{"paper":{"title":"Certified Minimal-Prime Branch Closures for Odd Perfect Numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Marco Mantovanelli","submitted_at":"2026-07-05T15:47:30Z","abstract_excerpt":"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 $\\sigma(N)=2N$ 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-sma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04365","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.04365/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}