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  "problem_id": "vibemathed:entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a",
  "title": "Entropy of Bernoulli Measures Conditioned on Affine Subspaces and a Problem of Ancheta-Massey",
  "canonical_statement": "A textbook result in information theory is that linear encoders achieve the entropy for lossless compression of Bernoulli source with parameter $p$. For lossy compression, however, linearity is known to incur strict suboptimality compared to the rate-distortion function. Massey asked whether the optimal rate for linear encoding is achieved simply by compressing a fraction of the bits linearly and losslessly and estimating the rest by zero. For $p=\\frac{1}{2}$, Ancheta answered this question affirmatively. This note extends Ancheta's result to all $p<\\frac{1}{2}$. The key argument is to bound the entropy of the posterior distribution conditioned on an affine subspace in terms of its marginals. The proof was discovered by GPT-5.6 Sol in an interactive process guided by the author.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
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      "problem_id": "vibemathed:entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-24T00:00:00.000Z",
      "title": "arXiv",
      "summary": "For a Bernoulli$(p)$ source with $0<p<\\frac12$, the paper proves that the best lossy compression achievable by a linear encoder is exactly the simple time-sharing strategy that losslessly compresses a fraction of the bits and estimates the rest as zero. Equivalently, every full-row-rank $H\\in\\mathbb F_2^{k\\times n}$ satisfies\n$$\\frac{k}{n}\\ge h(p)\\left(1-\\frac{D(H)}p\\right).$$\nThis resolves Massey's question affirmatively for all $p<\\frac12$; Ancheta had already proved the $p=\\frac12$ case.",
      "ai_contribution": "ai_discovered",
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        "vibemathed:entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a:attempt"
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      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-24T00:00:00.000Z",
      "human_collaborators": [
        "Yihong Wu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "GPT-5.6 Sol discovered the first version of the proof in an interactive process guided by Yihong Wu. Wu subsequently simplified the argument and developed the self-contained proof in the paper. A later literature search aided by Codex identified that several proof ingredients had appeared previously or could be deduced from earlier work. The author wrote the final paper and assumes responsibility for its technical content.",
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.22837",
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        {
          "label": "VibeMathed: Entropy of Bernoulli Measures Conditioned on Affine Subspaces and a Problem of Ancheta-Massey",
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      "note": "Unreviewed. A short preprint with a self-contained argument, by a researcher who works on exactly this. The author reports that a later literature search found several ingredients had appeared before or followed from earlier work, and says so in the paper; that is a caution about novelty of the components, not about the result. No independent check.",
      "sources": [
        {
          "label": "VibeMathed: Entropy of Bernoulli Measures Conditioned on Affine Subspaces and a Problem of Ancheta-Massey",
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    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}