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 \(Z_3\)-graded Lie algebra \(g=sl(5)\) 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 \(R^3\). The paper contains three structural results and one bookkeeping correction. The algebra. The cubic bracket used in earlier versions, \(12(ψχ\!+(ψ\!·\!χ)1_n)\), does not satisfy the graded Jacobi identity. We prove that for the one-parameter family of actions \([A,ψ]=Aψ+α\,tr(A)ψ\) the identity holds if and only if \(c(1+nα)=1-α\), with canonical representatives \((α,c)=(0,1)\) and \((1,0)\); the two are isomorphic, and both algebras are \(sl(n+1)\). Hence \(g=sl(5)\), and the failure of the earlier \(19\)-dimensional truncation is structural: \(19≠ n^2+2n\).(ii) The lattice. The closure does not saturate at \(44\). It is infinite: \[ C_∞=\{e_1,e_2,e_3\}∪\{± d\}∪k≥03^kR∪k≥03^kH∪k≥03^k(1,1,1). \] The \(44\)-vector set is exactly the shell cut \(|v|^2≤486\); 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 \(44\).(iii) The numbers. With \(ε_q=_1/=1/(n+2)=16\) (the same Frobenius projection lemma that gives the \(u(3)\) dilution \(19\)), with \(λ_0=ε_qC_2(3)=29\), and with the second-order isotropy factor \(1{72}\), the Cabibbo angle is \(λ=73/324=0.22530864\) (\(+0.46σ\)). The PMNS sector follows from one complex breaking parameter fixed by \(Z_3\) and by the geometry of the tetrahedral frame: \(sin^2θ₁₃=16/729\), \(sin^2θ₁₂=0.31436\), \(sin^2θ₂₃=0.45022\), \(δCP=240^∘\), \(JCP=-0.0292\), all within \(0.9σ\). The neutrino sector gives normal ordering with \(m_1=0\), \(∑ m_ν=58.7\)~meV and \(|mₑₑ|∈[1.6,3.8]\)~meV.We also prove three no-go theorems: graded phases force \(JCKM≡0\) (so \(δCKM\) is not derivable); \(|Vcb|\) and \(|Vub|\) 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.