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  },
  "exact_identity": "f_(dA)(x) = f_A(d*x)",
  "exact_negative_depth_both": "9/8",
  "exact_rms_both": "1",
  "scope": "Common frequency scaling preserves the full-period minimum. Minima over one fixed short observation window need not match.",
  "plot_arithmetic": "float64 curves; stated invariance follows from the exact identity and periodicity"
}