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"boson_gap": 1.000000100000015e-11, "fermion_fixed_number_gap": 1.4142137037944657e-11 }, { "x": 0.01, "direction": [ 0.3244428422615251, 0.48666426339228763, 0.8111071056538127 ], "energy": 5.857863962055494e-12, "radial": -3.5355342594860635e-08, "fermion_contact": 0.0, "fermion_spectral_curvature": -3.5355342594861635e-08, "boson_adiabatic_inertia": 249999974999993.72, "fermion_adiabatic_inertia": 176776677618967.25, "boson_gap": 1.000000100000015e-11, "fermion_fixed_number_gap": 1.4142137037944657e-11 }, { "x": 0.01, "direction": [ 0.3333333333333333, -0.6666666666666666, -0.6666666666666666 ], "energy": 5.857863962055494e-12, "radial": -3.5355342594860635e-08, "fermion_contact": 0.0, "fermion_spectral_curvature": -3.5355342594861635e-08, "boson_adiabatic_inertia": 249999974999993.72, "fermion_adiabatic_inertia": 176776677618967.25, "boson_gap": 1.000000100000015e-11, "fermion_fixed_number_gap": 1.4142137037944655e-11 }, { "x": 0.1, "direction": [ 1.0, 0.0, 0.0 ], "energy": 9.501243738915481e-12, "radial": -9.85185435360106e-11, "fermion_contact": 0.0, "fermion_spectral_curvature": -9.85185435360116e-11, "boson_adiabatic_inertia": 249999974999.9998, "fermion_adiabatic_inertia": 9754309290.327114, "boson_gap": 1.0000001000000102e-10, "fermion_fixed_number_gap": 1.0049876626108554e-10 }, { "x": 0.1, "direction": [ 0.3244428422615251, 0.48666426339228763, 0.8111071056538127 ], "energy": 9.501243738915481e-12, "radial": -9.851854353601061e-11, "fermion_contact": 0.0, "fermion_spectral_curvature": -9.851854353601161e-11, "boson_adiabatic_inertia": 249999974999.9998, "fermion_adiabatic_inertia": 9754309290.327118, "boson_gap": 1.0000001000000102e-10, "fermion_fixed_number_gap": 1.0049876626108552e-10 }, { "x": 0.1, "direction": [ 0.3333333333333333, -0.6666666666666666, -0.6666666666666666 ], "energy": 9.501243738915546e-12, "radial": -9.851854353601061e-11, "fermion_contact": 0.0, "fermion_spectral_curvature": -9.851854353601161e-11, "boson_adiabatic_inertia": 249999974999.9998, "fermion_adiabatic_inertia": 9754309290.327126, "boson_gap": 1.0000001000000102e-10, "fermion_fixed_number_gap": 1.0049876626108546e-10 }, { "x": 1.0, "direction": [ 1.0, 0.0, 0.0 ], "energy": 9.950001244937827e-12, "radial": -9.998501187328064e-14, "fermion_contact": 0.0, "fermion_spectral_curvature": -9.998501187328164e-14, "boson_adiabatic_inertia": 249999975.0, "fermion_adiabatic_inertia": 99974.9943768426, "boson_gap": 1.00000010000001e-09, "fermion_fixed_number_gap": 1.0000500987550722e-09 }, { "x": 1.0, "direction": [ 0.3244428422615251, 0.48666426339228763, 0.8111071056538127 ], "energy": 9.950001244937206e-12, "radial": -9.99850118732824e-14, "fermion_contact": 0.0, "fermion_spectral_curvature": -9.998501187328339e-14, "boson_adiabatic_inertia": 249999975.0, "fermion_adiabatic_inertia": 99974.99437684426, "boson_gap": 1.00000010000001e-09, "fermion_fixed_number_gap": 1.0000500987550726e-09 }, { "x": 1.0, "direction": [ 0.3333333333333333, -0.6666666666666666, -0.6666666666666666 ], "energy": 9.950001244937827e-12, "radial": -9.998501187328096e-14, "fermion_contact": 0.0, "fermion_spectral_curvature": -9.998501187328195e-14, "boson_adiabatic_inertia": 249999975.0, "fermion_adiabatic_inertia": 99974.99437684292, "boson_gap": 1.00000010000001e-09, "fermion_fixed_number_gap": 1.0000500987550722e-09 } ], "scope": "Rotationally invariant polynomial source couplings. Spectral weight and gap hierarchy are inputs; a symmetry protecting their matching is not established." }, "bands": { "physical_basis_checks": [ { "k": -1.2, "x": 0.0, "errors": [ 1.0658141036401503e-14, 5.603965943075893e-17, 0.0 ] }, { "k": -1.2, "x": 0.1, "errors": [ 1.4210854715202004e-14, 2.636779683484747e-16, 2.220446049250313e-16 ] }, { "k": -1.2, "x": 1.0, "errors": [ 1.4210854715202004e-14, 4.163336342344337e-16, 6.661338147750939e-16 ] }, { "k": -1.2, "x": 4.25, "errors": [ 1.7053025658242404e-13, 3.174543961037557e-16, 1.1102230246251565e-16 ] }, { "k": -1.2, "x": 12.0, "errors": [ 4.334310688136611e-13, 1.4582519219930035e-17, 2.7755575615628914e-17 ] }, { "k": -0.1, "x": 0.0, "errors": [ 1.0658141036401503e-14, 4.301759993491944e-17, 5.329070518200751e-15 ] }, { "k": -0.1, "x": 0.1, "errors": [ 7.105427357601002e-15, 4.996003610813204e-16, 8.881784197001252e-15 ] }, { "k": -0.1, "x": 1.0, "errors": [ 1.0658141036401503e-14, 1.2212453270876722e-15, 1.0658141036401503e-14 ] }, { "k": -0.1, "x": 4.25, "errors": [ 1.1368683772161603e-13, 2.0174834025610267e-15, 7.771561172376096e-16 ] }, { "k": -0.1, "x": 12.0, "errors": [ 1.8189894035458565e-12, 2.1575623232461538e-16, 8.326672684688674e-17 ] }, { "k": 0.3, "x": 0.0, "errors": [ 1.4210854715202004e-14, 3.074963059362588e-16, 2.842170943040401e-14 ] }, { "k": 0.3, "x": 0.1, "errors": [ 1.4210854715202004e-14, 9.71445146547012e-16, 3.552713678800501e-15 ] }, { "k": 0.3, "x": 1.0, "errors": [ 2.1316282072803006e-14, 2.4147350785597155e-15, 1.0658141036401503e-14 ] }, { "k": 0.3, "x": 4.25, "errors": [ 8.171241461241152e-14, 5.620504062164855e-16, 1.3322676295501878e-15 ] }, { "k": 0.3, "x": 12.0, "errors": [ 9.094947017729282e-13, 9.849434635944565e-16, 2.220446049250313e-16 ] }, { "k": 1.1, "x": 0.0, "errors": [ 3.552713678800501e-15, 3.0421287470418795e-16, 3.552713678800501e-14 ] }, { "k": 1.1, "x": 0.1, "errors": [ 1.0658141036401503e-14, 2.2690183065776637e-15, 3.552713678800501e-14 ] }, { "k": 1.1, "x": 1.0, "errors": [ 1.4210854715202004e-14, 1.5543122344752192e-15, 0.0 ] }, { "k": 1.1, "x": 4.25, "errors": [ 3.708144902248023e-14, 5.857293816635689e-15, 1.7763568394002505e-15 ] }, { "k": 1.1, "x": 12.0, "errors": [ 1.4317436125566019e-12, 5.452994826515845e-16, 2.498001805406602e-16 ] } ], "independent_zone_and_orientation_quadratures": [ { "radial_order": 64, "angle_order": 24, "orientation_order": 40, "x": 0.03, "mu": 0.02584128303261779, "radial": 0.05282385937719214, "difference_from_saved_table": [ -5.014610948705922e-11, -3.6430569672063484e-10 ] }, { "radial_order": 64, "angle_order": 24, "orientation_order": 40, "x": 0.3, "mu": 0.32195298295748953, "radial": 0.6710890000553182, "difference_from_saved_table": [ -2.8751894198819627e-08, -9.366750630679377e-08 ] }, { "radial_order": 64, "angle_order": 24, "orientation_order": 40, "x": 2.0, "mu": 0.9870103995024857, "radial": 1.0507836420308203, "difference_from_saved_table": [ -7.300681315047797e-08, 1.3750920646593556e-07 ] }, { "radial_order": 64, "angle_order": 24, "orientation_order": 40, "x": 6.0, "mu": 1.000718298458598, "radial": 0.9994557526436325, "difference_from_saved_table": [ -4.724446833748175e-09, 2.2368879970535716e-08 ] }, { "radial_order": 64, "angle_order": 24, "orientation_order": 40, "x": 20.0, "mu": 1.0000375369200942, "radial": 0.9999330995092964, "difference_from_saved_table": [ 1.2386247583151544e-10, -2.4389357200504946e-10 ] }, { "radial_order": 96, "angle_order": 32, "orientation_order": 64, "x": 0.03, "mu": 0.025841283032585816, "radial": 0.052823859377012616, "difference_from_saved_table": [ -5.0178083910168425e-11, -3.6448521978371673e-10 ] }, { "radial_order": 96, "angle_order": 32, "orientation_order": 64, "x": 0.3, "mu": 0.32195298295748687, "radial": 0.6710890000553121, "difference_from_saved_table": [ -2.8751896863354887e-08, -9.36675124130204e-08 ] }, { "radial_order": 96, "angle_order": 32, "orientation_order": 64, "x": 2.0, "mu": 0.9870103995024855, "radial": 1.0507836420308234, "difference_from_saved_table": [ -7.300681337252257e-08, 1.3750920957456003e-07 ] }, { "radial_order": 96, "angle_order": 32, "orientation_order": 64, "x": 6.0, "mu": 1.000718298458598, "radial": 0.9994557526436331, "difference_from_saved_table": [ -4.724446833748175e-09, 2.2368880525647228e-08 ] }, { "radial_order": 96, "angle_order": 32, "orientation_order": 64, "x": 20.0, "mu": 1.000037536919969, "radial": 0.9999330995089388, "difference_from_saved_table": [ 1.2373724267433772e-10, -2.4425117484128123e-10 ] } ], "scope": "The inherited eight-band response has explicit quadratic source couplings. Its source contact term is retained; the critical matching and interface population remain assumptions." }, "shape": { "frequency": 1.7, "metric_matrix": [ [ 0.08333333333333336, -0.041666666666666685, -0.041666666666666685, 0.0, 0.0, 0.0 ], [ -0.041666666666666685, 0.08333333333333337, -0.041666666666666685, 0.0, 0.0, 0.0 ], [ -0.041666666666666685, -0.041666666666666685, 0.08333333333333337, 0.0, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.12499999999999997, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.12499999999999997, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.12499999999999997 ] ], "inertia_matrix": [ [ 0.04901960784313727, -0.024509803921568638, -0.024509803921568638, 0.0, 0.0, 0.0 ], [ -0.024509803921568638, 0.04901960784313727, -0.024509803921568634, 0.0, 0.0, 0.0 ], [ -0.024509803921568638, -0.024509803921568634, 0.04901960784313727, 0.0, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.07352941176470587, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.07352941176470587, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.07352941176470587 ] ], "inertia_eigenvalues": [ -7.437768629024741e-18, 0.07352941176470587, 0.07352941176470587, 0.07352941176470587, 0.0735294117647059, 0.0735294117647059 ], "rank": 5, "volume_inertia": 0.0, "Gaussian_overlap_checks": [ { "background_log_shape": [ 0.0, 0.0, 0.0 ], "overlap_metric": 0.08333333467724913, "expected_metric": 0.08333333333333334 }, { "background_log_shape": [ 0.0, 0.0, 0.0 ], "overlap_metric": 0.08333333467724913, "expected_metric": 0.08333333333333334 }, { "background_log_shape": [ 0.0, 0.0, 0.0 ], "overlap_metric": 0.08333333467724913, "expected_metric": 0.08333333333333334 }, { "background_log_shape": [ 0.0, 0.0, 0.0 ], "overlap_metric": 0.12499999924031613, "expected_metric": 0.12499999999999997 }, { "background_log_shape": [ 0.0, 0.0, 0.0 ], "overlap_metric": 0.12499999924031613, "expected_metric": 0.12499999999999997 }, { "background_log_shape": [ 0.0, 0.0, 0.0 ], "overlap_metric": 0.12499999924031613, "expected_metric": 0.12499999999999997 }, { "background_log_shape": [ 0.4, -0.15, -0.25 ], "overlap_metric": 0.083333329126134, "expected_metric": 0.08333333333333334 }, { "background_log_shape": [ 0.4, -0.15, -0.25 ], "overlap_metric": 0.08333333467724913, "expected_metric": 0.08333333333333334 }, { "background_log_shape": [ 0.4, -0.15, -0.25 ], "overlap_metric": 0.08333334022836425, "expected_metric": 0.08333333333333334 }, { "background_log_shape": [ 0.4, -0.15, -0.25 ], "overlap_metric": 0.12499999924031613, "expected_metric": 0.12499999999999997 }, { "background_log_shape": [ 0.4, -0.15, -0.25 ], "overlap_metric": 0.12499999924031613, "expected_metric": 0.12499999999999997 }, { "background_log_shape": [ 0.4, -0.15, -0.25 ], "overlap_metric": 0.12499999646475857, "expected_metric": 0.12499999999999997 } ], "cell_effective_Lagrangian": "hbar/(16*omega) tr[(Q^-1 dQ/dtau)^2] = hbar/(4*omega) K_TF,ij K_TF^ij", "scope": "Decoupled three-mode Gaussian cells depend on unimodular shape. This derives a local rank-five inertia; spatial gauge constraints, the curvature term and a universal gravitational coupling are not derived." }, "clock": { "two_level_checks": [ { "volume": 0.3, "clock": 0.8, "gap": 3.0, "Berry_curvature": -1.5625, "Berry_loop_estimate": -1.5625000002452534, "metric": [ [ 0.4901960784313726, -7.558139031412388e-17 ], [ -7.558139031412388e-17, 1.2451171874999998 ] ], "inertia": [ [ 0.32679738562091504, -5.0387593542749255e-17 ], [ -5.0387593542749255e-17, 0.8300781249999999 ] ], "metric_determinant_minus_Berry_bound": 0.0 }, { "volume": 0.7, "clock": 1.3, "gap": 3.0, "Berry_curvature": -0.5917159763313607, "Berry_loop_estimate": -0.5917159775810685, "metric": [ [ 0.2747252747252747, 3.1822666071798506e-19 ], [ 3.1822666071798506e-19, 0.31861629494765575 ] ], "inertia": [ [ 0.18315018315018314, 2.1215110714532336e-19 ], [ 2.1215110714532336e-19, 0.21241086329843717 ] ], "metric_determinant_minus_Berry_bound": 0.0 }, { "volume": 1.4, "clock": 2.0, "gap": 3.0, "Berry_curvature": -0.2500000000000001, "Berry_loop_estimate": -0.24999999306177698, "metric": [ [ 0.29761904761904767, 6.300847170674081e-18 ], [ 6.300847170674081e-18, 0.05250000000000005 ] ], "inertia": [ [ 0.19841269841269846, 4.200564780449387e-18 ], [ 4.200564780449387e-18, 0.03500000000000003 ] ], "metric_determinant_minus_Berry_bound": 6.938893903907228e-18 } ], "clock_without_volume_inertia_obtained": false, "obstruction": "For a nondegenerate gapped ground state, M_VV=2*hbar^2 sum ||^2/(E_n-E_0). If M_VV vanishes, every off-diagonal volume derivative vanishes and Berry F_VT must vanish.", "pure_phase_check": "For |0>=exp[-i*b*V*q(T)]|fixed>, A=b*d[V*q(T)]. Keeping both A_V and A_T gives zero curvature; dropping A_T creates a spurious clock force.", "scope": "Restriction on a passive finite-gap adiabatic construction, not a no-go theorem for constrained Hilbert spaces, retained slow variables or gapless quantum systems." }, "response_and_scales": { "candidate": { "amplitude": -0.45, "position": 4.25, "IR_epsilon": 1e-09, "IR_cutoff": 0.01, "mu_at_zero": 1.0, "radial_stiffness_at_zero": 1.0, "numerical_force_splice_interval": [ 5e-05, 0.0002 ], "response": "mu_S=mu_previous_smoothed+B*(1/sqrt(x^2+epsilon^2)-1/sqrt(x^2+ell^2))", "IR_origin": "Ground energy of two harmonic modes minus a filled four-component fermionic doublet", "IR_spectral_coefficient": 1.00000010000001e-09, "shape_parameters_refitted": false, "metric_sector_derived_from_this_module": false, "galaxy_scale_m_s2": 1.5716639863145666e-10, "rar_rms_dex": 0.14717628635765218, "previous_rar_rms_same_grid_dex": 0.14717628635940785, "maximum_RAR_prediction_change_dex": 3.327009778786305e-10, "minimum_RAR_predicted_x": 0.07651872527871907, "rar_rows": 2693, "scale_refitted": false, "scope": "Frozen previous galaxy scale; full-sample spherical RAR proxy", "IR_parameter_acceleration_m_s2": 1.5716639863145667e-19, "IR_cutoff_acceleration_m_s2": 1.5716639863145666e-12 }, "minimum_mu_on_scan": 5.7237402431390594e-05, "minimum_radial_stiffness": 3.530039912784786e-07, "x_at_minimum_radial_stiffness": 2.0668617250744687e-07, "physical_matching": { "reference_energy_density_J_m3": 2.945125565794609e-11, "common_density_gap_product_s_minus2": 2.7483879829907663e-55, "maximum_possible_smaller_gap_common_density_rad_s": 5.24250701763075e-28, "gap_product_formula": "omega_s*omega_IR,0 = B*epsilon*(g_dark/c)^2 * (n_s/n_IR)", "examples_with_common_density": [ { "cell_length_m": 1.61625502392855e-35, "mode_density_m_minus3": 2.3684833522454663e+104, "shear_gap_rad_s": 2.3308840986373006e+44, "IR_boson_gap_at_zero_rad_s": 1.1791182515670987e-99 }, { "cell_length_m": 1e-12, "mode_density_m_minus3": 1e+36, "shear_gap_rad_s": 9.841251771634708e-25, "IR_boson_gap_at_zero_rad_s": 2.7927219491654547e-31 }, { "cell_length_m": 1e-06, "mode_density_m_minus3": 1.0000000000000001e+18, "shear_gap_rad_s": 9.841251771634708e-43, "IR_boson_gap_at_zero_rad_s": 2.792721949165455e-13 }, { "cell_length_m": 0.0001, "mode_density_m_minus3": 999999999999.9999, "shear_gap_rad_s": 9.841251771634705e-49, "IR_boson_gap_at_zero_rad_s": 2.7927219491654556e-07 }, { "cell_length_m": 0.001, "mode_density_m_minus3": 999999999.9999999, "shear_gap_rad_s": 9.841251771634707e-52, "IR_boson_gap_at_zero_rad_s": 0.0002792721949165456 } ], "cosmic_frequency_benchmark_rad_s": 1e-18, "minimum_density_ratio_for_both_gaps_above_cosmic_benchmark": 3.63849647935009e+18, "Mercury_mean_motion_rad_s": 8.266737939701787e-07, "minimum_density_ratio_for_both_gaps_above_Mercury_frequency": 2.486510514041006e+42, "scope": "Conditional matching of these specific Gaussian and spectral cells. Equal mode densities fail a common adiabatic interpretation; separate sectors can evade this bound and require their own microscopic account." } }, "all_macro_equations_derived": false, "source_sha256": "fcd0b60480958639d6a2614d85099e4f34e979db82e191d210fe067859b410de", "input_sha256": { "dark_micro_response.py": "190d43ba132c061b94a74723b97817465a149685562eb8a0f5c4a4eff7085b45", "dark_cmc_model.py": "63495c697f9521ed3f39be5785cb70964d1c45fc1bf9428210e6026955c5a50a", "dark_band_probe.py": "d7ce52e95d0b0677abbd342ef3bd28367afbe8b63bd70c68178f5d4c9be687fd", "dark_transition_tweak.py": "21ae0f598515c821ffb0df608b39c3217ce6d6d349c69ce1324e52e5610e17d8", "MICRO_MACRO_PROTOCOL.md": "8174967caf964f66e5541face5cad38be7943315385b050f59ed5a7f9e160469", "CMC_METRIC_DERIVATION.md": "1b55dcbd063c4ea0ca80aaea2517d50c049fe87ad819f9d0a7633d0f21c379a0" } }