Certified Minimal-Prime Branch Closures for Odd Perfect Numbers

Abstract

For an odd perfect number $N$, write $q=\min\{p:p\mid N\}$ for its smallest prime divisor. This paper proves a certified branch-closure theorem for the five minimal-prime branches $q=5,7,11,13,17$. The proof combines the exact $q$-adic valuation balance for $σ(N)=2N$ with lower-prime avoidance: primes below $q$ cannot occur in the support and therefore cannot divide any divisor-sum factor. These constraints reduce each branch to a finite first-input coverage split and then to terminal forced-or-pure cofactor records. The terminal records are checked by the frozen certificate release C-small-2026-07, consisting of JSONL certificate bundles, Python verifier scripts, expected terminal outputs, and SHA256 hashes. The $q=5$ branch is presented as the detailed audit model, while the branches $q=7,11,13,17$ are closed by the same forced-or-pure mechanism. The result is scoped: it does not prove nonexistence of odd perfect numbers, and the branches $q=3$ and $q\ge 19$ remain outside the paper.