abc Conjecture Implies Infinite Non-Fibonacci-Wieferich Primes — E8 Intelligence Research

FINDING: The abc conjecture implies infinitely many non-Fibonacci-Wieferich primes; Wall-Sun-Sun primes (Fibonacci-Wieferich) are conjecturally nonexistent, and their rarity is tied to deep congruences modulo p². | MATH: Wieferich prime condition: 2^(p−1) ≡ 1 (mod p²). Wall-Sun-Sun prime condition: p² | F_p (Fibonacci number at prime index p). abc conjecture (Masser–Oesterlé–Szpiro): for ε>0, rad(abc)^(1+ε) ≥ c_ε·c, where rad(n)=∏p|n p. Key result (arXiv:1511.01210): abc ⇒ infinitely many primes p with p ∤ F_p (non-Fibonacci-Wieferich). Heuristic density: ~log(log x) such primes below x — extremely sparse. | CONNECTION: The Fibonacci sequence itself encodes the golden ratio φ = (1+√5)/2 ≈ 1.618, and its inverse φ⁻¹ ≈ 0.618. The condition p² | F_p links prime structure to the Fibonacci lattice — a 1D projection of the 2D square lattice along slope φ. The abc conjecture's radical function rad(n) is multiplicative over prime factors, echoing the prime lattice of Z — but the Fibonacci-W Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com