Residue-Class Reduction of Collatz to 1 mod 4 Cases — E8 Intelligence Research

FINDING: The Collatz recurrence, when analyzed via residue classes modulo 4, reduces the conjecture to checking convergence only for integers congruent to 1 mod 4 — a structural simplification that parallels lattice/root-system reduction. The Borel sigma-algebra videos are pedagogical, not novel; the arXiv paper is the substantive finding. MATH: - Collatz map: \( F(n) = n/2 \) if \( n \) even; \( F(n) = (3n+1)/2 \) if \( n \) odd. - Residue classes mod 4: \( n \equiv 0,1,2,3 \pmod{4} \). - Key reduction: If \( n \equiv 3 \pmod{4} \), then \( F(n) \equiv 1 \pmod{4} \) after one step (since \( 3(4k+3)+1 = 12k+10 = 2(6k+5) \), and \( 6k+5 \equiv 1 \pmod{4} \)). - Thus the entire orbit's convergence is determined by the subsequence of terms \( \equiv 1 \pmod{4} \). - No new constants or ratios appear; the structure is purely modular arithmetic. CONNECTION: - The mod-4 reduction mirrors the **root system A₁×A₁** decomposition in crystallography: the parity (mod 2) and mod-4 cl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com