The Weak Simplex Conjecture

Among $d+1$ equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any $m \times m$ correlation matrix $R$ with $R - \mathbf{1}\mathbf{1}^{\mathsf T}/m \succeq 0$ and $X \sim \mathcal{N}(0,R)$ , the maximum of the $X_i$ is stochastically dominated by the maximum of $m$ independent standard Gaussians.

Result: Proved
Status: Resolved
AI contribution: AI-assisted
Method: Argument
Field: Information theory
Posed by: Claude Shannon; Balakrishnan; Weber
Year posed: 1965
Years open: 61y
Solved: 2026-07-15
Model: GPT-5.6 Pro, GPT-5.6 Sol Max
Vendor: OpenAI
Collaborators: Abhijeet Mulgund
Verification: Unreviewed
Publication: Preprint
Significance: 20 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

The disclosure says the models were used during the development of the work to assist with mathematical exploration, proof development and checking, literature discovery and organization, and drafting. It attributes no specific step, so the lowest tier applies.

Verification

Single-author arXiv preprint; not yet peer-reviewed.

Source

arXiv:2607.14087 - Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture

The Weak Simplex Conjecture

What the AI did

Verification

Source

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