GPT-5.6和Fable联手，解决了一道悬了25年的数学难题
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TL;DR - Microsoft Research principal researcher Dimitris Papailiopoulos used GPT-5.6 and Fable 5 to produce a proof that a simple polynomial-time algorithm achieves exact MIMO detection right at the maximum-likelihood threshold (SNR = 2logN), closing a gap open since ~2001. It matters as a concrete case of frontier LLMs contributing the core proof strategy to a long-standing open problem in information theory.
- The problem: recovering N transmitted bits from a noisy N×N channel. ML detection is exhaustive (2^N) and NP-hard in the worst case (Verdú, 1989); prior methods (sphere decoding — shown exponential by Jaldén & Ottersten 2005, SDP relaxation, bit-flipping, AMP, statistical physics) never matched the 2logN threshold, with box relaxation stuck at 4logN.
- The algorithm is two steps and O(N³): sign-rounded LMMSE estimation (proved to land within o(N) Hamming distance of the true bits), then greedy single-bit flipping, shown to converge in O(NlogN) steps.
- Proof structure: within a neighborhood of the start, every incorrect point has a flip that strictly decreases the cost by a non-vanishing amount, and the cost grows with Hamming distance — forming a barrier so greedy search can only halt at the true bit string. A converse is also proved: slightly below 2logN, even ML detection fails.
- Process notes: GPT-5.6 proposed an AMP-based route, Fable 5 proposed "sign LMMSE + greedy bit flipping"; the author picked Fable's path, had GPT patch gaps, then iterated for ~7 days having the models mutually simplify until he could hand-check line by line. He declined Lean formalization because he doesn't know Lean. Claims are from the author's X post, not (per the content) peer review.
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GPT-5.6和Fable联手，解决了一道悬了25年的数学难题
TL;DR - Microsoft Research principal researcher Dimitris Papailiopoulos used GPT-5.6 and Fable 5 to produce a proof that a simple polynomial-time algorithm achieves exact MIMO detection right at the maximum-likelihood threshold (SNR = 2logN), closing a gap open since ~2001. It matters as a concrete case of frontier LLMs contributing the core proof strategy to a long-standing open problem in information theory.
- The problem: recovering N transmitted bits from a noisy N×N channel. ML detection is exhaustive (2^N) and NP-hard in the worst case (Verdú, 1989); prior methods (sphere decoding — shown exponential by Jaldén & Ottersten 2005, SDP relaxation, bit-flipping, AMP, statistical physics) never matched the 2logN threshold, with box relaxation stuck at 4logN.
- The algorithm is two steps and O(N³): sign-rounded LMMSE estimation (proved to land within o(N) Hamming distance of the true bits), then greedy single-bit flipping, shown to converge in O(NlogN) steps.
- Proof structure: within a neighborhood of the start, every incorrect point has a flip that strictly decreases the cost by a non-vanishing amount, and the cost grows with Hamming distance — forming a barrier so greedy search can only halt at the true bit string. A converse is also proved: slightly below 2logN, even ML detection fails.
- Process notes: GPT-5.6 proposed an AMP-based route, Fable 5 proposed "sign LMMSE + greedy bit flipping"; the author picked Fable's path, had GPT patch gaps, then iterated for ~7 days having the models mutually simplify until he could hand-check line by line. He declined Lean formalization because he doesn't know Lean. Claims are from the author's X post, not (per the content) peer review.