{
  "what": "Independent exact verification of the claimed lower bound delta_3 >= 117/2192 (Theorem A: Q(phi) >= -5/137 for every measurable phi:[0,1]->[-1,1]), from the five claimant certificates, with a verifier written clean-room from THEOREM-EXACT.md and CERT-FORMAT.md only.",
  "decidedOn": "2026-10-05",
  "verifier": {
    "files": {
      "instruments/delta3/model.js": "65b68c49c45f8a6cf5045de9e632e2f67e8f8285de2d1a77e8b242c5bb4d440a",
      "instruments/delta3/psd.js": "0757eb59af84f3224cb9784add29d93e9368ced8700ccc42a4df8f83253ffae5",
      "instruments/delta3/verify.js": "a49e68a17e5e0a785ccd9a38a6679bbf8a5f2584f35021d139a608e6ccfccd7c",
      "instruments/delta3/lemmas.js": "bdbd0b706cb002ed27250dda08cd4a3a5f714f33ac6b1d9b8ec993b4fdc0e439",
      "instruments/delta3/battery.js": "cc19edd9f8e36ad6e49217177ae41b810e01ba2e0ce83832b40c3d19861afcba",
      "instruments/interval/rational.js": "59e97fc40328adc7866f148d8ee61156810dbbcd890b4361dca697aa58d71920"
    },
    "node": "v24.14.1",
    "method": {
      "areas": "exact: 4u^2*area(R ∩ I_i×I_j) by the trapezoid rule on the piecewise-linear section length (kinks enumerated), cross-checked pair by pair against Sutherland-Hodgman clipping + shoelace in rationals",
      "psd": "exact: dyadic rescaling, float Cholesky of A - 2^-k I rounded to a dyadic factor L, exact remainder A - LL^T shown diagonally dominant (Gershgorin); fallback exact witness / Bareiss leading minors"
    }
  },
  "phiStar": {
    "Q": "-5/137",
    "hIntegral": "5/137",
    "hZeros": [
      "7/137",
      "17/274",
      "31/274",
      "99/548",
      "79/274",
      "1/2",
      "195/274",
      "449/548",
      "243/274",
      "257/274",
      "130/137"
    ],
    "hMinOffZeros": {
      "value": "1/1096",
      "note": "least nonzero node value of h (h is linear between nodes)"
    },
    "slopesAtZeros": [
      {
        "at": "7/137",
        "left": "-1",
        "right": "1"
      },
      {
        "at": "17/274",
        "left": "-1",
        "right": "1"
      },
      {
        "at": "31/274",
        "left": "-3/2",
        "right": "3/2"
      },
      {
        "at": "99/548",
        "left": "-1",
        "right": "1"
      },
      {
        "at": "79/274",
        "left": "-3/2",
        "right": "3/2"
      },
      {
        "at": "1/2",
        "left": "-3/2",
        "right": "3/2"
      },
      {
        "at": "195/274",
        "left": "-3/2",
        "right": "3/2"
      },
      {
        "at": "449/548",
        "left": "-1",
        "right": "1"
      },
      {
        "at": "243/274",
        "left": "-3/2",
        "right": "3/2"
      },
      {
        "at": "257/274",
        "left": "-1",
        "right": "1"
      },
      {
        "at": "130/137",
        "left": "-1",
        "right": "1"
      }
    ]
  },
  "certificates": [
    {
      "file": "certs/delta3/inner-r20-g24_4_8.json",
      "sha256": "2edd22852d329458752a9e6d1495b3412fb3db2ec68fad591f9730fe99467c60",
      "kind": "inner",
      "region": {
        "lo": "0",
        "hi": "1/5"
      },
      "n": 130,
      "counts": {
        "fullPairs": 7718,
        "cutPairs": 318,
        "forms": 22808,
        "triangles": 0
      },
      "verdict": "VERIFIED",
      "why": null,
      "bound": "0",
      "boundDecimal": "0.0000000e+0",
      "claimMatchesFile": null,
      "seconds": 0.249,
      "psdMethod": "dyadic Cholesky of A − 2^−4I (rows rescaled by powers of 2), remainder diagonally dominant"
    },
    {
      "file": "certs/delta3/slab-1_5-3_10-g24_4_8.json",
      "sha256": "f8c569e4a47bc6d958f2515485c355a016609a5f27178962a71ff749dce80c99",
      "kind": "slab",
      "region": {
        "lo": "1/5",
        "hi": "3/10"
      },
      "n": 130,
      "counts": {
        "fullPairs": 7718,
        "cutPairs": 318,
        "forms": 13324,
        "triangles": 0
      },
      "verdict": "VERIFIED",
      "why": null,
      "bound": "6296384718336605890182449/22609041927129531731084058720",
      "boundDecimal": "2.7848967e-4",
      "claimMatchesFile": true,
      "seconds": 0.121,
      "psdMethod": "dyadic Cholesky of A − 2^−22I (rows rescaled by powers of 2), remainder diagonally dominant"
    },
    {
      "file": "certs/delta3/slab-2_5-9_20-g8_4_0.json",
      "sha256": "bc23d33cb0db48a29e7bde319ef2e20dcd56d18b8ab498c06fec52624537d2c3",
      "kind": "slab",
      "region": {
        "lo": "2/5",
        "hi": "9/20"
      },
      "n": 146,
      "counts": {
        "fullPairs": 10380,
        "cutPairs": 294,
        "forms": 1998,
        "triangles": 0
      },
      "verdict": "VERIFIED",
      "why": null,
      "bound": "50963377393403307885655938/185520059131196374275938963885",
      "boundDecimal": "2.7470548e-4",
      "claimMatchesFile": true,
      "seconds": 0.177,
      "psdMethod": "dyadic Cholesky of A − 2^−22I (rows rescaled by powers of 2), remainder diagonally dominant"
    },
    {
      "file": "certs/delta3/slab-3_10-4_10-g24_4_8.json",
      "sha256": "d9b37a69505a3e642b479b2bd57b655d5fa33a716cf3ffdcd6b3869af524fbce",
      "kind": "slab",
      "region": {
        "lo": "3/10",
        "hi": "2/5"
      },
      "n": 130,
      "counts": {
        "fullPairs": 7718,
        "cutPairs": 318,
        "forms": 9849,
        "triangles": 2204
      },
      "verdict": "VERIFIED",
      "why": null,
      "bound": "302061852991525508034881073/995093408858134763756936327093",
      "boundDecimal": "3.0355125e-4",
      "claimMatchesFile": true,
      "seconds": 0.159,
      "psdMethod": "dyadic Cholesky of A − 2^−21I (rows rescaled by powers of 2), remainder diagonally dominant"
    },
    {
      "file": "certs/delta3/slab-9_20-1_2-g8_4_0.json",
      "sha256": "b3f8615a9f1009624d87852859526056321eccc7b5fc5d57af502403ff5803d2",
      "kind": "slab",
      "region": {
        "lo": "9/20",
        "hi": "1/2"
      },
      "n": 146,
      "counts": {
        "fullPairs": 10380,
        "cutPairs": 294,
        "forms": 2332,
        "triangles": 470
      },
      "verdict": "VERIFIED",
      "why": null,
      "bound": "336509882373552357045412601/725496284546822070496074644287",
      "boundDecimal": "4.6383405e-4",
      "claimMatchesFile": true,
      "seconds": 0.124,
      "psdMethod": "dyadic Cholesky of A − 2^−21I (rows rescaled by powers of 2), remainder diagonally dominant"
    }
  ],
  "cover": {
    "intervals": [
      [
        "0",
        "1/5"
      ],
      [
        "1/5",
        "3/10"
      ],
      [
        "3/10",
        "2/5"
      ],
      [
        "2/5",
        "9/20"
      ],
      [
        "9/20",
        "1/2"
      ]
    ],
    "complete": true
  },
  "theorem": {
    "verdict": "VERIFIED",
    "Qstar": "-5/137",
    "delta3LowerBound": "117/2192"
  },
  "seconds": 0.846
}