{
  "title": "Trace-scaling cocycle stability: four exact views",
  "normalization": "All action integrals use Lebesgue ds, never ds/(2*pi).",
  "translation": {
    "theta": "theta_s f(q) = f(q+s)",
    "trace": "tau(f) = integral_R exp(q) f(q) dq",
    "interval": [0, 1],
    "shift_exact": "log(2)",
    "translated_interval_exact": ["-log(2)", "1-log(2)"],
    "areas_exact": ["e-1", "(e-1)/2"],
    "domain": [-1.08, 1.24],
    "proof_locators": ["CST6.1", "CST6.2", "CST6.3"]
  },
  "noncommuting_cocycle": {
    "X": [[0, 1], [1, 0]],
    "Y_real": [[0, 0], [0, 0]],
    "Y_imaginary": [[0, -1], [1, 0]],
    "Z": [[1, 0], [0, -1]],
    "v": "v(q)=exp(i q X) exp(i q Z)",
    "c": "c_s(q)=v(q)^* v(q+s)=exp(-i q Z) (exp(i s X) exp(i s Z)) exp(i q Z)",
    "times_pi_factors": [0.5, 0.25],
    "values_at_zero_exact": ["iY", "(I+iX+iY+iZ)/2"],
    "commutator_exact": "i(Z-X)",
    "commutator_real": [[0, 0], [0, 0]],
    "commutator_imaginary": [[1, -1], [-1, -1]],
    "operator_norm_exact": "sqrt(2), for every real q",
    "proof_locators": ["CST6.6", "CST6.7", "CST6.8", "CST6.9"]
  },
  "modulation": {
    "scope": "Actual density model for the identity cocycle; matrices displayed are integrands before integration, not L_r or R_r.",
    "A1": [[0, 0], [0, 1]],
    "A2": [[1, 1], [1, 1]],
    "K": "diag(exp(A1), exp(A2))",
    "Pi": "P_i(q)=A_i-qI",
    "k": "k(t)=1/sqrt(pi*(1+t*t))",
    "sample": {"q": 0, "s": 0, "r_exact": "pi"},
    "constants_exact": {"a": "1/sqrt(pi)", "b": "1/sqrt(2*pi)", "d": "1/sqrt(5*pi)"},
    "B_exact": [["a*(a+d)/2", "a*(d-a)/2"], ["b*(d-a)/2", "b*(a+d)/2"]],
    "left_unitary": [[1, 0], [0, -1]],
    "right_unitary": [[1, 0], [0, 1]],
    "products": ["exp(i*pi*A1)*B = diag(1,-1)*B", "B*exp(i*pi*A2) = B"],
    "display_precision": 3,
    "proof_locators": ["CST6.16", "CST6.17", "CST6.18", "CST6.19", "CST7.3"]
  },
  "localization": {
    "algebra": "product_{j>=1} M_2(C)",
    "trace": "rho(x)=sum_{j>=1} Tr_2(x_j)",
    "projections": {"a": "(e11)_j", "b": "(I_2)_j", "z_j": "central projection onto coordinate j"},
    "displayed_indices": [1, 2, 3, 4],
    "local_a_values": [1, 1, 1, 1],
    "local_b_values": [2, 2, 2, 2],
    "totals_exact": ["infinity", "infinity"],
    "scope": "Auxiliary projection-comparison example; it fails equality on central pieces and is not a linking-theorem counterexample.",
    "proof_locators": ["CST6.20", "CST6.21", "CST7.5"]
  },
  "human_source_context": {
    "trace_scaling_coordinate": "Takesaki, Theory of Operator Algebras II, XII.1, Lemma 1.2, printed pp.366-367; locator recorded in OA-FLOW-L19.md#l19-reading. No book figure is used.",
    "complete_stability_context": "OA-FLOW-CST.md#cst-reading"
  },
  "rights": "Original mathematical illustration and data: CC0-1.0 to the extent of rights held. DejaVu font license is retained separately."
}