RUN LOG -- verification scripts for J. Thrussell, 'Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere' (2026). DOI 10.5281/zenodo.23765515. Final run of every script in its deposited form, 29 October 2026. Two docstring-only edits followed some of these runs and touch no code: the author's archive labels were removed from the docstrings of 326, 329, 330, 332 and 333, and 338's docstring writes R^4 for E^4. Each script's docstring records its edits. test_344 was added after a second independent read of the draft, the same day. Python 3.13.16; sympy 1.14.0; numpy 2.5.3; scipy 1.18.1; mpmath 1.3.0 SUMMARY test_325_realizability_symbol.py 6 pass 0 fail exit 0 [81s] test_326_symbol_module_is_immersions.py 4 pass 0 fail exit 0 [18s] test_327_round_sphere_conformal_path.py 5 pass 0 fail exit 0 [485s] test_329_degenerate_locus.py 4 pass 0 fail exit 0 [2s] test_330_ma_small_identities.py 6 pass 0 fail exit 0 [5s] test_331_contracted_polynomial_form.py 6 pass 0 fail exit 0 [2s] test_332_theorem_C_exact.py 3 pass 0 fail exit 0 [9s] test_333_zeta_count_scope.py 4 pass 0 fail exit 0 [19s] test_334_spinors_and_the_chiral_form.py 6 pass 0 fail exit 0 [7s] test_335_reduction_and_symbol_against_F.py 5 pass 0 fail exit 0 [26s] test_336_round_sphere_linearisation.py 5 pass 0 fail exit 0 [63s] test_337_shell_metrics_first_order.py 9 pass 0 fail exit 0 [44s] test_338_swept_spheres.py 8 pass 0 fail exit 0 [87s] test_339_second_order_round_base.py 6 pass 0 fail exit 0 [32s] test_340_second_order_general_bases.py 11 pass 0 fail exit 0 [350s] test_341_polynomial_form_and_the_degenerate_symbol.py 3 pass 0 fail exit 0 [5s] test_342_rank_one_branch.py 18 pass 0 fail exit 0 [3s] test_343_chiral_form.py 5 pass 0 fail exit 0 [8s] test_344_second_read_checks.py 4 pass 0 fail exit 0 [48s] total checks passed: 118 ================================================================================================== test_325_realizability_symbol.py ================================================================================================== P1 the kernel at every nondegenerate point [PASS] sigma(xi) kills xi(.)v and Q_xi = P K P exactly (symbolic K, xi); rank 2 at random rational K and xi -- exact ranks [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] P2 where the symbol is elliptic: the first Newton tensor [PASS] TT-injective at xi iff xi^T T1 xi != 0; diag(2,3,-3) counterexample exact; characteristic covectors iff k1+k2 <= 0 <= k2+k3; conformal coincidence at fixed xi iff Q scalar -- TT singular value at null covectors 1.5e-14; min over definite-T1 cases 1.14e-01 P3 four identities among the nine components; five genuine conditions [PASS] rank tau_3 = 5; the cyclic identity and the three Bianchi functionals B_b(C) are independent and span the left null space -- rank tau_3 5, rank tau_3 5, rank tau_3 5 P4 Cartan's test at order 3 [PASS] (re-posed) rank tau_4 = 12, its left null space = 12 prolonged identities + 3 functionals xi ^ C; dim g_4 = 78 = 55 + 19 + 4: involutive at order 3 -- K=[1.3, 2.1, 0.8]: rank tau_4 12, dim g_4 78, Cartan sum 55 + 19 + 4 = 78; K=[2.0, 3.0, -3.0]: rank tau_4 12, dim g_4 78, Cartan sum 55 + 19 + 4 = 78 P5 four functions of three variables; the joint kernel is K [PASS] dim g_q has second differences 4 (four functions of three variables); joint kernel = span{K} -- K=[1.3, 2.1, 0.8]: dim g_q (q=3..9) [55, 78, 105, 136, 171, 210, 253], second differences [4, 4]; K=[2.0, 3.0, -3.0]: dim g_q (q=3..9) [55, 78, 105, 136, 171, 210, 253], second differences [4, 4]; joint-kernel alignment with K 1.000000000000 P6 the relations are exact identities (cyclic, contracted Bianchi, d_nabla Cod K = R.K = 0 under Gauss) [PASS] cyclic sum of Cod K = 0; div adj K = B(Cod K); cyclic nabla Cod K = -R.K (Ricci + Bianchi), exactly at a curved point; R.K = 0 under Gauss, symbolically -- residuals 0, 0, 0; Gauss cancellation True; div adj K at 0 = [-39037/25200, -91057/12600, -29783/3150] test_325: ALL PASS exit 0 [81s] ================================================================================================== test_326_symbol_module_is_immersions.py ================================================================================================== P1 the two lemmas behind the symbol theorem, exactly [PASS] sigma_2(DM)(xi) h = -1/2 adj_2(PhP), zero on gauge; D adj_K polynomial for every K, D adj_K(xi xi^T) = adj_2(PKP) P2 the immersions' symbol module M_K: dimension (q+2)(2q+5), inside g_q [PASS] rank M_K = (q+2)(2q+5) and tau_q M_K = 0 for q = 3, 4, 5 at all fifteen K -- q=3: rank M_K 55 = 55, tau_3 M_K = 0: True P3 Cartan's test at fifteen special shape operators; g_q = M_K [PASS] dim g_3 = 55, g_{3,1} = 19, g_{3,2} = 4, dim g_4 = 78 = Cartan sum, dim g_5 = 105 at every K: g_q = M_K, involutive -- (1, 1, 1):ok, (1, 1, 2):ok, (1, 1, -1):ok, (2, 3, -3):ok, (-1, 1, 2):ok, (1, 2, 3):ok, (1, 1, -2):ok, (-1, 2, 2):ok, (2, 2, -1):ok, (1, -2, -2):ok, (3, -1, -1):ok, (1, 2, -1):ok, (5, -2, 1):ok, (1, 1, 1/10):ok, (1, -1, 1):ok P4 the Kulkarni-Nomizu map with K is injective iff det K != 0 [PASS] det(A -> A (o) K on Sym^2) = 2 (det K)^2 -- 2*(w0*w1*w2 - w0*w5**2 - w1*w4**2 - w2*w3**2 + 2*w3*w4*w5)**2 test_326: ALL PASS exit 0 [18s] ================================================================================================== test_327_round_sphere_conformal_path.py ================================================================================================== P1 the implementation: F(gamma) = 0 and F(radial graph) = 0 through t^2 [PASS] F(gamma) = 0, R = 6; F((1+u)^2 gamma + du du) = 0 exactly through t^2 for a random cubic u -- radial-graph residual coefficients: {0} P2 the kernel at gamma: conformal and Lie directions [PASS] t-linear part of F(gamma + t h) is 0 for h = 2u gamma and h = L_X gamma (random cubic u, X) -- second-order part nonzero for the conformal one: True P3 the straight conformal path leaves at second order; the radial lift does not [PASS] F(e^{2tf} gamma) = -t^2 DF_gamma(df df) + O(t^3) with DF_gamma(df df) != 0; F(radial lift) = 0 through t^2 -- DF_gamma(df df) components at the point: [384/845, -55296/105625, 7962624/17850625, 884736/3570125] P4 the scalar-curvature detector on conformal directions [PASS] delta R(2u gamma) = -4(Delta + 3)u; 0 on a first harmonic; 4 lambda (2 cot^2 r - 1) cos theta on the shell dipole -- dipole value at the point: 5/26 = 5/26 P5 the count of ten [PASS] zeta(0) = 2[zeta_H(-2,3) - 4 zeta_H(0,3)] = 10 for lambda_n with an even-power expansion (test_333); 4 over n >= 1; scalars n >= 3: -5 -- exact 10, continuation 10.0 test_327: ALL PASS exit 0 [485s] ================================================================================================== test_329_degenerate_locus.py ================================================================================================== P1 the adjugate's ranks; a rank-2 M is never adj K [PASS] rank adj K = 3, 1, 0 for rank K = 3, 2, <= 1; adj of rank-2 diag(m1, m2, 0) is m1 m2 e3 e3^T P2 not vacuous where rank M = 2 (the audit's example) [PASS] M = diag(-1,-2,0), det M = 0, d_r det M = -12, P = diag(0,0,2), E_ryy = 24 and nothing else -- nonzero E components {(0, 2, 2): 24} P3 identically zero on a realizable rank-one point: the cone over a Clifford torus [PASS] M = diag(-1/t^2, 0, 0), det M = 0, P = 0 identically, E = 0 at the point P4 the Bianchi split on the cone [PASS] v = d_t geodesic (Gamma^i_tt = 0); div(mu v) = 0 -- div = 0 test_329: ALL PASS exit 0 [2s] ================================================================================================== test_330_ma_small_identities.py ================================================================================================== P1 det D adj_K = -2 (det K)^2 [PASS] det D adj_K on Sym^2 = -2 (det K)^2, general symmetric K P2 Theorem A(2) on the indefinite branch [PASS] adj(sqrt(det M) M^{-1}) = M for M of signature (+,-,-), det M > 0 P3 the Flamm hypersurface [PASS] metric, principal curvatures, M's eigenvalues, det M (1/256m^6 at the throat), finite distance, T_1 -- distance from 2m to 3m = m*(2*acosh(sqrt(6)/2) + sqrt(3)) P4 the count [PASS] n^2(n^2-1)/12 - n(n+1)/2 = n(n-3)(n+1)(n+2)/12, zero only at n = 3 P5 Lemma 7.3 [PASS] E'/E = -W.n, |n| = 1 P6 Proposition 7.6 [PASS] Q(V) = 1, -Q(V') = |W|^2, E = B(V,N)/2 with N null, Q((Phi V)') = Phi'^2 - Phi^2 |W|^2 test_330: ALL PASS exit 0 [5s] ================================================================================================== test_331_contracted_polynomial_form.py ================================================================================================== P1 phi (F M) = R on a generic metric with det M > 0 det M at the point = 171/200 [PASS] det M > 0; phi (F M)_ab^d = R_ab^d (nine components); d det M = tr(P nabla M); F != 0 there -- max |F| ~ 1.081e-1 P2 R = 0 on the round S^3 [PASS] M = I and R = 0 at the point P3 the audit's rank-2 point [PASS] M = diag(-1,-2,0); R_rx^x = -6, R_ry^y = 6 (and antisymmetric partners) -- nonzero {(0, 1, 1): -6, (0, 2, 2): 6, (1, 0, 1): 6, (2, 0, 2): -6} P4 the realizable rank-one cone [PASS] M = diag(-1/t^2, 0, 0) and R = 0 at the point P6 the identities of R [PASS] R_ab^b = 0 (three); cyclic sum of R_ab^d P_dc = 0 -- cyclic = 0 P5 a non-umbilic realizable graph [PASS] M = diag(6,3,2), nabla M != 0, F = 0, R = 0 test_331: ALL PASS exit 0 [2s] ================================================================================================== test_332_theorem_C_exact.py ================================================================================================== P1 five conditions at every nondegenerate K (symbolic, principal frame) [PASS] a 5x5 minor of the coordinate-covector images is a monomial; a 4x4 minor of the identities is +-k1k2k3; the identities kill tau_3 -- 5x5 minor (scaled by (det K)^10): k1**8*k2**8*k3**9/256 P2 Cartan's test in exact arithmetic at sixteen shape operators [PASS] dim g_3 = 55, g_{3,1} = 19, g_{3,2} = 4, dim g_4 = 78, dim g_5 = 105, exactly over Q, at all sixteen -- ('1', '1', '1'):ok, ('1', '1', '2'):ok, ('1', '1', '-1'):ok, ('2', '3', '-3'):ok, ('-1', '1', '2'):ok, ('1', '2', '3'):ok, ('1', '1', '-2'):ok, ('-1', '2', '2'):ok, ('2', '2', '-1'):ok, ('1', '-2', '-2'):ok, ('3', '-1', '-1'):ok, ('1', '2', '-1'):ok, ('5', '-2', '1'):ok, ('1', '1', '1/10'):ok, ('1', '-1', '1'):ok, ('13/10', '21/10', '4/5'):ok P3 the kernel is gauge + span{K}; sigma(xi) is injective on S_xi [PASS] sigma(xi) kills xi(.)v and K (direct, dim 4), rank 2, injective on S_xi; TT rank shown for comparison -- TT rank 1, S_xi rank 2; TT rank 2, S_xi rank 2; TT rank 2, S_xi rank 2; TT rank 2, S_xi rank 2; TT rank 2, S_xi rank 2; TT rank 2, S_xi rank 2 test_332: ALL PASS exit 0 [9s] ================================================================================================== test_333_zeta_count_scope.py ================================================================================================== P1 the continuation against direct sums [PASS] continuation = direct sum at s = 2, 3 for all five lambda -- n^3 + 7n, 2n^5 + n^3/2, n^3 + n^2, n^3 - n^2/2, n^3 + 2 P2 even powers: zeta(0) = 10 [PASS] n^3 + 7n and 2n^5 + n^3/2 give 10 -- 10.0, 10.0 P3 a first odd power changes the count [PASS] n^3 + n^2 gives 112/9; n^3 - n^2/2 gives 313/36 -- 12.4444444444444, 8.69444444444444 P4 a third odd power changes it too [PASS] n^3 + 2 gives 26/3 -- 8.66666666666667 test_333: ALL PASS exit 0 [19s] ================================================================================================== test_334_spinors_and_the_chiral_form.py ================================================================================================== P1 the two half-spin representations [PASS] rho_+ and rho_- are homomorphisms for one overall sign c; kernels 3 + 3 complementary; image elements square to -|v|^2/4 -- c = -1 P2 the Dirac operator on a generalized Killing spinor [PASS] sum_i e_i K(e_i) = -(tr K) 1 for symmetric K P3 end to end on a generic diagonal metric and generic K [PASS] spinor curvature = rho_s(curvature of Omega); tangential block = Riemann - k^k; mixed block = Codazzi of K -- (i) True, (ii) True, (iii) True P4 consequences [PASS] (a) paraboloid in R^4, both chiralities flat; (b) 1.1 K, both curved -- (a) True, (b) True [PASS] (c) round S^3: K = Id flat for both s (Killing spinors), K = 2 Id not flat [PASS] (d) K = Id on the generic metric: Codazzi holds, spinor curvature = rho_s(Gauss residual) != 0 for both s -- Codazzi True, Gauss-only True, curved True test_334: ALL PASS exit 0 [7s] ================================================================================================== test_335_reduction_and_symbol_against_F.py ================================================================================================== P1 det adj K = (det K)^2 [PASS] det adj K = (det K)^2, general symmetric K P2 adj K~ = M on det M > 0 [PASS] adj(sqrt(det M) M^{-1}) = M, M = diag(a, b, c/(ab)), c > 0, a, b any sign P3 K~(adj K) = sign(det K) K [PASS] K~(adj K) = sign(det K) K, all sign patterns (symbolic) and four rotated rational K P4 adj K > 0 iff K definite [PASS] adj K positive definite iff all k_i of one sign (eight sign patterns) P5 the symbol formula against the full operator [PASS] O(eps) part of F(g + eps (xi.x)^3/6 h0) at 0 = sigma(xi) h0, both branches, nonzero -- M(0) = diag(1, 2, 3): True; M0 = diag(1, 2, 3), xi = [3/5, 4/5, 0]: equal; M0 = diag(1, 2, 3), xi = [3/5, 4/5, 0]: equal; M0 = diag(1, 2, 3), xi = [2/3, 1/3, 2/3]: equal; M0 = diag(1, 2, 3), xi = [2/3, 1/3, 2/3]: equal; M(0) = diag(2, -1, -3): True; M0 = diag(2, -1, -3), xi = [3/5, 4/5, 0]: equal; M0 = diag(2, -1, -3), xi = [3/5, 4/5, 0]: equal; M0 = diag(2, -1, -3), xi = [2/3, 1/3, 2/3]: equal; M0 = diag(2, -1, -3), xi = [2/3, 1/3, 2/3]: equal test_335: ALL PASS exit 0 [26s] ================================================================================================== test_336_round_sphere_linearisation.py ================================================================================================== P1 DF_gamma h = Cod (1/2)(-Lap + 2) h on transverse-traceless jets 120 jet coefficients, 50 constraint equations, 70 free [PASS] O(eps) of F(gamma + eps h) at 0 = Cod((1/2)(-Lap h + 2h))(0) on three random TT jets; differs off TT -- trial 0: equal; trial 1: equal; trial 2: equal; control differs: True P2 level 3 on SU(2): constant trace-free tensors in a left-invariant frame [PASS] sectional curvature 1; div h = 0; -Lap h = 6h; |nabla h|^2 = 6|h|^2; |Cod h|^2 = 18|h|^2 = 2|nabla h|^2 + 6|h|^2 P3 the augmented symbol [PASS] (xi ^ H, H xi, (tr H) xi) injective in H; |xi ^ H|^2 = 2|xi|^2|H|^2 on {H xi = 0, tr H = 0} P4 the count of ten and the determinant [PASS] zeta(0) = 10 for both multipliers; ln det_zeta(n(n^2-1)) on TT: numerical = closed form = -14.0371968... -- zeta(0) = 10.0, 10.0; ln det = -14.0371968383834 vs -14.0371968383834 P5 the shell dipole's decomposition [PASS] 2 lambda cos(theta) dr^2 = 2 w gamma + L_eta gamma, w = (lambda + c sin r) cos(theta), all c, q0 test_336: ALL PASS exit 0 [63s] ================================================================================================== test_337_shell_metrics_first_order.py ================================================================================================== [PASS] P1 background M0 = diag(Q, P, P), K~0 = diag(P/sqrt Q, sqrt Q, sqrt Q) on every base [PASS] P2 first family C_{r th r} = -((Lam-2)/2) A^3/(Phi Z') dY (l = 2, 3) [PASS] P3 second family C_{th ph ph} = -((Lam-2)/2) Z'/(A Phi P) sin^2 th dY (l = 2, 3) [PASS] P4 third family = D T[Y]; D measured = closed form = curvature form; l- and angle-independent -- round: D = 0; B2: D = -17/47; B2': D = -73/122; B3: D = 314163*sqrt(161)/14168; B4: D = -25911*sqrt(455)/902720; BC: D = 0 [PASS] P5 the five components with an odd number of phi indices vanish [PASS] P6 kernel: l = 0 and l = 1 give C = 0 on every base; l = 2, 3 do not P7 two further first-order kernel directions [PASS] P7 (a) l = 1 with r-dependent amplitude, (b) off-diagonal beta(r) sin(theta): all O(eps) components vanish -- B2: True, True; B3: True, True; B4: True, True P8 symbolic, general base [PASS] P8 closed form = curvature form (general base); D = 0 on the round base and on the constant-ratio family P9 D on a constant-ratio base and on the others [PASS] P9 D = 0 on the round base and BC (k_mer/k_sph = 1/2); D != 0 on B2, B3, B4 -- round: 0, B2: -17/47, B2': -73/122, B3: 314163*sqrt(161)/14168, B4: -25911*sqrt(455)/902720, BC: 0 [43s] test_337: ALL PASS exit 0 [44s] ================================================================================================== test_338_swept_spheres.py ================================================================================================== P1 the swept-sphere construction reproduces A^2 = H^2 + (alpha + Phi b.n)^2, round shells, no cross terms [PASS] explicit embedding: g = diag(H^2 + (alpha + Phi b.n)^2, Phi^2/E^2, Phi^2/E^2), cross terms 0 -- max relative deviation 1.3e-10; chart flow is a CKF to 8.2e-11; g_rr 0.1253966056 vs 0.1253966056 P2 tilt with no chart drift, convex branch, F = 0 [PASS] tilt-only family: exact F = 0, M > 0, chart drift W = 0, radial coefficient with an l = 1 part -- point (1/2, 1/3, -1/5): residual 0 True, M eigenvalues [0.322338, 0.074767, 0.116838]; point (2/3, -1/2, 1/4): residual 0 True, M eigenvalues [0.283054, 0.059569, 0.083753] P3 the general swept family, and its Szekeres subfamily [PASS] general swept family F = 0 (det M > 0); Szekeres subfamily A^2 = H^2/(1-k) identically, and F = 0 -- general: M eigenvalues [0.133625, 0.341712, 0.285426]; Szekeres: M eigenvalues [0.4, 0.455917, 0.455917] P4 route 1 of the second-order removal is the tilt [PASS] round base: A = 1 + eps cos(theta) + eps^2 [cot^2 r/6 + (cot^2 r/3) P_2] + O(eps^3) P5 first order, general alpha [PASS] delta A / A0 = Phi (Phi' W + alpha b).n / A0^2; with W = 0, delta(A^2) = 2 alpha Phi b.n -- Phi*(Phi_p*Wn + alpha*bn)/(Phi_p**2 + alpha**2) P6 the axisymmetric case, off the round base [PASS] axisymmetric tilt, W = 0, non-round base: F = 0 exactly, M > 0 -- M eigenvalues [0.377669, 0.414567, 0.411746] P7 each shell is intrinsically round [PASS] Gauss curvature of Phi^2 E^{-2}(dp^2 + dq^2) = 1/Phi^2 -- K = Phi**(-2) P8 the drift in shell coordinates [PASS] g_rr = A^2 + 2c'Phi' cos th + c'^2, g_{r th} = -c' Phi sin th, angular block round; round base beta = -tan r test_338: ALL PASS exit 0 [87s] ================================================================================================== test_339_second_order_round_base.py ================================================================================================== [PASS] P1 the O(eps^2) obstruction: the four displayed components, the rest zero, O(eps) zero -- r0 = atan(4/3), th0 = acos(3/5): C_rthr = -27/32; r0 = atan(4/3), th0 = acos(8/17): C_rthr = -3375/4624 [PASS] P2 route 1 (the tilt's second-order correction) removes every component through O(eps^2) [PASS] P3 route 2: O(eps) zero for every s [PASS] P3 route 2: C_{r th r} = -(1 - s)^2 2 sin th cos th / sin^4 r, double root at s = 1 [PASS] P3 route 2: s = 1, mu = sec^2 r: every component vanishes [PASS] P3 route 2: the O(eps^2) residual does not depend on mu (sec^2 r against 0, s = 0, 2) [31s] test_339: ALL PASS exit 0 [32s] ================================================================================================== test_340_second_order_general_bases.py ================================================================================================== base-radius pairs on the positive branch: 8; constant-ratio configurations: 6 mode F done [173s] mode C done [340s] [PASS] P1 (F) O(eps) zero for every s; f_X(s) quadratic [PASS] P2 (F) f_X(1) = 0 for all four components, 8 base-radius pairs and three angular points [PASS] P3' (F) all four families have a double root at s = 1, every configuration [PASS] P5' (F) constant-ratio bases: all four families double root at s = 1 [PASS] P1 (C) O(eps) zero for every s; f_X(s) quadratic [PASS] P2 (C) f_X(1) = 0 for all four components, 8 base-radius pairs and three angular points [PASS] P3 (C) (Lambda-2) families double root, D family simple root [PASS] P4' (C) the D family's second root differs between bases -- B2@1/2: 0.376941, B2@1/3: 0.442371, B2@3/4: 0.252259, B3@1/2: 0.617772, B3@3/4: 0.350832, B4@1/2: 0.285611, B4@1/3: 0.394797, B4@3/4: 0.105186 [PASS] P5 (C) constant-ratio bases: D slots simple root, (Lambda-2) slots double root, root at s = 1 P7 the two parametrisations: beta' enters only the D family [PASS] P7' (Lambda-2) components agree between (F) and (C); D components differ for s = 2, 3 (every configuration) P6 (C) the eps^2 radial companion does not enter [PASS] P6 (C) mu scaled by 2, 1/2, 0: every f_X(s) unchanged [348s] test_340: ALL PASS exit 0 [350s] ================================================================================================== test_341_polynomial_form_and_the_degenerate_symbol.py ================================================================================================== P1 the branch-free variable P [PASS] adj adj K = (det K) K; even in K; P = sqrt(det M) K~; P > 0 iff K~ definite P2 T = 2 (det M)^{3/2} F at a point, both branches [PASS] T = 2 (det M)^{3/2} F exactly, convex and indefinite branches -- M(0) = diag(1, 2, 3): equal True, F != 0 True; M(0) = diag(2, -1, -3): equal True, F != 0 True P3 the symbol as k3 -> 0 [PASS] P3' rank 2; sv_max ~ 1/k3 always; sv_min bounded iff xi is orthogonal to e_3, else ~ 1/k3 -- perp [1, 0, 0]: sv_max k3 -> 0.35355, sv_min -> 0.50000; perp [0, 1, 0]: sv_max k3 -> 0.08839, sv_min -> 0.25000; perp [3/5, 4/5, 0]: sv_max k3 -> 0.17337, sv_min -> 0.35043; oblique [0, 0, 1]: sv_max k3 -> 0.59293, sv_min k3 -> 0.50000; oblique [2/3, 1/3, 2/3]: sv_max k3 -> 0.41962, sv_min k3 -> 0.28112; oblique [12/13, 0, 5/13]: sv_max k3 -> 0.38666, sv_min k3 -> 0.14569 test_341: ALL PASS exit 0 [5s] ================================================================================================== test_342_rank_one_branch.py ================================================================================================== [PASS] Q1 adj diag(k1, k2, 0) = diag(0, 0, k1 k2): rank one [PASS] Q1 adj M = 0 on the branch (adj M vanishes) [PASS] Q1 M's nonzero eigenvector spans ker K (the ruling) [PASS] Q2 along v: the Bianchi component is exactly div(mu v) [PASS] Q2 across v: the components are mu times the acceleration of the ruling [PASS] P1 Codazzi with v: b21 k11 + k12 (b22 - b11) - b12 k22 = 0 (-b11*k12 - b12*k22 + b21*k11 + b22*k12) [PASS] P1 equivalently B K = K B^T (the diagonal is identically zero) [PASS] P1 principal frame: k2 b12 = k1 b21 (-b12*k2 + b21*k1) [PASS] P2 k1 = k2 = k forces b12 = b21 (no twist) [PASS] P2 a pure rotation of the rulings forces tr K = 0 on the block (so mu = det < 0) (-c*(k11 + k22)) [PASS] P3 the pointwise system has a curve of solutions with Jacobian rank 2 (one free parameter) (4 solutions on distinct k12, ranks [2, 2, 2, 2]) [PASS] P4 transport + Jacobi: v(ln det K) = -tr B = -div v, i.e. nabla(mu v) = 0 (-b11 - b22) [PASS] P5 cone over a Clifford torus: N is the unit normal and K kills the ruling d_t [PASS] P5 the block is diag(tan al, -cot al)/t and B = I/t (B [1/t, 0, 0, 1/t]) [PASS] P5 frame parallel along the ruling, and d/dt block = -B K exactly [PASS] P5 cone class B = f I: P1's equation is empty there [PASS] P6 where det M > 0, sqrt(det M) M^-1 = sign(det K) K (20 random symmetric K) [PASS] P6 across a seam K~ flips sign; sign(det K) K~ = K is continuous, so K is fixed there (K~_22: -2 | 2; K_22: [2, 2]) test_342: ALL PASS exit 0 [3s] ================================================================================================== test_343_chiral_form.py ================================================================================================== P1 so(4) = su(2)+ (+) su(2)- [PASS] each half closes and has dimension 3; the halves commute and sum to so(4) P2 Theorem 9.1 on a generic metric and K [PASS] curvature of each half = projection of the curvature of Omega; blocks = Gauss and Codazzi residuals P3 a realizable hypersurface, and a scaled K [PASS] paraboloid in R^4: both halves flat; 1.1 K: Gauss block nonzero, Codazzi block zero, both halves curved P4 Corollary 9.2: K = K~(g) on a non-realizable metric [PASS] Gauss block 0; Codazzi block = Cod K~ != 0; the halves are equal and opposite (su(2) labels opposite) -- Gauss 0 True, Codazzi != 0 True, = Cod K~ True, opposite True P5 Proposition 9.3: reversing the normal [PASS] K -> -K: Gauss block unchanged, Codazzi block reversed, Omega -> R Omega R, the halves exchanged test_343: ALL PASS exit 0 [8s] ================================================================================================== test_344_second_read_checks.py ================================================================================================== P1 the chart's rotation [PASS] d_r n = -W + (W.n) n + rho_c x n; the trajectory condition holds with rho = (u2', -u1', 0)/u3, not with +rho_c P2 the shell variation, coordinate components [PASS] the five printed components are the coordinate components; the rest vanish; lowered M_1 is symmetric -- (M_1)_r_theta = -Derivative(Phi(r), r)*Derivative(Y(theta), theta)/Phi(r) P3 T and R vanish at a point where rank M <= 1 [PASS] P(0) = 0, d det M(0) = 0, T(0) = R(0) = 0 at rank 1 and rank 0 (rank >= 2 nearby); nonzero at rank 2 -- rank 1: silent; rank 0: silent; rank 2 control: nonzero P4 the flag bounds from the linearised immersions [PASS] free of the first coordinate: rank 19 = 3*5 + 4; of the first two: rank 4 = 3 + 1 (injective), at four K -- diag(1, 2, 3): 19/19, 4/4; diag(2, 3, -3): 19/19, 4/4; diag(1, 1, 1): 19/19, 4/4; diag(1, 1, -2): 19/19, 4/4 test_344: ALL PASS exit 0 [48s]