{
  "description": "Exact symbolic verification, not numerical sampling of arbitrary-group spectra.",
  "basis": "(E11,0),(E12,0),(E21,0),(E22,0),(0,E11),(0,E12),(0,E21),(0,E22)",
  "D": "Matrix([[1, 0], [0, -1]])",
  "W": "Matrix([[1, 0], [0, I]])",
  "e_in_each_summand": "Matrix([[1, 0], [0, 0]])",
  "alpha_generator": "Matrix([[0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0]])",
  "beta_generator": "Matrix([[0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, -I, 0, 0], [0, 0, 0, 0, 0, 0, I, 0], [0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0], [0, I, 0, 0, 0, 0, 0, 0], [0, 0, -I, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0]])",
  "alpha_characteristic_polynomial": "(lambda - 1)**2*(lambda + 1)**2*(lambda**2 + 1)**2",
  "beta_characteristic_polynomial": "(lambda - 1)**4*(lambda + 1)**4",
  "alpha_eigenvalues_with_multiplicity": {
    "-1": 2,
    "1": 2,
    "-I": 2,
    "I": 2
  },
  "beta_eigenvalues_with_multiplicity": {
    "-1": 4,
    "1": 4
  },
  "fixed_algebra_basis": [
    "(E11,E11)",
    "(E22,E22)"
  ],
  "fixed_projection_corners": [
    "e",
    "1-e",
    "1"
  ],
  "full_exact_checks": [
    "W^2=D",
    "W*D=W and DW=W*",
    "W=(1+i)I/2+(1-i)D/2",
    "alpha^4=id, alpha^2 is not id",
    "beta=Ad(W*,W*) alpha",
    "beta^2=id, beta is not id",
    "fixed space exactly span{(E11,E11),(E22,E22)}",
    "alpha eigenvalues 1,-1,i,-i",
    "beta eigenvalues 1,-1",
    "w extends v and is fixed (powers reduced using W^4=I)"
  ],
  "proof_of_action_spectrum_not_computation": "MF.MODEL uses M9 and the topological identification of the dual of Z with T. The full fixed-projection intersection gives Gamma={1,-1}.",
  "proof_of_kernel": "Integer powers of the exact nonidentity involution beta have kernel 2Z; all odd powers retain the central swap.",
  "passed": true
}