Projected Yukawa Operator and Chiral Polar Factor

This note takes the second step of the mass-sector frontier of the fermionic-matter sub-programme of Cosmochrony, and draws a sharp negative conclusion at its entry. The companion line note fixed the determinant line Lₘ = ^2 (S_{}) as the Yukawa coupling line and the generation-level assignment from E_{}^2|₂^{₃₆₄₍} = diag (1, 12 + u, 12 - u). We ask whether the squared projective residue determines the Yukawa morphism itself, and show that it does not. The generation-level observable produced by the Schur-residue sector is the positive hermitian square H_{}: = Y_{}^Y_{}, whose spectrum on C^{3₆₄₍} is fixed up to the overall Yukawa norm by H_{}|₂^{₃₆₄₍} = ₘ^2\, diag (1, 12 + u, 12 - u). The morphism Y_{} is then determined only up to a chiral polar factor, Y_{} = U_{}\, H_{}^1/2, with U_{} a unitary chiral map on the support of H_{}. Hence the squared residue fixes the squared Yukawa levels but not the morphism: U_{} carries the chiral orientation, the relative generation phases, and the mixing, while the positive scale ₘ is a separate normalisation of H_{} — none of it visible to the dimensionless level operator E_{}^2|₂^{₃₆₄₍}. Front 3b is therefore closed as a squared-Yukawa-levels result, not a mixing result. A further analysis characterises U_{} not as a canonical matrix but as a rephasing class, the double quotient U_{} U (1) ^3ₑ U (3) / U (1) ^3₋, whose invariants are the moduli | (U_{}) ₈₉| and a Jarlskog phase — three mixing angles and one CP phase — and whose non-triviality U_{} I is controlled exactly by the transverse component of the metaplectic generator. On the derived real cascade that transverse component is absent, so U_{} = I and no physical mixing is produced at this stratum; the only route left open to a non-trivial class is a genuine complex non-central metaplectic phase, not derived here, observable only relative to a second fermionic sector. Beyond the mixing question, the same stratum settles the generation-scale question negatively: every depth read from the static admissibility spectrum — crossing, saturation, capacity cell, quantised count, and quantised resolution density — yields only order-one ratios or re-introduces the target scale through an unforced normalisation, so the static spectral stratum fixes the generation structure and the order-one level split, but not the charged-fermion hierarchy. All operator identities are verified by exact symbolic computation, with no sampling.