Bad Primes and Arithmetic Desingularization: E8 as Prime Filter in the TTOE Cascade

Six rigorous connections between prime numbers and the E8 root system, all based on peer-reviewed mathematics. The central observation (Springer 1966): the bad primes of E8 are {2, 3, 5} — exactly the prime factors of the Coxeter number h = 30. This property, "bad primes = prime factors of h", holds for all five exceptional Lie algebras and fails for the classical families except trivially (8 of 55 cases of rank ≤ 15, all with h a power of two): it characterizes the exceptionals. The TTOE cascade E8 → E6 → SO(10) → SU(5) → SM eliminates bad primes monotonically ({2,3,5} → {2,3} → {2} → ∅ → ∅): the Standard Model has zero bad primes — its representation theory is smooth at every prime. The TTOE-BP conjecture (no maximal subgroup introduces new bad primes; for E8, every maximal subgroup strictly loses bad primes) is verified on the full Dynkin list of the 11 maximal subalgebras of E8 — the strict decrease turns out to be special to E8, since in lower exceptionals equality occurs (F4 ⊃ A1×G2, E6 ⊃ F4). Further connections: the Coxeter exponents of E8 are U(30) = Gal(Q(ζ30)/Q), with φ(30) = 8 = rk(E8) and the dyadic mirror structure 15 ± 2^j; complex phases in the character cascade come only from the prime 5; the E8 theta function is the Eisenstein series E4 and the Epstein zeta factorizes as 240·2^(−s)·ζ(s)ζ(s−3), linking the lattice spectrum to the Riemann Hypothesis; the binary icosahedral group (order 120 = 2³·3·5) and W(E8) as a Galois group over Q close the circle in both directions. All mathematical claims are derived (classical results, computationally re-verified); the physical interpretation — arithmetic desingularization as a guide to symmetry breaking — is explicitly classified as plausible, with no claimed dynamical mechanism.