{ "version": "2026-09-07", "scope": "Selected numerical controls and explicit adverse inputs; not verification of every claim in the guide.", "started_utc": "2026-09-07T07:28:13.150221+00:00", "environment": { "python": "3.12.9", "platform": "macOS-26.6.2-arm64-arm-64bit", "packages": { "mpmath": { "expected": "1.3.0", "actual": "1.3.0" }, "numpy": { "expected": "2.0.2", "actual": "2.0.2" }, "scipy": { "expected": "1.13.1", "actual": "1.13.1" }, "sympy": { "expected": "1.14.0", "actual": "1.14.0" } }, "differences": [] }, "manifest_sha256": "ccdcd032c6bfc0fe2cdee79066640e79cbb65812676748019f345bf4b9f2048c", "cases": [ { "id": "gauss", "purpose": "Connection coefficients, Abel identity, ODE residual, and basis phase at three points", "page": "/advanced-ode/start-here/gauss-connection/", "command": [ "python", "gauss-starter-check.py", "--json" ], "expected_exit": 0, "source_sha256": "b6c340a02891e08cd495c647c14b1ebeda4ccbfb494a5465160c24fb87db32e4", "exit": 0, "stdout": "{\n \"status\": \"PASS\",\n \"terms\": 240,\n \"dps\": 60,\n \"tolerance\": \"1.0e-30\",\n \"reference_A\": \"1.16278702942702179\",\n \"reference_B\": \"-0.375\",\n \"maximum_checked_error\": \"8.4508284e-41\",\n \"match_points\": [\n {\n \"z\": \"0.333333333333\",\n \"A\": \"1.16278702942702179\",\n \"B\": \"-0.375\",\n \"coefficient_relative_error\": \"8.5519947e-43\",\n \"abel_relative_error\": \"1.5756177e-42\",\n \"ode_scaled_residual\": \"8.4508284e-41\",\n \"basis_phase_error\": \"3.1346632e-61\"\n },\n {\n \"z\": \"0.5\",\n \"A\": \"1.16278702942702179\",\n \"B\": \"-0.375\",\n \"coefficient_relative_error\": \"3.1115076e-61\",\n \"abel_relative_error\": \"2.3336307e-61\",\n \"ode_scaled_residual\": \"6.6933164e-62\",\n \"basis_phase_error\": \"2.0593737e-62\"\n },\n {\n \"z\": \"0.666666666667\",\n \"A\": \"1.16278702942702179\",\n \"B\": \"-0.375\",\n \"coefficient_relative_error\": \"1.0935294e-44\",\n \"abel_relative_error\": \"0.0\",\n \"ode_scaled_residual\": \"4.0415898e-43\",\n \"basis_phase_error\": \"7.9553388e-62\"\n }\n ]\n}\n", "stderr": "", "scientific_status": "PASS", "elapsed_seconds": 0.083, "test_status": "PASS" }, { "id": "gauss-underresolved", "purpose": "Insufficient series order must be rejected", "page": "/advanced-ode/start-here/gauss-connection/", "command": [ "python", "gauss-starter-check.py", "--terms", "24", "--json" ], "expected_exit": 1, "source_sha256": "b6c340a02891e08cd495c647c14b1ebeda4ccbfb494a5465160c24fb87db32e4", "exit": 1, "stdout": "{\n \"status\": \"FAIL\",\n \"terms\": 24,\n \"dps\": 60,\n \"tolerance\": \"1.0e-30\",\n \"reference_A\": \"1.16278702942702179\",\n \"reference_B\": \"-0.375\",\n \"maximum_checked_error\": \"0.0005176834\",\n \"match_points\": [\n {\n \"z\": \"0.333333333333\",\n \"A\": \"1.16276842844700104\",\n \"B\": \"-0.374980883162825697\",\n \"coefficient_relative_error\": \"5.0978232e-5\",\n \"abel_relative_error\": \"9.9821721e-5\",\n \"ode_scaled_residual\": \"0.0005176834\",\n \"basis_phase_error\": \"1.6465599e-62\"\n },\n {\n \"z\": \"0.5\",\n \"A\": \"1.16278700125981103\",\n \"B\": \"-0.374999961762260758\",\n \"coefficient_relative_error\": \"1.019673e-7\",\n \"abel_relative_error\": \"1.062737e-7\",\n \"ode_scaled_residual\": \"1.1011029e-6\",\n \"basis_phase_error\": \"4.3180878e-62\"\n },\n {\n \"z\": \"0.666666666667\",\n \"A\": \"1.16278589883546727\",\n \"B\": \"-0.374998029237277551\",\n \"coefficient_relative_error\": \"5.2553673e-6\",\n \"abel_relative_error\": \"6.5751093e-12\",\n \"ode_scaled_residual\": \"2.0282694e-5\",\n \"basis_phase_error\": \"1.5855687e-62\"\n }\n ]\n}\n", "stderr": "", "scientific_status": "FAIL", "elapsed_seconds": 0.055, "test_status": "PASS" }, { "id": "weber", "purpose": "Exact Wronskian and independently refined finite-difference spectrum", "page": "/advanced-ode/start-here/weber-spectrum/", "command": [ "python", "weber-starter-check.py", "--json" ], "expected_exit": 0, "source_sha256": "69b52074ff4bfd74126e4921e2d14caed4148803d2004f6a2a4957e89b62a40e", "exit": 0, "stdout": "{\n \"status\": \"PASS\",\n \"hbar\": \"1.0\",\n \"half_width_in_oscillator_lengths\": 8.0,\n \"grid_intervals\": [\n 1200,\n 2400,\n 4800\n ],\n \"tolerance_in_hbar_units\": 2e-07,\n \"wronskian_checks\": [\n {\n \"E_over_hbar\": \"0\",\n \"scaled_wronskian_error\": \"8.9092157e-52\"\n },\n {\n \"E_over_hbar\": \"1.4\",\n \"scaled_wronskian_error\": \"8.3058227e-52\"\n },\n {\n \"E_over_hbar\": \"2.5\",\n \"scaled_wronskian_error\": \"7.7100207e-52\"\n },\n {\n \"E_over_hbar\": \"6.2\",\n \"scaled_wronskian_error\": \"1.0714775e-51\"\n }\n ],\n \"grid_spectra_in_hbar_units\": [\n [\n 0.9999888887648858,\n 2.999944443334502,\n 4.999855551233039,\n 6.999722210987527\n ],\n [\n 0.9999972222109081,\n 2.999986111039895,\n 4.999963888626226,\n 6.999930554849909\n ],\n [\n 0.9999993055773935,\n 2.99999652778056,\n 4.999990972212403,\n 6.999982638872925\n ]\n ],\n \"richardson_spectrum_in_hbar_units\": [\n 1.0000000000328886,\n 3.0000000000274483,\n 5.000000000074462,\n 7.00000000021393\n ],\n \"exact_spectrum_in_hbar_units\": [\n 1.0,\n 3.0,\n 5.0,\n 7.0\n ],\n \"maximum_spectral_error\": 2.1393020688265096e-10,\n \"richardson_refinement_change\": 2.590105907529505e-09,\n \"endpoint_refinement_change\": 6.454958789703369e-11\n}\n", "stderr": "", "scientific_status": "PASS", "elapsed_seconds": 1.584, "test_status": "PASS" }, { "id": "weber-wrong-domain", "purpose": "An inadequate endpoint distance must fail the spectral comparison", "page": "/advanced-ode/start-here/weber-spectrum/", "command": [ "python", "weber-starter-check.py", "--half-width", "2", "--json" ], "expected_exit": 1, "source_sha256": "69b52074ff4bfd74126e4921e2d14caed4148803d2004f6a2a4957e89b62a40e", "exit": 1, "stdout": "{\n \"status\": \"FAIL\",\n \"hbar\": \"1.0\",\n \"half_width_in_oscillator_lengths\": 2.0,\n \"grid_intervals\": [\n 1200,\n 2400,\n 4800\n ],\n \"tolerance_in_hbar_units\": 2e-07,\n \"wronskian_checks\": [\n {\n \"E_over_hbar\": \"0\",\n \"scaled_wronskian_error\": \"8.9092157e-52\"\n },\n {\n \"E_over_hbar\": \"1.4\",\n \"scaled_wronskian_error\": \"8.3058227e-52\"\n },\n {\n \"E_over_hbar\": \"2.5\",\n \"scaled_wronskian_error\": \"7.7100207e-52\"\n },\n {\n \"E_over_hbar\": \"6.2\",\n \"scaled_wronskian_error\": \"1.0714775e-51\"\n }\n ],\n \"grid_spectra_in_hbar_units\": [\n [\n 1.0749217517714826,\n 3.529626489572658,\n 6.79954710528412,\n 11.169187190880324\n ],\n [\n 1.074922252006135,\n 3.5296312805185144,\n 6.799569138017372,\n 11.169255416317835\n ],\n [\n 1.0749223769077823,\n 3.529632478483618,\n 6.799574646758238,\n 11.169272472315171\n ]\n ],\n \"richardson_spectrum_in_hbar_units\": [\n 1.0749224185416646,\n 3.5296328778053194,\n 6.799576483005193,\n 11.169278157647616\n ],\n \"exact_spectrum_in_hbar_units\": [\n 1.0,\n 3.0,\n 5.0,\n 7.0\n ],\n \"maximum_spectral_error\": 4.169278157647616,\n \"richardson_refinement_change\": 7.434044491105851e-10,\n \"endpoint_refinement_change\": 2.8007555204380115\n}\n", "stderr": "", "scientific_status": "FAIL", "elapsed_seconds": 1.556, "test_status": "PASS" }, { "id": "heun", "purpose": "Both Abel determinants and full connection-matrix drift", "page": "/advanced-ode/reference/appendix-c-computational-laboratory/#a-quick-connection-matrix-micro-lab", "command": [ "python", "general-heun-connection-check.py", "--no-table", "--verify" ], "expected_exit": 0, "source_sha256": "5675e36730d49f9cb9218421b50b76173910ef351e91c99e4a9fa2799ea8b5bc", "exit": 0, "stdout": "t=0.04, order=2, terms=120\ndirect Frobenius matrix\n 0.6852083269293459 0.5321827959047746\n 1.047109773636109 -0.6945959089055297\nclassical-block matrix\n 0.6852142054292528 0.5321841142916558\n 1.047098391355132 -0.6945967754832237\nmaximum relative entry error = 1.0870188841823e-5\n (finite-order block comparison; excluded from verification)\ndeterminants (direct, block, Abel) = -1.03319670758600654, -1.03319670758600654, -1.03319670758600654\n\nmaximum relative change between z=t/2 and z=t/3 = 2.2798812124209e-22\n\nNumerical checks (errors scaled by max(1, |reference|)); tolerance = 1.0e-12\n PASS: Abel determinant at z=t/2 = 3.0381862002693e-39\n PASS: Abel determinant at z=t/3 = 1.598258677289e-22\n PASS: two-point matrix drift = 1.5835961629381e-22\nverification: PASS\n", "stderr": "", "elapsed_seconds": 0.075, "test_status": "PASS" }, { "id": "heun-underresolved", "purpose": "Matrix error is detected even when a determinant alone appears converged", "page": "/advanced-ode/study-paths/#route-a-analytic-connection-methods", "command": [ "python", "general-heun-connection-check.py", "--t", "0.04", "--terms", "24", "--dps", "50", "--no-table", "--verify" ], "expected_exit": 1, "source_sha256": "5675e36730d49f9cb9218421b50b76173910ef351e91c99e4a9fa2799ea8b5bc", "exit": 1, "stdout": "t=0.04, order=2, terms=24\ndirect Frobenius matrix\n 0.685208335509995 0.532182798236554\n 1.047109746507397 -0.6945959170766861\nclassical-block matrix\n 0.6852142054292528 0.5321841142916558\n 1.047098391355132 -0.6945967754832237\nmaximum relative entry error = 1.0844280938967e-5\n (finite-order block comparison; excluded from verification)\ndeterminants (direct, block, Abel) = -1.03319670714922966, -1.03319670758600654, -1.03319670758600654\n\nmaximum relative change between z=t/2 and z=t/3 = 3.0840638920608e-5\n\nNumerical checks (errors scaled by max(1, |reference|)); tolerance = 1.0e-12\n FAIL: Abel determinant at z=t/2 = 4.2274320167639e-10\n FAIL: Abel determinant at z=t/3 = 2.144021116614e-5\n FAIL: two-point matrix drift = 2.1421781874291e-5\nverification: FAIL\nRefine --terms or move farther inside the convergence disks.\n", "stderr": "", "elapsed_seconds": 0.056, "test_status": "PASS" }, { "id": "heun-invalid-match", "purpose": "Reject a second point outside the Frobenius convergence disc", "page": "/advanced-ode/reference/appendix-c-computational-laboratory/", "command": [ "python", "general-heun-connection-check.py", "--t", "0.66", "--no-table", "--check-second-point" ], "expected_exit": 2, "source_sha256": "5675e36730d49f9cb9218421b50b76173910ef351e91c99e4a9fa2799ea8b5bc", "exit": 2, "stdout": "", "stderr": "usage: general-heun-connection-check.py [-h] [--t T] [--order {0,1,2}]\n [--terms TERMS] [--dps DPS]\n [--no-table] [--show-ablation]\n [--check-second-point] [--verify]\n [--tolerance TOLERANCE]\n [--show-reverse]\ngeneral-heun-connection-check.py: error: matching at z=t/3 requires 0 < t < 3/5: its distance 2*t/3 from t must be smaller than the Frobenius radius min(t, 1-t); more terms cannot repair an invalid match point\n", "elapsed_seconds": 0.048, "test_status": "PASS" }, { "id": "isomonodromy", "purpose": "Generic Fourier/Nystrom tau ratio, contour/order/precision refinements, and PVI sigma-form residual", "page": "/advanced-ode/isomonodromy/series-fredholm-representations-numerical-realizations/", "command": [ "python", "isomonodromy-fredholm-check.py", "--json" ], "expected_exit": 0, "source_sha256": "c7a4c25cae902f430f3c1e6a7011ab7e03e732872f8fbd8576a25884057498c6", "exit": 0, "stdout": "{\n \"status\": \"PASS\",\n \"source\": \"https://arxiv.org/pdf/1608.00958\",\n \"source_conventions\": \"residue eigenvalues +/-theta_GL; website theta=2*theta_GL\",\n \"parameters\": {\n \"theta0\": \"0.17\",\n \"thetat\": \"0.23\",\n \"theta1\": \"0.19\",\n \"thetainf\": \"0.31\",\n \"sigma\": \"0.27\",\n \"eta\": \"0.41\"\n },\n \"time\": \"0.12\",\n \"reference_time\": \"0.08\",\n \"normalization\": \"tau(t)/tau(reference), real Log(t) and Log(1-t)\",\n \"prefactor_exponents\": {\n \"t\": \"-0.0089\",\n \"1-t\": \"-0.0874\"\n },\n \"genericity_distance_from_integers\": \"0.13\",\n \"noncommuting_composite_trace_gap\": \"1.36736752162\",\n \"tolerance\": \"1.0e-9\",\n \"checks\": {\n \"fourier_last_refinement\": \"8.33565608022e-14\",\n \"sigma_form\": \"5.74979257047e-13\",\n \"nystrom_last_refinement\": \"1.88687359815e-13\",\n \"method_agreement\": \"1.17914778408e-15\",\n \"parametrix_normalization\": \"4.13922927759e-41\",\n \"contour_invariance\": \"1.48182220252e-14\",\n \"fourier_precision_change\": \"2.58427899334e-41\",\n \"nystrom_precision_change\": \"7.42205973378e-41\",\n \"finite_matrix_derivative_consistency\": \"8.74553994201e-62\"\n },\n \"maximum_checked_error\": \"5.74979257047e-13\",\n \"refinements\": [\n {\n \"method\": \"Fourier\",\n \"modes\": 8,\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354778442974302237\",\n \"imag\": \"0.00082276452057994390213\"\n },\n \"sigma_scaled_residual\": \"1.84077386538e-9\"\n },\n {\n \"method\": \"Fourier\",\n \"modes\": 10,\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354778441288198513\",\n \"imag\": \"0.00082276452060986609798\"\n },\n \"sigma_scaled_residual\": \"3.31894024639e-11\"\n },\n {\n \"method\": \"Fourier\",\n \"modes\": 12,\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354778441271580996\",\n \"imag\": \"0.00082276452060964783148\"\n },\n \"sigma_scaled_residual\": \"5.74979257047e-13\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 16,\n \"radius\": \"0.36\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.9935477841540052309\",\n \"imag\": \"0.00082276454735927056958\"\n },\n \"parametrix_normalization_error\": \"3.25482341519e-41\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 22,\n \"radius\": \"0.36\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354778441233884713\",\n \"imag\": \"0.00082276452064374473641\"\n },\n \"parametrix_normalization_error\": \"3.2677063843e-41\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 28,\n \"radius\": \"0.36\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354778441271345962\",\n \"imag\": \"0.00082276452060968863927\"\n },\n \"parametrix_normalization_error\": \"3.26598330207e-41\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 28,\n \"radius\": \"0.42\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354778441268394448\",\n \"imag\": \"0.0008227645206109204897\"\n }\n },\n {\n \"method\": \"Fourier\",\n \"modes\": 12,\n \"dps\": 60,\n \"tau_ratio\": {\n \"real\": \"0.99354778441271580996\",\n \"imag\": \"0.00082276452060964783148\"\n }\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 28,\n \"radius\": \"0.36\",\n \"dps\": 60,\n \"tau_ratio\": {\n \"real\": \"0.99354778441271345962\",\n \"imag\": \"0.00082276452060968863927\"\n }\n }\n ],\n \"scope\": \"Numerical convergence and sigma-form check; no rigorous enclosure, reconstructed Schlesinger transport, or scalar spectral assertion.\",\n \"elapsed_seconds\": 13.323\n}\n", "stderr": "", "scientific_status": "PASS", "elapsed_seconds": 13.374, "test_status": "PASS" }, { "id": "isomonodromy-underresolved", "purpose": "Insufficient Fourier/quadrature resolution must fail verification", "page": "/advanced-ode/isomonodromy/series-fredholm-representations-numerical-realizations/", "command": [ "python", "isomonodromy-fredholm-check.py", "--modes", "4", "--nodes", "8", "--json" ], "expected_exit": 1, "source_sha256": "c7a4c25cae902f430f3c1e6a7011ab7e03e732872f8fbd8576a25884057498c6", "exit": 1, "stdout": "{\n \"status\": \"FAIL\",\n \"source\": \"https://arxiv.org/pdf/1608.00958\",\n \"source_conventions\": \"residue eigenvalues +/-theta_GL; website theta=2*theta_GL\",\n \"parameters\": {\n \"theta0\": \"0.17\",\n \"thetat\": \"0.23\",\n \"theta1\": \"0.19\",\n \"thetainf\": \"0.31\",\n \"sigma\": \"0.27\",\n \"eta\": \"0.41\"\n },\n \"time\": \"0.12\",\n \"reference_time\": \"0.08\",\n \"normalization\": \"tau(t)/tau(reference), real Log(t) and Log(1-t)\",\n \"prefactor_exponents\": {\n \"t\": \"-0.0089\",\n \"1-t\": \"-0.0874\"\n },\n \"genericity_distance_from_integers\": \"0.13\",\n \"noncommuting_composite_trace_gap\": \"1.36736752162\",\n \"tolerance\": \"1.0e-9\",\n \"checks\": {\n \"fourier_last_refinement\": \"1.2819554479e-6\",\n \"sigma_form\": \"4.52180430107e-6\",\n \"nystrom_last_refinement\": \"8.54779169113e-6\",\n \"method_agreement\": \"1.08922927758e-6\",\n \"parametrix_normalization\": \"3.25482341519e-41\",\n \"contour_invariance\": \"3.38598265407e-7\",\n \"fourier_precision_change\": \"8.21915709076e-42\",\n \"nystrom_precision_change\": \"3.31879646609e-41\",\n \"finite_matrix_derivative_consistency\": \"5.0083183667e-62\"\n },\n \"maximum_checked_error\": \"8.54779169113e-6\",\n \"refinements\": [\n {\n \"method\": \"Fourier\",\n \"modes\": 2,\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99358389419899587052\",\n \"imag\": \"0.00082174433804131276923\"\n },\n \"sigma_scaled_residual\": \"0.000193096841733\"\n },\n {\n \"method\": \"Fourier\",\n \"modes\": 3,\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99355057225183762718\",\n \"imag\": \"0.00082271143065091204835\"\n },\n \"sigma_scaled_residual\": \"2.93636228905e-5\"\n },\n {\n \"method\": \"Fourier\",\n \"modes\": 4,\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354801710069363896\",\n \"imag\": \"0.00082276142415277327514\"\n },\n \"sigma_scaled_residual\": \"4.52180430107e-6\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 4,\n \"radius\": \"0.36\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.9933561255028110252\",\n \"imag\": \"0.00084601625188390916163\"\n },\n \"parametrix_normalization_error\": \"3.25482341519e-41\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 6,\n \"radius\": \"0.36\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99352894159815770898\",\n \"imag\": \"0.00082504467317553606067\"\n },\n \"parametrix_normalization_error\": \"2.29625414766e-41\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 8,\n \"radius\": \"0.36\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354585804854052473\",\n \"imag\": \"0.00082299295691398526399\"\n },\n \"parametrix_normalization_error\": \"3.25482341519e-41\"\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 8,\n \"radius\": \"0.42\",\n \"dps\": 40,\n \"tau_ratio\": {\n \"real\": \"0.99354518688162227563\",\n \"imag\": \"0.00082292101803248992797\"\n }\n },\n {\n \"method\": \"Fourier\",\n \"modes\": 4,\n \"dps\": 60,\n \"tau_ratio\": {\n \"real\": \"0.99354801710069363896\",\n \"imag\": \"0.00082276142415277327514\"\n }\n },\n {\n \"method\": \"Nystrom\",\n \"nodes\": 8,\n \"radius\": \"0.36\",\n \"dps\": 60,\n \"tau_ratio\": {\n \"real\": \"0.99354585804854052473\",\n \"imag\": \"0.00082299295691398526399\"\n }\n }\n ],\n \"scope\": \"Numerical convergence and sigma-form check; no rigorous enclosure, reconstructed Schlesinger transport, or scalar spectral assertion.\",\n \"elapsed_seconds\": 0.754\n}\n", "stderr": "", "scientific_status": "FAIL", "elapsed_seconds": 0.811, "test_status": "PASS" }, { "id": "scalar-schwarzschild", "purpose": "Separately initialized scalar continued fraction and complex-ray Riccati roots with refinements", "page": "/advanced-ode/black-holes-holography/wronskian-recurrence-qnm-conditions/#a-schwarzschild-mode-survives-two-independent-tests", "command": [ "python", "schwarzschild-scalar-lab.py", "--verify", "--profile", "full", "--json", "{result_json}" ], "expected_exit": 0, "source_sha256": "fa537ce57f2841405959ca77af85e7401bd8fcfc9bdb02a9d92aa9b54b8ea815", "exit": 0, "stdout": "massless scalar Schwarzschild ell=0, fundamental n=0\nexp(-i*omega*t); z=r/(2M); Omega=2M*omega; u=r*R\nEnvironment: {'Python': '3.12.9', 'mpmath': '1.3.0', 'numpy': '2.0.2', 'scipy': '1.13.1'}\nEach method starts independently from 0.22 -0.21i\n\nContinued fraction: zero terminal ratio; arbitrary-precision arithmetic\nN= 50 Omega=(0.22088411117941089290043303771697602849 - 0.20978491347964292777137766911585314969j) residual=8.2506e-62\nN= 100 Omega=(0.22091002426058629417292399826010345387 - 0.20979215236642305569920040287447750405j) residual=1.9447e-62\nN= 200 Omega=(0.22090987578642000692608220424516348607 - 0.20979143801099821132174549335164426385j) residual=5.9146e-62\nN= 400 Omega=(0.22090987815925762487310363358107945162 - 0.20979143417081678314406463129803795595j) residual=8.6969e-62\nN= 800 Omega=(0.22090987816083946978324255048513086187 - 0.20979143417376184114449413301636211581j) residual=8.3077e-62\nN= 1600 Omega=(0.22090987816083937175085889743607975931 - 0.20979143417376191756345768480103345097j) residual=8.6969e-62\nN= 3200 Omega=(0.22090987816083937175092301233596936219 - 0.20979143417376191756347813371074420398j) residual=7.0117e-62\nN= 6400 Omega=(0.22090987816083937175092301233587754315 - 0.20979143417376191756347813371075195441j) residual=2.917e-62\nPASS: CF last-depth change = 9.2145571e-32 (limit 1.0e-20)\nPASS: CF 30-digit versus requested-precision change = 8.2412653e-33 (limit 1.0e-20)\n\nIndependent Riccati roots: DOP853 binary64; all controls in JSON\nbaseline: Omega=0.22090987816388222-0.2097914341723593i residual=1.657e-12 local_delta=2.299e-12 solver_success=True\nlarger infinity cutoff: Omega=0.2209098781615587-0.2097914341731471i residual=5.478e-13 local_delta=7.600e-13 solver_success=True\nhigher infinity order: Omega=0.22090987816449797-0.2097914341714501i residual=8.524e-13 local_delta=1.183e-12 solver_success=True\ntighter ODE tolerances: Omega=0.22090987816345065-0.2097914341724763i residual=4.650e-13 local_delta=6.451e-13 solver_success=True\nshifted match point: Omega=0.22090987816078247-0.209791434173529i residual=2.921e-13 local_delta=7.395e-13 solver_success=True\nsmaller infinity cutoff: Omega=0.22090987830359152-0.2097914334467371i residual=7.378e-13 local_delta=1.024e-12 solver_success=True\nlower infinity order: Omega=0.22090987816120264-0.2097914341732776i residual=1.266e-12 local_delta=1.757e-12 solver_success=True\nsmaller horizon offset: Omega=0.22090987817065222-0.2097914341669829i residual=7.198e-12 local_delta=9.987e-12 solver_success=True\nlarger horizon offset: Omega=0.22090987816568983-0.2097914341716606i residual=5.022e-12 local_delta=6.968e-12 solver_success=True\nfewer horizon terms: Omega=0.22090987816388222-0.2097914341723593i residual=1.657e-12 local_delta=2.299e-12 solver_success=True\nmore horizon terms: Omega=0.22090987816388222-0.2097914341723593i residual=1.657e-12 local_delta=2.299e-12 solver_success=True\nlower contour angle: Omega=0.22090987816072027-0.2097914341734216i residual=9.788e-13 local_delta=1.358e-12 solver_success=True\nhigher contour angle: Omega=0.22090987816274055-0.209791434173742i residual=5.142e-13 local_delta=7.135e-13 solver_success=True\ninner match point: Omega=0.22090987816377597-0.2097914341730036i residual=9.635e-13 local_delta=6.719e-13 solver_success=True\nlooser ODE tolerances: Omega=0.2209098781652137-0.2097914341712828i residual=4.451e-12 local_delta=6.175e-12 solver_success=True\nPASS: ODE frequency refinement spread = 7.389494076276026e-10 (limit 1e-08)\nPASS: ODE local residual/derivative frequency estimate = 9.987338544164688e-12 (limit 1e-09)\nPASS: ODE derivative finite-difference refinement = 9.26822291022344e-07 (limit 0.001)\nPASS: independent CF versus Riccati frequency gap = 3.3505505e-12 (limit 1e-08)\n\nverification: PASS\nRefinements and cross-method agreement support numerical digits only; no interval certificate.\nFull record: {result_json}\n", "stderr": "", "scientific_record": { "model": "massless scalar Schwarzschild ell=0, fundamental n=0", "conventions": "exp(-i*omega*t); z=r/(2M); Omega=2M*omega; u=r*R", "environment": { "Python": "3.12.9", "mpmath": "1.3.0", "numpy": "2.0.2", "scipy": "1.13.1" }, "inputs": { "method": "all", "profile": "full", "verify": true, "json": "/var/folders/d4/pzcpjrzd60z6thz8cmnnmnmw0000gn/T/advanced-ode-result-62d7aa7_/scientific-result.json", "depths": null, "dps": 60, "seed_real": "0.22", "seed_imag": "-0.21", "cf_tolerance": "1e-20", "ode_tolerance": 1e-08, "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "initial_seed": { "real": "0.22", "imag": "-0.21" }, "evidence_status": "empirical convergence and independent-method comparison; no enclosure", "checks": [ { "name": "CF last-depth change", "value": "9.21455706062223811300396132392837816691172399562688231435895e-32", "tolerance": "1.0e-20", "passed": true }, { "name": "CF 30-digit versus requested-precision change", "value": "8.24126530745354004507961657382097945694507715103679324334374e-33", "tolerance": "1.0e-20", "passed": true }, { "name": "ODE frequency refinement spread", "value": "7.389494076276026e-10", "tolerance": "1e-08", "passed": true }, { "name": "ODE local residual/derivative frequency estimate", "value": "9.987338544164688e-12", "tolerance": "1e-09", "passed": true }, { "name": "ODE derivative finite-difference refinement", "value": "9.26822291022344e-07", "tolerance": "0.001", "passed": true }, { "name": "independent CF versus Riccati frequency gap", "value": "0.00000000000335055051067462402393315026981476228161221856654647367219989", "tolerance": "1e-08", "passed": true } ], "cf": { "rows": [ { "depth": 50, "dps": 60, "omega_re": "0.220884111179410892900433037716976028493515547834365226887944", "omega_im": "-0.209784913479642927771377669115853149689879634113089006213002", "residual": "8.25063056700581357234458580068146007425802132795868523918421e-62" }, { "depth": 100, "dps": 60, "omega_re": "0.220910024260586294172923998260103453874477867611678312395621", "omega_im": "-0.209792152366423055699200402874477504048484273492478765513514", "residual": "1.9446922743316067834825200168062888251844538027253647290941e-62" }, { "depth": 200, "dps": 60, "omega_re": "0.220909875786420006926082204245163486065716308944349535270947", "omega_im": "-0.209791438010998211321745493351644263848733497595051228269436", "residual": "5.91455064963236203125627804006429403428578505833728904403294e-62" }, { "depth": 400, "dps": 60, "omega_re": "0.22090987815925762487310363358107945162100951095907236461684", "omega_im": "-0.209791434170816783144064631298037955945605614851342023910231", "residual": "8.69692824144828433779738955758388759223266889697601346189823e-62" }, { "depth": 800, "dps": 60, "omega_re": "0.220909878160839469783242550485130861871074576842724960404109", "omega_im": "-0.209791434173761841144494133016362115810678513434220872978807", "residual": "8.30772903768965806948077108927357432007058950755650398743319e-62" }, { "depth": 1600, "dps": 60, "omega_re": "0.220909878160839371750858897436079759308775021611605183617665", "omega_im": "-0.209791434173761917563457684801033450966709517591360616560313", "residual": "8.69692824144828433779738955758388759223266889697601346189823e-62" }, { "depth": 3200, "dps": 60, "omega_re": "0.220909878160839371750923012335969362191182022832144227162678", "omega_im": "-0.209791434173761917563478133710744203979651769351060299831078", "residual": "7.01168771010129100468558194361689021277017365432015749545938e-62" }, { "depth": 6400, "dps": 60, "omega_re": "0.220909878160839371750923012335877543146538517965070922922792", "omega_im": "-0.209791434173761917563478133710751954413388043357637828518075", "residual": "2.91703841149741017522378002520943323777668070408804709364115e-62" } ], "precision_control": { "dps": 30, "depth": 6400, "omega_re": "0.220909878160839371750923012336", "omega_im": "-0.209791434173761917563478133711" }, "M_omega_re": "0.110454939080419685875461506167938771573269258982535461461396", "M_omega_im": "-0.104895717086880958781739066855375977206694021678818914259037" }, "riccati": { "rows": [ { "omega_re": "0.22090987816388222", "omega_im": "-0.20979143417235932", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 1.6566735046146185e-12, "derivative_modulus": 0.7207429484696906, "derivative_step": 1e-05, "derivative_refinement_relative": 1.5473654639023526e-07, "local_residual_frequency_estimate": 2.2985635976489705e-12, "case": "baseline" }, { "omega_re": "0.2209098781615587", "omega_im": "-0.20979143417314713", "controls": { "cutoff": 50, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 5.477867180196312e-13, "derivative_modulus": 0.7207430184761515, "derivative_step": 1e-05, "derivative_refinement_relative": 6.855377555961063e-07, "local_residual_frequency_estimate": 7.600305573237499e-13, "case": "larger infinity cutoff" }, { "omega_re": "0.22090987816449797", "omega_im": "-0.2097914341714501", "controls": { "cutoff": 40, "infinity_order": 31, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 8.524384509540073e-13, "derivative_modulus": 0.7207430248063781, "derivative_step": 1e-05, "derivative_refinement_relative": 7.043769619338265e-08, "local_residual_frequency_estimate": 1.182721749104694e-12, "case": "higher infinity order" }, { "omega_re": "0.22090987816345065", "omega_im": "-0.20979143417247625", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 9.999999999999999e-14, "atol": 1e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 4.649745811227504e-13, "derivative_modulus": 0.7207431998450176, "derivative_step": 1e-05, "derivative_refinement_relative": 8.916944791239921e-08, "local_residual_frequency_estimate": 6.451321097760403e-13, "case": "tighter ODE tolerances" }, { "omega_re": "0.22090987816078247", "omega_im": "-0.20979143417352902", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 5, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 10, "residual": 2.921131807494776e-13, "derivative_modulus": 0.3949955818958367, "derivative_step": 1e-05, "derivative_refinement_relative": 1.6125428723198833e-07, "local_residual_frequency_estimate": 7.395353116291564e-13, "case": "shifted match point" }, { "omega_re": "0.22090987830359152", "omega_im": "-0.20979143344673715", "controls": { "cutoff": 30, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 7.378401333412806e-13, "derivative_modulus": 0.7207429721897439, "derivative_step": 1e-05, "derivative_refinement_relative": 1.7058856316709175e-07, "local_residual_frequency_estimate": 1.0237215787197932e-12, "case": "smaller infinity cutoff" }, { "omega_re": "0.22090987816120264", "omega_im": "-0.20979143417327759", "controls": { "cutoff": 40, "infinity_order": 23, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 1.2662720143303126e-12, "derivative_modulus": 0.7207431946605488, "derivative_step": 1e-05, "derivative_refinement_relative": 3.682123939801179e-07, "local_residual_frequency_estimate": 1.7568976352620208e-12, "case": "lower infinity order" }, { "omega_re": "0.22090987817065222", "omega_im": "-0.2097914341669829", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 5e-05, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 7.198304712295683e-12, "derivative_modulus": 0.7207430368425274, "derivative_step": 1e-05, "derivative_refinement_relative": 6.445969969572932e-07, "local_residual_frequency_estimate": 9.987338544164688e-12, "case": "smaller horizon offset" }, { "omega_re": "0.22090987816568983", "omega_im": "-0.20979143417166063", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0002, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 5.021790105530142e-12, "derivative_modulus": 0.7207429412558436, "derivative_step": 1e-05, "derivative_refinement_relative": 1.860696600595726e-07, 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converged.", "function_evaluations": 7, "residual": 1.6566735046146185e-12, "derivative_modulus": 0.7207429484696906, "derivative_step": 1e-05, "derivative_refinement_relative": 1.5473654639023526e-07, "local_residual_frequency_estimate": 2.2985635976489705e-12, "case": "more horizon terms" }, { "omega_re": "0.22090987816072027", "omega_im": "-0.20979143417342158", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 41.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 9.787667390071394e-13, "derivative_modulus": 0.7207429998123778, "derivative_step": 1e-05, "derivative_refinement_relative": 8.279187593710237e-08, "local_residual_frequency_estimate": 1.357996871647633e-12, "case": "lower contour angle" }, { "omega_re": "0.22090987816274055", "omega_im": "-0.209791434173742", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 45.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 5.142296220686085e-13, "derivative_modulus": 0.7207428407154642, "derivative_step": 1e-05, "derivative_refinement_relative": 9.26822291022344e-07, "local_residual_frequency_estimate": 7.134717031086219e-13, "case": "higher contour angle" }, { "omega_re": "0.22090987816377597", "omega_im": "-0.20979143417300355", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 3, "angle_degrees": 43.5212526, "rtol": 3e-13, "atol": 3e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 9.634697154148845e-13, "derivative_modulus": 1.4340019263358785, "derivative_step": 1e-05, "derivative_refinement_relative": 1.2705697328081043e-07, "local_residual_frequency_estimate": 6.718747706822928e-13, "case": "inner match point" }, { "omega_re": "0.2209098781652137", "omega_im": "-0.20979143417128285", "controls": { "cutoff": 40, "infinity_order": 27, "horizon_offset": 0.0001, "horizon_terms": 30, "match": 4, "angle_degrees": 43.5212526, "rtol": 9e-13, "atol": 9e-14 }, "solver_success": true, "solver_message": "The solution converged.", "function_evaluations": 7, "residual": 4.45065739802863e-12, "derivative_modulus": 0.7207431495450539, "derivative_step": 1e-05, "derivative_refinement_relative": 1.8638621331882743e-07, "local_residual_frequency_estimate": 6.17509497084775e-12, "case": "looser ODE tolerances" } ], "refinement_spread": 7.389494076276026e-10, "local_estimate_maximum": 9.987338544164688e-12, "observed_frequency_scale": 7.389494076276026e-10, "qualification": "residual/derivative is local linearization; refinement spread is empirical and is not a rigorous bound" }, "independent_frequency_gap": "3.3505505106746240239e-12", "verification_passed": true, "exit_code": 0 }, "scientific_status": "PASS", "elapsed_seconds": 16.145, "test_status": "PASS" }, { "id": "quartic-tba", "purpose": "Refined TBA spectrum and independently recomputed Rayleigh-Ritz comparison", "page": "/advanced-ode/tba/pure-quartic-oscillator-worked-example/", "command": [ "python", "pure-quartic-tba-spectrum.py", "--high" ], "expected_exit": 0, "source_sha256": "3cd8a006941dd3f4e547022e9e6b3a68a5870675348c61f8dfa786255d554d96", "exit": 0, "stdout": "Pure-quartic folded-TBA audit\nPython: 3.12.9\nNumPy: 2.0.2\nb0=m2: 3.4960767390561593\nm1: 2.4720995697351622\nm2/m1 minus sqrt(2): 0.000e+00\nleft plateau: (0.6931471805599452, 1.0986122886681096)\nkappa_0 FFT versus dense max error: 1.665e-16\nkappa_1 FFT versus dense max error: 2.220e-16\nRayleigh-Ritz basis/frequency spread: 3.197e-14\nT=30, N=16385, h=3.662109e-03, iterations=69, residual=1.277e-13\nT=22.5, N=24577, h=1.831055e-03, iterations=68, residual=1.701e-13\nT=30, N=32769, h=1.831055e-03, iterations=69, residual=1.279e-13\nT=30, N=65537, h=9.155273e-04, iterations=98, residual=5.684e-14\n\nPure-quartic spectrum\nn parity X=hbar^-1 E_TBA E_direct |cutoff shift| |mesh shift| |TBA-direct|\n0 even/N 1.0449382925090 1.060362090484179 1.060362090484184 2.442e-15 4.774e-14 5.551e-15\n1 odd/D 2.7215083408354 3.799673029801393 3.799673029801394 0.000e+00 4.441e-14 4.441e-16\n2 even/N 4.5119702604656 7.455697937986746 7.455697937986748 0.000e+00 0.000e+00 1.776e-15\n3 odd/D 6.3037285199589 11.644745511378165 11.644745511378158 0.000e+00 0.000e+00 7.105e-15\n4 even/N 8.0979852755214 16.261826018850229 16.261826018850250 0.000e+00 0.000e+00 2.132e-14\n5 odd/D 9.8932943734278 21.238372918235950 21.238372918235942 0.000e+00 0.000e+00 7.105e-15\n\nFine-grid period audit\nn B=epsilon_2 A=median period parity correction EQC residual\n0 3.938425611354309 3.418934503841784 +0.277341850251995 -3.997e-15\n1 9.627034960524387 9.408539817364582 -0.016238143404796 +0.000e+00\n2 15.841285504565230 15.708689605564786 +0.000726337615808 +1.243e-14\n3 22.086115145332105 21.991116579471846 -0.000031995656710 +3.553e-15\n\nAll pure-quartic checks passed.\n", "stderr": "", "elapsed_seconds": 10.161, "test_status": "PASS" }, { "id": "ns-direct", "purpose": "Orders 0, 1, 2 of the finite NS approximation versus regenerated and independently refined grid/DCHE spectral controls", "page": "/advanced-ode/quantum-spectra/three-controlled-capstones/", "command": [ "python", "modified-mathieu-ns-capstone.py", "--recompute-direct", "--json", "{result_json}" ], "expected_exit": 0, "source_sha256": "fe37f95cf07bc3cdec3128e9b3c78acd348d148e653ef8411178e4827d7926ed", "exit": 0, "stdout": "Modified-Mathieu NS comparison with regenerated direct spectra\nLambda=1; hbar_s=1; kappa=4; levels=4\nQ=2x; E_s=(hbar_s^2/4) A; no stored targets are used.\nEnvironment: {'Python': '3.12.9', 'NumPy': '2.0.2', 'SciPy': '1.13.1'}\nComputing parity 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