Hulek-Verrill fourfold, S_6-invariant sector (Section 4; SM Section S7). Hodge numbers h31=6 h21=0 h11=106 h22=492, chi=720, chi/24=30 Gram5 (det 6480, signature (3, 2)): 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 W5 = diag(1, 6, 15, 6, 1); N_flux = Q(g)/2 = g1**2 + 30*g1*g3 + g1*g5 - 30*g2**2 + 6*g2*g4 + 90*g3**2 u = Gram5 g = [2*g1 + 30*g3 + g5, -60*g2 + 6*g4, 30*g1 + 180*g3, 6*g2, g1] on the cone g2 = g3 = 0: N_flux = g1*(g1 + g5) M_d (residual monodromy on g): 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 L5 = sum_j phi^j Q_j(theta), Q_j (coefficients of theta^0..theta^5): Q_0 = [0, 0, 0, 0, 0, 1] Q_1 = [-6, -40, -112, -168, -140, -56] Q_2 = [1020, 4628, 8548, 8076, 3920, 784] Q_3 = [-13824, -50688, -72576, -50688, -17280, -2304] symbol [1, -56, 784, -2304] = 1-4*phi * 1-16*phi * 1-36*phi; singular points ['0', '1/36', '1/16', '1/4', 'inf'] holomorphic series c_0.. = 1, 6, 66, 996, 18306, 384156, 8848236, 218040696, 5651108226, ... Cut: n2 <= 751 (mu_lo in [5756447315870245/72057594037927936, 5756447891589797/72057594037927936], 60/mu_lo in ['751.0631076140321', '751.0631827301049']), 1 <= N_flux <= 30, caps [27, 11, 7, 11, 27]; box R: Re phi in ['52/4096', '76/4096'], |Im phi| <= 16/4096 Frame map C (det 6), C^T Sigma_th C = Gram5, 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]]: 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 vanishing class at 1/36: g0 = [1, 0, 0, 0, 0], Q = 2, N_flux = 1, (2/Q) Gram5 g0 = [2, 0, 30, 0, 1] vanishing class at 1/16: g0 = [6, 1, 0, 0, 0], Q = 12, N_flux = 6, (2/Q) Gram5 g0 = [2, -10, 30, 1, 1] vanishing class at 1/4: g0 = [15, 5, 1, 0, 0], Q = 30, N_flux = 15, (2/Q) Gram5 g0 = [4, -20, 42, 2, 1] Counts (signed flux vectors / classes under ): n2 <= 751: 3509207 in the cut: 59360 / 58388 N_flux = 1,2,3: [626, 1996, 1480] / [626, 1994, 1428] N_flux <= 3: 4102 / 4048 (off cone 3872 / 3818, on cone 230 / 230) N_flux <= 30 off cone: 55516; classes with / without a vacuum in R: 28 / 58360 The 28 vacua in R (class rep g, N_flux, n2, Re z*, Im z*, ball radius, |W| >=, |z*-1/64|): [-2, 0, 0, -1, -9] 22 91 0.013643010725768541 -0.0023215845083362947 2.1492396836395457e-22 13.268674232095478 0.0030525458411816875 [-2, 0, 0, -1, -8] 20 74 0.014741063517051909 -0.0024940074138566797 1.8762747791637372e-22 11.853051935995653 0.0026460190260575844 [-2, 0, 0, -1, -7] 18 59 0.01598555308960638 -0.002673670620765638 1.674040488457888e-22 10.472737850780485 0.0026978719611519776 [-2, 0, 0, -1, -6] 16 46 0.01740225898219134 -0.0028531301774222684 1.4734502884025893e-22 9.13667391851169 0.0033613987117117514 [-2, 0, 0, 0, -9] 22 85 0.013622853029954215 2.230580304835705e-36 1.1638152498286067e-22 12.550705658591896 0.0020021469700457852 [-2, 0, 0, 0, -8] 20 68 0.014697869995381866 7.792420846878926e-37 1.0133255910727525e-22 11.063268777036747 0.0009271300046181343 [-2, 0, 0, 0, -7] 18 53 0.01590980506135646 -3.5458191343585443e-38 9.168508536900844e-23 9.593721932423692 0.0002848050613564616 [-2, 0, 0, 0, -6] 16 40 0.0172802004854237 -9.493835552934273e-38 8.14281235944792e-23 8.143986223411783 0.0016552004854237005 [-2, 0, 0, 1, -9] 22 91 0.013643010725768541 0.0023215845083362947 2.1492396836395457e-22 13.268674232095478 0.0030525458411816875 [-2, 0, 0, 1, -8] 20 74 0.014741063517051909 0.0024940074138566797 1.8762747791637372e-22 11.853051935995653 0.0026460190260575844 [-2, 0, 0, 1, -7] 18 59 0.01598555308960638 0.002673670620765638 1.674040488457888e-22 10.472737850780485 0.0026978719611519776 [-2, 0, 0, 1, -6] 16 46 0.01740225898219134 0.0028531301774222684 1.4734502884025893e-22 9.13667391851169 0.0033613987117117514 [-1, 0, 0, 0, -4] 5 17 0.014697869995381866 7.792420846878926e-37 1.0133255910727525e-22 5.531634388518373 0.0009271300046181343 [-1, 0, 0, 0, -3] 4 10 0.0172802004854237 -9.493835552934273e-38 8.14281235944792e-23 4.071993111705892 0.0016552004854237005 [1, 0, 0, 0, 3] 4 10 0.0172802004854237 -9.493835552934273e-38 8.14281235944792e-23 4.071993111705892 0.0016552004854237005 [1, 0, 0, 0, 4] 5 17 0.014697869995381866 7.792420846878926e-37 1.0133255910727525e-22 5.531634388518373 0.0009271300046181343 [2, 0, 0, -1, 6] 16 46 0.01740225898219134 0.0028531301774222684 1.4734502884025893e-22 9.13667391851169 0.0033613987117117514 [2, 0, 0, -1, 7] 18 59 0.01598555308960638 0.002673670620765638 1.674040488457888e-22 10.472737850780485 0.0026978719611519776 [2, 0, 0, -1, 8] 20 74 0.014741063517051909 0.0024940074138566797 1.8762747791637372e-22 11.853051935995653 0.0026460190260575844 [2, 0, 0, -1, 9] 22 91 0.013643010725768541 0.0023215845083362947 2.1492396836395457e-22 13.268674232095478 0.0030525458411816875 [2, 0, 0, 0, 6] 16 40 0.0172802004854237 -9.493835552934273e-38 8.14281235944792e-23 8.143986223411783 0.0016552004854237005 [2, 0, 0, 0, 7] 18 53 0.01590980506135646 -3.5458191343585443e-38 9.168508536900844e-23 9.593721932423692 0.0002848050613564616 [2, 0, 0, 0, 8] 20 68 0.014697869995381866 7.792420846878926e-37 1.0133255910727525e-22 11.063268777036747 0.0009271300046181343 [2, 0, 0, 0, 9] 22 85 0.013622853029954215 2.230580304835705e-36 1.1638152498286067e-22 12.550705658591896 0.0020021469700457852 [2, 0, 0, 1, 6] 16 46 0.01740225898219134 -0.0028531301774222684 1.4734502884025893e-22 9.13667391851169 0.0033613987117117514 [2, 0, 0, 1, 7] 18 59 0.01598555308960638 -0.002673670620765638 1.674040488457888e-22 10.472737850780485 0.0026978719611519776 [2, 0, 0, 1, 8] 20 74 0.014741063517051909 -0.0024940074138566797 1.8762747791637372e-22 11.853051935995653 0.0026460190260575844 [2, 0, 0, 1, 9] 22 91 0.013643010725768541 -0.0023215845083362947 2.1492396836395457e-22 13.268674232095478 0.0030525458411816875 Certified vacua (Table 5 / Table S16): 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* = 0.01728020048542370039409738415466206210358 +- 6.706196830553663e-23, |Im z*| <= 7.85295607930744e-23, |W| >= 4.071993111705892, e^-K = 95.5736478647722, z* - 19/1024 = -0.0012744870145762995 charge 3, above R: g = [1, 0, 0, 0, 2], u = [4, 0, 30, 0, 1], N_flux = 3, n2 = 5, Re z* = 0.02058287610565320248969098182774725634291 +- 5.633536193817335e-23, |Im z*| <= 6.273169879041021e-23, |W| >= 2.6566902537836627, e^-K = 81.44738082664485, z* - 19/1024 = 0.0020281886056532023 charge 2, above R: g = [1, 0, 0, 0, 1], u = [3, 0, 30, 0, 1], N_flux = 2, n2 = 2, Re z* = 0.02463659201867469650402740477419252692710 +- 5.856732031415693e-23, |Im z*| <= 6.137532849035242e-23, |W| >= 1.2969535535636363, e^-K = 68.61402034602041, z* - 19/1024 = 0.006081904518674697 Special points: conifolds ['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']; in R: ['1/64']; twelfth point phi = 1 (outside R). Field signs on R: Re A_1 >= 3231, Im A_2 >= 820.7, Re A_3 <= -125.5, Im A_4 <= -2022.6, Re A_5 <= -8619.2 r = 2 ray: min margin 675.9 over 48 blocks; r = 5/2 edge zero: Re z* in ['0.018831602234793067654712', '0.018831602234793067654849'], z* - 19/1024 in ['2.7691473479306765e-4', '2.7691473479306766e-4'] Charge 1: 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)