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 ${L_{Y}} = \\wedge^{2}({\\mathcal{S}_{\\Pi}})$ as the Yukawa coupling line and the generation-level assignment from ${E_{\\Pi}}^{2}|_{{\\mathbb{C}^{3}_{\\mathrm{gen}}}} = {\\operatorname{diag}}(1, \\tfrac12 + u, \\tfrac12 - u)$. 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 ${H_{\\Pi}}:= {Y_{\\Pi}}^{\\dagger}{Y_{\\Pi}}$, whose spectrum on ${\\mathbb{C}^{3}_{\\mathrm{gen}}}$ is fixed up to the overall Yukawa norm by ${H_{\\Pi}}|_{{\\mathbb{C}^{3}_{\\mathrm{gen}}}} = {\\lambda_{Y}}^{2}\\,{\\operatorname{diag}}(1, \\tfrac12 + u, \\tfrac12 - u)$. The morphism ${Y_{\\Pi}}$ is then determined only up to a chiral polar factor, ${Y_{\\Pi}} = {U_{\\Pi}}\\,{H_{\\Pi}}^{1/2}$, with ${U_{\\Pi}}$ a unitary chiral map on the support of ${H_{\\Pi}}$. Hence the squared residue fixes the squared Yukawa levels but not the morphism: ${U_{\\Pi}}$ carries the chiral orientation, the relative generation phases, and the mixing, while the positive scale ${\\lambda_{Y}}$ is a separate normalisation of ${H_{\\Pi}}$ — none of it visible to the dimensionless level operator ${E_{\\Pi}}^{2}|_{{\\mathbb{C}^{3}_{\\mathrm{gen}}}}$. Front 3b is therefore closed as a squared-Yukawa-levels result, not a mixing result. A further analysis characterises ${U_{\\Pi}}$ not as a canonical matrix but as a rephasing class, the double quotient $[{U_{\\Pi}}] \\in {\\mathrm{U}}(1)^{3}_{R} \\backslash {\\mathrm{U}}(3) / {\\mathrm{U}}(1)^{3}_{L}$, whose invariants are the moduli $|({U_{\\Pi}})_{ij}|$ and a Jarlskog phase — three mixing angles and one CP phase — and whose non-triviality $[{U_{\\Pi}}] \\neq [I]$ is controlled exactly by the transverse component of the metaplectic generator. On the derived real cascade that transverse component is absent, so $[{U_{\\Pi}}] = [I]$ 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.