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          "83.177661667193437130067854575222453722938476755",
          "83.177661667193437130067854575222453722938476756"
        ],
        [
          "170.94259732786260602844718135646958620521125353",
          "170.942597327862606028447181356469586205211253531"
        ]
      ]
    },
    {
      "name": "dual 1: eight exact generator inequalities",
      "passed": true,
      "slacks": [
        [
          "1.663553233343868742601357091502423431924499865",
          "1.663553233343868742601357091502423431924499866"
        ],
        [
          "0",
          "0"
        ],
        [
          "33.271064666877374852027141830048468638489997312",
          "33.271064666877374852027141830048468638489997313"
        ],
        [
          "1.3998342716497715E-29",
          "1.3998342716497716E-29"
        ],
        [
          "39.925277600252849822432570196058162366187996775",
          "39.925277600252849822432570196058162366187996776"
        ],
        [
          "1.663553233343868742601357091516421774640997581",
          "1.663553233343868742601357091516421774640997582"
        ],
        [
          "89.831874600568912100473282941228853722938476755",
          "89.831874600568912100473282941228853722938476756"
        ],
        [
          "163.87249608615116387239141371758778620521125353",
          "163.872496086151163872391413717587786205211253531"
        ]
      ]
    },
    {
      "name": "published rung 1: archived cap 215",
      "passed": true
    },
    {
      "name": "published rung 3: archived cap 336",
      "passed": true
    },
    {
      "name": "candidate rung 2: archived cap 272.5",
      "passed": true
    },
    {
      "name": "candidate rung 4: archived cap 375.5",
      "passed": true
    }
  ],
  "proof": {
    "zeta3": [
      "1.202056903159594285399738161511449990764986292",
      "1.202056903159594285399738161511449990764986293"
    ],
    "precisionDigits": 60,
    "tail": "For one feasible dual, f=G_required-G_lower has f<0 and df/dA4<0 at the first excluded lattice point. Its second derivative is -1/(2*(6*mu+A4))-3*mu/(6*mu+A4)^2<0. Hence every larger A4 is excluded.",
    "massWindow": "dG_required/dmu=18*mu/(6*mu+A4)>0; using mh=127 bounds the whole [123,127] GeV window.",
    "zeroRung": "6*mu+A4 >= 6*mu_min+A4_gauge >0. Identity II therefore forbids k<=0 at any finite nonzero x in this mass window, within the small-angle model.",
    "missing": "No remainder enclosure relating identities I/II to the full Fourier potential; no global-vacuum, anomaly, or collider certificate. No bound for unlisted rungs."
  },
  "sources": [
    "https://arxiv.org/abs/2503.04090",
    "Part VII identities I/II and the archived exact rational moment-cone duals"
  ]
}
