Overview
This companion note takes the second step of the mass-sector frontier of the fermionic matter sub-programme, and draws a sharp negative conclusion at its entry. The line note fixed the determinant line $L_Y = \wedge^2(\mathcal{S}_\Pi)$ as the Yukawa coupling line and the generation levels from $E_\Pi^2|_{C^3_{\mathrm{gen}}} = \mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$. Here we ask whether the squared residue determines the Yukawa morphism itself, and show that it does not.
The squared residue is hermitian on a single chirality, while $Y_\Pi$ ties the two chiralities; the chiral connecting datum is therefore invisible to $E_\Pi^2$. What survives is the hermitian square $H_\Pi = Y_\Pi^\dagger Y_\Pi$; the morphism is then free up to a unitary chiral polar factor.
Core results
- Squared Yukawa observable. The generation-level observable of the Schur-residue sector is the positive hermitian square $H_\Pi := Y_\Pi^\dagger Y_\Pi$, with $\mathrm{Spec}(H_\Pi)|_{C^3_{\mathrm{gen}}} = \lambda_Y^2\,\{1, \tfrac12+u, \tfrac12-u\}$. Thus $E_\Pi^2|_{C^3_{\mathrm{gen}}}$ determines the squared Yukawa levels, not the morphism itself.
- Chiral polar ambiguity (partial no-go). By polar decomposition $Y_\Pi = U_\Pi H_\Pi^{1/2}$, where $U_\Pi$ is a unitary chiral map on the support of $H_\Pi$. Since $H_\Pi$ is blind to any right-carrier unitary ($Y_\Pi \mapsto V Y_\Pi$ leaves $Y_\Pi^\dagger Y_\Pi$ unchanged), the squared residue does not fix $Y_\Pi$.
- What lives in $U_\Pi$. The chiral orientation and the complex mixing phase reside in $U_\Pi$. The positive Yukawa norm $\lambda_Y$ is a separate scale of $H_\Pi$ — a unitary cannot carry it — and is also invisible to the dimensionless level operator $E_\Pi^2$.
- Sign of $u$. In an oriented $J_3$-labelled basis $(e_0, e_+, e_-)$ the level operator already sees the relative sign of $u$; without that orientation choice the exchange $e_+ \leftrightarrow e_-$ sends $u \mapsto -u$ and leaves the unordered squared spectrum unchanged. The open datum is the physical orientation branch, not a scalar visible to $E_\Pi^2$ alone.
Diagonal CP-real branch
The CP-even real reduction imposes $v = 0$ (no off-diagonal mixing for phase-free data). In the present language this is the branch $U_\Pi = \mathbf{1}$, where the Yukawa operator is the positive root itself, $Y_\Pi = \lambda_Y\,\mathrm{diag}(1, \sqrt{\tfrac12+u}, \sqrt{\tfrac12-u})$, real and diagonal with no mixing. This is a choice of branch, not a derived fact: a non-trivial $U_\Pi$ injects off-diagonal and complex structure that $E_\Pi^2$ cannot register, and its first-principles origin is the complex metaplectic phase of the Lorentzian completion.
Status and open deliverable
Established. The squared observable $H_\Pi = Y_\Pi^\dagger Y_\Pi$ with $\mathrm{Spec}(H_\Pi)|_{C^3_{\mathrm{gen}}} = \lambda_Y^2\{1, \tfrac12+u, \tfrac12-u\}$, and the polar decomposition $Y_\Pi = U_\Pi H_\Pi^{1/2}$. Both are structural and unconditional given the carrier identifications of the Schur reduction, the Born–Infeld even-sector closure, and the line note. All operator identities are verified by exact symbolic computation.
Open. The chiral polar factor $U_\Pi$ (chiral orientation and complex mixing phase), the separate positive norm $\lambda_Y$, and hence the absolute masses, the generation mixing, and the CP structure. Front 3b is closed as a squared-Yukawa-levels result, not a mixing result: $E_\Pi^2$ can close $H_\Pi = Y_\Pi^\dagger Y_\Pi$ but cannot close $Y_\Pi$ unless $U_\Pi$ is fixed.
Relation to the Cosmochrony programme
This note belongs to the fermionic matter sub-programme. It follows the projected Yukawa line note, using the squared-residue normal form of the Schur reduction ($E_\Pi = -M^\dagger M$ negative semi-definite) and the Born–Infeld even-sector closure. The chiral polar factor $U_\Pi$ — the mixing and the chiral orientation — is the next downstream frontier, to be derived or constrained from the complex metaplectic phase.
References
Jérôme Beau. Projected Yukawa Operator and Chiral Polar Factor. Working paper, 2026. 10.5281/zenodo.20767498