PASS (P0) chart OFF: m1(N) = m2(N) = 0, L(N) = 1, det(m1,m2,L) = 1 (unit); v o N^5 = 5 v(N) N^4, so (v,0) in Syz iff v in N_N  [0.0s]
PASS (P0) chart ON1: m1(N) = m2(N) = 0, L(N) = 1, det(m1,m2,L) = -1 (unit); v o N^5 = 5 v(N) N^4, so (v,0) in Syz iff v in N_N  [0.0s]
PASS (P0) chart ON2: m1(N) = m2(N) = 0, L(N) = 1, det(m1,m2,L) = 1 (unit); v o N^5 = 5 v(N) N^4, so (v,0) in Syz iff v in N_N  [0.0s]
   (P0) consequence: for V = <N^5, F> with r = 3, dim Syz = 6 - r = 3, Syz meets S_1 x 0 in N_N x 0 (dim 2), so Syz =
        <(m1,0), (m2,0), (alpha, v)> with v != 0 (else Syz would lie in S_1 x 0).  This is the only input from V.
PASS (U0) chart OFF: Bq vanishes at N and is a basis of I(N)_2 for every N (unit ideal of 5-minors)  [0.0s]
PASS (U1) chart OFF: A k(phi) = 0 identically, k(phi) != 0 iff rank(Phi K0) = 2 (k from the 2x2 minors)  [0.1s]
PASS (U2a) chart OFF: B_v involves only n1, n2, v  [0.1s]
PASS (U2) chart OFF: B_v (E2 on c_3, 15 x 5) has rank 5 iff v != 0: v0, v1, v2 in rad(I_5(B_v)) (Rabinowitsch)  [0.5s]
PASS (U2b) chart OFF: E2 does not involve c_1, c_2  [0.5s]
   (U3) chart OFF: nonzero rows of C: 10 of 15
PASS (U3) chart OFF: C K0 = 0 with K0 constant of rank 3, and 100 random 7-minors of C generate the unit ideal of QQ[n]: rank C = 7 for every N  [2.6s]
PASS (U4) chart OFF: Phi K0 = [[p2,p3,p5],[-p1,-p2,-p4]] = [phi(m2 l); -phi(m1 l)]_(l = m1,m2,L); hence for v != 0, dim K_V = 3 - rank(Phi K0)  [2.6s]
PASS (G) chart OFF: rank A over QQ(n, phi, alpha, v) = 14 (so dim K_V = 1 generically) [0.0s]  [2.7s]
PASS (U0) chart ON1: Bq vanishes at N and is a basis of I(N)_2 for every N (unit ideal of 5-minors)  [2.7s]
PASS (U1) chart ON1: A k(phi) = 0 identically, k(phi) != 0 iff rank(Phi K0) = 2 (k from the 2x2 minors)  [2.7s]
PASS (U2a) chart ON1: B_v involves only n1, n2, v  [2.7s]
PASS (U2) chart ON1: B_v (E2 on c_3, 15 x 5) has rank 5 iff v != 0: v0, v1, v2 in rad(I_5(B_v)) (Rabinowitsch)  [2.9s]
PASS (U2b) chart ON1: E2 does not involve c_1, c_2  [2.9s]
   (U3) chart ON1: nonzero rows of C: 10 of 15
PASS (U3) chart ON1: C K0 = 0 with K0 constant of rank 3, and 6 random 7-minors of C generate the unit ideal of QQ[n]: rank C = 7 for every N  [3.0s]
PASS (U4) chart ON1: Phi K0 = [[p2,p3,p5],[-p1,-p2,-p4]] = [phi(m2 l); -phi(m1 l)]_(l = m1,m2,L); hence for v != 0, dim K_V = 3 - rank(Phi K0)  [3.0s]
PASS (G) chart ON1: rank A over QQ(n, phi, alpha, v) = 14 (so dim K_V = 1 generically) [0.0s]  [3.0s]
PASS (U0) chart ON2: Bq vanishes at N and is a basis of I(N)_2 for every N (unit ideal of 5-minors)  [3.0s]
PASS (U1) chart ON2: A k(phi) = 0 identically, k(phi) != 0 iff rank(Phi K0) = 2 (k from the 2x2 minors)  [3.0s]
PASS (U2a) chart ON2: B_v involves only n1, n2, v  [3.0s]
PASS (U2) chart ON2: B_v (E2 on c_3, 15 x 5) has rank 5 iff v != 0: v0, v1, v2 in rad(I_5(B_v)) (Rabinowitsch)  [3.5s]
PASS (U2b) chart ON2: E2 does not involve c_1, c_2  [3.6s]
   (U3) chart ON2: nonzero rows of C: 7 of 15
PASS (U3) chart ON2: C K0 = 0 with K0 constant of rank 3, and 6 random 7-minors of C generate the unit ideal of QQ[n]: rank C = 7 for every N  [3.6s]
PASS (U4) chart ON2: Phi K0 = [[p2,p3,p5],[-p1,-p2,-p4]] = [phi(m2 l); -phi(m1 l)]_(l = m1,m2,L); hence for v != 0, dim K_V = 3 - rank(Phi K0)  [3.6s]
PASS (G) chart ON2: rank A over QQ(n, phi, alpha, v) = 14 (so dim K_V = 1 generically) [0.0s]  [3.6s]
   (U2)-(U4) chain: for v != 0, E2 forces c_3 = 0 (U2, U2b); then E1 reads C (c_1,c_2) = 0, whose solutions are K0 lambda for
   lambda in QQ^3 (U3: rank C = 7, K0 of rank 3 in ker C); the phi-conditions on c_3 are void and on c_1, c_2 read (Phi K0) lambda = 0.
PASS (U5) chart OFF: Q = m S_1 + C q, m in N_N, q in I(N)_2 symbolic: generic dim S_1 Q = 8 <= 8 (closed condition: everywhere)  [3.6s]
PASS (U5) chart ON1: Q = m S_1 + C q, m in N_N, q in I(N)_2 symbolic: generic dim S_1 Q = 8 <= 8 (closed condition: everywhere)  [3.6s]
PASS (U5) chart ON2: Q = m S_1 + C q, m in N_N, q in I(N)_2 symbolic: generic dim S_1 Q = 8 <= 8 (closed condition: everywhere)  [3.6s]
PASS (U5b) sanity: 19 of 19 k = 2 hyperplanes (I_Gamma)_2 through N = Z have dim S_1 Q = 8 and rank(Phi K0) <= 1  [3.7s]
   (SP) OFF, v = x, alpha = 0 (binary type (4)): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = x, alpha = -y/5 (non-binary type (4) syzygy): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = m1, alpha general: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = m2, alpha = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = L, alpha = L: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v general, alpha = v: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v in N_N general, alpha in N_N general: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = x, alpha = 0, p4 = p5 = 0 (Q > L N_N): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = x, alpha = 0, p1 = p2 = 0 (Q > m1 N_N): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = x, alpha = 0, p2 = p4 = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) OFF, v = x, alpha = 0, p1 = p3 = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = x, alpha = 0 (binary type (4)): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = x, alpha = -y/5 (non-binary type (4) syzygy): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = m1, alpha general: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = m2, alpha = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = L, alpha = L: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v general, alpha = v: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v in N_N general, alpha in N_N general: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = x, alpha = 0, p4 = p5 = 0 (Q > L N_N): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = x, alpha = 0, p1 = p2 = 0 (Q > m1 N_N): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = x, alpha = 0, p2 = p4 = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON1, v = x, alpha = 0, p1 = p3 = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = x, alpha = 0 (binary type (4)): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = x, alpha = -y/5 (non-binary type (4) syzygy): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = m1, alpha general: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = m2, alpha = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = L, alpha = L: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v general, alpha = v: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v in N_N general, alpha in N_N general: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = x, alpha = 0, p4 = p5 = 0 (Q > L N_N): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = x, alpha = 0, p1 = p2 = 0 (Q > m1 N_N): generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = x, alpha = 0, p2 = p4 = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
   (SP) ON2, v = x, alpha = 0, p1 = p3 = 0: generic rank A = 14, generic rank(Phi K0) = 2 -> dim K_V = 1
PASS (SP) 33 special strata (each generic on the stratum): dim K_V = 1 on all  [4.9s]
PASS (RP) 554 of 554 random rational points (30% zero coordinates): dim K_V = 3 - rank(Phi K0); histogram (dim K, predicted): [((1, 1), 462), ((2, 2), 92)]  [7.9s]
PASS (P1) chart OFF, V = <N^5, F>, F binary symbolic: (m1,0),(m2,0),(0,x) in Syz identically; (x I(N)_4) o V = 0; generic r = 3  [8.0s]
PASS (P1) chart ON1, V = <N^5, F>, F binary symbolic: (m1,0),(m2,0),(0,x) in Syz identically; (x I(N)_4) o V = 0; generic r = 3  [8.0s]
PASS (P1) chart ON2, V = <N^5, F>, F binary symbolic: (m1,0),(m2,0),(0,x) in Syz identically; (x I(N)_4) o V = 0; generic r = 3  [8.0s]
   (P1) so on the r = 3 locus Syz = <(m1,0),(m2,0),(0,x)> (alpha = 0, v = x), and (U) gives dim K_V = 1 at every point.
PASS (P2) (N_t^5 - Y^5)/(5t) is the polynomial sum_k C(5,k) t^(k-1) X^k Y^(5-k) / 5, equal to XY^4 at t = 0  [8.0s]
PASS (P2) (x - ty, 0), (z, 0), (-y/5, x) in Syz(V_t) identically in (t, G); N_(N_t) = <x - ty, z>  [8.0s]
PASS (P2) r_t <= 3 identically (generic rank 3 over QQ(t,G), a closed condition) and generic r_0 = 3  [8.0s]
PASS (P2) at t = 0: r = 3 iff (g2,g3,g4,g5) != 0, i.e. z o G not in C Y^4 (radical of the 3-minors = (g2,g3,g4,g5))  [8.7s]
PASS (P2) sigma_t(I(Y)_2) = I(N_t)_2 (5 independent quadrics vanishing at N_t, all t)  [8.7s]
PASS (P2) (x I(N_t)_4) o V_t = 0 identically: D_4(t) = S_1 o V_t inside D_3'(t)  [8.7s]
PASS (P2) rho_t over QQ(t, G, phi) = 11 and rho_0 over QQ(G, phi) = 11 (both 11)  [8.7s]
   (P2) every t with r_t = 3 is covered by (U) (N_t in chart OFF after scaling for t != 0, chart ON1 at t = 0): rho_t = 11.
PASS (P2) sample G: r_t < 3 only at the roots of 1 (finite, t = 0 not among them)  [9.0s]
total time 9.0 s
ALL CHECKS PASSED
