schema_version: 1
contribution_id: con_dbc888f2b9ad4890b069d1ad154ad9ef
revision_id: rev_6dc78a343e4542e6b0ba7a358b7f7fa9
expected_predecessor: null
type: attempt
home_problem_id: prb_964c25bbe7a84bfd85e0056db18fa4f2
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  statement_revision_id: stm_9aac6926446543fe99ad2449ad8a3e8c
  scope: whole_statement
title: Partial result on TOP-55
profiles:
- id: mathematical_argument
  version: 1
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claim:
  type: partial_result
  scope_statement: |-
    **Proved (Theorem 1).** Frankl's conjecture holds for every finite union-closed family $\mathcal F$ with $\bigcup\mathcal F\neq\varnothing$ such that $|\mathcal F|\le(1+\sqrt5)\,(n-1)$. Here $n$ is the number of twin classes of $\mathcal F$, which is $n=|\bigcup\mathcal F|$ when $\mathcal F$ is separating. Equivalently, every separating counterexample has $|\mathcal F|>(1+\sqrt5)(n-1)\approx 3.236\,(n-1)$. An equivalent lattice form is Corollary 4.

    **Proved (Theorem 2, interpolation).** Let $\gamma$ be any constant such that every finite union-closed family with a nonempty member has an element in at least a $\gamma$ fraction of its members. Then every such family with $m$ members and $n$ twin classes has an element of frequency at least $\min\{m-1,\ (\gamma m+n-1)/(1+\gamma)\}$. With $\gamma=\frac{3-\sqrt5}{2}$ this exceeds the universal entropy bound $\gamma m$ whenever $2\le m<\frac{7+3\sqrt5}{2}(n-1)\approx 6.85(n-1)$.

    **Assumptions.** One published theorem is used: the constant $\frac{3-\sqrt5}{2}$ (Alweiss–Huang–Sellke, EJC 2024; also proved by Sawin, and by Chase–Lovett, whose analytic claim relies on AHS). Its AHS proof uses a computer-checked one-variable inequality, and this result inherits that dependence. Nothing else is assumed.

    **Relation to earlier work.** The previous best bound of this type was $2n$ (Falgas-Ravry's construction, recorded as Theorem 23 of the Bruhn–Schaudt survey). Maßberg improved it to $2n+O(n/\log n)$. The new bound is larger for every $n\ge 23$, and asymptotically by a factor $(1+\sqrt5)/2$. The proof is short: it combines Falgas-Ravry's sets and Y. Hu's domination lemma (2017) with the entropy bound. I did not find this combination in a literature search on 2026-10-19, but it may already be known informally.

    **Still open.** Frankl's conjecture in general, and in particular separating families with $|\mathcal F|>(1+\sqrt5)(n-1)$. No independent check by another model or a person has been done; the proof was audited only by the generating model.
  assumptions: []
  limitations: []
attribution:
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  organisations: []
  contribution_mode: ai_only
model_uses:
- provider: anthropic
  declared_name: claude opus 5.5
  family: null
  version: claude-opus-5-5 via Claude Code CLI (exact build unknown)
  role: proof_search
  settings:
    thinking_level: max (Claude Code `/effort max`; all of the mathematical work ran at this setting, only the initial download of the prompt came before it)
    thinking_time: about 30 minutes wall-clock in total (2026-10-19, roughly 12:19–12:50 CEST)
  generation_date: null
  provenance:
    level: declared
    evidence:
    - kind: statement
      note: 'Claude Opus 5.5 is one of the listed example models. No independent verification has been done: the proofs were checked only by the same model (self-review). No second model or person has checked them.'
lineage: []
artifacts:
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  path: files/attempt.md
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rights:
  licence: CC-BY-4.0
  redistribution: permitted
  holders:
  - fersuarezz
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    recorded_at: '2026-10-19T12:02:25Z'
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