{
  "license": "CC0-1.0 to the extent of rights held; numerical facts are not claimed as exclusive property.",
  "model": {
    "G": "Z/4Z",
    "H": "Z/4Z",
    "pairing": "chi_j(s)=i^(j*s)",
    "beta": "beta_k(X)_j=X_(j-k)",
    "eigenunitaries": "u_s|block_j=i^(j*s) I2",
    "coefficient_algebra": "constant M2 tuples",
    "recovered_action": "trivial",
    "spectral_algebra": "scalar-block tuples",
    "J": "J(f)_j=f_j I2"
  },
  "Haar": {
    "original_singleton_mass": "3",
    "dual_singleton_mass": "1/12",
    "dual_total_mass": "1/3",
    "Plancherel_identity_indicator_squared_norm": "3=4*(1/12)*3^2"
  },
  "exact_projection_values": [
    [
      1,
      0,
      0,
      0
    ],
    [
      0,
      1,
      0,
      0
    ],
    [
      0,
      0,
      1,
      0
    ],
    [
      0,
      0,
      0,
      1
    ]
  ],
  "exact_phase_covariance_checked_for_all_64_triples": true,
  "matrix_entries_format": "Each scalar is [real integer, imaginary integer].",
  "positive_input_blocks": [
    [
      [
        [
          2,
          0
        ],
        [
          0,
          0
        ]
      ],
      [
        [
          0,
          0
        ],
        [
          0,
          0
        ]
      ]
    ],
    [
      [
        [
          3,
          0
        ],
        [
          3,
          0
        ]
      ],
      [
        [
          3,
          0
        ],
        [
          3,
          0
        ]
      ]
    ],
    [
      [
        [
          0,
          0
        ],
        [
          0,
          0
        ]
      ],
      [
        [
          0,
          0
        ],
        [
          5,
          0
        ]
      ]
    ],
    [
      [
        [
          7,
          0
        ],
        [
          0,
          -7
        ]
      ],
      [
        [
          0,
          7
        ],
        [
          7,
          0
        ]
      ]
    ]
  ],
  "rank_one_coefficients": [
    2,
    3,
    5,
    7
  ],
  "rank_one_vectors": [
    [
      [
        1,
        0
      ],
      [
        0,
        0
      ]
    ],
    [
      [
        1,
        0
      ],
      [
        1,
        0
      ]
    ],
    [
      [
        0,
        0
      ],
      [
        1,
        0
      ]
    ],
    [
      [
        1,
        0
      ],
      [
        0,
        1
      ]
    ]
  ],
  "sum_numerator": [
    [
      [
        12,
        0
      ],
      [
        3,
        -7
      ]
    ],
    [
      [
        3,
        7
      ],
      [
        15,
        0
      ]
    ]
  ],
  "average_denominator": 12,
  "spectral_input_values": [
    1,
    0,
    2,
    4
  ],
  "spectral_average": "7/12 I2",
  "identity_average": "1/3 I2",
  "unital_expectation": "3 E_beta",
  "average_eigenvalue_samples": [
    0.47815938765583244,
    1.7718406123441675
  ],
  "numeric_residuals": {
    "positive_factorization_X0": 0.0,
    "selfadjoint_X0": 0.0,
    "positive_factorization_X1": 0.0,
    "selfadjoint_X1": 0.0,
    "positive_factorization_X2": 0.0,
    "selfadjoint_X2": 0.0,
    "positive_factorization_X3": 0.0,
    "selfadjoint_X3": 0.0,
    "general_average_output_block_0": 0.0,
    "spectral_average_output_block_0": 1.1102230246251565e-16,
    "general_average_output_block_1": 0.0,
    "spectral_average_output_block_1": 1.1102230246251565e-16,
    "general_average_output_block_2": 0.0,
    "spectral_average_output_block_2": 1.1102230246251565e-16,
    "general_average_output_block_3": 0.0,
    "spectral_average_output_block_3": 1.1102230246251565e-16,
    "eigencharacter_k0_s0": 0.0,
    "eigencharacter_k0_s1": 0.0,
    "eigencharacter_k0_s2": 0.0,
    "eigencharacter_k0_s3": 0.0,
    "eigencharacter_k1_s0": 0.0,
    "eigencharacter_k1_s1": 0.0,
    "eigencharacter_k1_s2": 0.0,
    "eigencharacter_k1_s3": 0.0,
    "eigencharacter_k2_s0": 0.0,
    "eigencharacter_k2_s1": 0.0,
    "eigencharacter_k2_s2": 0.0,
    "eigencharacter_k2_s3": 0.0,
    "eigencharacter_k3_s0": 0.0,
    "eigencharacter_k3_s1": 0.0,
    "eigencharacter_k3_s2": 0.0,
    "eigencharacter_k3_s3": 0.0,
    "constant_coefficient_normalized_average": 0.0,
    "identity_average": 0.0,
    "normalized_expectation_identity": 0.0
  },
  "numeric_tolerance": 1e-12,
  "scope": "Exact finite example with bounded output in M_2+. The general extended-positive target is proved in OA-FLOW-L31, Sections 4–6."
}