High-Order Dispersion Isotropy on the Discrete E₈ Root Lattice and Exact Sparse Zero-Divisor Kernel Projectors in 𝕋₃₂

## Abstract We analyze two foundational structural problems in discrete hypercomplex quantum gravity frameworks: (1) directional dispersion anisotropy induced by spatial discreteness, and (2) cross-sector amplitude leakage induced by non-associative zero divisors in Cayley–Dickson algebras beyond octonions ($\mathbb{T}_{32}$). First, we prove that the 240 roots of the exceptional Lie group $E_8$ form an exact **spherical 7-design** in $\mathbb{R}^8$. Applying the Chevalley–Shephard–Todd theorem to the $E_8$ Weyl group $W(E_8)$, we demonstrate that the algebra of invariant polynomials has fundamental generators with degrees $d_i \in \{2, 8, 12, 14, 18, 20, 24, 30\}$. Because there exist **no independent invariants of degree 4 or degree 6**, all directional spherical harmonic components of degree $\le 7$ vanish identically when summed over the 240 roots. Consequently, the discrete nearest-neighbour Graph Laplacian dispersion moments $M_2, M_4, M_6$ are strictly isotropic on $S^7$ to machine precision ($< 10^{-15}$). While a standard hypercubic lattice $\mathbb{Z}^8$ exhibits a massive $188.9\%$ directional anisotropy in its fourth-order moment $M_4$, directional anisotropy on the $E_8$ root lattice is delayed to degree 8 ($O(k^8)$). We show that while isotropic, momentum-dependent dispersion shifts remain present at lower orders (modifying phase and group velocity uniformly), directional variations in group velocity are suppressed to $\Delta v_g / c \approx \xi_{6, g}(\hat{\mathbf{n}}) (E/M_{\text{Planck}})^6 \approx 7.9 \times 10^{-90}$ for $100\text{ TeV}$ photons. Second, we analyze the 32-dimensional trigintaduonionic algebra $\mathbb{T}_{32}$. We prove that naive operator products $L(A)L(B)$ across primitive zero divisors ($A \cdot B = 0$) fail to isolate sectors, acting non-trivially on 16 out of 32 dimensions via non-zero associators $[A, B, X] \neq 0$. Through an exhaustive combinatorial audit of all 465 primitive sparse binomials $A = e_i + e_j$ ($1 \le i < j \le 31$), we identify exactly 294 zero-divisor pairs partitioned into three nullity strata: nullity 4 (84 pairs), nullity 8 (84 pairs), and nullity 12 (126 pairs). We prove the **Universal Spectrum Theorem for the Sparse Binomial Family**: for every such zero divisor, the orthogonal signed-permutation matrix $Q = L(e_i)L(e_j)$ satisfies $Q^4 = I_{32}$, forcing the symmetric anticommutator $\{L(e_i), L(e_j)\}$ to have eigenvalues in $\{-2, 0, 2\}$. This analytically restricts the non-zero eigenvalues of $M_A = L(A)^T L(A)$ strictly to $\lambda \in \{2, 4\}$. We construct the exact quadratic Lagrange spectral projector:$$P_{\ker}(A) = \frac{1}{8}\left(L(A)^T L(A) - 2I_{32}\right)\left(L(A)^T L(A) - 4I_{32}\right)$$We verify that $P_{\ker}(A)$ is strictly idempotent ($P^2 = P$), symmetric ($P^T = P$), has trace equal to the kernel dimension ($\operatorname{Tr}(P) = \operatorname{dim}(\ker L(A))$), and satisfies $L(A) P_{\ker}(A) = 0$ to exact machine zero ($0.0 \times 10^{-16}$). Finally, we delineate the physical boundary between static algebraic kernel projection and dynamical superselection, identifying the requirement $[H, P_{\ker}(A)] = 0$ as the open condition for full sector preservation under Hamiltonian evolution. ================================================================================MAYA E8 / T_32 COMPUTATIONAL RESEARCH AUDIT — S3P15 HARNESS================================================================================ [+] Building Cayley-Dickson multiplication table for dim 32 (T_32)... Table built successfully: shape 32x32. [+] Auditing all 465 sparse binomials A = e_i + e_j (1 <= i < j <= 31)... Census of sparse binomials: - Nullity 0 (Invertible) : 171 pairs - Nullity 4 stratum : 84 pairs - Nullity 8 stratum : 84 pairs [S_16 embedded handles] - Nullity 12 stratum : 126 pairs Total zero-divisor pairs : 294 / 465 [+] Integer Matrix Algebraic Identity Checks: - Q^4 == I on all zero divisors : 294 / 294 (Exact) - M_A (M_A - 2I)(M_A - 4I) == 0 : 465 / 465 (Exact) - R_A^2 == 8 R_A : 465 / 465 (Exact) - L(A) R_A == 0 : 465 / 465 (Exact) --> PASS: All exact algebraic integer identities confirmed across all pairs. [+] Testing Primitive Zero-Divisor Pair A = e_1 + e_10, B = e_4 - e_15: Direct multiplication ||A * B|| : 0.00e+00 Product Frobenius Norm ||L(A) L(B)||_F : 9.7980 (||LA LB||_F^2 = 96.0) Rank of Operator Product L(A) L(B) : 16 / 32 dimensions [+] Auditing Universal Spectral Projector P_ker(A) = 1/8 (M_A - 2 I)(M_A - 4 I): Idempotence error max|P^2 - P| : 0.00e+00 Hermiticity error max|P^T - P| : 0.00e+00 Trace (Kernel Dimension) : 8.0 Projected Product Leakage ||LA P|| : 0.00e+00 --> PASS: Projector is exact, hermitian, idempotent, and satisfies L(A)P = 0. ================================================================================PART 2: E_8 ROOT LATTICE DISPERSION & DIRECTIONAL ISOTROPY AUDIT================================================================================ [+] Generating the 240 roots of E_8 in R^8... Total roots generated: 240 (112 vector roots + 128 spinor roots). All roots verified with ||alpha||^2 = 2.0. [+] Sampling 5,000 unit vectors uniformly over S^7... --- Comparative Spherical Dispersion Moments Audit ---Order Manifold Mean Value Max Anisotropy Status--------------------------------------------------------------------k^2 Z^8 Cubic 2.000000 5.551e-16 ISOTROPICk^2 E_8 Root 60.000000 4.737e-16 EXACT ZERO (< 10^-15)--------------------------------------------------------------------k^4 Z^8 Cubic 0.600871 1.889e+00 ANISOTROPIC (188.9%)k^4 E_8 Root 36.000000 1.184e-15 EXACT ZERO (< 10^-15)--------------------------------------------------------------------k^6 Z^8 Cubic 0.251114 5.412e+00 ANISOTROPIC (541.2%)k^6 E_8 Root 30.000000 1.658e-15 EXACT ZERO (< 10^-15)-------------------------------------------------------------------- [+] Analytical Directional Values of M_8 on E_8: - Coordinate Axis (1,0,...) : M_8 = 28.5 (delta = -1.5) - Hyper-Diagonal (1/sqrt(8),...) : M_8 = 39.0 (delta = +9.0) - Normalized Root Direction : M_8 = 39.0 (delta = +9.0) - Exact Spherical Mean : M_8 = 30.0 - Maximum Analytical Deviation : max|delta M_8| = 9.0 (Relative spread: 35.0%) [+] Directional Group Velocity Anisotropy Evaluation: Probe Energy : 100 TeV (1.0e+14 eV) Planck Scale M_P : 1.22e+28 eV Ratio (E / M_P) : 8.197e-15 Raw Kinematic Ratio (E / M_P)^6 : 3.033e-85 Max Group Velocity Anisotropy : Delta v_g / c ~ (1/38400)*(E/M_P)^6 ~ 7.898e-90 Astrophysical Comparison : Birefringence suppressed by > 60 orders of mag below current limits. ================================================================================ALL MATHEMATICAL AND COMPUTATIONAL AUDITS PASSED WITH ZERO FREE PARAMETERS.================================================================================