Homological percolation in characteristic two: the Ising gauge case

Duncan and Schweinhart proved that the i-dimensional plaquette random-cluster model on the 2i-dimensional torus has a sharp homological percolation transition at the self-dual point p_sd = √q/(1+√q). Their proof assumes that the homology coefficient field does not have characteristic 2, because it uses the irreducibility of H_i(T²ⁱ; F) under the point group of the lattice, which holds exactly when char F ≠ 2. The excluded case contains the most important example: q = 2 with F₂ coefficients, which is the cellular representation of the (i−1)-form Ising lattice gauge theory (Wegner's model when i = 2). We prove the theorem for q = 2^r and F = F₂, for every r ≥ 1. The representation-theoretic step is replaced by a transfer-matrix inequality: adding a "temporal" 't Hooft twist to a "spatial" one cannot increase the twisted partition function. This is the classical positivity of 't Hooft's electric-flux sectors. Together with the FKG inequality it yields a "spin-up" bound that needs no representation theory. As a corollary, every 't Hooft twist free energy of the four-dimensional Ising gauge theory on T⁴_N tends to 0 for J < J_sd and to ∞ for J > J_sd, where J is the Ising coupling and J_sd = ½ ln(1+√2) is Wegner's self-dual point. For Bernoulli plaquette percolation (q = 1) on T⁴_N we prove that near p = 1/2 the classes of the integral giant cycles are divisible by any fixed integer M. There is w_M > 0 such that for p ∈ (1/2, 1/2 + w_M), with probability tending to 1, the lattice of classes of integral giant cycles has full rank and lies in M·H₂(T⁴; Z). Moreover, for p < 1/2 + w₂ no giant mod-2 cycle lifts to a mod-8 cycle; for each odd prime ℓ, just above 1/2 there are giant F_ℓ-cycles in every class but none lifts to a Z/ℓ²-cycle; and just below 1/2 the torsion subgroup of H₁ has order at least M⁶. The proof is an Aizenman–Grimmett strict-shift argument. Its input, the conversion of a pivotal plaquette into an enhanceable one by a bounded local change, is proved for every configuration by an explicit construction in a fixed region around the pivotal plaquette, which attaches a certified comb-shaped network. This rests on one finite certificate for a capsule-shaped region, verified exactly by computer, and on an induction over the length of the region. For q = 1 the sharp threshold with F₂ coefficients remains open. We reduce it to a single inequality, which holds cell by cell on all the small tori we enumerated but does not follow from reflection positivity. We also show that below p = 1/2 the classes of all giant cycles lie in a fixed 4-dimensional subspace on which the duality bound gives nothing, and that every periodic lattice has such classes; that box crossings are sharp away from aspect ratio 1; and that the threshold lies in [√2 − 1, 2 − √2]. Mathematics Subject Classification (2020): Primary 60K35; Secondary 82B43, 82B20, 82B26, 81T25, 55N10.