{
  "table": "T558",
  "status": "built; private and unoffered; independent critique required",
  "indices": [
    0,
    1,
    2,
    3,
    4,
    5
  ],
  "stored_entries": 6,
  "type": "Z[]",
  "rigour": "exact",
  "precision": "Exact integer coefficients in two indeterminates; no rounded arguments or significant-digit claim.",
  "skill_manifest": "agents/runs/20261028T224818Z-repair-skills/manifest.json",
  "skill_sha256": {
    "local": "28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199",
    "live": "28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199"
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  "skill_manifest_sha256": "f3ffe644f05a2d8ba38978976adde3737ab5de3d1ebeb214066f311233b3080e",
  "source_checks": "sources-checked.md",
  "all_entry_readback": {
    "source": "live authenticated numberdb.table readback",
    "all_entry_verify_ok": true,
    "independent_checks": {
      "entries_checked": 6,
      "indices": [
        0,
        1,
        2,
        3,
        4,
        5
      ],
      "independent_wronskians": 6,
      "published_explicit_polynomials": 5,
      "recurrence_degree_monicity_integrality": true,
      "mu_zero_and_reverse_bessel_specializations": true,
      "painleve_III_and_log_derivative_identities": true,
      "known_and_perturbed_controls": true,
      "parity_and_mu_shift_bilinear_identity": true,
      "measure": {
        "entries": 6,
        "longest": 898,
        "longest_at": 5,
        "block_kb": 1.650390625
      }
    },
    "stored_program_agrees": true,
    "environment": {
      "python": "3.12.3",
      "numberdb": "0.1.10",
      "sage": "10.9"
    }
  },
  "checks_coverage": {
    "independent_wronskians": "all n=0..5; n=6 supplies recurrence boundary",
    "printed_source_polynomials": "all n=1..5 in SIGMA Table 1",
    "symbolic_PIII_and_log_derivative_identity": "all n=0..5, generic symbolic m",
    "monicity_integrality_degrees_parity_specializations_parameter_shift": "all n=0..5",
    "program": "all n=0..5 and next n=6",
    "live_T116_bessel_specialization": "n=0..5",
    "growth_measurement": "n=0..9",
    "extra_research_checks": "trial n=0..6 independently checked, including neighbor S7; excluded from final selection"
  },
  "sizes": {
    "value_bytes": 1424,
    "max_entry_bytes": 898,
    "numbers_block_yaml_bytes": 1678,
    "api_document_bytes": 7657
  },
  "audit": {
    "tid": "T558",
    "title": "Umemura polynomials for Painlev\u00e9 III",
    "findings": [],
    "clean": true
  },
  "rendering": {
    "url": "https://numberdb.org/T558",
    "http_status": 200,
    "browser_version": "156.0.8078.4",
    "playwright_version": "1.64.1",
    "page_errors": [],
    "request_failures": [],
    "math_errors": 0,
    "math_expressions": 55,
    "unknown_key": false,
    "no_entries_message": false,
    "document_width": 1440,
    "viewport_width": 1440,
    "number_section_text": "Polynomials\n\ud835\udc5b\n\u00a0\t\n\ud835\udc46\n\ud835\udc5b\n(\n\ud835\udc65\n;\n\ud835\udc5a\n)\n\n0:\t1\n1:\tx + m\n2:\tx^3 + 3*x^2*m + 3*x*m^2 + m^3 - m\n3:\tx^6 + 6*x^5*m + 15*x^4*m^2 + 20*x^3*m^3 - 5*x^3*m + 15*x^2*m^4 - 15*x^2*m^2 + 6*x*m^5 - 15*x*m^3 + 9*x*m + m^6 - 5*m^4 + 4*m^2\n4:\tx^10 + 10*x^9*m + 45*x^8*m^2 + 120*x^7*m^3 - 15*x^7*m + 210*x^6*m^4 - 105*x^6*m^2 + 252*x^5*m^5 - 315*x^5*m^3 + 63*x^5*m + 210*x^4*m^6 - 525*x^4*m^4 + 315*x^4*m^2 + 120*x^3*m^7 - 525*x^3*m^5 + 630*x^3*m^3 - 225*x^3*m + 45*x^2*m^8 - 315*x^2*m^6 + 630*x^2*m^4 - 360*x^2*m^2 + 10*x*m^9 - 105*x*m^7 + 315*x*m^5 - 220*x*m^3 + m^10 - 15*m^8 + 63*m^6 - 85*m^4 + 36*m^2\n5:\tx^15 + 15*x^14*m + 105*x^13*m^2 + 455*x^12*m^3 - 35*x^12*m + 1365*x^11*m^4 - 420*x^11*m^2 + 3003*x^10*m^5 - 2310*x^10*m^3 + 252*x^10*m + 5005*x^9*m^6 - 7700*x^9*m^4 + 2695*x^9*m^2 + 6435*x^8*m^7 - 17325*x^8*m^5 + 12915*x^8*m^3 - 2025*x^8*m + 6435*x^7*m^8 - 27720*x^7*m^6 + 36540*x^7*m^4 - 15255*x^7*m^2 + 5005*x^6*m^9 - 32340*x^6*m^7 + 67620*x^6*m^5 - 51310*x^6*m^3 + 11025*x^6*m + 3003*x^5*m^10 - 27720*x^5*m^8 + 85554*x^5*m^6 - 100905*x^5*m^4 + 40068*x^5*m^2 + 1365*x^4*m^11 - 17325*x^4*m^9 + 74970*x^4*m^7 - 127050*x^4*m^5 + 68040*x^4*m^3 + 455*x^3*m^12 - 7700*x^3*m^10 + 44940*x^3*m^8 - 104825*x^3*m^6 + 79730*x^3*m^4 - 12600*x^3*m^2 + 105*x^2*m^13 - 2310*x^2*m^11 + 17640*x^2*m^9 - 55230*x^2*m^7 + 66255*x^2*m^5 - 26460*x^2*m^3 + 15*x*m^14 - 420*x*m^12 + 4095*x*m^10 - 16935*x*m^8 + 31290*x*m^6 - 26145*x*m^4 + 8100*x*m^2 + m^15 - 35*m^13 + 427*m^11 - 2305*m^9 + 5768*m^7 - 6160*m^5 + 2304*m^3"
  },
  "versions": {
    "python": "3.12.3",
    "numberdb": "0.1.10",
    "sage": "10.9"
  },
  "provenance": {
    "account": "zeta3",
    "assisted_by": "codex-cli",
    "transport": "authenticated NumberDB API",
    "builder_choices": "single-letter m=mu storage notation, final range, prose, generator repair, exact checks"
  },
  "limitations": [
    "Finite initial segment n=0..5; selection cutoff is noncanonical.",
    "No independent critique actor has yet reviewed this completed draft.",
    "Draft entries are intentionally absent from public value search; live format requests returned HTTP 200 and no T558 matches.",
    "Primary-source unsigned parity typo was rejected; see sources-checked.md."
  ],
  "files_sha256": {
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}