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.

Disclosure

“. The residual branch q ≥ 19 requires a separate treatment of the possible first q-adic input orders and is left for future work. Acknowledgements I thank Benoît Cloître for the very helpful feedback in the development of this paper. ChatGPT 5.5 (OpenAI) was used for assistance with LaTeX formatting, language refinement, literature search, proof presentation, and coding support. The mathematical content, including all results, proofs, computations, and conjectures, remains the”

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Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-24T00:34:05.733Z

Structural counts