# Original contributor disclosures

Transcribed from the original public submission. Editor normalization changes only the structured parser representation; these original statements are preserved.

### Who did the work

LLM only (people only prompted or submitted)

### Models used

Anthropic Claude Opus 5.5

### Thinking level / time

Max

### Other LLMs

_No response_

### Per-model versions, thinking and roles

Claude Opus 5.5 | version: claude-opus-5-5 via Claude Code CLI (exact build unknown) | thinking: max (Claude Code `/effort max`; all of the mathematical work ran at this setting, only the initial download of the prompt came before it) | time: about 30 minutes wall-clock in total (2026-10-19, roughly 12:19–12:50 CEST) | role: literature search, found the argument, wrote the proofs and exposition, self-audit


### Model eligibility and verification evidence

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.


### People and roles

@fersuarezz: prompted the model and submitted the issue; no mathematical contribution.

### References

H. Bruhn, O. Schaudt, The journey of the union-closed sets conjecture, Graphs Combin. (2015), doi:10.1007/s00373-014-1515-0; arXiv:1309.3297. (Used: statement and separating reduction; Theorems 22–24; the discussion of improving the factor 2.)
V. Falgas-Ravry, Minimal weight in union-closed families, Electron. J. Combin. 18 (2011), #P95. (Used: Lemmas 1–2, the construction of the sets A_i.)
Y. Hu, On the union-closed sets conjecture, arXiv:1706.06167 (2017). (Used: Lemma 1 / Corollary 1, the domination lemma; Theorems 1–2 for comparison.)
J. Maßberg, The union-closed sets conjecture for small families, arXiv:1508.05718 (2015). (Used: Theorem 2.1 and inequality (13) for comparison; its summary of the known cases n ≤ 12 and m ≤ 50.)
R. Alweiss, B. Huang, M. Sellke, Improved lower bound for Frankl's union-closed sets conjecture, Electron. J. Combin. 31(3) (2024), P3.35; arXiv:2211.11731. (Used: Theorem 2 with Claim 4, the constant (3−√5)/2, which is input (E).)
Z. Chase, S. Lovett, Approximate union closed conjecture, arXiv:2211.11689 (2022). (Used: independent statement of the constant (3−√5)/2.)
W. Sawin, An improved lower bound for the union-closed set conjecture, arXiv:2211.11504. (Used: Theorem 1, an independent statement of the constant (3−√5)/2.)
J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055 (2022). (Used: the entropy framework and the nondegeneracy convention.)
Y. Jiang, Entropy bounds and global couplings for union-closed families, arXiv:2609.08291 (2026). (Used: only the claimed constant 0.38288525 from the abstract; not verified.)
S. Cambie, Progress on the union-closed conjecture and offsprings in winter 2022–2023, arXiv:2306.12351. (Used: literature check for novelty only.)
we-solve contribution Math-415-A1 (TOP-55-A1, r1). (Read to avoid repeating it; not used in the proofs.)


### Declarations

- [x] I have the right to publish this material under CC BY 4.0 (software Apache-2.0), and third-party material is credited.
- [x] I understand that this issue and its contents are public immediately.
- [x] Every LLM and person involved is listed, with versions or "unknown".