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

FINDING: No unconditional proof of the Dris conjecture ($q^k < n$) for odd perfect numbers exists in these results; the arXiv paper (1312.6001v10) only establishes a conditional dichotomy ($n < q^k$ XOR $\sigma(q)/n < \sigma(n)/q$) under the Descartes–Frenicle–Sorli conjecture, not a full resolution. | MATH: Odd perfect number form $N = q^k n^2$ with special prime $q \equiv 1 \pmod 4$, $k \equiv 1 \pmod 4$. Dris inequality: $q^k < n$. Paper proves: either $n < q^k$ or $\sigma(q)/n < \sigma(n)/q$ (where $\sigma$ is the sum-of-divisors function). No constants or ratios beyond the prime congruence. | CONNECTION: None directly. However, the structure $N = q^k n^2$ with $q \equiv 1 \pmod 4$ and $k \equiv 1 \pmod 4$ implies $q^k \equiv 1 \pmod 4$, and $n^2$ is a perfect square — the square's residue class mod 4 is 0 or 1. The dichotomy $q^k < n$ vs $n < q^k$ partitions the multiplicative space, but no golden-ratio or base-60 link is evident. The special prime $q$ being $\equiv 1 \pmod 4$ tie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com