An upper bound for four-uniform three-sunflower-free families

This research report by A1K Agentic SuperMath presents an ordinary computer-assisted proof that every finite simple four-uniform family with no three-sunflower has at most 67 members. The common core may be empty. With M(4, 3) denoting the maximum number of members, the result is 54 ≤ M(4, 3) ≤ 67; the known lower bound 54 is unchanged. Equivalently, every such family with at least 68 members contains a three-sunflower. The argument excludes every possible maximum point degree. Its new low-degree ingredient proves that every sunflower-free family of fifteen triples has a member disjoint from at least seven others. The proof combines arbitrary-support reductions, structural arguments, exact integer certificates, acyclic coverage records, deletion-cover checks and graph-fibre reconstructions. The result imports the established intersecting bound H27, Jin’s graph theorem and specified prior structural results. The complete historical H27 computation has not been replayed locally. Lean sources and audit receipts support the component results identified in the companion; the full upper-67 theorem is not an end-to-end Lean formalization. The general Erdős–Rado sunflower conjecture remains outside the claim. AI assistance: OpenAI Codex and autonomous AI agents were used for literature search, mathematical proof exploration, proof-source and checker generation, certificate and finite-check workflows, manuscript preparation and the recorded agent reviews. The archived source, status and verification records document these workflows and their limits. Agent reviews do not constitute independent human peer review or certification. The main PDF adds a bibliographic cover to the unchanged original manuscript. The original PDF, HTML and Markdown are also supplied, together with a frozen GitHub repository snapshot containing the accompanying proof sources, mathematical certificates, verification scripts, evidence, status records and indexes. The README and SHA-256 manifest identify the archived files and source revision. Historical drafts and negative controls retain their recorded status. Third-party sources without established redistribution permission remain external references with pinned identifiers and hashes. This work is published by A1K. Repository: https://github.com/0xagi-bsky-social/A1K. Archived source revision: 424f6716ea580863011739517503ebea61fd7e40.