{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R410",
  "content_hash": null,
  "slug": "ipte11-artifact-size12-control-check",
  "type": "artifact",
  "title": "The supplied size-twelve control witness is valid and primitive",
  "summary": "Exact integer arithmetic verifies eleven equal moments, distinct disjoint sets, centering, primitive gcd one, and the first unequal moment.",
  "relevance": "For An ideal Prouhet-Tarry-Escott solution of size eleven, record ipte11-artifact-size12-control-check (“The supplied size-twelve control witness is valid and primitive”) supplies evidence or a replay used to check the packet. The record states: Exact integer arithmetic verifies eleven equal moments, distinct disjoint sets, centering, primitive gcd one, and the first unequal moment.",
  "relevance_source": "recorded",
  "body": "The candidate's control pair is\n\\[\nA=\\{\\pm22,\\pm61,\\pm86,\\pm127,\\pm140,\\pm151\\},\n\\]\n\\[\nB=\\{\\pm35,\\pm47,\\pm94,\\pm121,\\pm146,\\pm148\\}.\n\\]\nThe replay checks that both sides contain twelve distinct integers and are disjoint. Their sums vanish, and the gcd of all twenty-four coordinates is one, so this representative is centered and primitive. Direct summation gives equal powers for \\(1\\leq k\\leq11\\). At \\(k=12\\), the difference \\(\\sum_Aa^{12}-\\sum_Bb^{12}\\) is\n\\[\n809283534716112648192000.\n\\]\nThe two monic root polynomials differ by the nonzero constant\n\\[\n-67440294559676054016000.\n\\]\nThis certifies a nontrivial ideal solution of size twelve. It supplies a control for the definitions and gives no size-eleven witness.\n\nThe four-line output has SHA-256 digest `0617ec2157d54c020ce602649b765768866531635ca4c54ea3484e0ebebe2fb6`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the size-twelve symmetric ideal PTE witness displayed in the candidate record, including moments 1 through 12 and affine-normalization checks",
    "bounds": {
      "size": {
        "min": 12,
        "max": 12
      },
      "moments_checked": {
        "min": 1,
        "max": 12
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "runnable",
    "kind": "inline_python_computation",
    "command": "python3 check.py",
    "runtime": "CPython 3 standard library",
    "citation": {
      "url": "https://doi.org/10.1090/mcom/3917",
      "locator": "Coppersmith, Mossinghoff, Scheinerman, and VanderKam, Mathematics of Computation 93 (2024), equation (5) on page 2475; executable reproduction in this record"
    },
    "inline_source": "from functools import reduce\nfrom hashlib import sha256\nfrom math import gcd\n\nhalf_a=(22,61,86,127,140,151)\nhalf_b=(35,47,94,121,146,148)\nA=tuple(sorted(half_a+tuple(-x for x in half_a)))\nB=tuple(sorted(half_b+tuple(-x for x in half_b)))\nassert len(A)==len(set(A))==len(B)==len(set(B))==12\nassert set(A).isdisjoint(B)\nassert sum(A)==sum(B)==0\nassert reduce(gcd,(abs(x) for x in A+B))==1\n\ndiffs=tuple(sum(x**k for x in A)-sum(x**k for x in B) for k in range(1,13))\nassert diffs[:11]==(0,)*11\nassert diffs[11]==809283534716112648192000\n\ndef root_polynomial(roots):\n    coefficients=[1]\n    for root in roots:\n        updated=[0]*(len(coefficients)+1)\n        for degree,coefficient in enumerate(coefficients):\n            updated[degree]-=root*coefficient\n            updated[degree+1]+=coefficient\n        coefficients=updated\n    return coefficients\npa=root_polynomial(A)\npb=root_polynomial(B)\npolynomial_difference=tuple(x-y for x,y in zip(pa,pb))\nassert polynomial_difference[0]==-67440294559676054016000\nassert polynomial_difference[1:]==(0,)*12\n\nout=(\n    'size=12 disjoint=True distinct=True centered=True primitive_gcd=1\\n'\n    'moment_differences_k1_to_k11=0,0,0,0,0,0,0,0,0,0,0\\n'\n    'k12_difference=809283534716112648192000\\n'\n    'polynomial_difference_constant=-67440294559676054016000\\n'\n)\nassert sha256(out.encode()).hexdigest()=='0617ec2157d54c020ce602649b765768866531635ca4c54ea3484e0ebebe2fb6'\nprint(out,end='')\n",
    "missing": [
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1090/mcom/3917",
    "locator": "Coppersmith, Mossinghoff, Scheinerman, and VanderKam, Mathematics of Computation 93 (2024), equation (5) on page 2475; executable reproduction in this record"
  },
  "relations": [
    {
      "slug": "R411",
      "title": "Integral ideal solutions of size eleven remain unknown",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "ideal-pte-size-11",
      "title": "ideal pte size 11",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}