Derivation of Standard Model Mixing Angles from a 44-Vector Discrete Vacuum Lattice

We derive the Standard-Model mixing angles and CP phases from a finite algebraic structure: the \(\mathbb{Z}_3\)-graded Lie algebra \(\mathfrak{g}=\mathfrak{sl}(5)\) of dimension \(24=15+4+4+1\), graded by the mod-\(3\) reduction of \((1,1,1,1,-4)\), and realized on a discrete vacuum lattice in \(\mathbb{R}^3\). The paper contains three structural results and one bookkeeping correction. The algebra. The cubic bracket used in earlier versions, \(\tfrac12(\psi\chi^{\!\top}+(\psi\!\cdot\!\chi)\mathbf{1}_n)\), does \emph{not} satisfy the graded Jacobi identity. We prove that for the one-parameter family of actions \([A,\psi]=A\psi+\alpha\,\operatorname{tr}(A)\psi\) the identity holds if and only if \(c(1+n\alpha)=1-\alpha\), with canonical representatives \((\alpha,c)=(0,1)\) and \((1,0)\); the two are isomorphic, and both algebras are \(\mathfrak{sl}(n+1)\). Hence \(\mathfrak{g}=\mathfrak{sl}(5)\), and the failure of the earlier \(19\)-dimensional truncation is structural: \(19\neq n^2+2n\).\emph{(ii) The lattice.} The closure does \emph{not} saturate at \(44\). It is infinite: \[ \mathcal{C}_\infty=\{e_1,e_2,e_3\}\cup\{\pm d\}\cup\bigcup_{k\ge0}3^kR\cup\bigcup_{k\ge0}3^kH\cup\bigcup_{k\ge0}3^k(1,1,1). \] The \(44\)-vector set is exactly the shell cut \(|v|^2\le486\); it consists of \(3\) flavour vectors, \(5\) democratic vectors, \(18\) root-like and \(18\) hybrid vectors. The finite invariant is the set of \(17\) directions, reached after two iterations. No prediction of this paper depends on \(44\).\emph{(iii) The numbers.} With \(\varepsilon_q=\dim\mathfrak{g}_1/\dim\mathfrak{g}=1/(n+2)=\frac16\) (the same Frobenius projection lemma that gives the \(\mathfrak{u}(3)\) dilution \(\frac19\)), with \(\lambda_0=\varepsilon_qC_2(\mathbf{3})=\frac29\), and with the second-order isotropy factor \(\frac1{72}\), the Cabibbo angle is \(\lambda=73/324=0.22530864\) (\(+0.46\sigma\)). The PMNS sector follows from one complex breaking parameter fixed by \(\mathbb{Z}_3\) and by the geometry of the tetrahedral frame: \(\sin^2\theta_{13}=16/729\), \(\sin^2\theta_{12}=0.31436\), \(\sin^2\theta_{23}=0.45022\), \(\delta_{CP}=240^\circ\), \(J_{CP}=-0.0292\), all within \(0.9\sigma\). The neutrino sector gives normal ordering with \(m_1=0\), \(\sum m_\nu=58.7\)~meV and \(|m_{ee}|\in[1.6,3.8]\)~meV.We also prove three no-go theorems: graded phases force \(J_{\mathrm{CKM}}\equiv0\) (so \(\delta_{\mathrm{CKM}}\) is not derivable); \(|V_{cb}|\) and \(|V_{ub}|\) require the mass spectrum; and no dimensionless algebra can fix a mass scale. The paper closes with an explicit list of what is a theorem, what is a discrete choice, and what is an assumption.