Courtade and Kumar's Coordinate-wise Mutual Information Question
Statement
The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp bound of on the sum of coordinate-wise mutual informations for arbitrary bias - and extends the proven high-noise range of the main conjecture via optimal entropy bounds.
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No person has examined this. Nothing here has been checked at all. say whether it holds →
proof attempt · #1
Adel Javanmard and David P. Woodruff, using Gemini Deep Think (larger internal version)That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
"The results in this paper were obtained with significant interaction with a larger version of Google's Deep Think Gemini-based model. The authors verified the entire paper and take full responsibility." The acknowledgments thank the Deep Think team by name.
Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.
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