{
  "banco": "lagrangiana",
  "verdict": "CONFERMATO",
  "claim": "Principio Lagrangiano DOPPIO: 1/phi ha V_c \u2248 1.035 (rank 2/14) + massima stabilita' (CV=4.6%, rank 1/6)",
  "N": 500,
  "n_frequencies": 14,
  "phi_is_global_min": false,
  "phi_is_local_min": true,
  "phi_most_stable": true,
  "phi_cv": 4.619236344648908,
  "phi_mean_Vc": 1.0353538839226095,
  "correlation_Vc_irrationality": 0.5183286707144003,
  "ranking_Vc": [
    [
      "1/e",
      0.9709773528270232
    ],
    [
      "1/phi",
      0.9924688485952523
    ],
    [
      "1/(phi+2)",
      1.0382631350786256
    ],
    [
      "1/(phi+1)",
      1.1524202395708136
    ],
    [
      "sqrt(5)/3",
      1.1597944601988184
    ],
    [
      "sqrt(2)-1",
      1.2071377093975362
    ],
    [
      "ln(2)",
      1.2099750616611578
    ],
    [
      "sqrt(2)/2",
      1.2479666294195715
    ],
    [
      "sqrt(3)-1",
      1.2958252344376
    ],
    [
      "sqrt(3)/3",
      1.3129333104842749
    ],
    [
      "pi-3",
      2.4881125034430083
    ],
    [
      "sqrt(7)/3",
      2.5315066215319506
    ],
    [
      "liouville",
      2.8200624514848753
    ],
    [
      "random",
      4.0
    ]
  ],
  "ranking_stability": [
    [
      "1/phi",
      4.619236344648908
    ],
    [
      "1/(phi+2)",
      5.745797935724093
    ],
    [
      "1/(phi+1)",
      5.924873916071753
    ],
    [
      "sqrt(2)-1",
      11.58216566147367
    ],
    [
      "1/e",
      13.402418274460008
    ],
    [
      "sqrt(5)/3",
      20.815207085809753
    ]
  ],
  "stability": {
    "1/e": {
      "mean": 1.1315000745985246,
      "std": 0.15164837277352128,
      "cv": 13.402418274460008,
      "vals": [
        1.1326166788736978,
        1.1007920094025443,
        0.9888613578600762,
        0.9709773528270232,
        1.164528206067208,
        1.4312248425605976
      ]
    },
    "1/phi": {
      "mean": 1.0353538839226095,
      "std": 0.04782544290188725,
      "cv": 4.619236344648908,
      "vals": [
        0.9817129673101963,
        1.1171675853240184,
        1.0787562052408854,
        0.9924688485952523,
        1.0179789616511417,
        1.0240387354141627
      ]
    },
    "1/(phi+2)": {
      "mean": 1.0788237612471623,
      "std": 0.06198703340384047,
      "cv": 5.745797935724093,
      "vals": [
        0.997590558956831,
        1.109845479329427,
        1.0727746034279848,
        1.0382631350786256,
        1.1947778555063104,
        1.0596909351837942
      ]
    },
    "1/(phi+1)": {
      "mean": 1.0722254696055356,
      "std": 0.06352800717013624,
      "cv": 5.924873916071753,
      "vals": [
        1.0877054459009416,
        1.02450197660006,
        1.1513518895858372,
        1.1524202395708136,
        1.0302635388496593,
        0.9871097271259015
      ]
    },
    "sqrt(5)/3": {
      "mean": 1.3769609630617323,
      "std": 0.28661727595205994,
      "cv": 20.815207085809753,
      "vals": [
        1.8816979432717351,
        1.1413285084259819,
        1.0602381388346354,
        1.1597944601988184,
        1.5028506156725765,
        1.515856111966647
      ]
    },
    "sqrt(2)-1": {
      "mean": 1.208573757073818,
      "std": 0.13997901468538596,
      "cv": 11.58216566147367,
      "vals": [
        1.2106988760141224,
        1.2274508353991387,
        0.9286313179211739,
        1.2071377093975362,
        1.2890821799253809,
        1.3884416237855568
      ]
    }
  },
  "results": {
    "liouville": {
      "alpha": 0.110001,
      "cf": [
        0,
        9,
        11,
        99,
        1,
        10,
        9,
        122965,
        1,
        3
      ],
      "irrationality": 11.43731047327559,
      "Vc": 2.8200624514848753,
      "r_at_Vc": 0.500000004310532
    },
    "pi-3": {
      "alpha": 0.14159265358979312,
      "cf": [
        0,
        7,
        15,
        1,
        292,
        1,
        1,
        1,
        2,
        1
      ],
      "irrationality": 3.4314100725342627,
      "Vc": 2.4881125034430083,
      "r_at_Vc": 0.4999999911465766
    },
    "1/(phi+2)": {
      "alpha": 0.276393202250021,
      "cf": [
        0,
        3,
        1,
        1,
        1,
        1,
        1,
        1,
        1,
        1
      ],
      "irrationality": 1.0595260647382752,
      "Vc": 1.0382631350786256,
      "r_at_Vc": 0.5000001898621559
    },
    "1/e": {
      "alpha": 0.36787944117144233,
      "cf": [
        0,
        2,
        1,
        2,
        1,
        1,
        4,
        1,
        1,
        6
      ],
      "irrationality": 1.8251211957688878,
      "Vc": 0.9709773528270232,
      "r_at_Vc": 0.4999997461691081
    },
    "1/(phi+1)": {
      "alpha": 0.38196601125010515,
      "cf": [
        0,
        2,
        1,
        1,
        1,
        1,
        1,
        1,
        1,
        1
      ],
      "irrationality": 1.0371550444461919,
      "Vc": 1.1524202395708136,
      "r_at_Vc": 0.4999999526832591
    },
    "sqrt(2)-1": {
      "alpha": 0.41421356237309515,
      "cf": [
        0,
        2,
        2,
        2,
        2,
        2,
        2,
        2,
        2,
        2
      ],
      "irrationality": 2.0,
      "Vc": 1.2071377093975362,
      "r_at_Vc": 0.49999996259932433
    },
    "sqrt(3)/3": {
      "alpha": 0.5773502691896257,
      "cf": [
        0,
        1,
        1,
        2,
        1,
        2,
        1,
        2,
        1,
        2
      ],
      "irrationality": 1.3886511426146562,
      "Vc": 1.3129333104842749,
      "r_at_Vc": 0.4999999438153375
    },
    "1/phi": {
      "alpha": 0.6180339887498948,
      "cf": [
        0,
        1,
        1,
        1,
        1,
        1,
        1,
        1,
        1,
        1
      ],
      "irrationality": 1.0,
      "Vc": 0.9924688485952523,
      "r_at_Vc": 0.49999998627890974
    },
    "ln(2)": {
      "alpha": 0.6931471805599453,
      "cf": [
        0,
        1,
        2,
        3,
        1,
        6,
        3,
        1,
        1,
        2
      ],
      "irrationality": 1.6460874159288834,
      "Vc": 1.2099750616611578,
      "r_at_Vc": 0.49999976497795734
    },
    "sqrt(2)/2": {
      "alpha": 0.7071067811865476,
      "cf": [
        0,
        1,
        2,
        2,
        2,
        2,
        2,
        2,
        2,
        2
      ],
      "irrationality": 1.9283519958849902,
      "Vc": 1.2479666294195715,
      "r_at_Vc": 0.5000001544035095
    },
    "random": {
      "alpha": 0.7291836475,
      "cf": [
        0,
        1,
        2,
        1,
        2,
        3,
        1,
        24,
        1,
        3
      ],
      "irrationality": 2.4533562016398798,
      "Vc": 4.0,
      "r_at_Vc": 0.6310785118618308
    },
    "sqrt(3)-1": {
      "alpha": 0.7320508075688772,
      "cf": [
        0,
        1,
        2,
        1,
        2,
        1,
        2,
        1,
        2,
        1
      ],
      "irrationality": 1.3886511426146562,
      "Vc": 1.2958252344376,
      "r_at_Vc": 0.4999999876809873
    },
    "sqrt(5)/3": {
      "alpha": 0.7453559924999299,
      "cf": [
        0,
        1,
        2,
        1,
        12,
        1,
        2,
        2,
        2,
        1
      ],
      "irrationality": 2.055844179515702,
      "Vc": 1.1597944601988184,
      "r_at_Vc": 0.4999998286018067
    },
    "sqrt(7)/3": {
      "alpha": 0.8819171036881969,
      "cf": [
        0,
        1,
        7,
        2,
        7,
        2,
        7,
        2,
        7,
        2
      ],
      "irrationality": 2.9846215904406677,
      "Vc": 2.5315066215319506,
      "r_at_Vc": 0.4999999954516829
    }
  }
}