{ "format": "mbs-controlled-manufacturing-tilt-ladder-v1", "created_at_utc": "2026-08-16T00:28:34.501648+00:00", "outcome": "controlled tilt sensitivity experiment; classifications are numerical outcomes, not a priori assumptions", "scope": "three angular degrees of freedom; uniform prescribed rotating field; no translation or force balance", "design": { "controlled_variable": "orientation of fixed-principal-moment axisymmetric inertia tensor relative to magnetic body z", "tilts_deg": [ 0.0, 1.0, 2.121, 5.476228 ], "tilt_rotation_axis_azimuth_parameter_deg": 135.2348, "principal_moments_kg_m2": [ 1e-07, 1e-07, 1.4811621398856308e-07 ], "common_state": [ 0.03816346508367067, -0.9334806171853058, 0.35559682139034077, 0.02661555727829521, 4.833488455540749, 5.167977623933744, -0.6780922038662158 ], "common_state_sha256": "32b14e9d705ccfc118c4a0c8eab51937be51d242d154b363cbe47755ef30ba84", "field_frequency_hz": 60.0, "magnetic_coupling_J": 0.00218476083, "field_tilt_deg": 4.570079, "damping_input_per_s": 0.0982788, "spin_acceleration_rad_s2": 0.0, "duration_s": 200.0, "saved_samples_per_s": 600.0, "strict_maximum_step_s": 0.00020833333333333335, "solver": "LSODA", "relative_tolerance": 1e-10, "absolute_tolerance": 1e-12 }, "fixed_state_comparability": { "all_common_state_hashes_identical": true, "max_quaternion_component_difference": 0.0, "max_omega_body_component_difference_rad_s": 0.0, "max_omega_lab_component_difference_rad_s": 0.0, "max_principal_moment_difference_kg_m2": 5.293955920339377e-23, "initial_physical_axis_spin_is_not_controlled": true, "reason": "The common q and omega are expressed in the fixed magnetic body frame. Rotating the physical cylinder axis within that frame changes its omega projection and physical cylinder-axis direction in the lab frame.", "engine_damping_coefficient_range_Nm_s": [ 9.82788e-09, 9.84941357897372e-09 ], "engine_damping_coefficient_relative_span": 0.002189650681010981, "damping_note": "xi is exactly fixed. The engine's scalar c=xi*(I11+I22)/2 changes slightly when the fixed-eigenvalue tensor is rotated." }, "zero_tilt_analytic_check": { "equation": "I_parallel ds/dt = -c s because a dot tau_B = 0 identically", "initial_spin_rad_s": -0.6780922038662158, "decay_time_constant_s": 15.071023861561505, "maximum_abs_sampled_magnetic_axial_torque_Nm": 0.0, "maximum_abs_spin_error_vs_exponential_rad_s": 1.2634338020234281e-13, "terminal_spin_numeric_rad_s": -1.1694575651839572e-06, "terminal_spin_analytic_rad_s": -1.1694575649556802e-06, "conclusion": "Exact alignment cannot receive magnetic torque about the physical cylinder axis in this ideal point-dipole model; only damping acts on axial spin." }, "cases": [ { "tilt_deg": 0.0, "classification": "statistically stationary large-angle branch", "initial_state": { "common_state_sha256": "32b14e9d705ccfc118c4a0c8eab51937be51d242d154b363cbe47755ef30ba84", "quaternion_magnetic_body_to_lab": [ 0.03816346508367067, -0.9334806171853058, 0.35559682139034077, 0.02661555727829521 ], "omega_magnetic_body_rad_s": [ 4.833488455540749, 5.167977623933744, -0.6780922038662158 ], "omega_lab_rad_s": [ 0.1781060204919033, -7.106164596513176, 0.033397320206941056 ], "magnetic_axis_lab": [ -0.02254859991575866, 0.09017852501585819, -0.995670324087147 ], "physical_cylinder_axis_body": [ 0.0, 0.0, 1.0 ], "physical_cylinder_axis_lab": [ -0.02254859991575866, 0.09017852501585819, -0.995670324087147 ], "physical_cylinder_axis_spin_rad_s": -0.6780922038662158, "body_speed_rad_s": 7.108474689289518, "theta_deg": 174.6663790777206, "kinetic_energy_J": 2.5375827554212255e-06, "potential_energy_J": 0.0003697166050588937, "total_energy_J": 0.0003722541878143149 }, "geometry": { "inertia_tensor_magnetic_body_frame_kg_m2": [ [ 1e-07, 0.0, 0.0 ], [ 0.0, 1e-07, 0.0 ], [ 0.0, 0.0, 1.4811621398856308e-07 ] ], "eigenvalues_kg_m2": [ 1e-07, 1e-07, 1.4811621398856308e-07 ], "axis_from_tensor": [ 0.0, 0.0, 1.0 ], "physical_cylinder_axis_magnetic_body_frame": [ 0.0, 0.0, 1.0 ], "projected_physical_axis_azimuth_deg": null, "parameter_azimuth_note": "135.2348 deg is the Rodrigues tilt-rotation-axis convention used by index19; for nonzero tilt the physical cylinder-axis projection is at 45.2348 deg.", "height_over_diameter": 0.5125605301591867, "axis_over_transverse_inertia": 1.481162139885631, "eigenvalue_max_abs_error_from_fixed_principals_kg_m2": 0.0 }, "damping": { "input_xi_per_s": 0.0982788, "engine_scalar_coefficient_Nm_s": 9.82788e-09, "zero_tilt_axial_decay_rate_per_s": 0.06635249264985174, "note": "The engine keeps xi fixed as requested and computes c=xi*(I11+I22)/2; rotating the tensor therefore causes a small deterministic change in c." }, "windows": { "0_10": { "range_s": [ 0.0, 10.0 ], "drive_cycles": 600, "theta_mean_deg": 99.22462292265507, "theta_min_deg": 13.32630874299211, "theta_max_deg": 179.39574771059063, "theta_span_deg": 166.06943896759853, "theta_slope_deg_s": -0.1936239360903781, "cylinder_theta_mean_deg": 99.22462292265507, "symmetry_spin_mean_rad_s": -0.49561447790310337, "symmetry_spin_abs_mean_rad_s": 0.49561447790310337, "symmetry_spin_std_rad_s": 0.09460247099310785, "symmetry_spin_slope_rad_s2": 0.032646573325655175, "body_speed_mean_rad_s": 103.52608830039215, "body_speed_slope_rad_s2": 0.03422341819422152, "u_mean": 0.1406491578186439, "u_positive_fraction": 0.6825529078486919, "kinetic_mean_J": 0.0005950597378081295, "potential_mean_J": 0.00030728477077466144, "total_mean_J": 0.0009023445085827909, "energy_change_rate_W": 0.00024979839654750227, "field_power_mean_W": 0.0003628407357714216, "damping_loss_mean_W": 0.00011696231006005655, "power_balance_W": 0.0002458784257113651, "power_closure_error_W": 3.919970836137191e-06, "symmetry_torque_mean_Nm": 0.0, "symmetry_spin_power_mean_W": 0.0, "physical_magnetic_torque_mean_Nm": 0.0, "physical_magnetic_torque_rms_Nm": 0.0, "physical_magnetic_torque_max_abs_Nm": 0.0, "physical_magnetic_torque_impulse_Nm_s": 0.0, "physical_spin_dot_mean_rad_s2": 0.03288525600223367, "physical_spin_dot_rms_rad_s2": 0.03347877923227848, "physical_spin_endpoint_delta_rad_s": 0.32885255968724625, "physical_spin_integrated_derivative_delta_rad_s": 0.32885256002233676, "physical_spin_integral_closure_rad_s": -3.3509051089453123e-10, "axial_equation_residual_max_abs_Nm": 2.1684126167771344e-19, "axial_equation_residual_rms_Nm": 4.579593259937001e-20, "theta_q05_deg": 22.10194027450104, "theta_median_deg": 97.3103820908076, "theta_q95_deg": 174.93827620357297, "ap_error_mean_deg": 81.8156158689723, "ap_error_median_deg": 78.516928745499 }, "80_120": { "range_s": [ 80.0, 120.0 ], "drive_cycles": 2400, "theta_mean_deg": 96.82695064305288, "theta_min_deg": 22.06727628803733, "theta_max_deg": 169.24850346674717, "theta_span_deg": 147.18122717870983, "theta_slope_deg_s": -0.001331267953632591, "cylinder_theta_mean_deg": 96.82695064305288, "symmetry_spin_mean_rad_s": -0.0011758816111838248, "symmetry_spin_abs_mean_rad_s": 0.0011758816111838248, "symmetry_spin_std_rad_s": 0.0008544344239431835, "symmetry_spin_slope_rad_s2": 7.017010951332916e-05, "body_speed_mean_rad_s": 103.32437069139574, "body_speed_slope_rad_s2": -0.0005309987734974352, "u_mean": 0.1413793522770908, "u_positive_fraction": 0.6161409941252448, "kinetic_mean_J": 0.0005979688846890044, "potential_mean_J": 0.0003088800710257593, "total_mean_J": 0.0009068489557147637, "energy_change_rate_W": 5.465577123546492e-05, "field_power_mean_W": 0.0001705740258056285, "damping_loss_mean_W": 0.00011753532883915707, "power_balance_W": 5.303869696647142e-05, "power_closure_error_W": 1.6170742689934972e-06, "symmetry_torque_mean_Nm": 0.0, "symmetry_spin_power_mean_W": 0.0, "physical_magnetic_torque_mean_Nm": 0.0, "physical_magnetic_torque_rms_Nm": 0.0, "physical_magnetic_torque_max_abs_Nm": 0.0, "physical_magnetic_torque_impulse_Nm_s": 0.0, "physical_spin_dot_mean_rad_s2": 7.802267595925381e-05, "physical_spin_dot_rms_rad_s2": 9.644349418495646e-05, "physical_spin_endpoint_delta_rad_s": 0.003120907035349368, "physical_spin_integrated_derivative_delta_rad_s": 0.0031209070383701534, "physical_spin_integral_closure_rad_s": -3.0207854447017457e-12, "axial_equation_residual_max_abs_Nm": 2.1684043772827515e-19, "axial_equation_residual_rms_Nm": 4.640899421475638e-20, "theta_q05_deg": 26.03837766208505, "theta_median_deg": 95.31187784842237, "theta_q95_deg": 167.50428591051144, "ap_error_mean_deg": 81.68483124902481, "ap_error_median_deg": 74.92769314091336 }, "120_160": { "range_s": [ 120.0, 160.0 ], "drive_cycles": 2400, "theta_mean_deg": 96.8601017420341, "theta_min_deg": 22.071404832650128, "theta_max_deg": 169.22874303610828, "theta_span_deg": 147.15733820345815, "theta_slope_deg_s": -0.003449469417809682, "cylinder_theta_mean_deg": 96.8601017420341, "symmetry_spin_mean_rad_s": -8.273756628977231e-05, "symmetry_spin_abs_mean_rad_s": 8.273756628977231e-05, "symmetry_spin_std_rad_s": 6.011984890041515e-05, "symmetry_spin_slope_rad_s2": 4.937320247239082e-06, "body_speed_mean_rad_s": 103.32037024002213, "body_speed_slope_rad_s2": 0.0011861503315125296, "u_mean": 0.14145758573749476, "u_positive_fraction": 0.6154743552351986, "kinetic_mean_J": 0.0005979587802912447, "potential_mean_J": 0.0003090509924256453, "total_mean_J": 0.00090700977271689, "energy_change_rate_W": -1.6452868203987847e-05, "field_power_mean_W": 0.00010179980003184572, "damping_loss_mean_W": 0.00011753334275292489, "power_balance_W": -1.573354272107917e-05, "power_closure_error_W": -7.193254829086757e-07, "symmetry_torque_mean_Nm": 0.0, "symmetry_spin_power_mean_W": 0.0, "physical_magnetic_torque_mean_Nm": 0.0, "physical_magnetic_torque_rms_Nm": 0.0, "physical_magnetic_torque_max_abs_Nm": 0.0, "physical_magnetic_torque_impulse_Nm_s": 0.0, "physical_spin_dot_mean_rad_s2": 5.489843759993674e-06, "physical_spin_dot_rms_rad_s2": 6.7859722620001375e-06, "physical_spin_endpoint_delta_rad_s": 0.0002195937501353488, "physical_spin_integrated_derivative_delta_rad_s": 0.0002195937503997469, "physical_spin_integral_closure_rad_s": -2.6439809543143955e-13, "axial_equation_residual_max_abs_Nm": 2.1684043490099767e-19, "axial_equation_residual_rms_Nm": 4.644078035122292e-20, "theta_q05_deg": 26.030269226367455, "theta_median_deg": 95.34840639569833, "theta_q95_deg": 167.50517327845844, "ap_error_mean_deg": 81.67949589036637, "ap_error_median_deg": 74.94174159518423 }, "160_200": { "range_s": [ 160.0, 200.0 ], "drive_cycles": 2400, "theta_mean_deg": 96.87128052064732, "theta_min_deg": 22.071685550866523, "theta_max_deg": 169.22717515780352, "theta_span_deg": 147.15548960693698, "theta_slope_deg_s": 0.0017134722271962772, "cylinder_theta_mean_deg": 96.87128052064733, "symmetry_spin_mean_rad_s": -5.8215936120393485e-06, "symmetry_spin_abs_mean_rad_s": 5.8215936120393485e-06, "symmetry_spin_std_rad_s": 4.230162234729369e-06, "symmetry_spin_slope_rad_s2": 3.4740050137671787e-07, "body_speed_mean_rad_s": 103.30764363882068, "body_speed_slope_rad_s2": 0.00015781171957796573, "u_mean": 0.14142292534574324, "u_positive_fraction": 0.6158076746802217, "kinetic_mean_J": 0.0005977981304433452, "potential_mean_J": 0.00030897526775939415, "total_mean_J": 0.0009067733982027393, "energy_change_rate_W": -9.075644815017979e-05, "field_power_mean_W": 2.8918173771785274e-05, "damping_loss_mean_W": 0.00011750176580443064, "power_balance_W": -8.858359203264536e-05, "power_closure_error_W": -2.1728561175344277e-06, "symmetry_torque_mean_Nm": 0.0, "symmetry_spin_power_mean_W": 0.0, "physical_magnetic_torque_mean_Nm": 0.0, "physical_magnetic_torque_rms_Nm": 0.0, "physical_magnetic_torque_max_abs_Nm": 0.0, "physical_magnetic_torque_impulse_Nm_s": 0.0, "physical_spin_dot_mean_rad_s2": 3.8627724738975306e-07, "physical_spin_dot_rms_rad_s2": 4.774756442473285e-07, "physical_spin_endpoint_delta_rad_s": 1.54510898777336e-05, "physical_spin_integrated_derivative_delta_rad_s": 1.5451089895590122e-05, "physical_spin_integral_closure_rad_s": -1.785652178963419e-14, "axial_equation_residual_max_abs_Nm": 2.1684043452234444e-19, "axial_equation_residual_rms_Nm": 4.587387801238236e-20, "theta_q05_deg": 26.032433881049222, "theta_median_deg": 95.35842528211963, "theta_q95_deg": 167.50708150354265, "ap_error_mean_deg": 81.68037169443824, "ap_error_median_deg": 74.98625076918415 }, "180_200": { "range_s": [ 180.0, 200.0 ], "drive_cycles": 1200, "theta_mean_deg": 96.89993454907724, "theta_min_deg": 22.071685550866523, "theta_max_deg": 169.22711698338077, "theta_span_deg": 147.15543143251423, "theta_slope_deg_s": 0.0031807659400298377, "cylinder_theta_mean_deg": 96.89993454907726, "symmetry_spin_mean_rad_s": -2.4409682702630138e-06, "symmetry_spin_abs_mean_rad_s": 2.4409682702630138e-06, "symmetry_spin_std_rad_s": 9.219210865136859e-07, "symmetry_spin_slope_rad_s2": 1.5740257680168306e-07, "body_speed_mean_rad_s": 103.29793465516539, "body_speed_slope_rad_s2": 0.0014373899843072308, "u_mean": 0.14144354080097363, "u_positive_fraction": 0.6158653445546205, "kinetic_mean_J": 0.0005977265928837091, "potential_mean_J": 0.00030902030759847406, "total_mean_J": 0.000906746900482183, "energy_change_rate_W": -0.00010151484639783545, "field_power_mean_W": 1.840227718014999e-05, "damping_loss_mean_W": 0.00011748770455339888, "power_balance_W": -9.908542737324888e-05, "power_closure_error_W": -2.4294190245865673e-06, "symmetry_torque_mean_Nm": 0.0, "symmetry_spin_power_mean_W": 0.0, "physical_magnetic_torque_mean_Nm": 0.0, "physical_magnetic_torque_rms_Nm": 0.0, "physical_magnetic_torque_max_abs_Nm": 0.0, "physical_magnetic_torque_impulse_Nm_s": 0.0, "physical_spin_dot_mean_rad_s2": 1.619643299406365e-07, "physical_spin_dot_rms_rad_s2": 1.7312923777995072e-07, "physical_spin_endpoint_delta_rad_s": 3.239286580792051e-06, "physical_spin_integrated_derivative_delta_rad_s": 3.23928659881273e-06, "physical_spin_integral_closure_rad_s": -1.8020679315910917e-14, "axial_equation_residual_max_abs_Nm": 2.1684043450341177e-19, "axial_equation_residual_rms_Nm": 4.5509456197399744e-20, "theta_q05_deg": 26.032433881049222, "theta_median_deg": 95.39395738285788, "theta_q95_deg": 167.50708150354265, "ap_error_mean_deg": 81.67850457163588, "ap_error_median_deg": 74.98625076918415 } }, "adaptive_power_audit_180_200": { "method": "strict LSODA reintegration plus composite trapezoid work/impulse quadrature at 80 saved points per drive cycle", "range_s": [ 180.0, 200.0 ], "drive_cycles": 1200, "relative_tolerance": 1e-10, "absolute_tolerance": 1e-12, "maximum_step_s": 0.00020833333333333335, "quadrature_samples_per_s": 4800, "quadrature_samples_per_drive_cycle": 80, "nfev": 250196, "initial_spin_rad_s": -4.408744145976008e-06, "final_spin_rad_s": -1.1694575651825565e-06, "spin_change_rad_s": 3.239286580793452e-06, "field_work_J": 0.0003202024160810625, "damping_loss_J": 0.0023497460718360594, "mechanical_energy_change_J": -0.0020302965690754036, "full_energy_closure_J": 7.529133204064141e-07, "full_energy_closure_mean_power_W": 3.76456660203207e-08, "field_power_mean_W": 1.6010120804053126e-05, "damping_loss_mean_W": 0.00011748730359180298, "mechanical_energy_change_rate_W": -0.00010151482845377018, "axial_torque_impulse_Nm_s": 0.0, "axial_drag_impulse_Nm_s": -4.797908643975647e-13, "axial_momentum_change_Nm_s": 4.797908643710838e-13, "axial_momentum_closure_Nm_s": 2.64808867632424e-23, "axial_torque_work_J": 0.0, "axial_drag_work_J": 1.338185110454199e-18, "axial_spin_energy_change_J": -1.3381851103165132e-18, "axial_spin_energy_closure_J": -1.3768588728221012e-28, "quadrature_refinement_40_to_80_points_per_cycle": { "field_work_difference_J": -2.2603475208343975e-06, "damping_loss_difference_J": -3.7617060768407873e-10, "axial_torque_impulse_difference_Nm_s": 0.0, "axial_drag_impulse_difference_Nm_s": 2.292053661692651e-23 }, "initial_energy_J": { "kinetic_J": 0.0005373835694964248, "potential_J": 3.0739075721742636e-05, "total_J": 0.0005681226452181675 }, "final_energy_J": { "kinetic_J": 0.00039080931806183706, "potential_J": -0.001852983241919073, "total_J": -0.001462173923857236 }, "terminal_state_difference_norm_vs_uninterrupted_saved_run": 5.30075486415235e-05, "note": "The saved 600 Hz trapezoid residual is not an acceptance test. This strict 4800 Hz reintegration is the precision energy/power audit; refinement differences quantify remaining quadrature sensitivity." }, "late_stationarity": { "theta_mean_delta_160_200_minus_120_160_deg": 0.011178778613214035, "theta_span_delta_deg": -0.0018485965211709754, "physical_spin_mean_delta_rad_s": 7.691597267773296e-05, "total_energy_relative_delta": -0.0002606085637232857 } }, { "tilt_deg": 1.0, "classification": "statistically stationary large-angle branch", "initial_state": { "common_state_sha256": "32b14e9d705ccfc118c4a0c8eab51937be51d242d154b363cbe47755ef30ba84", "quaternion_magnetic_body_to_lab": [ 0.03816346508367067, -0.9334806171853058, 0.35559682139034077, 0.02661555727829521 ], "omega_magnetic_body_rad_s": [ 4.833488455540749, 5.167977623933744, -0.6780922038662158 ], "omega_lab_rad_s": [ 0.1781060204919033, -7.106164596513176, 0.033397320206941056 ], "magnetic_axis_lab": [ -0.02254859991575866, 0.09017852501585819, -0.995670324087147 ], 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0.9998476951563913 ], "projected_physical_axis_azimuth_deg": 45.2348, "parameter_azimuth_note": "135.2348 deg is the Rodrigues tilt-rotation-axis convention used by index19; for nonzero tilt the physical cylinder-axis projection is at 45.2348 deg.", "height_over_diameter": 0.5125605301591867, "axis_over_transverse_inertia": 1.481162139885631, "eigenvalue_max_abs_error_from_fixed_principals_kg_m2": 2.6469779601696886e-23 }, "damping": { "input_xi_per_s": 0.0982788, "engine_scalar_coefficient_Nm_s": 9.828600164872374e-09, "zero_tilt_axial_decay_rate_per_s": 0.06635249264985174, "note": "The engine keeps xi fixed as requested and computes c=xi*(I11+I22)/2; rotating the tensor therefore causes a small deterministic change in c." }, "windows": { "0_10": { "range_s": [ 0.0, 10.0 ], "drive_cycles": 600, "theta_mean_deg": 99.27954949535743, "theta_min_deg": 13.251061823613984, "theta_max_deg": 179.6853228123854, "theta_span_deg": 166.43426098877143, "theta_slope_deg_s": -0.19258212212668163, 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0.00010463441361103244, "physical_magnetic_torque_max_abs_Nm": 0.00020843154480414087, "physical_magnetic_torque_impulse_Nm_s": -1.8318962212998335e-06, "physical_spin_dot_mean_rad_s2": -0.16730938130326675, "physical_spin_dot_rms_rad_s2": 706.4343208237501, "physical_spin_endpoint_delta_rad_s": -3.420869139790379, "physical_spin_integrated_derivative_delta_rad_s": -3.346187626065335, "physical_spin_integral_closure_rad_s": -0.07468151372504384, "axial_equation_residual_max_abs_Nm": 7.318364664277155e-19, "axial_equation_residual_rms_Nm": 1.085134596714838e-19, "theta_q05_deg": 34.22613369852828, "theta_median_deg": 99.78184893948414, "theta_q95_deg": 175.9084676220929, "ap_error_mean_deg": 81.43232115169234, "ap_error_median_deg": 80.22070534441501 } }, "adaptive_power_audit_180_200": { "method": "strict LSODA reintegration plus composite trapezoid work/impulse quadrature at 80 saved points per drive cycle", "range_s": [ 180.0, 200.0 ], "drive_cycles": 1200, "relative_tolerance": 1e-10, "absolute_tolerance": 1e-12, "maximum_step_s": 0.00020833333333333335, "quadrature_samples_per_s": 4800, "quadrature_samples_per_drive_cycle": 80, "nfev": 227766, "initial_spin_rad_s": -3.6213174693399632, "final_spin_rad_s": -7.042187885808107, "spin_change_rad_s": -3.420870416468144, "field_work_J": 0.00106921604829394, "damping_loss_J": 0.0023786633033849335, "mechanical_energy_change_J": -0.0013094577018294734, "full_energy_closure_J": 1.0446738479916867e-08, "full_energy_closure_mean_power_W": 5.223369239958434e-10, "field_power_mean_W": 5.3460802414697004e-05, "damping_loss_mean_W": 0.00011893316516924668, "mechanical_energy_change_rate_W": -6.547288509147368e-05, "axial_torque_impulse_Nm_s": -1.8427853967238616e-06, "axial_drag_impulse_Nm_s": -1.3362710488327526e-06, "axial_momentum_change_Nm_s": -5.066863746327405e-07, "axial_momentum_closure_Nm_s": 1.7202674163152714e-10, "axial_torque_work_J": 1.300342293255463e-05, "axial_drag_work_J": 1.0303146038403304e-05, "axial_spin_energy_change_J": 2.7015264346383946e-06, "axial_spin_energy_closure_J": -1.2495404870683103e-09, "quadrature_refinement_40_to_80_points_per_cycle": { "field_work_difference_J": -3.052827325829288e-08, "damping_loss_difference_J": 1.3569346644579094e-10, "axial_torque_impulse_difference_Nm_s": -5.16267696248872e-10, "axial_drag_impulse_difference_Nm_s": 2.4623771666560388e-14 }, "initial_energy_J": { "kinetic_J": 0.0011190423093081546, "potential_J": 0.0012780964844601756, "total_J": 0.00239713879376833 }, "final_energy_J": { "kinetic_J": 0.0004424636609136748, "potential_J": 0.000645217431025182, "total_J": 0.0010876810919388568 }, "terminal_state_difference_norm_vs_uninterrupted_saved_run": 2.8716053369349407e-05, "note": "The saved 600 Hz trapezoid residual is not an acceptance test. This strict 4800 Hz reintegration is the precision energy/power audit; refinement differences quantify remaining quadrature sensitivity." }, "late_stationarity": { "theta_mean_delta_160_200_minus_120_160_deg": -0.00039111599107855, "theta_span_delta_deg": -0.1263055595112803, "physical_spin_mean_delta_rad_s": -0.0007116037403012143, "total_energy_relative_delta": 1.5614930379352136e-05 } } ], "caveats": [ "The ladder fixes the magnetic-frame state rather than forcing equal initial physical cylinder-axis spin or equal initial kinetic energy.", "At 0 deg the azimuth is geometrically undefined even though the numerical Rodrigues parameter is retained in metadata.", "The model omits translational MBS dynamics, field gradients, gravity, and rotation-translation feedback.", "A statistically stationary branch need not be an exact periodic orbit.", "Saved-trace trapezoid power and torque-impulse closures are sampling diagnostics only: 600 Hz gives ten saved samples per drive cycle. Precision energy/power and axial-impulse closure use the separate strict augmented-state audit over 180--200 s." ] }