The bunkbed conjecture, posed by Pieter Kasteleyn in the mid-1980s, concerns paired graphs connected by vertical posts — imagine two copies of a network stacked like bunk beds with ladders between them. Each edge is independently kept or removed at random. The conjecture: the probability of finding a path between two vertices on the same level is always at least as large as the probability of finding a path from one to the other's copy on the opposite level. Staying on one floor should be at least as easy as crossing between floors.
For forty years, every graph anyone checked confirmed the conjecture. The intuition was compelling — crossing levels introduces constraints that can only reduce connectivity. Noga Alon summarized it as a statement that “intuitively looks very likely to be true.”
In 2024, Gladkov, Pak, and Zimin (PNAS, 2025) proved the conjecture false. Their counterexample is a connected planar graph with 7,222 vertices and 14,442 edges. The cross-level probability exceeds the same-level probability by approximately 10^(-6500) percent.
The number requires attention. 10^(-6500) is not a small number in the way that laboratory precision limits are small. It is smaller than any physically meaningful quantity in the observable universe. No experiment, simulation, or finite computation could detect the violation. The conjecture is false by an amount that is mathematically rigorous and empirically invisible.
The counterexample's construction explains the magnitude. In June 2024, Lawrence Hollom disproved the bunkbed conjecture for hypergraphs — where edges can connect any number of vertices, not just pairs. His counterexample was small: three vertices. Gladkov and Zimin then translated Hollom's hypergraph into a standard graph by replacing each multi-vertex edge with a large cluster of ordinary edges. The translation preserved the sign of the violation — the inequality still pointed the wrong way — but attenuated the magnitude exponentially. A clear signal in a 3-vertex hypergraph became a 10^(-6500) whisper in a 7,222-vertex graph.
The structural lesson concerns what translations preserve. Moving from hypergraphs to graphs is not an approximation — it is an exact mathematical embedding. The qualitative fact (the conjecture is false) survives perfectly. The quantitative fact (by how much) is destroyed. The truth crosses the translation boundary; the evidence does not. Anyone who checked only the graph, without knowing the hypergraph origin, would never find the violation by computation. The counterexample was found by theory, not by search — by understanding the structure well enough to know where the sign flips, even when the magnitude vanishes.