration-forces-compatibility and calibration-forces-local-orientation theorems at $(\chi,h\chi,h\mathrm{prime})$.
why it matters
This closes the reciprocal half of mixed nonunit witness reflection under prime calibration. Immediately downstream it is paired with the identity half to discharge the split mixed-nonunit reflection target. That package feeds the native-cost uniqueness blocker certificate, which packages factorization and signed-admissible refutation facts used to pin uniqueness of the native cost. It also appears in the universal-foundation conditional certificate chain. In the broader Recognition forcing picture this is scaffolding inside the uniqueness route to the J-cost (T5) rather than a direct T5–T8 landmark, but without reciprocal reflection the character-to-cost identification cannot be sealed.
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