A finite set forge with an entropy ledger
The Half Entropy Could Not Reach
Build a family of sets, close it under unions, and count which element appears most often. A second live ledger follows the entropy calculation from the sharp independent-sampling threshold to a slightly higher published refereed floor, while keeping Frankl's one-half target visibly open.
Choose sets. The browser checks every ordered pair. If a required union is absent, close the family with one press. Only then does the theorem badge switch on.
Make union closure bite
Ground set and presets
Every selected tile is one distinct set.
Checking closure
| element | integer frequency | fraction | from refereed floor | from one half | share |
|---|
Let every union report where it went
Draw two sets independently and uniformly from your family. The table enumerates the exact distribution of their union. The marginal calculation below it is the independent-coupling mechanism that produced the first sharp threshold.
Exact union histogram
| union | count | probability | entropy contribution |
|---|
Independent element marginals
| element | p | q = 2p - p² | h(p) | h(q) |
|---|
Coupling threshold instrument
Move off equality, then solve back to it
The check
Recomputing the complete receipt
Show every ordered pair and its union
| A | B | A ∪ B | present |
|---|
Choices, conventions, and approximations
- The family is a set of distinct subsets. Pressing a selected tile removes it; duplicates never enter the count.
- Including the empty set is a free choice. The theorem requires at least one nonempty set, whether or not the empty set is present.
- The named ground set is a free choice capped at six elements. This keeps the complete subset grid and all ordered pairs operable in a browser.
- Sets use bit masks internally, with A at bit zero. Frequencies are integer counts divided by the number of distinct sets.
- All displayed entropies use base-two logarithms. A zero-probability entropy term is defined by continuity as zero.
- Root solving uses bisection with IEEE 754 double precision and a stopping width set in the page code. Residuals are printed, so numerical agreement is visible.
- Loading the published refereed certificate
- Loading the preprint precision note
- Loading the conditional result note
What remains open
Frankl's assertion that some element occurs in at least half of every finite nontrivial union-closed family remains unresolved in the cited peer-reviewed paper published on 1 April 2026. The cited sources supply neither an accepted general proof nor a counterexample. Liu's larger value remains conditional, and Cambie's last reported digits still depend on numerical and graphical global-minimum confirmation rather than a complete calculus or interval proof.
Primary sources and status
- Justin Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055, submitted 16 November 2022 and revised 28 November 2022. preprint
- Will Sawin, An improved lower bound for the union-closed set conjecture, arXiv:2211.11504, submitted 21 November 2022 and revised 19 June 2023. preprint
- Zachary Chase and Shachar Lovett, Approximate union closed conjecture, arXiv:2211.11689, submitted 21 November 2022. preprint
- Ryan Alweiss, Brice Huang, and Mark Sellke, Improved Lower Bound for Frankl's Union-Closed Sets Conjecture, The Electronic Journal of Combinatorics 31(3), P3.35, published 20 September 2024, DOI:10.37236/12232. peer-reviewed
- Lei Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture, Entropy 25(5), 767, published 8 May 2023, DOI:10.3390/e25050767. peer-reviewed
- Stijn Cambie, Better bounds for the union-closed sets conjecture using the entropy approach, arXiv:2212.12500v2, submitted 23 December 2022 and revised 16 February 2025. preprint
- Jingbo Liu, Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling, arXiv:2306.08824, submitted 15 June 2023. preprint
- Rein van der Hout and Kees Roos, Some results and conjectures related to Frankl's union closed conjecture, Journal of Applied and Numerical Optimization 8(1), 11 to 18, published 1 April 2026, DOI:10.23952/jano.8.2026.1.02. peer-reviewed
Offline differential verifier: node research/the-half-entropy-could-not-reach/verify-the-half-entropy-could-not-reach.mjs