Bad Primes and Arithmetic Desingularization: E8 as Prime Filter in the TTOE Cascade
Authors/Creators
Description
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.
Notes (English)
This work is a modest contribution from an independent researcher. It does not claim to constitute a Theory of Everything. Each result is classified by its epistemic status — derived, conditional, or open — and should be regarded as a hypothesis requiring independent verification. No result should be treated as established physics until it has been independently checked by the scientific community.
If any researcher discovers results — experimental, theoretical, or computational — that relate to this work or the broader TTOE framework, whether confirming or refuting any part of it, I would be grateful to be informed.
Computational materials (Python scripts, JSON outputs, analysis notebooks) are not included in this deposit but are available upon request. Researchers who wish to inspect, reproduce, or challenge any result may contact me at:
https://orcid.org/0009-0004-8792-9396 foksus@proton.me
I wish to express my gratitude to all the scientists, mathematicians, physicists, biologists, philosophers, and thinkers — past and present — whose discoveries, ideas, and tireless pursuit of truth have built the edifice of knowledge upon which this work rests. No single contribution can emerge in isolation; even the solitary researcher draws from a collective intellectual heritage that spans centuries. Any errors or shortcomings are entirely my own.
E.U.O.