αc = 2 from Gardner uses the replica method — a brilliant statistical-physics trick that was non-rigorous for decades (since proven in cases). Strong result; one of its two proofs is heuristic. Hence: middling, not bedrock.
Take d weights and N random points, each given a random ±1 label (load factor α = N/d). Cover counted exactly what fraction of all possible labellings a perceptron can separate: P = 2·Σk=0..d−1 C(N−1, k) / 2N It's 1 for small α and crashes to 0 past the wall — passing through exactly 0.5 at α = 2. Drag d up and watch the slope sharpen into a vertical cliff: in the limit, a true phase transition.
What's rock-solid: Cover's count is a finite combinatorial fact — the curve above is not a fit or a simulation, it's the exact formula. The capacity of 2d for random points in general position is proven.
Where it softens: (1) Gardner's elegant αc=2 derivation runs on the replica method, which assumes a symmetry that wasn't rigorously justified for decades (and only proven in special cases since). (2) It's capacity for random patterns — real data has structure, and structured patterns change the number. (3) The leap from "a perceptron stores 2d" to "this is why big neural nets generalize" is an active research frontier (double descent, the interpolation threshold) — suggestive, not settled. The wall is real; its reach into modern deep learning is the open part.