Dancing with frustration: short-range Ising spin glasses, what is proved, what is believed, and where proofs break

The Edwards–Anderson model, Ising spins on ℤ^d with independent symmetric random nearest-neighbour couplings, was introduced in 1975 as the simplest short-range model of a spin glass. Physicists expect a spin-glass transition at positive temperature in three dimensions and above, and none in two. Mathematically, no proof of spin-glass order is known for symmetric independent couplings in any finite dimension, and for undiluted laws such as ±J or Gaussian there is no proof that the two-dimensional model is disordered at all positive temperatures. This review records, as of November 2026, what is proved about short-range Ising spin glasses, what is believed, and where proofs break. It covers the competing physical pictures (replica symmetry breaking, droplets, trivial–nontrivial and chaotic pairs) and the numerics that calibrate them; metastates and the two-dimensional fluctuation bounds; ground states, disorder chaos and Chatterjee's zero-temperature theorems; the cluster representations proposed for frustrated systems, and two-replica blue percolation; high-temperature methods, the comparison with the random-field Ising model, and a comparison with spanning forests and electrical networks; and hierarchical lattices, diluted lattices and the Nishimori line. Results from several programmes are restated in one notation, and the arguments that teach something are proved or sketched. Route warnings record attractive lines of argument together with the lemma each one would need and the cases in which that lemma fails. A final part collects the barriers that any proof of order or non-order must respect, small test cases for candidate inequalities, and open problems. Results of the author's own programme are marked with their status, and each computer-assisted proof states what its certificate checks.