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    -1.1468563519975012,
    -1.1251209212169164,
    -1.1032816211450802,
    -1.0813387382035529,
    -1.0592925617693654,
    -1.037143384170083,
    -1.0148915006786114,
    -0.9925372095077414,
    -0.9700808118051524,
    -0.9475226116480076,
    -0.9248629160370702,
    -0.9021020348916718,
    -0.8792402810438097,
    -0.8562779702321457,
    -0.8332154210967339,
    -0.8100529551726184,
    -0.7867908968842482,
    -0.7634295735389037,
    -0.739969315321123,
    -0.7164104552859907,
    -0.6927533293530158,
    -0.6689982762996416,
    -0.6451456377549235,
    -0.6211957581926754,
    -0.5971489849250052,
    -0.5730056680954517,
    -0.5487661606723874,
    -0.5244308184416224,
    -0.5
  ],
  "energy_formula": "sum(cos(a*(2*pi*j+phi)/q), a=1..N)",
  "exact_zero_twist": {
    "energy_N_multiplicity": 1,
    "energy_minus_half_multiplicity": 24
  },
  "exact_recovery": "min_{phi in [0,2*pi]} min_j E_j(phi) = -K({1,...,N})",
  "plot_arithmetic": "float64; endpoint root-of-unity values set by the exact identity",
  "scope": "Auxiliary lattice spectrum. Boundary phase is not a second free phase for each original harmonic."
}