MERLIN SCIENCE — Fortuin–Kasteleyn random-cluster unification extended to plaquette lat — E8 Intelligence Research

Here is the finding in one sentence. The Fortuin-Kasteleyn random-cluster representation, which already unifies percolation, Ising, and Potts models through a single measure, extends to plaquette variables, and that extension is dual to a Potts lattice gauge theory. Three frameworks, one partition function. Let me give you the context. In 1972 Fortuin and Kasteleyn showed that the Potts partition function equals a sum over edge subsets, weighted by q to the number of connected clusters times v to the number of edges, with v equal to e to the J minus one. That identity turned percolation, magnetism, and cluster geometry into the same object. What Schweinhart now does is lift the sum from edges to two-cells. You sum over plaquette subsets, weight each by q raised to the first Betti number of the plaquette subcomplex, times v to the plaquette count. The Betti number counts independent cycles, the holes. That is the natural higher-dimensional FK model. The reasoning is checkable. On a d- Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com