{ "description": "Data for the Hulek-Verrill fourfold sector (main text Section 4, Eqs. (44)-(56), Proposition 4.1, Table 5; SM Section S7, Eqs. (S7.1)-(S7.13), Tables S12-S16). Fluxes in the S_6-invariant sublattice are 5-tuples g in Z^5 in the orbit basis (orbit sizes 1,6,15,6,1), G = B g; n_2(g) = g^T W5 g with W5 = diag(1,6,15,6,1); N_flux = Q(g)/2 = (1/2) g^T Gram5 g; u = Gram5 g is the flux in the period frame; the F-flatness equation on the slice is G_g(phi) = sum_a u_a A_a(phi) = 0 with A_a = Pi'_a - Pi_a bilinears in the periods (not intersection numbers). The cut is n_2 <= 751 and 1 <= N_flux <= 30; the box R is Re phi in [13/1024, 19/1024], |Im phi| <= 1/256. Classes are orbits of the residual monodromy (M_d acting on g) that meet the cut. Decimal strings are the stored values (Python repr of the stored double, or the stored decimal string); the Krawczyk certificates themselves (period evaluation with tail bounds) are data here and are not re-run by verify_hv4.py.", "fourfold": { "equation": "(X_1+...+X_6)(phi_1/X_1+...+phi_6/X_6) = 1 in T^5, diagonal slice phi_i = phi", "hodge_numbers": { "h31": 6, "h21": 0, "h11": 106, "h22": 492, "chi": 720 }, "chi_over_24": 30, "horizontal_lattice": { "rank": 29, "det": "2^16 * 3^6", "det_value": 47775744, "signature": [ 17, 12 ] } }, "invariant_lattice": { "orbit_sizes": [ 1, 6, 15, 6, 1 ], "W5_diag": [ 1, 6, 15, 6, 1 ], "Gram5": [ [ 2, 0, 30, 0, 1 ], [ 0, -60, 0, 6, 0 ], [ 30, 0, 180, 0, 0 ], [ 0, 6, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ] ], "det_Gram5": 6480, "signature_Gram5": [ 3, 2 ], "N_flux_polynomial": "g1**2 + 30*g1*g3 + g1*g5 - 30*g2**2 + 6*g2*g4 + 90*g3**2", "u_of_g": "[2*g1 + 30*g3 + g5, -60*g2 + 6*g4, 30*g1 + 180*g3, 6*g2, g1]", "n2_polynomial": "g1**2 + 6*g2**2 + 15*g3**2 + 6*g4**2 + g5**2", "on_cone_N_flux": "g1*(g1 + g5)", "family": { "g": "(N,1,0,0,3N)", "N_flux": "4*N**2 - 30", "u": "[5*N, -60, 30*N, 6, N]" }, "M_d": [ [ 1, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 1, 2, 1, 0, 0 ], [ -20, -60, -60, 1, 0 ], [ 30, 120, 180, -6, 1 ] ], "M_d_note": "residual monodromy generator on g: M_d^T Gram5 M_d = Gram5, (M_d - 1)^5 = 0 != (M_d - 1)^4" }, "picard_fuchs_L5": { "form": "L5 = sum_{j=0}^{3} phi^j Q_j(theta), theta = phi d/dphi; Q_j given as coefficient lists in theta (c_0..c_5)", "Q_theta": [ [ 0, 0, 0, 0, 0, 1 ], [ -6, -40, -112, -168, -140, -56 ], [ 1020, 4628, 8548, 8076, 3920, 784 ], [ -13824, -50688, -72576, -50688, -17280, -2304 ] ], "symbol": [ 1, -56, 784, -2304 ], "symbol_factors": [ "1-4*phi", "1-16*phi", "1-36*phi" ], "singular_points": [ "0", "1/36", "1/16", "1/4", "inf" ], "local_exponents": { "0": [ "0", "0", "0", "0", "0" ], "1/36": [ "0", "1", "3/2", "2", "3" ], "1/16": [ "0", "1", "3/2", "2", "3" ], "1/4": [ "0", "1", "3/2", "2", "3" ], "inf": [ "1", "1", "3/2", "2", "2" ] }, "apparent_singularities": "NONE (leading D-form coeff = phi^5 * symbol)", "holomorphic_series_c0_c40": [ "1", "6", "66", "996", "18306", "384156", "8848236", "218040696", "5651108226", "152254667436", "4229523740916", "120430899525096", "3499628148747756", "103446306284890536", "3102500089343886696", "94219208840385966096", "2892652835496484004226", "89662253086458906345036", "2802887038905690746642916", "88285112311736445352115976", "2799753247995266147347843956", "89333847417312523954034046936", "2866354076060681446555954224696", "92437346938232317505833680390576", "2994900985489170290757597878329836", "97447591912171214828256883106131656", "3183236280393766829982694868375178936", "104363235523568961506091130323484381296", "3433150098404204615367747409340231222376", "113292804630449390414523077344383926930736", "3749593498015432616816199839014231473497136", "124438685799375152191173444580928148226331616", "4140386890068440610567742586524925029208842626", "138093273693648760389487232466632008487098268556", "4616237101589363619424107315336055982588227061316", "154643238715090776461769052462229535066917009716296", "5190976376129449809203046819701042142643950898251876", "174580102876035759598849980550598438577853599389481336", "5881987600620241901798300190848926683935796208144613016", "198516508257702381370914801510101357882654405176272740976", "6710808924308403895947675999397468718194852387440533897076" ], "series_note": "c_n = sum over n_1+...+n_6 = n of (n!/(n_1! ... n_6!))^2", "attribution": "Jockers-Kotlewski-Kuusela arXiv:2312.07611; all-orders annihilation of the fundamental period is taken from there (hypothesis (O) of Proposition 4.1)" }, "cut_and_box": { "mu_lo_dyadic_interval": { "lower": "5756447315870245/72057594037927936", "upper": "5756447891589797/72057594037927936" }, "sixty_over_mu_lo_interval": [ "751.0631076140321", "751.0631827301049" ], "n2_cut": 751, "N_flux_range": [ 1, 30 ], "component_caps": [ 27, 11, 7, 11, 27 ], "box_R": { "Re_phi": [ "52/4096", "76/4096" ], "Im_phi": [ "-16/4096", "16/4096" ] }, "box_R_note": "R contains 1/64 = 16/1024, the position of the (5,0,1) vacuum of arXiv:2404.12422", "mu_lo_note": "mu_lo is a certified lower bound, on a 21-point grid spanning R, for the smallest eigenvalue of the Hodge norm relative to n_2 on invariant fluxes; at an F-flat point Q(G) = ||G||^2_Hodge >= mu_lo n_2, and Q <= 60 = 2 chi/24, so n_2 <= floor(60/mu_lo) = 751 on that grid; the cut is a hypothesis of Proposition 4.1" }, "frame": { "note": "Frame map to the rational period basis Pi_th of the thesis cited in SM Section S7.3 (Piribauer 2026): Pi_th = C p with p the period vector of the frame of SM Section S7.1; C is integral with det 6 and C^T Sigma_th C = Gram5, so the invariant lattice is an index-6 sublattice; flux components transform by the same matrix (T = C). M_th are the printed monodromies about 0, 1/36, 1/16, 1/4, infinity in that basis; M_g their pull-backs to g (integral, Gram5-orthogonal); at each conifold point M_g is the reflection x -> x - (2/Q(g0)) (x^T Gram5 g0) g0 in the vanishing class g0.", "C": [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 6, 0, 0 ], [ 0, -5, 0, 1, 0 ], [ 1, 0, 30, 0, 1 ] ], "det_C": 6, "Sigma_th": [ [ 0, 0, 0, 0, 1 ], [ 0, 0, 0, 6, 0 ], [ 0, 0, 5, 0, 0 ], [ 0, 6, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ] ], "det_Sigma_th": 180, "M_th": { "0": [ [ 1, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 6, 12, 1, 0, 0 ], [ -25, -60, -10, 1, 0 ], [ 60, 150, 30, -6, 1 ] ], "1/36": [ [ 0, 0, 0, 0, -1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ -1, 0, 0, 0, 0 ] ], "1/16": [ [ -5, 30, 0, -6, -6 ], [ -1, 6, 0, -1, -1 ], [ 0, 0, 1, 0, 0 ], [ 5, -25, 0, 6, 5 ], [ -6, 30, 0, -6, -5 ] ], "1/4": [ [ -44, 150, -30, -30, -15 ], [ -15, 51, -10, -10, -5 ], [ -18, 60, -11, -12, -6 ], [ 75, -250, 50, 51, 25 ], [ -135, 450, -90, -90, -44 ] ], "inf": [ [ -99, 300, -90, -36, -10 ], [ -45, 136, -40, -15, -4 ], [ -84, 252, -71, -24, -6 ], [ 245, -735, 210, 76, 20 ], [ -490, 1470, -420, -150, -39 ] ] }, "M_g": { "0": [ [ 1, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 1, 2, 1, 0, 0 ], [ -20, -60, -60, 1, 0 ], [ 30, 120, 180, -6, 1 ] ], "1/36": [ [ -1, 0, -30, 0, -1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ], "1/16": [ [ -11, 60, -180, -6, -6 ], [ -2, 11, -30, -1, -1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ], "1/4": [ [ -59, 300, -630, -30, -15 ], [ -20, 101, -210, -10, -5 ], [ -4, 20, -41, -2, -1 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ], "inf": [ [ -109, 480, -840, -36, -10 ], [ -49, 211, -360, -15, -4 ], [ -15, 62, -101, -4, -1 ], [ 20, -60, 60, 1, 0 ], [ 30, -120, 180, 6, 1 ] ] }, "vanishing_classes": [ { "point": "1/36", "g0": [ 1, 0, 0, 0, 0 ], "Q": 2, "N_flux": 1, "two_over_Q_Gram5_g0": [ 2, 0, 30, 0, 1 ] }, { "point": "1/16", "g0": [ 6, 1, 0, 0, 0 ], "Q": 12, "N_flux": 6, "two_over_Q_Gram5_g0": [ 2, -10, 30, 1, 1 ] }, { "point": "1/4", "g0": [ 15, 5, 1, 0, 0 ], "Q": 30, "N_flux": 15, "two_over_Q_Gram5_g0": [ 4, -20, 42, 2, 1 ] } ] }, "counts": { "unit_A_signed_flux_vectors": { "ellipsoid_points_n2_le_751": 3509207, "in_cut": 59360, "offcone_Nflux_le_30": 55516, "offcone_Nflux_le_3": 3872, "oncone_Nflux_le_3": 230, "Nflux_le_3": 4102, "offcone_Nflux_le_30_g1_abs_le_2": 12082, "per_Nflux_1_2_3": [ 626, 1996, 1480 ] }, "unit_B_classes_under_Md": { "in_cut": 58388, "per_Nflux_1_to_30": { "1": 626, "2": 1994, "3": 1428, "4": 1946, "5": 826, "6": 4502, "7": 650, "8": 1994, "9": 1814, "10": 2508, "11": 656, "12": 4344, "13": 728, "14": 1854, "15": 1720, "16": 1978, "17": 676, "18": 5116, "19": 654, "20": 2248, "21": 1414, "22": 1830, "23": 622, "24": 4530, "25": 762, "26": 1960, "27": 1734, "28": 1842, "29": 582, "30": 4850 }, "Nflux_le_3": 4048, "Nflux_le_3_oncone_offcone": [ 230, 3818 ], "with_vacuum_in_R": 28, "without_vacuum_in_R": 58360 } }, "on_cone_Nflux_le_3": { "n": 230, "description": "(+-1,0,0,g4,{0,+-1,+-2}) with tied signs, (+-2,0,0,g4,-+1), (+-3,0,0,g4,-+2), |g4| <= 11: 2*23*3 + 2*23 + 2*23 = 230; all have r = g5/g1 <= 2; 228 are excluded by the field-sign table, the two on the r = 2 ray by the 48-block first-order certificate" }, "field_signs_on_R": [ { "component": "Re A_1", "bound": ">= 3231" }, { "component": "Im A_2", "bound": ">= 820.7" }, { "component": "Re A_3", "bound": "<= -125.5" }, { "component": "Im A_4", "bound": "<= -2022.6" }, { "component": "Re A_5", "bound": "<= -8619.2" } ], "scan_margins": { "worst_margin_Nflux_le_3_offcone": { "margin": "22794.4498", "g": [ -3, -1, 0, -4, 0 ] }, "worst_margin_Nflux_le_30_offcone": { "margin": "5063.7822", "g": [ -2, 1, 0, 9, -1 ] } }, "r2_ray": { "u": [ 4, 0, 30, 0, 1 ], "statement": "the r = 2 ray (u = Gram5 (1,0,0,0,2)) has no F-flat point in R: first-order exclusion on all 48 blocks", "min_margin": "675.9", "max_margin": "7604.3", "precision_bits": 220, "n_blocks": 48 }, "r52_edge_zero": { "g": [ 2, 0, 0, 0, 5 ], "u": [ 9, 0, 60, 0, 2 ], "Re_z_interval": [ "0.018831602234793067654712", "0.018831602234793067654849" ], "Im_z_abs_bound": "7.77e-23", "z_minus_box_top_interval": [ "2.7691473479306765e-4", "2.7691473479306766e-4" ], "statement": "the r = 5/2 zero of the family lies above R" }, "vacua_in_R": { "n_vacua": 28, "n_positions": 12, "per_Nflux": { "4": 2, "5": 2, "16": 6, "18": 6, "20": 6, "22": 6 }, "max_ball_radius": "2.1492396836395457e-22", "min_dist_to_1_64": "0.0002848050613564616", "absW_lower_bound_range": [ "4.071993111705892", "13.268674232095478" ], "rows": [ { "g": [ -2, 0, 0, -1, -9 ], "N_flux": 22, "Q": 44, "n2": 91, "Re_z": "0.013643010725768541", "Im_z": "-0.0023215845083362947", "ball_radius": "2.1492396836395457e-22", "absW_lower_bound": "13.268674232095478", "dist_to_1_64": "0.0030525458411816875", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, -1, -8 ], "N_flux": 20, "Q": 40, "n2": 74, "Re_z": "0.014741063517051909", "Im_z": "-0.0024940074138566797", "ball_radius": "1.8762747791637372e-22", "absW_lower_bound": "11.853051935995653", "dist_to_1_64": "0.0026460190260575844", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, -1, -7 ], "N_flux": 18, "Q": 36, "n2": 59, "Re_z": "0.01598555308960638", "Im_z": "-0.002673670620765638", "ball_radius": "1.674040488457888e-22", "absW_lower_bound": "10.472737850780485", "dist_to_1_64": "0.0026978719611519776", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, -1, -6 ], "N_flux": 16, "Q": 32, "n2": 46, "Re_z": "0.01740225898219134", "Im_z": "-0.0028531301774222684", "ball_radius": "1.4734502884025893e-22", "absW_lower_bound": "9.13667391851169", "dist_to_1_64": "0.0033613987117117514", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 0, -9 ], "N_flux": 22, "Q": 44, "n2": 85, "Re_z": "0.013622853029954215", "Im_z": "2.230580304835705e-36", "ball_radius": "1.1638152498286067e-22", "absW_lower_bound": "12.550705658591896", "dist_to_1_64": "0.0020021469700457852", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 0, -8 ], "N_flux": 20, "Q": 40, "n2": 68, "Re_z": "0.014697869995381866", "Im_z": "7.792420846878926e-37", "ball_radius": "1.0133255910727525e-22", "absW_lower_bound": "11.063268777036747", "dist_to_1_64": "0.0009271300046181343", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 0, -7 ], "N_flux": 18, "Q": 36, "n2": 53, "Re_z": "0.01590980506135646", "Im_z": "-3.5458191343585443e-38", "ball_radius": "9.168508536900844e-23", "absW_lower_bound": "9.593721932423692", "dist_to_1_64": "0.0002848050613564616", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 0, -6 ], "N_flux": 16, "Q": 32, "n2": 40, "Re_z": "0.0172802004854237", "Im_z": "-9.493835552934273e-38", "ball_radius": "8.14281235944792e-23", "absW_lower_bound": "8.143986223411783", "dist_to_1_64": "0.0016552004854237005", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 1, -9 ], "N_flux": 22, "Q": 44, "n2": 91, "Re_z": "0.013643010725768541", "Im_z": "0.0023215845083362947", "ball_radius": "2.1492396836395457e-22", "absW_lower_bound": "13.268674232095478", "dist_to_1_64": "0.0030525458411816875", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 1, -8 ], "N_flux": 20, "Q": 40, "n2": 74, "Re_z": "0.014741063517051909", "Im_z": "0.0024940074138566797", "ball_radius": "1.8762747791637372e-22", "absW_lower_bound": "11.853051935995653", "dist_to_1_64": "0.0026460190260575844", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 1, -7 ], "N_flux": 18, "Q": 36, "n2": 59, "Re_z": "0.01598555308960638", "Im_z": "0.002673670620765638", "ball_radius": "1.674040488457888e-22", "absW_lower_bound": "10.472737850780485", "dist_to_1_64": "0.0026978719611519776", "W_nonzero_certified": true }, { "g": [ -2, 0, 0, 1, -6 ], "N_flux": 16, "Q": 32, "n2": 46, "Re_z": "0.01740225898219134", "Im_z": "0.0028531301774222684", "ball_radius": "1.4734502884025893e-22", "absW_lower_bound": "9.13667391851169", "dist_to_1_64": "0.0033613987117117514", "W_nonzero_certified": true }, { "g": [ -1, 0, 0, 0, -4 ], "N_flux": 5, "Q": 10, "n2": 17, "Re_z": "0.014697869995381866", "Im_z": "7.792420846878926e-37", "ball_radius": "1.0133255910727525e-22", "absW_lower_bound": "5.531634388518373", "dist_to_1_64": "0.0009271300046181343", "W_nonzero_certified": true }, { "g": [ -1, 0, 0, 0, -3 ], "N_flux": 4, "Q": 8, "n2": 10, "Re_z": "0.0172802004854237", "Im_z": "-9.493835552934273e-38", "ball_radius": "8.14281235944792e-23", "absW_lower_bound": "4.071993111705892", "dist_to_1_64": "0.0016552004854237005", "W_nonzero_certified": true }, { "g": [ 1, 0, 0, 0, 3 ], "N_flux": 4, "Q": 8, "n2": 10, "Re_z": "0.0172802004854237", "Im_z": "-9.493835552934273e-38", "ball_radius": "8.14281235944792e-23", "absW_lower_bound": "4.071993111705892", "dist_to_1_64": "0.0016552004854237005", "W_nonzero_certified": true }, { "g": [ 1, 0, 0, 0, 4 ], "N_flux": 5, "Q": 10, "n2": 17, "Re_z": "0.014697869995381866", "Im_z": "7.792420846878926e-37", "ball_radius": "1.0133255910727525e-22", "absW_lower_bound": "5.531634388518373", "dist_to_1_64": "0.0009271300046181343", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, -1, 6 ], "N_flux": 16, "Q": 32, "n2": 46, "Re_z": "0.01740225898219134", "Im_z": "0.0028531301774222684", "ball_radius": "1.4734502884025893e-22", "absW_lower_bound": "9.13667391851169", "dist_to_1_64": "0.0033613987117117514", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, -1, 7 ], "N_flux": 18, "Q": 36, "n2": 59, "Re_z": "0.01598555308960638", "Im_z": "0.002673670620765638", "ball_radius": "1.674040488457888e-22", "absW_lower_bound": "10.472737850780485", "dist_to_1_64": "0.0026978719611519776", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, -1, 8 ], "N_flux": 20, "Q": 40, "n2": 74, "Re_z": "0.014741063517051909", "Im_z": "0.0024940074138566797", "ball_radius": "1.8762747791637372e-22", "absW_lower_bound": "11.853051935995653", "dist_to_1_64": "0.0026460190260575844", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, -1, 9 ], "N_flux": 22, "Q": 44, "n2": 91, "Re_z": "0.013643010725768541", "Im_z": "0.0023215845083362947", "ball_radius": "2.1492396836395457e-22", "absW_lower_bound": "13.268674232095478", "dist_to_1_64": "0.0030525458411816875", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 0, 6 ], "N_flux": 16, "Q": 32, "n2": 40, "Re_z": "0.0172802004854237", "Im_z": "-9.493835552934273e-38", "ball_radius": "8.14281235944792e-23", "absW_lower_bound": "8.143986223411783", "dist_to_1_64": "0.0016552004854237005", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 0, 7 ], "N_flux": 18, "Q": 36, "n2": 53, "Re_z": "0.01590980506135646", "Im_z": "-3.5458191343585443e-38", "ball_radius": "9.168508536900844e-23", "absW_lower_bound": "9.593721932423692", "dist_to_1_64": "0.0002848050613564616", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 0, 8 ], "N_flux": 20, "Q": 40, "n2": 68, "Re_z": "0.014697869995381866", "Im_z": "7.792420846878926e-37", "ball_radius": "1.0133255910727525e-22", "absW_lower_bound": "11.063268777036747", "dist_to_1_64": "0.0009271300046181343", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 0, 9 ], "N_flux": 22, "Q": 44, "n2": 85, "Re_z": "0.013622853029954215", "Im_z": "2.230580304835705e-36", "ball_radius": "1.1638152498286067e-22", "absW_lower_bound": "12.550705658591896", "dist_to_1_64": "0.0020021469700457852", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 1, 6 ], "N_flux": 16, "Q": 32, "n2": 46, "Re_z": "0.01740225898219134", "Im_z": "-0.0028531301774222684", "ball_radius": "1.4734502884025893e-22", "absW_lower_bound": "9.13667391851169", "dist_to_1_64": "0.0033613987117117514", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 1, 7 ], "N_flux": 18, "Q": 36, "n2": 59, "Re_z": "0.01598555308960638", "Im_z": "-0.002673670620765638", "ball_radius": "1.674040488457888e-22", "absW_lower_bound": "10.472737850780485", "dist_to_1_64": "0.0026978719611519776", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 1, 8 ], "N_flux": 20, "Q": 40, "n2": 74, "Re_z": "0.014741063517051909", "Im_z": "-0.0024940074138566797", "ball_radius": "1.8762747791637372e-22", "absW_lower_bound": "11.853051935995653", "dist_to_1_64": "0.0026460190260575844", "W_nonzero_certified": true }, { "g": [ 2, 0, 0, 1, 9 ], "N_flux": 22, "Q": 44, "n2": 91, "Re_z": "0.013643010725768541", "Im_z": "-0.0023215845083362947", "ball_radius": "2.1492396836395457e-22", "absW_lower_bound": "13.268674232095478", "dist_to_1_64": "0.0030525458411816875", "W_nonzero_certified": true } ], "positions": [ { "Re_z": 0.01362285303, "Im_z": 0.0, "fluxes": [ [ -2, 0, 0, 0, -9 ], [ 2, 0, 0, 0, 9 ] ] }, { "Re_z": 0.013643010726, "Im_z": -0.002321584508, "fluxes": [ [ -2, 0, 0, -1, -9 ], [ 2, 0, 0, 1, 9 ] ] }, { "Re_z": 0.013643010726, "Im_z": 0.002321584508, "fluxes": [ [ -2, 0, 0, 1, -9 ], [ 2, 0, 0, -1, 9 ] ] }, { "Re_z": 0.014697869995, "Im_z": 0.0, "fluxes": [ [ -2, 0, 0, 0, -8 ], [ -1, 0, 0, 0, -4 ], [ 1, 0, 0, 0, 4 ], [ 2, 0, 0, 0, 8 ] ] }, { "Re_z": 0.014741063517, "Im_z": -0.002494007414, "fluxes": [ [ -2, 0, 0, -1, -8 ], [ 2, 0, 0, 1, 8 ] ] }, { "Re_z": 0.014741063517, "Im_z": 0.002494007414, "fluxes": [ [ -2, 0, 0, 1, -8 ], [ 2, 0, 0, -1, 8 ] ] }, { "Re_z": 0.015909805061, "Im_z": 0.0, "fluxes": [ [ -2, 0, 0, 0, -7 ], [ 2, 0, 0, 0, 7 ] ] }, { "Re_z": 0.01598555309, "Im_z": -0.002673670621, "fluxes": [ [ -2, 0, 0, -1, -7 ], [ 2, 0, 0, 1, 7 ] ] }, { "Re_z": 0.01598555309, "Im_z": 0.002673670621, "fluxes": [ [ -2, 0, 0, 1, -7 ], [ 2, 0, 0, -1, 7 ] ] }, { "Re_z": 0.017280200485, "Im_z": 0.0, "fluxes": [ [ -2, 0, 0, 0, -6 ], [ -1, 0, 0, 0, -3 ], [ 1, 0, 0, 0, 3 ], [ 2, 0, 0, 0, 6 ] ] }, { "Re_z": 0.017402258982, "Im_z": -0.002853130177, "fluxes": [ [ -2, 0, 0, -1, -6 ], [ 2, 0, 0, 1, 6 ] ] }, { "Re_z": 0.017402258982, "Im_z": 0.002853130177, "fluxes": [ [ -2, 0, 0, 1, -6 ], [ 2, 0, 0, -1, 6 ] ] } ] }, "certified_vacua": [ { "name": "charge 4, inside R (Proposition 4.1)", "g": [ 1, 0, 0, 0, 3 ], "u": [ 5, 0, 30, 0, 1 ], "N_flux": 4, "n2": 10, "Re_z_mid": "0.01728020048542370039409738415466206210358", "Re_z_rad": "6.706196830553663e-23", "Im_z_mid": "0", "Im_z_rad": "7.85295607930744e-23", "Im_enclosure_contains_0": true, "krawczyk_first_radius": "1/10000000000000", "uniqueness_box_radius": "1/15625", "absW_lower_bound": "4.071993111705892", "W_nonzero": true, "e_minus_K": "95.5736478647722", "z_minus_box_top": "-0.0012744870145762995", "above_R_certified": false, "dist_to_1_64": "0.0016552004854237005" }, { "name": "charge 3, above R", "g": [ 1, 0, 0, 0, 2 ], "u": [ 4, 0, 30, 0, 1 ], "N_flux": 3, "n2": 5, "Re_z_mid": "0.02058287610565320248969098182774725634291", "Re_z_rad": "5.633536193817335e-23", "Im_z_mid": "0", "Im_z_rad": "6.273169879041021e-23", "Im_enclosure_contains_0": true, "krawczyk_first_radius": "1/10000000000000", "uniqueness_box_radius": "1/15625", "absW_lower_bound": "2.6566902537836627", "W_nonzero": true, "e_minus_K": "81.44738082664485", "z_minus_box_top": "0.0020281886056532023", "above_R_certified": true, "dist_to_1_64": "0.004957876105653202" }, { "name": "charge 2, above R", "g": [ 1, 0, 0, 0, 1 ], "u": [ 3, 0, 30, 0, 1 ], "N_flux": 2, "n2": 2, "Re_z_mid": "0.02463659201867469650402740477419252692710", "Re_z_rad": "5.856732031415693e-23", "Im_z_mid": "0", "Im_z_rad": "6.137532849035242e-23", "Im_enclosure_contains_0": true, "krawczyk_first_radius": "1/10000000000000", "uniqueness_box_radius": "1/15625", "absW_lower_bound": "1.2969535535636363", "W_nonzero": true, "e_minus_K": "68.61402034602041", "z_minus_box_top": "0.006081904518674697", "above_R_certified": true, "dist_to_1_64": "0.009011592018674696" } ], "charge_1": { "g": [ 1, 0, 0, 0, 0 ], "u": [ 2, 0, 30, 0, 1 ], "N_flux": 1, "n2": 1, "statement": "inside R: excluded (one of the 228 field-table exclusions); outside R on the slice: undecided (a floating-point sign scan finds no crossing, which is an observation, not a certificate)" }, "special_points": { "conifold_points": [ "1/36", "1/16", "1/4" ], "gvdh_vacuum_positions": [ "1/16", "(3sqrt3-5)/4", "-(5+3sqrt3)/4", "1/4-sqrt3/8", "1/4+sqrt3/8", "1/(22+9sqrt6)", "1/4", "-(22+9sqrt6)/2", "-1/2", "1/64" ], "n_catalogue": 11, "in_R": [ "1/64" ], "note": "the comparison catalogue: the three conifold points of the slice and the diagonal positions of the ten exact vacua of Grimm-van de Heisteeg (arXiv:2404.12422, Table 5.1), two of which are conifold points; eleven distinct points, of which only 1/64 = 16/1024 lies in R. The slice has a twelfth special point with exact algebraic phi outside the catalogue, phi = 1 (Jockers-Kotlewski-Kuusela, arXiv:2312.07611, Secs. 3.4 and 4), also outside R." }, "gvdh_table_5_1_crosscheck": { "rows": [ { "abc": [ 1, 0, 1 ], "Lhat": 1, "N_flux": 6, "n2": 6, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 2, 0, 1 ], "Lhat": 2, "N_flux": 12, "n2": 30, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 7, 1, 2 ], "Lhat": 2, "N_flux": 12, "n2": 468, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 3, 0, 1 ], "Lhat": 3, "N_flux": 18, "n2": 78, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 5, 1, 3 ], "Lhat": 3, "N_flux": 18, "n2": 198, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 4, 0, 1 ], "Lhat": 4, "N_flux": 24, "n2": 150, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 4, 1, 4 ], "Lhat": 4, "N_flux": 24, "n2": 120, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 13, 2, 4 ], "Lhat": 4, "N_flux": 24, "n2": 1596, "in_invariant_sublattice": false, "n2_le_751": false, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 8, 1, 2 ], "Lhat": 4, "N_flux": 24, "n2": 624, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false }, { "abc": [ 5, 0, 1 ], "Lhat": 5, "N_flux": 30, "n2": 246, "in_invariant_sublattice": false, "n2_le_751": true, "Nflux_in_1_30": true, "admissible_in_DIAG_SLICE_census": false } ], "misses": 10, "finding": "10/10 MISS on invariant-sublattice membership: Table 5.1 fluxes are S6-BREAKING by construction (ansatz (5.73), sum n_I=0, rank(n_IJ)=6 required for full stabilization (5.83)); an S6-invariant flux has constant G^I, and constant n with sum 0 forces n=0 => the invariant sublattice meets the GvdH family only at G=0. NOT a bound error: convention/scope finding \u2014 the DIAG-SLICE invariant-flux census and GvdH Table 5.1 probe DISJOINT flux sectors. Bound sanity where applicable: 9/10 vacua satisfy n2<=751; (13,2,4) has n2=1596>751 (moot: N2_ISD-DIAG floor is only proven on the invariant sector); all 10 satisfy 0