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.

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MAYA E8 / T_32 COMPUTATIONAL RESEARCH AUDIT — S3P15 HARNESS
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[+] 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.

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PART 2: E_8 ROOT LATTICE DISPERSION & DIRECTIONAL ISOTROPY AUDIT
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[+] 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            ISOTROPIC
k^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.

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ALL MATHEMATICAL AND COMPUTATIONAL AUDITS PASSED WITH ZERO FREE PARAMETERS.
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