Projected Yukawa Operator and 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.

Polar class (Front 3c)

The polar factor $U_\Pi$ is not a canonical observable but a rephasing class, the double quotient $[U_\Pi] \in U(1)^3_R \backslash U(3) / U(1)^3_L$: the distinct generation levels pin each chiral carrier to its level eigenbasis up to a diagonal phase, so the admissible chiral basis changes are diagonal rephasings $U_\Pi \mapsto V_R U_\Pi V_L^{-1}$. Its rephasing invariants are the moduli $|(U_\Pi)_{ij}|$ and a Jarlskog phase — three mixing angles and one CP phase.

Non-triviality. The class moves off the identity iff the metaplectic generator has a non-zero transverse (off-diagonal) component; the detectors are the off-diagonal moduli and the Jarlskog invariant, whose leading order is $\gamma^3 \propto \Im(A_{12}A_{23}A_{31})$. On the derived real cascade that transverse component is absent ($v = 0$), so $[U_\Pi] = [I]$ and no physical mixing is produced at this stratum. A non-trivial class needs a genuine complex metaplectic phase, not derived here, and becomes a physical mixing matrix only relative to a second fermionic sector. All identities are verified by exact symbolic computation.

Two transverse blocks (bridge to the rigidity programme). The transverse datum splits into an internal block $e_0 \leftrightarrow e_\pm$ and an external block $e_+ \leftrightarrow e_-$ ($R_{\mathrm{mix}}$). The correct chiral generator is the anti-hermitian part of the $J_\Pi$-odd component; it vanishes on real data. A genuinely complex non-central $\mathfrak{sl}_2$ coefficient sources the internal block, whereas the external block is never in the image of the $\mathrm{Sym}^2(\mathfrak{sl}_2)$ lift and would need a new tower stratum. Hence the oriented area $N_A$ (the real Cartan channel) can fix the diagonal split normalisation without predicting the polar mixing class: the $N_A/\varepsilon$ and CKM/PMNS data separate.

Reality structure (Front 3e): negative closure at the present stratum. The chiral polar generator depends only on the imaginary non-central coefficients: $[U_\Pi] = [I]$ iff $\Im p = \Im q = 0$, the imaginary Cartan part being a class-trivial $J_3$ rephasing, and a complex spinorial frame leaves the class unchanged (matrix complexity is not coefficient complexity). At the present stratum these coefficients are forced real — the cascade step is real and the inherited lift carries only the central cyclotomic phase $\zeta_q$ (no metaplectic Gauss phase) — so $[U_\Pi] = [I]$: the corpus derives the diagonal generation split but not the mixing. A non-trivial class needs a genuinely complex non-central metaplectic phase, available only through a new tower stratum (the same rigidity gate as $\varepsilon=1/10$).

Status and open deliverable

Established. The squared observable $H_\Pi = Y_\Pi^\dagger Y_\Pi$ with $\mathrm{Spec}(H_\Pi)|_{C^3_{\mathrm{gen}}} = \lambda_Y^2\{1, \tfrac12+u, \tfrac12-u\}$, the polar decomposition $Y_\Pi = U_\Pi H_\Pi^{1/2}$, and the polar-class characterisation $[U_\Pi] \in U(1)^3_R \backslash U(3) / U(1)^3_L$ with its transverse-component non-triviality criterion. All are structural and unconditional given the carrier identifications of the Schur reduction, the Born–Infeld even-sector closure, and the line note, and are verified by exact symbolic computation.

Conditional. On the derived real cascade the transverse channel is absent ($v = 0$), so $[U_\Pi] = [I]$ and no mixing is produced at this stratum.

Open. The separate positive norm $\lambda_Y$ and a non-trivial polar class $[U_\Pi] \neq [I]$ — which requires a genuinely complex non-central metaplectic phase, available only through a new tower stratum — and hence the absolute masses, the generation mixing, and the CP structure. Front 3b is closed as a squared-Yukawa-levels result and Front 3c characterises the polar class; $E_\Pi^2$ can close $H_\Pi = Y_\Pi^\dagger Y_\Pi$ but cannot close $Y_\Pi$ unless the class $[U_\Pi]$ is fixed.

Arithmetic gate for the diagonal split. The split parameter $u$ and the angular datum $\varepsilon$ are not independent invariants: with $R_\Pi = (\tfrac12+u)/(\tfrac12-u)$ and $\varepsilon(R) = (R-1)/(2(R+1))$, one has $\varepsilon(R_\Pi) = u$ identically. The diagonal split $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ commutes with $J_3$ for every $u$, so $\mathrm{Sym}^2(\mathbb{C}^2)$ leaves the value of $u$ free. The equality $u = \varepsilon = 1/10$ is therefore exactly the dictionary gate $R_\Pi = R_{\mathrm{ADE}} = 3/2$ (the order-five Cayley ratio), not produced by the carrier-preserving recursion. Status: $u \sim_{\mathrm{dict}} \varepsilon$ — same functional, distinct carriers.

The split is not the mass hierarchy. Even granting the gate, the squared Yukawa levels $\lambda_Y^2\{1, \tfrac35, \tfrac25\}$ are order-one ratios and cannot be the observed charged-fermion hierarchy by direct assignment; the single-level cascade factor is itself order one. The hierarchy is therefore carried, if at all, by generation-dependent cascade depths $n_g$ in the projective amplification law $\Lambda_{\mathrm{proj}}(n)$ (exponential growth regime): the cascade exponent fixes the amplification rate, while the generation stabilisation depths remain the next structural datum — with $\lambda_Y$ excluded as a hierarchy carrier. The linear-growth route to these depths is now closed: every depth read off the growth law $N(\lambda;n)\sim F_{\mathrm{KM}}(\lambda)\,n$ — crossing depth, saturation rank, or any intermediate fraction — is a function of the same Kesten–McKay coefficients $c_g(p)$, inheriting the band-edge divergence and the symmetry pin $c_2=\tfrac12$, so it cannot be both cascade-stable and correctly ordered. The remaining positive route — a decaying capacity-controlled per-level profile $\sigma_{\lambda_g}(n)$ transported onto the cascade — has now also been closed as a static route: quantising the per-level resolution density removes the band-edge denominator obstruction (the depth gap becomes independent of $c_g(p)$), but an exact audit shows the Weil–Schur transport is scale-free, so the required stabilisation window is reached only by re-inserting the Laplacian valency $|S|$ — the count normalisation — which returns to the already closed law $n_g=\Delta I_g/c_g(p)$. The static admissibility spectrum therefore fixes the generation structure and the order-one level split, but not the charged-fermion hierarchy. The remaining dynamical route — reading the depths off the relaxation trajectory rather than the static spectrum — has now also been tested at first order and reproduces the same obstruction: a parallel-threshold reading makes the depth gap independent of the band-edge coefficient but stays on the order-one level-crossing scale, while the sequential Born–Infeld and renewed-floor readings reach the hierarchy scale only through the cascade amplification itself or an unpinned ceiling-to-floor ratio, both free normalisations. The hierarchy is thus carried by no forced static or first-order dynamical quantity of the projective cascade, which fixes the generation structure, the order-one splitting, and a band-edge-independent order-one depth gap, but not the charged-fermion hierarchy at this stratum.

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 $U_\Pi$ is now characterised as a rephasing class whose non-triviality is the transverse metaplectic datum, collapsing to the identity on the derived real cascade; a physical mixing matrix remains the next downstream frontier, requiring a genuine complex metaplectic phase and a second sector.

References

Jérôme Beau. Projected Yukawa Operator and Chiral Polar Factor. Working paper, 2026. 10.5281/zenodo.20767498