The Universal SDRIS Zeta-Regularization Theorem: Isoclinic ϕ-Rotations, p-adic Stability, Geometric Duality, and Hierarchical Structure in Coxeter and Kac-Moody Lattices

We establish a universal closed-form expression for the optimal spectral zeta-target \ (c (N, W) \) governing the regularized sum of Laplacian eigenvalues of any projected Coxeter root lattice after isoclinic \ (\) -rotation: (N, W) = -² (W) 4 + (² (W) 4 - 112) N^-1/2, \ (² (W) \) is the modified variance determined by the Coxeter exponents of the (finite, affine or hyperbolic) group \ (W\). This formula interpolates exactly between the discrete Casimir point \ (c (N 0, W) = -1/12\) and the continuous Wigner bulk \ (c (N, W) = -² (W) /4\), satisfies a universal quadratic Stieltjes equation, and is strictly monotonic and convex. A geometric duality relates the Casimir regime to the unit-square diagonal \ (2\) and the Wigner bulk to the unit-circle quadrature \ (\). The target remains stable under arbitrary prime-adic completions, converging hierarchically to the eighth shell \ (S₈\) of the \ (p\) -adic ultrametric topology. The result holds uniformly for all irreducible Coxeter groups and their Kac--Moody extensions. The dual formulation obtained by the substitution \ (M=N^-1\), ^* (M, W) = -112 + (² (W) 4 - 112) M^1/2, an alternating transformation between the discrete Casimir vacuum and the continuous macroscopic field, thereby unifying spectral graph theory, number theory, and exceptional algebraic geometry. Data Availability: The complete Python source code used for the numerical validation of the scaling collapse, moment recursion, p-adic trajectories, and 8-shell hierarchy is permanently archived on Zenodo and can be accessed via DOI: 10. 5281/zenodo. 1958148.