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  "date": "2026-10-08",
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              "2426125879704193768793244203598/50561060209264429106754833069"
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            ],
            "curvature": [
              "1164680402449906684592712542056/24461644989694597878656175043",
              "3317756646435056297132723449304/69098873298548621150991633361"
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            "curvature": [
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              "224919988286144602155837283380/4681427487984725196437010373"
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          }
        ],
        "claim": "The mean-value theorem bounds |alpha_full-alpha_approx| by epsilon/curvatureFloor on the certified convex interval. Globality is a separate claim."
      }
    }
  ],
  "momentWitness": {
    "seed": "published",
    "coordinateOrder": [
      "2A4",
      "8D",
      "2U",
      "V",
      "2W"
    ],
    "contents": [
      {
        "name": "A",
        "bulk": [
          {
            "rep": "7",
            "parities": [
              1,
              -1
            ],
            "multiplicity": 1
          },
          {
            "rep": "28",
            "parities": [
              1,
              1
            ],
            "multiplicity": 1
          },
          {
            "rep": "84",
            "parities": [
              1,
              1
            ],
            "multiplicity": 1
          }
        ],
        "terms": [
          [
            "25/2",
            -1,
            1
          ],
          [
            "1",
            -1,
            2
          ],
          [
            "1",
            -1,
            3
          ],
          [
            "4",
            1,
            1
          ],
          [
            "4",
            1,
            2
          ]
        ],
        "coordinates": [
          "136",
          "7",
          "347",
          "0",
          "19"
        ],
        "localRoot": [
          "80538649474890611628670515491451810888892172552151/1000000000000000000000000000000000000000000000000000",
          "80540649474890611628670515491451810888892172552151/1000000000000000000000000000000000000000000000000000"
        ],
        "numericalLocalAlpha": "0.080539649474890611628670515491451811",
        "approximateRoot": [
          "506204229784878366929792750687522828423633517951794341/6250000000000000000000000000000000000000000000000000000",
          "506204229797378366929792750687522828423633517951794341/6250000000000000000000000000000000000000000000000000000"
        ],
        "endpointMinusLocalEnergy": [
          "353651722664871206651112903769/18515871158368318820876302405",
          "810644447226258537243737057310/42442287645434725055683756879"
        ],
        "endpointOrdering": [
          "490459970273753897269179510515/25697433552153243911434859281",
          "462897687216443727640200624789/24253319903050473089145712013"
        ],
        "localMinimumCertified": true,
        "lowerEndpointCertified": false
      },
      {
        "name": "B",
        "bulk": [
          {
            "rep": "7",
            "parities": [
              1,
              1
            ],
            "multiplicity": 1
          },
          {
            "rep": "28",
            "parities": [
              1,
              -1
            ],
            "multiplicity": 6
          },
          {
            "rep": "48",
            "parities": [
              1,
              1
            ],
            "multiplicity": 3
          },
          {
            "rep": "48",
            "parities": [
              1,
              -1
            ],
            "multiplicity": 1
          }
        ],
        "terms": [
          [
            "59/2",
            -1,
            1
          ],
          [
            "7",
            -1,
            2
          ],
          [
            "36",
            1,
            1
          ],
          [
            "2",
            1,
            2
          ]
        ],
        "coordinates": [
          "136",
          "7",
          "347",
          "0",
          "-13"
        ],
        "localRoot": [
          "80719883172784698313265506226896741967270537099993/1000000000000000000000000000000000000000000000000000",
          "80721883172784698313265506226896741967270537099993/1000000000000000000000000000000000000000000000000000"
        ],
        "numericalLocalAlpha": "0.080720883172784698313265506226896742",
        "approximateRoot": [
          "506204229784878366929792750687522828423633517951794341/6250000000000000000000000000000000000000000000000000000",
          "506204229797378366929792750687522828423633517951794341/6250000000000000000000000000000000000000000000000000000"
        ],
        "endpointMinusLocalEnergy": [
          "-723634859547903148419880209901/55473035903964183836690899382",
          "-318689876593214705648733577537/24430408147049490642010374089"
        ],
        "endpointOrdering": [
          "-315268823724024120510191012449/24142234237734875333585008377",
          "-622905199636680025443968939462/47700001097145539173616016457"
        ],
        "localMinimumCertified": true,
        "lowerEndpointCertified": true
      }
    ],
    "claim": "The two contents have identical D,A4,G exactly, different full potentials and opposite endpoint ordering. Content B has a certified local minimum with a strictly lower competitor at alpha=1.",
    "scope": "One-loop mathematical witness; perturbative and phenomenological viability are not assessed. Global minima of the pair are not certified here."
  },
  "sources": [
    {
      "url": "https://arxiv.org/abs/2503.04090",
      "role": "Model: Yuzuho Komori and Nobuhito Maru."
    },
    {
      "url": "https://arxiv.org/abs/hep-ph/0411250",
      "role": "Kernel expansions: Naoyuki Haba, Kazunori Takenaga and Toshifumi Yamashita."
    },
    {
      "url": "https://arxiv.org/abs/hep-th/0609067",
      "role": "Three-coefficient potential: Makoto Sakamoto and Kazunori Takenaga."
    },
    {
      "url": "https://github.com/karlesmarin/su7-compactification-bound",
      "role": "SU(7) moment coordinates and interpretation: Carles Marin, Part VII."
    }
  ],
  "inputs": {
    "reconstruction.json": "2c6163f34c39da8e85cd756f1faf89d852671d16ab1440f4e19dac0442247c69",
    "global_certificates.json": "7bd87afde56f0e6a818c6961c98a83fda743d182fd8c2ac68e2d9e16792c05d8",
    "sage_verification.json": "8e5140574a67e23a1eb0846671048604ae73cd90220b96596417a3ded18628c4"
  },
  "modelSHA256": "17b8d5d7c92310c3646a54486d41c8bffc3327551cd0cb5135a34c82fe6427f8",
  "scriptSHA256": "875736294a801c245fa1187eb472f1478363f5a547e4b67f1ce4ea7e4f69f4e8",
  "checks": [
    {
      "name": "existing Arb certificate input reconstruction.json",
      "passed": true
    },
    {
      "name": "existing Arb certificate input global_certificates.json",
      "passed": true
    },
    {
      "name": "(1) printed: inherited global certificate",
      "passed": true
    },
    {
      "name": "(1) printed: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(1) printed: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(1) printed: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(1) printed: approximate root bracket",
      "passed": true
    },
    {
      "name": "(1) printed: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(1) printed: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(1) candidate: inherited global certificate",
      "passed": true
    },
    {
      "name": "(1) candidate: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(1) candidate: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(1) candidate: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(1) candidate: approximate root bracket",
      "passed": true
    },
    {
      "name": "(1) candidate: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(1) candidate: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(2) printed: inherited global certificate",
      "passed": true
    },
    {
      "name": "(2) printed: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(2) printed: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(2) printed: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(2) printed: approximate root bracket",
      "passed": true
    },
    {
      "name": "(2) printed: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(2) printed: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(2) candidate: inherited global certificate",
      "passed": true
    },
    {
      "name": "(2) candidate: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(2) candidate: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(2) candidate: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(2) candidate: approximate root bracket",
      "passed": true
    },
    {
      "name": "(2) candidate: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(2) candidate: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(3) printed: inherited global certificate",
      "passed": true
    },
    {
      "name": "(3) printed: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(3) printed: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(3) printed: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(3) printed: approximate root bracket",
      "passed": true
    },
    {
      "name": "(3) printed: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(3) printed: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(3) candidate: inherited global certificate",
      "passed": true
    },
    {
      "name": "(3) candidate: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(3) candidate: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(3) candidate: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(3) candidate: approximate root bracket",
      "passed": true
    },
    {
      "name": "(3) candidate: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(3) candidate: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(4) printed: inherited global certificate",
      "passed": true
    },
    {
      "name": "(4) printed: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(4) printed: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(4) printed: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(4) printed: approximate root bracket",
      "passed": true
    },
    {
      "name": "(4) printed: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(4) printed: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(4) candidate: inherited global certificate",
      "passed": true
    },
    {
      "name": "(4) candidate: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(4) candidate: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(4) candidate: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(4) candidate: approximate root bracket",
      "passed": true
    },
    {
      "name": "(4) candidate: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(4) candidate: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(5) printed: inherited global certificate",
      "passed": true
    },
    {
      "name": "(5) printed: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(5) printed: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(5) printed: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(5) printed: approximate root bracket",
      "passed": true
    },
    {
      "name": "(5) printed: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(5) printed: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "(5) candidate: inherited global certificate",
      "passed": true
    },
    {
      "name": "(5) candidate: independently replayed full root bracket",
      "passed": true
    },
    {
      "name": "(5) candidate: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "(5) candidate: two real Lambert branches",
      "passed": true
    },
    {
      "name": "(5) candidate: approximate root bracket",
      "passed": true
    },
    {
      "name": "(5) candidate: positive approximate curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "positive mass curvature",
      "passed": true
    },
    {
      "name": "(5) candidate: full potential convex between the roots",
      "passed": true
    },
    {
      "name": "witness A: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "witness A: two real Lambert branches",
      "passed": true
    },
    {
      "name": "witness A: approximate root bracket",
      "passed": true
    },
    {
      "name": "witness A: positive approximate curvature",
      "passed": true
    },
    {
      "name": "witness A: local root near approximation",
      "passed": true
    },
    {
      "name": "witness A: full derivative sign bracket",
      "passed": true
    },
    {
      "name": "witness A: full local curvature",
      "passed": true
    },
    {
      "name": "witness B: Lambert minimum signs",
      "passed": true
    },
    {
      "name": "witness B: two real Lambert branches",
      "passed": true
    },
    {
      "name": "witness B: approximate root bracket",
      "passed": true
    },
    {
      "name": "witness B: positive approximate curvature",
      "passed": true
    },
    {
      "name": "witness B: local root near approximation",
      "passed": true
    },
    {
      "name": "witness B: full derivative sign bracket",
      "passed": true
    },
    {
      "name": "witness B: full local curvature",
      "passed": true
    },
    {
      "name": "witness: exact equality of four local coordinates",
      "passed": true
    },
    {
      "name": "witness: exact endpoint difference",
      "passed": true
    },
    {
      "name": "witness: opposite endpoint ordering",
      "passed": true
    },
    {
      "name": "witness B: endpoint below the certified local minimum",
      "passed": true
    }
  ],
  "passed": 110,
  "scope": "Conditional on the declared one-loop potentials and fixed mass inputs. Computer-assisted interval proofs, not Lean formalization or a claim of scientific originality."
}