{
  "title": "Canonical implementation: phases and strong convergence",
  "license": "CC0-1.0 to the extent of rights held; font terms separate",
  "matrix": {
    "h": [
      [
        1,
        0
      ],
      [
        0,
        4
      ]
    ],
    "v": [
      [
        0,
        1
      ],
      [
        1,
        0
      ]
    ],
    "transported_h": [
      [
        4,
        0
      ],
      [
        0,
        1
      ]
    ],
    "weight": "Tr(h*x)",
    "weight_of_identity": 5,
    "cocycle": "diag(exp(i*t*log(4)),exp(-i*t*log(4)))",
    "V": "V(r)A=h^(-i*r)*v*h^(i*r)*A*v*",
    "entries": "V(r)[[a,b],[c,d]]=[[z*d,z*c],[z^-1*b,z^-1*a]], z=exp(i*r*log(4))",
    "r0": "pi/(2*log(4))",
    "cocycle_at_r0": [
      [
        "i",
        0
      ],
      [
        0,
        "-i"
      ]
    ],
    "V_E11_at_r0": "-i E22",
    "F": "F(r)A=h^(i*r)*A",
    "F_at_r0": "left diag(1,i)",
    "F_V_Fstar": "constant U_v: A -> v*A*v*",
    "F_pi_x_Fstar": "constant left x",
    "F_lambda_t_Fstar": "left h^(i*t) times T_t",
    "T_t": "xi(r)->xi(r-t)",
    "full_domain": "L2(R, Hilbert-Schmidt M2)"
  },
  "phase": {
    "canonical_square": "I",
    "canonical_E11_orbit": [
      "E11",
      "E22",
      "E11"
    ],
    "noncanonical_square": "-I",
    "noncanonical_E11_orbit": [
      "E11",
      "i E22",
      "-E11"
    ],
    "cone_test": "i U(I2)=i I2 is not positive",
    "matrix_rephasing_notice": "v->i v leaves U_v and V_v unchanged"
  },
  "infinite": {
    "K": "ell2(N0)",
    "H": "Hilbert-Schmidt operators on K",
    "n_range": "n>=1",
    "trace": "full faithful normal semifinite Tr",
    "v_n": "I-2*|e_n><e_n|",
    "cocycle": "1 from exact whole-positive weight preservation and BC22",
    "canonical_U_n": "A->v_n*A*v_n",
    "V_n": "constant U_n on all L2 sections",
    "fixed_vector": "A=sqrt(3)*sum_(j>=1) 2^-j E_j0",
    "fixed_vector_norm": 1,
    "fixed_error": "2*sqrt(3)*2^-n",
    "operator_norm_error": 2,
    "moving_witness": "E_n0",
    "moving_witness_error": 2,
    "section_witness": "1_[0,1](r) E_n0",
    "section_witness_norm": 1,
    "fixed_shift": "S e_j=e_(j+1)",
    "fixed_shift_operator_error": 2,
    "sample": [
      {
        "n": 1,
        "fixed_error_exact": "2*sqrt(3)/2^1",
        "operator_norm_error": 2
      },
      {
        "n": 2,
        "fixed_error_exact": "2*sqrt(3)/2^2",
        "operator_norm_error": 2
      },
      {
        "n": 3,
        "fixed_error_exact": "2*sqrt(3)/2^3",
        "operator_norm_error": 2
      },
      {
        "n": 4,
        "fixed_error_exact": "2*sqrt(3)/2^4",
        "operator_norm_error": 2
      },
      {
        "n": 5,
        "fixed_error_exact": "2*sqrt(3)/2^5",
        "operator_norm_error": 2
      },
      {
        "n": 6,
        "fixed_error_exact": "2*sqrt(3)/2^6",
        "operator_norm_error": 2
      },
      {
        "n": 7,
        "fixed_error_exact": "2*sqrt(3)/2^7",
        "operator_norm_error": 2
      },
      {
        "n": 8,
        "fixed_error_exact": "2*sqrt(3)/2^8",
        "operator_norm_error": 2
      }
    ],
    "sample_notice": "Only n=1,...,8 is plotted. Analytic full-H and u proofs are retained in CIM6.s-v."
  },
  "proof_locators": [
    "CIM6.d-n",
    "CIM6.o-p",
    "CIM6.s-v",
    "CIM7.a-f"
  ]
}