Exact percolation laboratory
The Other Bunk Wins by One
On every small bunk you try, the nearer copy seems at least as easy to reach. Enumerate the hidden six-hyperedge core and the count reverses. Then operate the weighted graph gadget that carries that tiny reversal into an ordinary planar graph.
Every probability and count below is computed in this browser. The first instrument is a sandbox, not evidence for every graph. The second is an exact hypergraph result. The third shows the sufficient inequality used in the peer-reviewed ordinary graph proof.
Layer oneBuild a bunk that behaves
Choose a base graph, change its edges, choose the always-open posts, and set the endpoints. The browser enumerates every open or closed state of every horizontal edge on both levels.
Exact small-graph sandbox
Preset, then edit any edge or post.
Horizontal edges in each copy
Always-open vertical posts
Enumerating live.
This census fixes T = {2,3}, u = 1, v = 4, and p = 1/2, then generates every labelled connected graph on four vertices.
Layer twoReveal the hidden core
The proof starts from Lawrence Hollom's alternative bunkbed hypergraph model. Each of six three-vertex hyperedges is placed on exactly one of the two levels by a fair bit. This is not ordinary bond percolation on a graph.
Six hyperedges, every state
Posts are fixed at u2, u7, and u9.
The exact count will be generated from the incidence list below.
Enumerate to inspect a state.
"Count-breaking" means the thirteenth cross success in ascending mask order. That ordering is a display convention, not part of the theorem.
The mechanism survivesTurn a hyperedge into a graph
For each triple {a,b,c}, the paper replaces the hyperedge with a fan graph Gn. Its spine edges open with probability p. Its star edges from a open with probability 1-p. Only at p = 1/2 is the gadget uniform.
Gadget terminal probabilities
Five mutually exclusive terminal partitions, computed from the paper's recurrences.
Computing.
The base graph is duplicated only after this bookkeeping. Three always-open posts are then added between its two levels.
This meter uses two independent binomial estimates and their worst-case one-standard-error bound. It is an order-of-magnitude diagnostic, not a proof method.
The check
These are fresh recomputations from the active controls. Published values are named separately.
Uncertainty, choices, approximations, and conventions
- The small sandbox depends freely on the base graph, edge set, post set T, endpoints u and v, and p. It uses vertex labels starting at 1 and independent horizontal edges on both levels. Posts are always open. One degenerate case worth naming, because the instrument will not warn you: if either endpoint is itself a post, the difference is exactly zero by definition rather than by evidence. If v is a post then the post edge is always open, so (v,0) and (v,1) are always in one component and the two events being compared are literally the same event. If u is a post then (u,0) and (u,1) are always in one component, and swapping the two levels is a symmetry of the remaining randomness, so the two probabilities are equal for that reason instead. Selecting every vertex as a post makes the whole comparison vacuous in this way, and a run of exact zeros there is not support for the conjecture. The presets deliberately use partial post sets.
- The small sandbox and six-hyperedge census are exhaustive finite sums. They are not Monte Carlo. Decimal formatting rounds the displayed probabilities, while the state census remains integer-exact.
- In the alternative hypergraph core, each base hyperedge has exactly one active level. That model is visually and mathematically separate from ordinary bond percolation.
- For Gn at general p, spine edges use p and star edges use 1-p. The five displayed probabilities use IEEE 754 arithmetic. Extremely small a|bc values are shown in logarithmic form to avoid pretending that floating-point underflow is zero.
- The exact normalized robust test is evaluated in a cancellation-free form. The simpler coefficient is only the paper's sufficient lower bound. The formal parameter is a free construction choice that clears the required threshold.
- The formal theorem proves a negative gap whose magnitude is published only as less than 10-4331. That is an upper bound on magnitude, not the gap and not a lower bound.
- The reported 82-vertex variant and its gap on the order of 10-47 are computer-assisted. This page does not present them as the formal theorem.
- The page's publication date uses the build date because the supplied spec did not contain a separate LANDING date.
What is proved, and what remains open
The universal fixed-post bunkbed conjecture is false. Gladkov, Pak, and Zimin proved a counterexample at p = 1/2 on a connected planar base graph. The minimum size of an unweighted Bernoulli bond-percolation counterexample is not known. Which restricted graph families satisfy related inequalities is not fully classified. The monotonicity of percolation two-point connection probabilities with distance on grid-like graphs also remains unresolved.
Primary sources and review status
- Gladkov, Pak, and Zimin, "The bunkbed conjecture is false", PNAS 122(24), e2420725122, DOI 10.1073/pnas.2420725122, published 13 June 2025. Peer-reviewed journal article. Used for the theorem, gadget recurrences, graph counts, and gap bound.
- Gladkov, Pak, and Zimin, arXiv:2410.02545, submitted 3 October 2024. Preprint version, used to cross-check the accessible formulas and version history.
- Hollom, "The bunkbed conjecture is not robust to generalisation", European Journal of Combinatorics 128, 104188, DOI 10.1016/j.ejc.2025.104188. Peer-reviewed journal article. The preprint is arXiv:2406.01790, submitted 3 June 2024. Used for the alternative hypergraph mechanism.
- van Hintum and Lammers, "The bunkbed conjecture on the complete graph", European Journal of Combinatorics 76, 175-177, DOI 10.1016/j.ejc.2018.10.002. Peer-reviewed journal article. Used only to identify a surviving restricted positive result.
- Hutchcroft, Kent, and Nizić-Nikolac, "The bunkbed conjecture holds in the p tending to 1 limit", Combinatorics, Probability and Computing 32(3), 363-369, DOI 10.1017/S096354832200027X, published online 14 December 2022. Peer-reviewed journal article. Used only to identify a surviving asymptotic positive result.
- Ayyer, Linusson, and Ravichandran, "The bunkbed problem and the random cluster model", arXiv:2509.18788, submitted 23 September 2025. Preprint, not treated here as peer-reviewed. Used only as a current pointer to extensions.