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2)*(315*J*R**7 - 8028*J*R**6 + 75915*J*R**5 - 352660*J*R**4 + 866250*J*R**3 - 1081080*J*R**2 + 540540*J*R - 7560*R**6 + 165726*R**5 - 1404270*R**4 + 5993988*R**3 - 13756680*R**2 + 16232580*R - 7741440)/R**16" ], "requirements_sha256": "96464f405b58137236b9c10c698be08a780ff7d062dd918ac40c0460f1b0831f", "scope": { "CF_boundary_condition": "Leaver RW outgoing prefactor and minimal series, zero ratio terminal approximation", "CF_deep_check_depth": 4096, "CF_deep_check_dps": 100, "CF_depths": [ 512, 1024, 2048 ], "CF_mode_inputs": [ { "ell": 2, "imag_selection_window": [ "-0.12", "-0.05" ], "n": 0, "real_selection_window": [ "0.30", "0.45" ], "seed": [ "0.374", "-0.089" ] }, { "ell": 10, "imag_selection_window": [ "-0.12", "-0.05" ], "n": 0, "real_selection_window": [ "1.90", "2.10" ], "seed": [ "1.997", "-0.096" ] } ], "CF_precisions_dps": [ 50, 80 ], "Fourier_convention": "exp(-i*omega*t)", "WKB_orders": [ 1, 2, 3 ], "WKB_overtone_inputs": [ 0 ], "derivative_rule": "W_j=(f*d/dR)^j W before peak substitution, j=0,...,6", "dimensionless_variables": "b=GM, R=r/b, x=r_star/b, w=b*omega, W=b^2*V, c=1", "limitations": [ "Finite WKB order is a local asymptotic approximation, not a certified error bound.", "The general-n exact oscillator identity is not a numerical overtone survey.", "CF depth/precision and ratio checks share one RW ODE; they are not an independent continuum or Kerr calculation.", "The general static F,H counterexample is kinematic, not asserted to solve vacuum Einstein equations.", "Pöschl--Teller check assumes V0>alpha^2/4 and is not a Schwarzschild spectrum.", "The semiclassical h expansion is formal bookkeeping; h=1 is not automatically a small parameter.", "Decreasing WKB errors in these two cases do not establish general monotonic convergence.", "110-place strings round-trip retained values; they are not 110 certified spectral digits. A zero depth drift can be a precision plateau." ], "model": "Schwarzschild vacuum axial gravitational RW potential, ell=2 and ell=10" }, "source_conventions": { "Iyer_1987_II": "Phys.Rev.D35,3632; p.3633 Eqs.(1.2),(1.3a),(1.3b), expanded omega_squared with exp(-i omega t)", "corrections": "D=sqrt(-2 V2)>0; E1=V0-i nu D; E2=E1+B2; E3=E2-i nu D Omega3; Omega3=B3/D^2", "counterexample": "F=1-2b/r, H=F[1+eta(3b/r)^2], eta>=0; rc=3b, Omega_c^2=1/(27b^2), lambda_t^2=(1+eta)/(27b^2)", "general_static_metric": "ds^2=-F dt^2+dr^2/H+r^2 dOmega^2; U=F/r^2; lambda_t^2=-Fc Hc Urr/(2 Uc)" }, "source_sha256": "cbff127da3f8467431234a5c31cec6c8a5253a88c85f77c398bb292ca9b53d7d", "status": "finite reproducible comparison; not a certified resonance-error bound", "versions": { "mpmath": "1.3.0", "sympy": "1.14.0" } }