Lower Bound for Primes Missed by Ramanujan Tau Function Up to 8×10²⁵ — E8 Intelligence Research

FINDING: The most substantive result is the explicit lower bound for primes missed by the Ramanujan tau function: every prime \(p \leq 8.0 \times 10^{25}\) is **not** of the form \(\tau(n) = \pm p\). | MATH: \(\tau(n)\) is the Ramanujan tau function (modular discriminant coefficient: \(q\prod_{k=1}^\infty (1-q^k)^{24} = \sum_{n=1}^\infty \tau(n) q^n\)). The bound: \(\forall p \le 8.0\times 10^{25},\ \tau(n) \neq \pm p\) for any \(n\). This is a non-vanishing statement: \(\tau(n) \neq 0\) is already known (Lehmer's conjecture is open, but this is about prime values). The proof likely uses congruences (e.g., \(\tau(n) \equiv \sigma_{11}(n) \mod 691\)) and explicit computation of \(\tau(n)\) mod small primes to eliminate all primes below the bound. | CONNECTION: The tau function is deeply tied to the **Leech lattice** (24-dimensional, the densest sphere packing in 24D) and the **Monster group** (via moonshine). The exponent 24 in the generating function is the kissing number dimension of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com