{
  "title": "Repair every frequency; preserve original central support",
  "license": "CC0-1.0",
  "model": {
    "hilbert_space": "L2(R,dq) tensor ell2(N)",
    "lambda": "(lambda_r f)(q)=f(q-r)",
    "modulation": "(V_t f)(q)=exp(i*t*q)f(q)",
    "modular_frequency": "V_t lambda_r V_t*=exp(i*r*t)lambda_r",
    "weight_density": "exp(Q) tensor 1",
    "weight_scaling": "nu Ad(lambda_r tensor 1)=exp(r)nu"
  },
  "selector": {
    "good_set": "R minus {0}",
    "bad_value": "X0(0)=lambda_1 tensor 1",
    "K": ["1/4", "3/4"],
    "epsilon": "1/2",
    "a": "max(1/4,1/4+r)",
    "b": "a(r)-r",
    "overlap_measure": "max(0,1/2-abs(r))",
    "sample_local_r": "1/3",
    "sample_local_a": "7/12",
    "sample_local_b": "1/4",
    "power_n": "1+floor(2*abs(r))",
    "sample_global_r": "7/6",
    "sample_n": 3,
    "sample_delta": "7/18",
    "sample_a_delta": "23/36",
    "sample_b_delta": "1/4",
    "frequency_positions": ["0", "7/18", "7/9", "7/6"],
    "proof_locators": ["OA-FLOW-DS#ds-3: DSEL12-DSEL16", "OA-FLOW-DS#ds-8: DSM7-DSM11"]
  },
  "support": {
    "original_algebra": "B direct-sum B",
    "psi": "nu direct-sum 0",
    "Phi": "nu direct-sum nu",
    "psi_support": [1, 0],
    "psi_central_support": [1, 0],
    "Phi_support": [1, 1],
    "partial_isometry_components": "a=(a1,a2)",
    "obstruction": "a*a=(1,0) forces a2=0, hence aa* cannot equal (1,1)",
    "display_meaning": "Each cell is one original central summand; area denotes neither dimension nor trace.",
    "proof_locators": ["OA-FLOW-DS#ds-6: DSEL39", "OA-FLOW-DS#ds-8: DSM16-DSM17"]
  },
  "human_source_context": "M. Takesaki, Theory of Operator Algebras II, Theorem XII.4.18, Lemma XII.4.19, Definition XII.4.20, printed pp. 417-418. The coordinate and support drawings are original models of the proofs in this lesson, not figures reproduced from that source.",
  "render": {"width_inches": 14, "height_inches": 10.5, "dpi": 180, "font": "DejaVu Sans", "svg_fonttype": "path"}
}