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. All operator identities are verified by exact symbolic computation, with no sampling.