{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R144",
  "content_hash": null,
  "slug": "ch31-artifact-interval-certificate",
  "type": "artifact",
  "title": "Sage interval certificate for both numerical endpoints",
  "summary": "The script makes the last coordinate enforce exact rational zero mean, encloses the witness ratio, and bounds the convolution row norm.",
  "relevance": "For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-artifact-interval-certificate (“Sage interval certificate for both numerical endpoints”) supplies evidence or a replay used to check the packet. The record states: The script makes the last coordinate enforce exact rational zero mean, encloses the witness ratio, and bounds the convolution row norm.",
  "relevance_source": "recorded",
  "body": "The first thirty coordinates are terminating decimals and therefore exact rationals. The final coordinate is defined as minus their exact sum. This removes any ambiguity caused by decimal rounding while preserving the displayed incumbent to far more precision than the certified lower endpoint needs.\n\nSage's `RealIntervalField(200)` encloses pi, every sine, the kernel entries, the transformed vector, and both fourth-power sums with directed rounding. The assertions certify a ratio above 1.5693, a kernel row norm below 2.707, and the final interpolated upper bound below 1.6453. The calculation certifies this witness and the analytic bound. It does not search all stationary points.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the order-31 kernel and one explicitly listed rational zero-mean witness, evaluated with 200-bit real intervals",
    "bounds": {
      "group_order": {
        "min": 31,
        "max": 31
      },
      "vector_coordinates": {
        "min": 31,
        "max": 31
      },
      "interval_precision_bits": {
        "min": 200,
        "max": 200
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_sagemath_interval_certificate",
    "entrypoint": "Join source_lines with LF characters, save as ch31.sage, and run sage ch31.sage",
    "runtime": "SageMath 10 or later",
    "citation": {
      "locator": "Self-contained SageMath interval program prepared from candidate record 160"
    },
    "inline_source": [
      "R = RealIntervalField(200)",
      "N = 31",
      "q = [QQ(s) for s in [",
      "    '0.5239134102516253','-0.48305993958895138','0.544403090028724','-0.50789308153175805',",
      "    '0.57816179856704941','-0.54666547924403952','0.6340979224878549','-0.60848357008706655',",
      "    '0.7368616038435335','-0.71839885717983865','0.99999987686128378','-0.99125223859886824',",
      "    '-0.9912523263587808','1','-0.71839900260621581','0.73686180173971261',",
      "    '-0.60848495885175369','0.63409898889914618','-0.54666555312102838','0.57816186539458969',",
      "    '-0.50789310240507879','0.54440313437174892','-0.48305994891673687','0.52391343257901168',",
      "    '-0.46807942981202094','0.51277431670549334','-0.46099481157583144','0.50923098786852994',",
      "    '-0.460994810441896','0.5127743064614968'",
      "]]",
      "q.append(-sum(q))",
      "assert len(q) == N and sum(q) == 0",
      "h = [sum((R(2)/N) * sin(R(2)*R.pi()*k*j/N) for k in range(1,16)) for j in range(N)]",
      "g = [sum(h[(x-y) % N] * R(q[y]) for y in range(N)) for x in range(N)]",
      "ratio = (sum(v^4 for v in g) / sum(R(v)^4 for v in q))^(R(1)/4)",
      "row_l1 = sum(abs(v) for v in h)",
      "upper = sqrt(row_l1)",
      "assert ratio.lower() > R('1.5693')",
      "assert row_l1.upper() < R('2.707')",
      "assert upper.upper() < R('1.6453')",
      "print('ratio =', ratio)",
      "print('row_l1 =', row_l1)",
      "print('upper =', upper)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Self-contained SageMath interval program prepared from candidate record 160"
  },
  "models": [],
  "relations": [
    {
      "slug": "R146",
      "title": "The sharp fourth-power norm lies between 1.5693 and 1.6453",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "cyclic-hilbert-l4-norm-31",
      "title": "cyclic hilbert l4 norm 31",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}