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    "id": "hyra-claimed-beurling-ahlfors-bellman-bound-2026",
    "title": "Hyra: claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles",
    "summary": "Presented a heat-extension/Bellman-function argument claiming ||B||p ≤ 1.523958(p*−1) for every 1<p<∞, using a two-sided drift constraint, two explicit generalized-polynomial profiles and exact rational endpoint comparisons. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.",
    "boundary": "Source-claimed proof, not independently verified by WRS. The coefficient is not the conjectured sharp value 1. The complete analytic argument and any formal coverage have not been independently reconstructed by WRS.",
    "area": "Mathematics",
    "topic": "Mathematics",
    "year": 2026,
    "lane": "AI",
    "result": "CANDIDATE_PROOF",
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        "id": "cur20261017-hyra-beurling-ahlfors",
        "title": "claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles",
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        "title": "Hyra Results: original artifact index and source-reported agent provenance",
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      "points": "180000",
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      "scope": "Source-claimed proof, not independently verified by WRS. The coefficient is not the conjectured sharp value 1. The complete analytic argument and any formal coverage have not been independently reconstructed by WRS."
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          "source": {
            "citation": "claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles",
            "url": "https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/AI4Science/beurling_ahlfors_bellman/manuscript.pdf"
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            "citation": "Hyra Results: original artifact index and source-reported agent provenance",
            "url": "https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/README.md"
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      "scope": {
        "description": "Source-claimed proof, not independently verified by WRS. The coefficient is not the conjectured sharp value 1. The complete analytic argument and any formal coverage have not been independently reconstructed by WRS.",
        "domain": "Harmonic analysis and AI-assisted proof discovery"
      },
      "sources": [
        {
          "citation": "claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles",
          "url": "https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/AI4Science/beurling_ahlfors_bellman/manuscript.pdf"
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        {
          "citation": "Hyra Results: original artifact index and source-reported agent provenance",
          "url": "https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/README.md"
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      "statement": "# Hyra: claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles\n\n## Control a transformation, not one picture\n\nA mathematical operator takes an input function and produces another function. The Beurling-Ahlfors operator is important in complex and harmonic analysis, where researchers study how such transformations stretch function size. The paper claims a better uniform upper bound on that stretching. Uniform means the stated rule works throughout the specified range of p, rather than only for a handful of test images or sampled functions.\n\n## Read the coefficient through an example\n\nThe reported bound uses 1.523958 times (p-star minus one), where p-star is the larger of p and p/(p-1). At p = 4, p-star is four, so the factor is 4.571874. The earlier coefficient 1.575 gives 4.725 at the same p. This multiplication is a classroom reading check. It does not measure the true operator norm or independently prove either analytic upper bound.\n\n## What became stronger?\n\nThe proposed coefficient improves a longstanding uniform estimate within a heat-extension and Bellman-function framework. A Bellman function is an auxiliary mathematical function designed to control the quantities in an argument. Here the claimed new constraints and explicit profiles are part of the proposed mechanism. Reusing an established framework can still produce a major advance. Its value depends on what the new argument actually controls, rather than whether every ingredient is newly invented.\n\n## The sharp problem stays open\n\nThe conjectured uniform coefficient is one. Moving from 1.575 to 1.523958 does not establish that sharp value, and a valid upper bound need not be attained by any chosen input. The manuscript and accompanying finite or numerical formal material have different scopes. WRS has not independently reconstructed the complete analytic proof or its formal coverage. The public record therefore keeps the result as a source-reported claim, even while showing a provisional significance assessment.\n\n## One attainable next step\n\nVerify p-star and the two displayed factors, including the conjugate exponent p = 4/3, which gives the same p-star. Then read the exact main theorem before following any claim about formal verification. Note which continuous inequalities, endpoint comparisons and assumptions each component actually addresses. A careful source-to-statement comparison is useful work. It should be submitted with its own limits, rather than described as a proof of the operator-norm conjecture.\n\n## Scientific scope\n\nSource-claimed proof, not independently verified by WRS. The coefficient is not the conjectured sharp value 1. The complete analytic argument and any formal coverage have not been independently reconstructed by WRS.\n\n## Sources for this explanation\n\n- [claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles](https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/AI4Science/beurling_ahlfors_bellman/manuscript.pdf): Original abstract, main stated result and evidence boundaries; full proof not independently reconstructed by WRS.\n- [Hyra Results: original artifact index and source-reported agent provenance](https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/README.md): Original abstract, main stated result and evidence boundaries; full proof not independently reconstructed by WRS.\n",
      "summary": "Presented a heat-extension/Bellman-function argument claiming ||B||p ≤ 1.523958(p*−1) for every 1<p<∞, using a two-sided drift constraint, two explicit generalized-polynomial profiles and exact rational endpoint comparisons. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.",
      "title": "Hyra: claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles"
    },
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    "id": "hyra-claimed-beurling-ahlfors-bellman-bound-2026",
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    "sections": [
      {
        "title": "Control a transformation, not one picture",
        "text": "A mathematical operator takes an input function and produces another function. The Beurling-Ahlfors operator is important in complex and harmonic analysis, where researchers study how such transformations stretch function size. The paper claims a better uniform upper bound on that stretching. Uniform means the stated rule works throughout the specified range of p, rather than only for a handful of test images or sampled functions."
      },
      {
        "title": "Read the coefficient through an example",
        "text": "The reported bound uses 1.523958 times (p-star minus one), where p-star is the larger of p and p/(p-1). At p = 4, p-star is four, so the factor is 4.571874. The earlier coefficient 1.575 gives 4.725 at the same p. This multiplication is a classroom reading check. It does not measure the true operator norm or independently prove either analytic upper bound."
      },
      {
        "title": "What became stronger?",
        "text": "The proposed coefficient improves a longstanding uniform estimate within a heat-extension and Bellman-function framework. A Bellman function is an auxiliary mathematical function designed to control the quantities in an argument. Here the claimed new constraints and explicit profiles are part of the proposed mechanism. Reusing an established framework can still produce a major advance. Its value depends on what the new argument actually controls, rather than whether every ingredient is newly invented."
      },
      {
        "title": "The sharp problem stays open",
        "text": "The conjectured uniform coefficient is one. Moving from 1.575 to 1.523958 does not establish that sharp value, and a valid upper bound need not be attained by any chosen input. The manuscript and accompanying finite or numerical formal material have different scopes. WRS has not independently reconstructed the complete analytic proof or its formal coverage. The public record therefore keeps the result as a source-reported claim, even while showing a provisional significance assessment."
      },
      {
        "title": "One attainable next step",
        "text": "Verify p-star and the two displayed factors, including the conjugate exponent p = 4/3, which gives the same p-star. Then read the exact main theorem before following any claim about formal verification. Note which continuous inequalities, endpoint comparisons and assumptions each component actually addresses. A careful source-to-statement comparison is useful work. It should be submitted with its own limits, rather than described as a proof of the operator-norm conjecture."
      }
    ],
    "references": [
      {
        "title": "claimed uniform Beurling–Ahlfors bound via explicit Bellman profiles",
        "url": "https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/AI4Science/beurling_ahlfors_bellman/manuscript.pdf",
        "locator": "Original abstract, main stated result and evidence boundaries; full proof not independently reconstructed by WRS."
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      {
        "title": "Hyra Results: original artifact index and source-reported agent provenance",
        "url": "https://raw.githubusercontent.com/Tencent-Hunyuan/Hyra-results/09b2d2e3c362c67a78f544c836bd8e4b54a05527/README.md",
        "locator": "Original abstract, main stated result and evidence boundaries; full proof not independently reconstructed by WRS."
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}