The CKM sector of the exceptional-Jordan programme: finite-Dirac mass moduli, two conditional angle estimates, and the weak-to-mass bridge

Abstract

The Standard Model does not determine quark masses or the CKM matrix. In the exceptional-Jordan programme, square-root masses occupy short ${\mathrm{Sym}}^3(3)$ chains. Compressing the symmetric-cube lift onto occupied nodes gives a root-mass operator whose square yields the proposed mass ratios, packaging that spectrum and relative left frames in one finite Dirac operator without deriving the frames. Conditional on the transport, virtual-node-amplitude, real-$(2,3)$ and balanced-quadrature choices, the no-fit layer gives $|V_{us}|=0.2371$ ($5.3\%$ high) and $|V_{cb}|=0.0422$ ($0.8\%$ high) at $M_Z$. The relation $|V_{ub}|/|V_{cb}|=\sqrt{m_u/m_c}$ is a factor two low; one complex long edge is fitted. The two balanced orientations give branches $(\epsilon ,\ddot{I}‰)=(0.002079,288.2^{\circ })$ and $(0.005358,202.2^{\circ })$, so the former $\epsilon$ ratio is not robust and raw $\ddot{I}‰$ is convention-dependent. For an adopted $F_1$--$F_2$ family embedding, we construct an exact Peirce-changing Albert lift in ${\mathfrak{f}}_4(\mathbb{C})$, reproducing conjugate up/anti-down transport and $\ddot{I}{†}_{12}=-2\ddot{I}‡$. For an adopted cyclic Majorana placement and real-linear projection, its completion has real $(e_4,e_3,e_6)$ support while the $e_1$ quadrature vanishes; $J_{\ell }=0$ and $\hat{I}{´}_{CP}^{\ell }=0$ or $\ddot{I}€$ are compatibility results in this class. The minimal radial-quartic cyclic truncation has equal-magnitude full-rank extrema or a flat direction; a mixed Albert cubic gives stable alignment only in a chosen three-edge subspace. Six diagonal mass links plus three directed Peirce links would form a connected one-cycle nine-link graph if the Yukawa block is linear in the projected bridge. This is a candidate Arkani-Hamed et al. texture skeleton, not a derivation: family-vacuum selection, chiral projection and right-frame locking remain open.

33 pages, 2 tables; ancillary verification script included. Results v1-corrected relative to Zenodo record 10.5281/zenodo.21196380 (4 July 2026): sign error in the texture scan fixed, Layer-2 fit solved exactly on a coherent PDG 2024 target set; see footnote 1 for the change log