Modular Obstructions in the Erdős–Straus Conjecture for n ≡ 1 (mod 4) — E8 Intelligence Research

FINDING: The Erdős–Straus conjecture (4/n = 1/a + 1/b + 1/c for all n≥2) remains open; modular arithmetic obstructions, particularly for n ≡ 1 (mod 4), are central to the proof strategy. | MATH: Conjecture: ∀n≥2, ∃a,b,c∈ℕ⁺: 4/n = 1/a + 1/b + 1/c. Known: true for all n except possibly n ≡ 1, 17, 25, 41, 49, 73, 89, 97, 121, 169 (mod 840) — the "exceptional" residue classes. Key modular obstruction: if n ≡ 1 (mod 4), the standard 3-term Egyptian fraction decomposition requires at least one denominator divisible by 4, forcing a specific parity structure. | CONNECTION: The exceptional residues mod 840 = 2³·3·5·7 — a highly composite modulus. The obstruction classes (1, 17, 25, 41, 49, 73, 89, 97, 121, 169 mod 840) are all ≡ 1 (mod 8) and ≡ 1 (mod 3) — they form a lattice structure in ℤ/840ℤ. The ratio 1/4 (0.25) appears as the target fraction; the harmonic mean of the three denominators must be exactly n/4. No direct golden ratio or base-60 link is evident from the search results. | DEPTH: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com