{
  "title": "The outer sequence: order, phases, and quotient choices",
  "conventions": {
    "composition": "alpha beta = alpha composed beta; rightmost acts first",
    "inner": "Ad(a)(x)=a x a*",
    "translation": "u_s xi(q)=xi(q+s)",
    "dual": "delta_t(u_s)=exp(-i*s*t)u_s",
    "Haar": "crossed-product ds; canonical dual average dt/(2*pi)"
  },
  "translation_system": {
    "K": "ell^2(N_0)",
    "N": "B(K) tensor L-infinity(R)",
    "tau": "integral_R exp(q) Tr_K(x(q)) dq",
    "theta": "theta_s x(q)=x(q+s)",
    "P": "B(K tensor L^2(R)) in the proved normal regular coordinates",
    "Phi": "Tr_D with D=1_K tensor exp(Q), on the whole positive cone"
  },
  "normalization": {
    "v": "1_K tensor M_exp(iq)",
    "w": "1_K tensor F_minus",
    "alpha": "Ad(v* w)",
    "first": "gamma=alpha Ad(w*)=Ad(v*)",
    "cocycle": "d_s=v*theta_s(v)=exp(is)1",
    "second": "Ad(v) gamma=id",
    "wrong_first": "Ad(w*) alpha=Ad(u_-1)",
    "wrong_weight": "Phi Ad(u_-1)=e Phi",
    "diagram_boxes": [[0.075, 0.758, 0.355, 0.070], [0.075, 0.646, 0.355, 0.070], [0.075, 0.534, 0.355, 0.070]],
    "proof_locators": ["OES7.11", "OES7.12", "OES7.13", "OES8.4", "OES8.5"]
  },
  "finite_weight_witness": {
    "xi": "1_[0,1](q)e_0/sqrt(e-1)",
    "operator": "A=theta_(xi,xi); not asserted to be a projection",
    "correct_support": [0, 1],
    "wrong_support": [1, 2],
    "values_exact": ["1", "e"],
    "proof_locators": ["OES7.13", "OES8.5"]
  },
  "phase_cancellation": {
    "a": 1,
    "b": "b(q)=exp(iq)h, h any constant unitary in B(K)",
    "k_b": "exp(is)",
    "Ad_b_on_u_s": "exp(-is)u_s",
    "correct_product": "Ad(b) alpha_(k_b) fixes u_s",
    "central_implementer": "z(q)=exp(-iq), so partial z=k_b and b z=h",
    "plot_domain_exact": ["-pi", "pi"],
    "plotted_real_phases": ["s", "-s", "0"],
    "sample_s_exact": "pi/2",
    "sample_unitary_scalars": ["i", "-i", "1"],
    "proof_locators": ["OES7.14", "OES7.15", "OES7.16", "OES8.6"]
  },
  "quotient_fiber": {
    "hypothesis": "A given properly infinite factor coefficient N with faithful normal semifinite tau and normal continuous theta satisfying tau theta_s=exp(-s) tau.",
    "qualification": "Conditional calculation in every such system; no claim that an arbitrary factor admits such an action.",
    "cohomology": "H^1_theta=R via a -> [exp(ias)]",
    "parameter": 1,
    "b_1": "A CST implementer with theta_s(b_1)=exp(is)b_1; no continuous family of choices asserted.",
    "beta_1": "Ad(b_1)|N lies in I but not I_0",
    "upper_points": ["[id]_I0", "[beta_1]_I0"],
    "upper_lifts": ["1", "[delta_-1] != 1"],
    "lower_point": "[id]_I=[beta_1]_I",
    "coordinates_are_schematic": true,
    "exact_claim": "The representative lift beta -> [Gamma_beta] does not descend to G/I; this does not establish nonsplitting of the extension.",
    "proof_locators": ["OES7.17", "OES7.18", "OES8.7", "OES8.10"]
  },
  "human_source_context": ["OA-FLOW-OES.md#oes-reading", "OA-FLOW-RCC.md#rcc-reading"],
  "rights": "Original illustration, exact data and renderer: CC0-1.0 to the extent of rights held; DejaVu font license retained separately."
}