Projected Yukawa Operator and Chiral Polar Factor

Premises, none supplied by a source

Four premises are named, and each statement of the paper is tagged by the ones it uses. [H-Res]: the model operator $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ is the restriction of $E_\Pi^2$ to $C^3_{\mathrm{gen}}$. [H-Sq]: the Yukawa square is the residue square, $H_\Pi = \lambda_Y^2\,E_\Pi^2|_{C^3_{\mathrm{gen}}}$. [H-Fac]: a weak linking carrier $K$ with a non-zero space of invariant couplings exists (F1) and that space has dimension at most one (F2); the two conditions are independent, and together they let the results on the generation block reach the full morphism $Y_\Pi$. [H-Grad]: each generation carrier has its own $J_3$ grading. The chiralities of the paper are Lorentz chiralities of the spinor factor under [H-Spin]; they are not the $V-A$ selection, which is the representation content of [H-Weak].

Core results

• The square fixes the levels, not the morphism. Under [H-Res] and [H-Sq], $H_\Pi$ is positive hermitian with $\mathrm{Spec}(H_\Pi)|_{C^3_{\mathrm{gen}}} = \lambda_Y^2\{1, \tfrac12+u, \tfrac12-u\}$. The polar decomposition $Y^{\mathrm{gen}}_\Pi = U_\Pi H_\Pi^{1/2}$ is linear algebra and uses no premise: a hermitian square determines the morphism only up to a unitary polar factor.
• What lives in $U_\Pi$. The chiral orientation and the complex mixing phase reside in $U_\Pi$. The positive norm $\lambda_Y$ is a separate scale of $H_\Pi$, which a unitary cannot carry and which $E_\Pi^2$ does not see either.
• Rephasing class (under [H-Grad]). The polar factor is not canonical but a class $[U_\Pi] \in U(1)^3_R \backslash U(3) / U(1)^3_L$, the two groups being the commutants of the $J_3$ gradings of the two carriers. For $0 < u < \tfrac12$ the class space has dimension $9 - 5 = 4$: three mixing angles and one Jarlskog phase $\mathcal{J}_{\mathrm{CP}}$.
• First-order criterion. For a family of morphisms, $[U_\Pi] \neq [I]$ at first order if and only if the polar generator $\Omega$ has a non-zero transverse (off-diagonal) part; the Jarlskog invariant has leading order $\gamma^3 \propto \Im(A_{12}A_{23}A_{31})$. A vanishing tangent does not exclude an exit at higher order.
• Sign of $u$. In an oriented $J_3$-labelled basis $(e_0, e_+, e_-)$ the level operator 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.
• No mechanism for a non-zero split. On the orientation-compatible branch of Q14, an input, in the framework of PRS (under its generation-reading hypothesis) a non-zero diagonal split requires a locking operator that breaks $J_\Pi$-equivariance, and no mechanism for it is derived; the branch is not the $V-A$ structure.

The polar generator versus $A_\Pi^{\mathrm{odd}}$

The note separates two objects that must not be identified. The polar generator of a real-parameter family of invertible morphisms $Y(\gamma) = U P$ between fixed orthonormal frames, with $P > 0$, is the anti-hermitian operator $\Omega = U^\dagger\,\mathrm{d}U/\mathrm{d}\gamma$. It is the unique solution of the Sylvester equation $\Omega P + P\Omega = U^\dagger\,\mathrm{d}Y - (\mathrm{d}Y)^\dagger U$. Singular supports and moving frames are not covered. At the identity, for $\mathrm{d}Y$ the $\mathrm{Sym}^2$ lift $L(M)$ of $M = pE + qF + rH \in \mathfrak{sl}_2(\mathbb{C})$, one has $\Omega(0) = $ the anti-hermitian part of $L(M)$: it is $J_\Pi$-even and generally non-zero, with internal entry $\tfrac{\sqrt2}{2}(q - \bar p)$, non-zero for real $p \neq q$.

The projection $A_\Pi^{\mathrm{odd}}$ is defined in the paper as the anti-hermitian part of the $J_\Pi$-odd part of that lift, for the internal antilinear parity $J_\Pi$ of the generation copy. It vanishes identically for every complex coefficient, because with an antilinear $J_\Pi$ the $J_\Pi$-odd part of the lift is its hermitian part; the $J_\Pi$-odd anti-hermitian operators lie in the spin-zero and spin-two components, outside the image of $\mathfrak{sl}_2$. This is a statement about $A_\Pi^{\mathrm{odd}}$ as defined: it is not the polar generator, it does not determine the polar factor, and no statement on $[U_\Pi]$ follows from $A_\Pi^{\mathrm{odd}} = 0$.

Exact counterexample. For the real cascade step $g = e^{tE}e^{sF}$, the polar factor of $\mathrm{Sym}^2 g$ has external $(e_+, e_-)$ entry $(t-s)^2/\big((2+ts)^2 + (t-s)^2\big)$, equal to $1/5$ at $(t, s) = (1, 0)$ and to $1/17$ at $(2, 1)$, although the external tangent at the identity vanishes: an infinitesimal generator without external entry does not exclude one in the finite factor. The factor is real, so its Jarlskog invariant is zero; non-zero mixing moduli with real data do not establish CP violation. $\mathrm{Sym}^2 g$ is not a Yukawa morphism, and the square $H_\Pi$ does not determine the polar data: $Y_0 = D$ and $Y_1 = VD$ have the same square for every unitary $V$.

Diagonal CP-real branch

The diagonal CP-real branch is defined by $U_\Pi = \mathbf{1}$. There the Yukawa block is the positive root itself, $Y^{\mathrm{gen}}_\Pi = \lambda_Y\,\mathrm{diag}(1, \sqrt{\tfrac12+u}, \sqrt{\tfrac12-u})$, real and diagonal, with no mixing. This is a choice of branch, not a derived fact: it is not implied by the vanishing of the off-diagonal channel $R_{\mathrm{mix}}$ of $E_\Pi^2$ ($v = 0$), which concerns $E_\Pi^2$ and not the polar class. The external block $R_{\mathrm{mix}}$ has no matrix element in the image of $\mathfrak{sl}_2(\mathbb{C})$ in the step model of Q14 (Remark 6.4, a statement on matrix elements of the lift, not on the entries of a polar factor); Q14 excludes the internal block $e_0 \leftrightarrow e_\pm$ only by hypothesis (Proposition 6.3), and no physical exclusion of the internal block or of mixing is claimed.

Status and open deliverable

Established (for the generation block, with the premises each statement names): the polar ambiguity; the spectrum of $H_\Pi$ under [H-Res] and [H-Sq]; the rephasing class and its four parameters under [H-Grad]; the first-order non-triviality criterion; the Sylvester characterisation of the polar generator; the vanishing of $A_\Pi^{\mathrm{odd}}$ as defined and its distinction from the polar generator; the exact counterexample; the representation-theoretic reading of [H-Fac]. The identities are checked in exact arithmetic in the repository scripts.

Open. The corpus supplies no family of Yukawa morphisms whose polar factor is $U_\Pi$, so the physical polar class $[U_\Pi]$ is open, together with the bridge from a metaplectic step to a Yukawa morphism (normalisation, carrier identification, hermitian metrics, parameter and frame transport, relative data of the two sectors). Defining a generator through the $J_\Pi$-odd part, as $A_\Pi^{\mathrm{odd}}$ does, is a modelling choice that no source justifies. The positive norm $\lambda_Y$, the four premises (including the weak linking carrier $K$ and the construction of $E_{\mathrm{weak}}$), the mixing matrix, which is observable only relative to a second fermionic sector, and hence the absolute masses, the generation mixing and the CP structure remain open.

Arithmetic gate for the diagonal split. With $R_\Pi = (\tfrac12+u)/(\tfrac12-u)$ and $\varepsilon(R) = (R-1)/(2(R+1))$, one has $\varepsilon(R_\Pi) = u$ identically. The split $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ commutes with $J_3$ for every $u$, so $\mathrm{Sym}^2(\mathbb{C}^2)$ leaves $u$ free: the equality $u = \varepsilon = 1/10$ is exactly the dictionary gate $R_\Pi = R_{\mathrm{ADE}} = 3/2$, not produced by the recursion of admissible operations (carrier-preserving relative to [M] of the AOG-Note, with [H-WS] for $R_b$).

The split is not the mass hierarchy. At the gate, the squared 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 of order one. The hierarchy would have to be carried by generation-dependent cascade depths, which are not derived: the level-crossing depths, the depths of the linear growth law, the quantised resolution-density route and the renewed-floor multi-cycle route are excluded within the families tested, each reintroducing a free normalisation or an order-one gap. The depth hierarchy therefore remains an open datum, and the charged-lepton target is recorded separately as the unit-dispersion Koide constraint.

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 (the compression remainder $-N^\dagger N \preceq 0$ of a locking operator, which is the projective endomorphism only under a symbol identity) and the typing of Q14 ([H-Spin], [H-Weak]). A physical mixing matrix would be observable only as a relative polar class between two fermionic sectors sharing a generation carrier, and remains the downstream open frontier.

References

Jérôme Beau. Projected Yukawa Operator and Chiral Polar Factor. Working paper, 2026. 10.5281/zenodo.20767498. Code: github.com/Cosmochrony/projected-yukawa-operator.