RUN LOG -- verification scripts for J. Thrussell, 'Closed three-manifolds as hypersurfaces of Euclidean four-space: compactness, the homogeneous geometries, and total absolute curvature' (2026). Version 2. Final runs of every script in its deposited form: tests 345-352 on 30 October 2026 (for version 1, unchanged since), and test_353, added for version 2, on 30 October 2026. No script was edited after its run. Python 3.13.16; sympy 1.14.0; mpmath 1.3.0 SUMMARY test_345_compact_realizations.py 7 pass 0 fail exit 0 [3s] test_346_homogeneous_geometries.py 22 pass 0 fail exit 0 [11s] test_347_degree_identity.py 8 pass 0 fail exit 0 [63s] test_348_second_variation_conformal.py 11 pass 0 fail exit 0 [585s] test_349_embedded_indefinite.py 6 pass 0 fail exit 0 [11s] test_350_fold_birth.py 4 pass 0 fail exit 0 [4s] test_351_convex_branch.py 6 pass 0 fail exit 0 [41s] test_352_round_characterisations.py 6 pass 0 fail exit 0 [273s] test_353_mb_v2_review_items.py 6 pass 0 fail exit 0 [8s] total checks passed: 76 ================================================================================================== test_345_compact_realizations.py ================================================================================================== P1 the ellipsoid at its farthest point [PASS] K(vertex) = diag(a1/a_j^2) >= 1/a1; det K = a1^3/(a2 a3 a4)^2 >= a1^-3, equality iff round -- K = [a1/a2**2, a1/a3**2, a1/a4**2] P2 non-convex star-shaped hypersurfaces: the bound at the farthest point [PASS] at the maximum of r every principal curvature is >= 1/R, on three non-convex examples -- ex0: R = 1.35044, min kappa = 1.48125 vs 1/R = 0.740499, indefinite somewhere: True; ex1: R = 1.35, min kappa = 1.49246 vs 1/R = 0.740741, indefinite somewhere: True; ex2: R = 1.35008, min kappa = 1.50845 vs 1/R = 0.740699, indefinite somewhere: True P3 Jung's constant in R^4 [PASS] circumradius / edge = sqrt(2/5); on the unit S^3 the bound 125/(8 pi^6) < 1 = det M -- ratio sqrt(10)/5; bound 0.0162525 P4 the degenerate products, and the space forms [PASS] det M = 0 identically on S^2 x S^1, H^2 x S^1, T^3; M = Id on the round S^3 (so on every S^3/Gamma) -- S^2(r) x S^1: det M = 0; H^2 x S^1: det M = 0; flat T^3: det M = 0; round S^3: M = Id: True P5 the radial graphs r = 1 + eps cos 2chi: where definiteness is lost [PASS] (a) k_mer and k_sph formulas agree with the level-set second fundamental form -- chi=0.4: err 2.98e-40; chi=1.1: err 2.53e-40; chi=1.4: err 3.44e-40 [PASS] (b) at the equator k_sph = 1/(1 - eps) > 0 and the k_mer numerator is (1 - eps)(1 - 5 eps) -- numerator (epsilon - 1)*(5*epsilon - 1), k_sph 1/sqrt((epsilon - 1)**2) [PASS] (c) definite with det K > 0 below eps = 1/5; det K = 0 first at the equator at eps = 1/5; indefinite only past a zero of det K -- eps=0.1: min det K 0.76208; eps=0.15: min det K 0.47892; eps=1/5: min det K 0.0 at chi = 1.5708; eps=3/10: det K(equator) -2.0825; definite for chi < 1.2959 test_345: ALL PASS exit 0 [3s] ================================================================================================== test_346_homogeneous_geometries.py ================================================================================================== P1 the engine against Milnor's formula; the Milnor-frame table [PASS] Milnor frame: Ric = diag(2 mu_j mu_k), M_i = mu_i l_i - mu_j mu_k (symbolic) [PASS] S^3 Id/4; SL2 det -7/64; Nil (-3/4, 1/4, 1/4) det -3/64; Sol diag(-1,-1,1) det +1 -- (1, 1, 1): [1/4, 1/4, 1/4], det 1/64; (1, 1, -1): [1/4, 1/4, -7/4], det -7/64; (1, 0, 0): [-3/4, 1/4, 1/4], det -3/64; (1, -1, 0): [-1, -1, 1], det 1 [PASS] Bianchi V (H^3): M = -a^2 Id, det M = -a^6 -- M = [-a**2, -a**2, -a**2] [PASS] Jacobi holds iff n a = 0 (one random example of each) P2 Sol^3 in coordinates; the Lie engine; positive controls [PASS] Christoffels: exactly G^x_xz = 1, G^y_yz = -1, G^z_xx = -e^2z, G^z_yy = e^-2z [PASS] Ric = diag(0,0,-2), M = diag(-1,-1,1), K~_ab = diag(-e^2z, -e^-2z, 1) [PASS] nonzero Codazzi (a < b): exactly C_xzx = 2e^2z, C_yzy = -2e^-2z; sum of squares 16 cosh 4z; invariant norm 16 -- components {(0, 2, 0): 2*exp(2*z), (1, 2, 1): -2*exp(-2*z)}; invariant norm 16 [PASS] Lie engine on (1,-1,0): orthonormal components of modulus 2, invariant norm 16 -- moduli [2] [PASS] control (a), unit S^3: Ric = 2 gamma, M = Id, residual zero (expand_trig) and < 1e-30 at 5 points -- max |C| = 0 [PASS] control (b), Flamm metric: M = (2m, -m, -m)/rho^3, K~ = -diag(k_mer, k_sph, k_sph) (MA Ex. 3.3), Cod K~ = 0 -- M = [2*m/rho**3, -m/rho**3, -m/rho**3] P3 the Minkowski-Hsiung formulas on radial graphs [PASS] int (1 - H1 s) dV = int (H1 - H2 s) dV = 0 on a convex and an indefinite graph; integrands not zero -- eps=0.1: -1.73e-44, -1.93e-44; int|1-H1 s| = 1.947; eps=0.3: -3.14e-44, -4.52e-44; int|1-H1 s| = 4.43 [PASS] 18 (H1^2 - H2) = sum_{i l1 = l2 = l3 -- det M on H = 0: -8*s1**4*s2**4*(s1 + s2)**4 P6 E(2) and E(1,1) [PASS] R = -(l1-l2)^2/2; det M = -(l1-l2)^4 (3l1+l2)(l1+3l2)/64; rank 2 on the boundary lines -- det M = -(l1 - l2)**4*(l1 + 3*l2)*(3*l1 + l2)/64 P7 the degenerate left-invariant metrics [PASS] unimodular: two M-eigenvalues vanish together only where M = 0 (flat) [PASS] class B: det M = -PQST/64; rank M <= 1 iff (n2, n3) = (-a, a), (a, -a); there M ~ (0,0,-4a^2) and the -4a^2 eigenvector is parallel (H^2 x R) -- solutions [{n2: -a, n3: a}, {n2: a, n3: -a}] P8 class B: test_215's algebra from the engine [PASS] all components divisible by (n2-n3); C_231 = -(n2-n3)^2(n2+n3)PQ/8; on n3=-n2 C_122 = 4a n2^2(n2^2-a^2), det M = -(a^2-n2^2)^3; det M = -a^6 at n2 = n3 -- 5 nonzero components P9 random left-invariant metrics, exact [PASS] Jacobi for all; every det M > 0, non-scalar sample has Cod(adj M) != 0 exactly; >= 20 of each kind with det M > 0 -- det M > 0 (scalar): {'uni': [112, 0], 'B': [67, 0], 'line': [100, 0]} [PASS] near-round: |Cod|^2/d^2 agrees at d = 1e-3, 1e-6 to 1%; zero at d = 0 -- 0.283518 vs 0.281252 P10 the product geometries [PASS] H^2 x R: M rank 1, image d_t, d_t parallel; rank-2 K: K adj K = 0, im adj K = ker K; rank adj = 3, 1, 0; cylinder: adj K = M, Cod K = 0 -- (rank K, rank adj K) = [(3, 3), (2, 1), (1, 0)] test_346: ALL PASS exit 0 [11s] ================================================================================================== test_347_degree_identity.py ================================================================================================== P1 the revolution reduction, intrinsic and extrinsic [PASS] M = diag((1-rho'^2)/rho^2, -rho''/rho, -rho''/rho); det M rho^4 = sin^4 psi psi'^2 when rho' = cos psi [PASS] extrinsic: principal curvatures psi', sin psi/rho (twice); det K rho^2 = psi' sin^2 psi P2 corrugation [PASS] (a) signed integral = 2 pi^2 for every amplitude (exact antiderivative; 1e-25 at the samples) [PASS] (b) unsigned = 2 pi^2 iff 2kA <= 1, exceeding it beyond -- (k=2, A=0.2, 2kA=0.8): unsigned/2pi^2 = 1.0; (k=2, A=0.25, 2kA=1.0): unsigned/2pi^2 = 1.0; (k=2, A=0.3, 2kA=1.2): unsigned/2pi^2 = 1.049426786; (k=2, A=0.4, 2kA=1.6): unsigned/2pi^2 = 1.224939924; (k=3, A=0.15, 2kA=0.9): unsigned/2pi^2 = 1.0; (k=3, A=0.2, 2kA=1.2): unsigned/2pi^2 = 1.049426786; (k=3, A=0.3, 2kA=1.8): unsigned/2pi^2 = 1.327793992 [PASS] (c) each profile closes, rho(pi) = 0, and rho/sin > 0.8 inside [PASS] (d) det M = 0 at the first zero of psi' (outside U); J from the intrinsic M = the unsigned integral -- det M(s0) = 4.41e-82, J/2pi^2 = 1.04942678646 P3 J on the Berger family; scale invariance [PASS] J(Berger r) = 2 pi^2 r^3 sqrt(4 - 3r^2), maximal (= 2 pi^2) only at r = 1 -- critical points [1] [PASS] J is scale invariant: det M(c^2 g) = c^-6 det M(g), dV(c^2 g) = c^3 dV test_347: ALL PASS exit 0 [63s] ================================================================================================== test_348_second_variation_conformal.py ================================================================================================== P1 TT eigentensors at levels 3-7, and a mirror [PASS] h_n traceless, divergence-free, nabla* nabla h_n = (n^2 - 3) h_n, n = 3..7 [PASS] the orientation-reversed h_4 is TT with eigenvalue 13, with a nonzero theta^1 theta^1 component -- 3s P2 J along gamma + t h_n, directly [PASS] J1 = 0 and Q/||h||^2 = -(mu+2)(mu+3)/16 at n = 3..7 and for the mirror -- n=3: -9/2; n=4: -15; n=5: -75/2; n=6: -315/4; n=7: -147; n=4 mirror: -15 [PASS] ingredients: M1 = -(mu+2) h/2 pointwise; int R2 = -(mu-2) ||h||^2/4 -- 231s P3 the radical: Lie and conformal directions, and the mixed terms [PASS] Q(L_Y gamma) = 0, Q(u gamma) = 0, Q(h_4 + u gamma + L_Y gamma) = Q(h_4), J1 = 0 -- Q(h_4) = -30*pi**2; 427s P4 the conformal sector [PASS] (a) e^{2u} M(e^{2u} gamma) = (1 - Delta u) Id + Hess u - du du, exactly [PASS] (b) c1 = -q, c2 = (sigma2(H) - |p|^2)/2, c3 = (det H + H(p,p))/2 [PASS] (c) div T2(Hess f) = -T1(Hess f) grad f on the unit S^3 [PASS] (d) int c1 = int c2 = int c3 = 0 for four f, harmonic and not -- x0x1: [1, 0, 0, 0, -1/24]; x0x1x2: [1, 0, 0, 0, -3/160]; x0x1 + x0x2/2: [1, 0, 0, 0, -25/384]; x0^3 + x1^2 x3: [1, 0, 0, 0, -19/40] [PASS] (e) quartic: -1/24, -3/160, -25/384; zonal x0^2 (through t^6) and x0^3, and the first harmonic x1: 0 -- x0^3+x1^2x3: -19/40; 511s P5 the deficit through the full engine; the radial control [PASS] conformal path: J/2pi^2 = 1 - t^4/24 + O(t^5); radial path: 1 + O(t^5); det > 0 at t = 1/5 -- conformal [1, 0, 0, 0, -1/24], radial [1, 0, 0, 0, 0]; 583s test_348: ALL PASS exit 0 [585s] ================================================================================================== test_349_embedded_indefinite.py ================================================================================================== P1 SO(3)-symmetric hypersurfaces [PASS] k_mer = -rho''/sqrt(1-rho'^2), k_sph = sqrt(1-rho'^2)/rho; adj K = diag((1-rho'^2)/rho^2, -rho''/rho, -rho''/rho) P2 radial graphs: the intrinsic M against the embedding [PASS] intrinsic M = diag(k_sph^2, k_mer k_sph, k_mer k_sph) for every r(chi) P3 the family r = 1 + eps cos 2chi [PASS] (a)-(c) embedded; k_sph numerator 1 - eps + 6eps z; k_mer numerator N(z) with N(0) = (1-eps)(1-5eps) -- N = -12*epsilon**2*z**2 + 12*epsilon**2*z + 5*epsilon**2 + 12*epsilon*z - 6*epsilon + 1 [PASS] (d) exactly one root z* in (0,1): indefinite band, two degenerate spheres, definite caps; eps = 1/5 semidefinite -- eps=3/10: z* = 0.076123586, band |chi - pi/2| < 0.279531; eps=1/2: z* = 0.085786438, band |chi - pi/2| < 0.297251; eps=9/10: z* = 0.017196610, band |chi - pi/2| < 0.131515 P4 the band is a neck of the orbit radius [PASS] sign k_mer = -sign Phi_ss at eps = 3/10 (the band is where the orbit radius is strictly convex) -- 399 points P5 a realizable path from the round sphere through the degenerate locus [PASS] embedded throughout; definite, then degenerate at t = 2/3, then indefinite; J = 2 pi^2 until 2/3, then above -- t=0.5: J/2pi^2 = 1.0, beta'(equator) = 0.294; t=0.667: J/2pi^2 = 1.0, beta'(equator) = -1.79e-41; t=0.833: J/2pi^2 = 1.123730778, beta'(equator) = -0.333; t=1.0: J/2pi^2 = 1.313753191, beta'(equator) = -0.714 test_349: ALL PASS exit 0 [11s] ================================================================================================== test_350_fold_birth.py ================================================================================================== P1 the Gauss map [PASS] beta' = (r^2+2r'^2-rr'')/(r^2+r'^2) = k_mer ds/dchi; k_sph = sin beta/(r sin chi); det K dV = 4pi sin^2 beta dbeta; beta(pi - chi) = pi - beta(chi) P2 the cubic normal form [PASS] beta - pi/2 = a1 x + a3 x^3 + O(x^5), a1 = (1-5eps)/(1-eps), a3 = 8eps(6eps^2+eps+1)/(3(1-eps)^3); cusp conditions -- a1 = (5*epsilon - 1)/(epsilon - 1), a3 = -8*epsilon*(6*epsilon**2 + epsilon + 1)/(3*(epsilon - 1)**3) P3 the excess of absolute curvature, exactly [PASS] J_abs - 2pi^2 = 8pi[G(beta(a)) - G(beta(pi-a))] = 2 int_{det K<0}|det K| = quadrature; the archive's values; 2pi^2 at eps <= 1/5 -- eps=1/4: fold at chi = 1.344115125, J_abs/2pi^2 = 1.123730778; eps=3/10: fold at chi = 1.291265172, J_abs/2pi^2 = 1.313753191; eps=2/5: fold at chi = 1.263167523, J_abs/2pi^2 = 1.726191266 P4 the scaling laws [PASS] x0 ~ (5/sqrt18) delta^1/2, Dbeta ~ (125/(9sqrt2)) delta^3/2, J_abs/2pi^2 - 1 ~ (500/(9sqrt2 pi)) delta^3/2, each within 4 delta, deviations falling tenfold per decade -- delta=0.001: 0.9968148918, 0.9973694549, 0.997369447; delta=0.0001: 0.9996806492, 0.9997361948, 0.9997361948; delta=1.0e-5: 0.9999680565, 0.9999736119, 0.9999736119 test_350: ALL PASS exit 0 [4s] ================================================================================================== test_351_convex_branch.py ================================================================================================== P1 the support-function parametrisation [PASS] X_b(x_h) = sum_a (Hess h + h Id)_ab X_a(nu): metric L^2, shape operator L^-1, Gauss map nu P2 Gauss for g_h through the curvature engine [PASS] M(g_h) = L_h / det L_h through t^2, exactly P3 the straight-line contraction [PASS] L along the line is (1-s) L_h + s Id; positive at 500 points for s in [0, 1]; centring kept; O(4)-equivariant P4 Huisken's linearisation at the round sphere [PASS] Delta Y_l = -l(l+2) Y_l and tr L^-1 = 3/R + eps (l(l+2) - 3) Y_l/R^2, l = 1..4: rates -[l(l+2)-3]/R^2 P5 the cross-curvature threshold on Bianchi IX [PASS] M > 0 iff |e| < (sqrt2 - 1)/2: zero exactly there, positive at 0.2, negative at 0.21 -- M at e*: [0, -48/7 + 40*sqrt(2)/7, 80/7 - 48*sqrt(2)/7] P6 the revolution class is convex in u = rho^2 [PASS] u_s > 0 inside, nonzero slope at the poles, for s in [0, 1]; no neck at s = 0, a neck at s = 1 -- max rho_zz near the equator: s=0 -1.0, s=1 1.327 test_351: ALL PASS exit 0 [41s] ================================================================================================== test_352_round_characterisations.py ================================================================================================== P1 parallel hypersurfaces [PASS] H(0) = p1, H'(0) = -p2, H''(0) = 2p3; Newton gives sigma_1..3 P2 Steiner's formula [PASS] sigma_k of the parallel surface times the Jacobian = sum C(j,k) d^(j-k) sigma_j; dW_k = (k+1) W_{k+1} P3 Schneider's normalisation; the capsule [PASS] W0 = 4V1, W1 = 12V2, W2 = 12V3, W3 = 4V4; capsule W_k; W2^2 - 3W1W3 = 64 pi^2 t^2 = 144(V3^2 - V2V4) P4 Minkowski's inequality [PASS] W1 = 3 W3^(1/3) W0^(2/3) on the ball, strictly larger on the capsule -- t/r=1/10: 0.101044; t/r=1: 7.10326; t/r=10: 222.287 P5 the cubic bound [PASS] AM-GM on 10^4 triples; int |A|^3 = 6 sqrt3 pi^2 on the sphere, larger at eps = 1/10 and 3/10 -- ratios to 6sqrt3 pi^2: 1.0663272, 1.5342908 P6 the corrugated chain [PASS] int sigma_2 = 4pi int (1 + cos^2 psi) = 6 pi^2 at every amplitude; W1's A^2 coefficient as stated -- k=2: A^2 coefficient -208*pi**2/225; k=3: A^2 coefficient -1188*pi**2/1225 test_352: ALL PASS exit 0 [273s] ================================================================================================== test_353_mb_v2_review_items.py ================================================================================================== P1 the Taylor coefficients of the conformal integrand [PASS] c1 = -q, c2 = (sigma2 - |p|^2)/2, c3 = (det H + H(p,p))/2, c4 = q(det H + H(p,p))/2 - (sigma2 + |p|^2)^2/8 - |Hp|^2/2 P2 the quartic coefficients of Proposition 5.7(3), through c4 and exact moments [PASS] int c1 = int c2 = int c3 = 0, and int c4/(2 pi^2) = -1/24, -3/160, -25/384, -19/40; 0 for zonal x0, x0^2, x0^3 -- x0*x1: -1/24; x0*x1*x2: -3/160; x0*x1 + x0*x2/2: -25/384; x0**3 + x1**2*x3: -19/40; x0: 0; x0**2: 0; x0**3: 0 P3 Example 5.3: the corrugated profiles stay off the axis for |A| < 2/pi [PASS] cos psi(pi - s) = -cos psi(s) (k = 2..6); rho >= sin sigma - |A| sigma > 0 on (0, pi/2] for |A| <= 0.63 < 2/pi -- min rho on the grids = 0.0004; 2/pi = 0.6366; largest amplitude used in MB 0.4 P4 Theorem 6.4(3): beta stays in (0, pi), folds once, and the belt has exactly one det K < 0 preimage [PASS] k_sph numerator 1 - eps + 6 eps cos^2 chi > 0; beta in (0, pi); two turning points; 3 preimages (one with det K < 0) inside the belt, 1 outside -- eps=0.25: belt (1.5222, 1.6194); eps=0.3: belt (1.4470, 1.6946); eps=0.4: belt (1.2773, 1.8643); eps=0.9: belt (0.4043, 2.7372) P5 Proposition 7.4: the radial graphs are smooth at the poles in the (zeta, u) description [PASS] Z = (1 - eps + 2 eps c^2) c, u = r^2 (1 - c^2) polynomial in c; dZ/dc = 1 - eps + 6 eps c^2 > 0; u(+-1) = 0, du/dc(+-1) = -+2(1 + eps)^2 P6 the algebra of Theorem 5.5(2)'s proof [PASS] sqrt det expansions; int R^(2) = -(mu - 2)/4; Q = -(mu + 2)(mu + 3)/16 [7s] test_353: ALL PASS exit 0 [8s]