Equality of quadratic-structure function spaces in Green Problem 57
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For a finite abelian group G, let Φ(G) be the convex hull of functions g↦E_{x₁+x₂+x₃=g}f₁(x₂,x₃)f₂(x₁,x₃)f₃(x₁,x₂), with complex-valued fᵢ bounded by one. Let Φ′(G) require f₃ to depend only on x₁+x₂. Must these two notions of quadratic structure coincide?
Why it matters and what counts as progress
Why it matters: The two spaces encode competing notions of quadratic structure; their distinction matters specifically for the intended complex-valued formulation.
What counts as progress: Prove equality in a specified group class or exhibit a complex-valued function separating the spaces; the real-valued variant is a different question.
How the problem developed
- 2018 · Open problem [1]
Ben Green Included Problem 57 in his circulated list, asking whether two spaces of complex-valued functions on every finite abelian group coincide.
- 21 May 2026 · Counterexample [2]
Google DeepMind AI-driven formal proof-search agent and the paper’s authors The research paper reports that its agent found a finite-group counterexample to equality, after Green clarified the complex-valued target; the counterexample is on the group of order three.
AI contribution
Advancing Mathematics Research with AI-Driven Formal Proof Search
A counterexample on Z/3Z disproves the intended complex-valued equality of two quadratically structured function spaces in Ben Green’s Problem 57.
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