For sums S = X₁ + ⋯ + Xₙ of independent random variables each bounded relative to σ(S), the anticoncentration bound conjectured by Feige holds: P(|S − E S| ≤ t) is controlled uniformly, giving optimal constants in load-balancing and scheduling analyses.
Circulated by Feige around 2010 in work on randomized allocation; resisted a decade and a half of attack. One researcher had used it informally as a personal benchmark for LLM ability since 2024 — always unsuccessfully.
Proved August 2026, with the initial proof found by ChatGPT 5.6 Pro; two other teams posted independent resolutions the same day. A controlled experiment run before publication showed the decisive factor: instructing the model to search the literature broadly succeeded in 3 of 4 runs, while omitting that single sentence failed 4 of 4. The proof chains a very recent AI-assisted resolution of Gaffke's conjecture through Grünbaum's classical theorem — machine literature-searching connecting two continents of mathematics that no human had thought to bridge.
Context: the classical ancestor is the Littlewood–Offord problem (1943) on concentration of weighted Bernoulli sums; Feige's conjecture asks for the sharp uniform version with optimal constants. That the proof routes through Gaffke and Grünbaum — geometry of numbers meeting convex geometry — shows the conjecture lived at an intersection nobody was watching.
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