   (M1) chart OFF: ok0 (kernel K0) = True, ok1 (unit 13-minor ideal) = True
PASS (M1) chart OFF, Syz = <(x,0),(0,x),(y,z)>: E-part rank 13 for every N (12 13-minors, unit ideal), kernel K0 = {(-xyl,-xzl,x^2 l): l in N_N} of rank 2 everywhere, I_2(Phi K0) = I_2(PK): K_V = 0 on the k = 1 locus  [0.4s]
PASS (M1) chart OFF: symbolic (a,b): rank over QQ(n,phi,a,b) = 15 (K_V = 0, rho = 12), random point 15  [0.4s]
   (M1) chart ON1: ok0 (kernel K0) = True, ok1 (unit 13-minor ideal) = True
PASS (M1) chart ON1, Syz = <(x,0),(0,x),(y,z)>: E-part rank 13 for every N (12 13-minors, unit ideal), kernel K0 = {(-xyl,-xzl,x^2 l): l in N_N} of rank 2 everywhere, I_2(Phi K0) = I_2(PK): K_V = 0 on the k = 1 locus  [0.5s]
PASS (M1) chart ON1: symbolic (a,b): rank over QQ(n,phi,a,b) = 15 (K_V = 0, rho = 12), random point 15  [0.5s]
   (M1) chart ON2: ok0 (kernel K0) = True, ok1 (unit 13-minor ideal) = True
PASS (M1) chart ON2, Syz = <(x,0),(0,x),(y,z)>: E-part rank 13 for every N (12 13-minors, unit ideal), kernel K0 = {(-xyl,-xzl,x^2 l): l in N_N} of rank 2 everywhere, I_2(Phi K0) = I_2(PK): K_V = 0 on the k = 1 locus  [0.6s]
PASS (M1) chart ON2: symbolic (a,b): rank over QQ(n,phi,a,b) = 15 (K_V = 0, rho = 12), random point 15  [0.6s]
PASS (M1) 169 of 169 random rational points: dim K_V = 2 - rank(PK) (= 0 iff k = 1)  [1.6s]
PASS (M2) (x,0),(y,0),(0,x) in Syz<Z^5, F> identically; generic r = 3  [1.6s]
PASS (M2) generic Q: rho_t over QQ(t,phi,f) = 12, rho_0 = 12  [1.6s]
PASS (M2) the E-part on (x I(N_t)_2)^3 has rank 13 for EVERY t (13-minors with gcd 1 in QQ[t])  [1.7s]
PASS (M2) K0(t) = {(xyl, -x^2 l, 0): l in N_(N_t)} (Syz order (x,0),(y,0),(0,x)) lies in the kernel and has rank 2 for every t  [1.8s]
   (M2) M_t = [p4, p5, -t*p4 - p1, -t*p5 - p2] ; M_0 = [p4, p5, -p1, -p2]
PASS (M2) M_t = E(t) M_0 (or M_0 E(t)) with E(t) unimodular: rank M_t = rank M_0 for every t and phi, so rho_t = rho_0 for all t  [1.8s]
PASS (M2) stratum Q > l<x,y>: rho_t over QQ(t, l, Q, f) = 11, rho_0 = 11  [1.8s]
PASS (M4) the divided differences f[-t,t], f[-t,t,2t] are polynomial in (t, c)  [1.9s]
PASS (M4) V_0 = <5Y^4Z, 5cXY^4 + 10Y^3Z^2>  [1.9s]
PASS (M4) N_t = P(2t) = (3ct^2, 1, 2t), N_(N_t) = <x - 3ct^2 y, z - 2ty>  [1.9s]
PASS (M4) (x I(N_t)_4) o V_t = 0 identically: D_4(t) inside D_3'(t)  [1.9s]
PASS (M4) r_t = 3 over QQ(t,c) and r_0 = 3 over QQ(c)  [1.9s]
PASS (M4) r_0 = 3 for EVERY c (gcd of the 3-minors of S_1 o V_0 is 1); r_t <= 3 as S_1 o V_t lies in nu_4(Gamma(t))  [4.4s]
PASS (M4) cleared Syz(V_t) basis lies in Syz identically  [4.4s]
   (M4) Syz(V_t) cleared basis at t = 0: [(x, 0), (y, -z), (c*z, -x)]
PASS (M4) Syz(V_t) has a basis polynomial in (t,c) whose value at t = 0 is independent for every c != 0 (gcd of the 3-minors: 1)  [4.5s]
PASS (M4) (x,0), (-5y,z), (-5cz,x) in Syz(V_0) for the basis (Y^4Z, 5cXY^4 + 10Y^3Z^2) of the table (the text writes 5Y^4Z: same pencil)  [4.5s]
PASS (M4) t = 0: K_0 (w_j in x I(Y)_2) = C(5x(xy - cz^2), x^3, -x^2 z) for every c != 0 (14-minors gcd c)  [5.0s]
   (M4) so at t = 0: dim K_V = 1 iff phi(xy - cz^2) = phi(x^2) = phi(xz) = 0, i.e. Q > (I_Gamma_c)_2; else K_V = 0 (rho_0 = 11 / 12)
PASS (M4) subcase 1: Q_t in I(N_t)_2  [5.0s]
PASS (M4) subcase 1: rho_t over QQ(t,c,phi) = 12, rho_0 over QQ(c,phi) = 12  [5.2s]
PASS (M4) subcase 2: (I_Gamma(t))_2 basis vanishes on Gamma(t)  [5.2s]
PASS (M4) subcase 2: Q_t polynomial in t, Q_0 = (I_Gamma_c)_2 + C q4, Q_t in I(N_t)_2  [5.2s]
PASS (M4) subcase 2: rho_t over QQ(t,c,q4) = 11, rho_0 over QQ(c,q4) = 11  [5.3s]
   (M4) kernel vector k(t,c): primitive in QQ[t,c]; gcd of the entries of k(0,c): 1
PASS (M4) subcase 2: dim ker((I_Gamma(t))_2 (x) Syz(V_t) -> S_3^2) = 1 over QQ(t,c), with a primitive polynomial generator k(t,c), k(0,c) != 0 for every c != 0  [5.3s]
PASS (T1) binary type (4), N = (0,1,n2) with n2 SYMBOLIC (the whole line minus Z), random phi, F: dim D3 = 6, r = 3, tails = 36 - 23 = 13 = 24 - rho (rho = 11) [0.8s]  [6.1s]
PASS (T1) binary type (4), N = Y, random phi, F: dim D3 = 6, r = 3, tails = 36 - 23 = 13 = 24 - rho (rho = 11) [0.1s]  [6.2s]
PASS (T1) binary type (4), N = X (off the line), random phi, F: dim D3 = 6, r = 3, tails = 36 - 23 = 13 = 24 - rho (rho = 11) [0.0s]  [6.2s]
PASS (T2) non-binary type (4) curve, symbolic t, random G, phi: tails = 13 = 24 - rho (rho = 11) [0.4s]  [6.6s]
PASS (T2) non-binary type (4) curve, t = 0 (collision), random G, phi: tails = 13 = 24 - rho (rho = 11) [0.2s]  [6.8s]
PASS (T3) move (ii), t = t: tails = 12 = 24 - rho (rho = 12)  [7.1s]
PASS (T3) move (ii), t = 0: tails = 12 = 24 - rho (rho = 12)  [7.3s]
PASS (T3) move (iv) subcase 1, c = 2, t = t: tails = 12 = 24 - rho (rho = 12)  [7.8s]
PASS (T3) move (iv) subcase 1, c = 2, t = 0: tails = 12 = 24 - rho (rho = 12)  [8.1s]
PASS (T3) move (iv) subcase 2, c = 2, t = t: tails = 13 = 24 - rho (rho = 11)  [8.7s]
PASS (T3) move (iv) subcase 2, c = 2, t = 0: tails = 13 = 24 - rho (rho = 11)  [8.8s]
PASS (C) type (4) binary, N = Y: rho = 11, tails = 13; 14 algebras (each basis tail + a random one, random E): closed, H = (1,3,6,6,3,2)  [9.2s]
PASS (C) type (4) binary, N = Y + 2Z: rho = 11, tails = 13; 14 algebras (each basis tail + a random one, random E): closed, H = (1,3,6,6,3,2)  [9.5s]
PASS (C) type (4) binary family, N = X + 2Y - Z (off): rho = 11, tails = 13; 14 algebras (each basis tail + a random one, random E): closed, H = (1,3,6,6,3,2)  [10.0s]
PASS (C) type (4) non-binary, t = 0 (collision): rho = 11, tails = 13; 14 algebras (each basis tail + a random one, random E): closed, H = (1,3,6,6,3,2)  [10.2s]
PASS (C) type (4) non-binary curve, t = 1/3: rho = 11, tails = 13; 14 algebras (each basis tail + a random one, random E): closed, H = (1,3,6,6,3,2)  [10.5s]
PASS (C) type (5)(ii), Q > (I_Gamma_c)_2, t = 0: rho = 11, tails = 13; 14 algebras (each basis tail + a random one, random E): closed, H = (1,3,6,6,3,2)  [10.8s]
total time 10.8 s
ALL CHECKS PASSED
