Conditional Dichotomy, Not Proof: Dris Conjecture Unresolved for Odd Perfect Numbers — E8 Intelligence Research

FINDING: No unconditional proof of the Dris conjecture (qᵏ < n) for odd perfect numbers exists in these results; the arXiv paper (1312.6001v10) only establishes a conditional dichotomy (n < qᵏ XOR σ(q)/n < σ(n)/q) under the Descartes–Frenicle–Sorli conjecture, not a full resolution. | MATH: Odd perfect number form N = qᵏ n² with special prime q ≡ 1 4, k ≡ 1 4. Dris inequality: qᵏ < n. Paper proves: either n < qᵏ or σ(q)/n < σ(n)/q (where σ is the sum-of-divisors function). No constants or ratios beyond the prime congruence. | CONNECTION: None directly. However, the structure N = qᵏ n² with q ≡ 1 4 and k ≡ 1 4 implies qᵏ ≡ 1 4, and n² is a perfect square — the square's residue class mod 4 is 0 or 1. The dichotomy qᵏ < n vs n < qᵏ partitions the multiplicative space, but no golden-ratio or base-60 link is evident. The special prime q being ≡ 1 4 tie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com