check_note.py: OWR-14601-005 (Python 3.13.5)
A1   PASS  Raicu's sum (2): grouped evaluation == term-by-term evaluation for 0 <= r <= 14
A2   PASS  closed form (4) of the multiplicities == Raicu's case statement, r <= 120
A3   PASS  telescoped form (5) == (2) for r <= 120
A4   PASS  inclusion-exclusion of Step 3 for S(r,g), r in {7,12,25,40,61}, 1 <= g <= 10
     HF(0..12) = [1, 32, 528, 5216, 35539, 184096, 774368, 2768160, 8686997, 24508704, 63288368, 151659872, 340929511]
A5   PASS  HF(r) == P(r) for all 0 <= r <= 120 (first failures: [])
A6   PASS  Step 4: HF == P at >= 12 values of each class with r <= 48 (odd: 24, 2 mod 4: 12, 0 mod 4: 12)
A7   PASS  (1-t)^12 * sum_{r<=120} HF(r) t^r = [1, 20, 210, 772, 1250, 772, 210, 20, 1] + 0*t^9 + ... + 0*t^120
A8   PASS  h symmetric, h(1) = 3256 = deg Y
A9   PASS  P(-4-k) == -P(k) for |k| <= 60; P(-1) = P(-2) = P(-3) = 0; P(-4) = -1
A10  PASS  consistency: 12th differences of HF vanish on each class r mod 4 (2 <= r <= 120)
B1   PASS  26 generators g_W = (1; chi_W), |W| >= 2; (0;e_i) and (1;0) are integer combinations
B2   PASS  Prop 2.2(a): the 11 inequalities hold on the generators, each is a facet (rank 5), and the vertices of Q are exactly the 26 points chi_W (all 26)
B3   PASS  Prop 2.2(b): lattice points of the cone at heights k <= 6 = sums of k generators [26, 192, 802, 2438, 6054, 13069]
B4   PASS  Prop 2.2(c): <u, m0> = 1 for all 11 normals; interior lattice points (heights <= 8) = m0 + cone lattice points
B5   PASS  Theorem 2.1(a) (MOZ Lemma 3.1): z_I(p(t,a,b)) = t(1-t)(1-2t)^(|I|-2) prod(b_i - a_i) for all 26 sets |I| >= 2, and z_i = (1-t)a_i + t b_i
B6   PASS  Lemma 4.2: x_empty(phi(P0)) = 0, x_[5](phi(P0)) = 274, rank of the 32x20 Jacobian at P0 = 12 (at 3 random points: [12, 12, 12])
B7   PASS  Lemma 4.4(a): pull-back of s_0 = +-t(1-t) dt da db at 3 random rational points (ratios ['-1', '-1', '-1']); z_I = t_0 prod t_i at these points
B8   PASS  Lemma 4.4(b): sigma_K(p(t,a,b)) = N_K p(t~,a~,b~) for all 32 sets K (random rational points)
B9   PASS  Lemma 4.4(c): pull-back of s_K = N_K^(-4) * pull-back of s_0 for all 32 sets K, computed from the cumulant formulas
C1   PASS  r = 2: certified 528 <= HF(2) <= 528 in all 6 weight orbits (Raicu: 528)
C2   PASS  r = 3: certified 5216 <= HF(3) <= 5216 in all 6 weight orbits (Raicu: 5216); dim I_3 = 768
C3   PASS  r = 4: certified 35539 <= HF(4) <= 35539 in all 21 weight orbits (Raicu: 35539)
C4   PASS  r = 5: certified 184096 <= HF(5) <= 184096 in all 21 weight orbits (Raicu: 184096)
D1   PASS  Theorem B(i): (n.c + 41505)^2 - |n|^2 rho^2 = 61643/4 > 0; the six F_i lie in the ball on n.z = h
D2   PASS  Theorem B(ii): the lattice points of the ball on n.z = h are exactly F_0..F_5 (slice radius^2 = 270429/43372)
D3   PASS  Theorem B(ii),(iii): F_i affinely independent; w = (413, 528, -572, 656, 472, -12) is integral, n.w = 3h, w is not a sum of three F_j; 0, e_1..e_6 in the ball
E1   PASS  Remark 7.1: for f = x0^(p-1) x1 over F_p (p <= 13) and x = (a:b), a != 0: grad f(x) = (-b:a) != 0 and x lies on the hyperplane grad f(x)
SUMMARY: 27 checks, 0 failed; ALL PASS
(running time 10 s)
