PRCCharacterMixedNonunitWitnessesReflectPrimeWitnessesSplit_of_reflects
plain-language theorem explainer
From the composite mixed-nonunit reflection property for a ratio-orbit character, one obtains the split form in which identity and reciprocal nonunit witnesses each separately force a prime-axis witness. Anyone equating the bundled and branched reflection statements cites this direction. The proof is a two-branch constructor that projects the composite conclusion under a shared mixed antecedent.
Claim. Let $\chi$ be a map on rational orbits. Suppose that whenever mixed nonunit witnesses exist for $\chi$ (an identity-oriented nonunit orbit together with a reciprocal-oriented nonunit orbit), mixed prime-axis witnesses already exist. Then the same mixed-nonunit antecedent forces the identity prime-axis conclusion and the reciprocal prime-axis conclusion separately.
background
In the primitive recognition calculus, a rational orbit is an integer numerator over a nonzero distinction-nat denominator. Characters act on these orbits. Directional witnesses record whether $\chi$ fixes an orbit direction (identity orientation) or sends it to its reciprocal.
Mixed nonunit witnesses mean there is some nonunit, nonzero orbit that is identity-oriented and some (possibly different) nonunit orbit that is reciprocal-oriented. The composite reflection property says that such a mixed pair already forces mixed prime-axis witnesses: a prime orbit fixed by $\chi$ and a prime orbit sent to its reciprocal. That composite is the reverse direction not supplied by product propagation.
The split form factors the same idea into two independent implications, each still under the joint mixed-nonunit antecedent: identity nonunit data pulls back to a prime identity witness, and reciprocal nonunit data pulls back to a prime reciprocal witness.
proof idea
One-line structural split. Apply constructor to the conjunction that defines the split property. On each branch, introduce the mixed-nonunit hypothesis, apply the composite reflection hypothesis, and project the resulting pair with .1 (identity prime witness) or .2 (reciprocal prime witness). No arithmetic or orbit algebra is used.
why it matters
This is the forward half of the equivalence between composite and split mixed-nonunit reflection. Downstream, that equivalence is packaged as the iff theorem equating the two formulations, so later developments may switch packaging without changing content.
It also feeds the prime-calibration forcing lemma that lifts a composite reflection target to the corresponding split target: once calibration forces the bundled reflection, this projection yields the branched form used in native-cost uniqueness arguments. In the broader PRC uniqueness chain, reflection closes the reverse bridge from mixed nonunit data back to the prime axis, complementary to product propagation along the orbit monoid.
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