The Embedding Required by the Koide Saturation Lemma A Clarification Question for the A3 Jones Subfactor

NOTE ON THE EARLIER FORMULATION: An earlier formulation of the open step in this programme imposed full S₃-invariance on a single defect operator, which forces equal channel weights and zero fluctuation; the correct object is an S₃-covariant family. With that correction in place, the conditional Koide bridge of TSCT primer v2 reduces a structural account of Koide's lepton mass relation to a single open step: a Saturation Lemma pinning the amplitude of a standard S₃-fluctuation in an equal-trace three-channel algebra B = C³ to the value A² = 2. This note observes that the lemma as stated is not yet a well-formed mathematical question. The norm lives in B, whereas the Jones index M: N = 2 belongs to the separate inclusion N ⊂ M of the A₃ subfactor. With no morphism between them, the proposed identity is a numerical coincidence rather than a derivable statement. The substantive question is whether there exists a natural construction - planar-algebraic, standard-invariant, or conditional-expectation in form - connecting B to the A₃ tower in a way that picks out A² = 2 from the subfactor-theoretic structure rather than choosing it by hand. This is the right specialist question. The note is offered as a clarification of the open step, not as a classification of which constructions succeed. Comments, references, and resolutions welcome. Contact: david. m. sparks@proton. me