check_note.py -- OWR-9790358-011; certificates read from certificates
Part A: Proposition 5.1 (n = 3)
A1   PASS  Prop. 5.1: sum_k w_k x_k^alpha = int x^alpha W exactly in Q(sqrt2,sqrt3) for all 35 monomials of degree <= 4
A2   PASS  degree exactly 4: 6 of the 21 monomials of degree 5 are not integrated exactly: [(0, 0, 5), (0, 2, 3), (0, 4, 1), (2, 0, 3), (2, 2, 1), (4, 0, 1)]
A3   PASS  10 nodes, pairwise distinct: True; weights > 0: True; sum of weights = 1; all coordinates in (-1,1) (exact signs): True
A4   PASS  the 10 D4-invariant polynomials of degree <= 4 are integrated exactly; integrals = rule values: z^0: 1, z^1: 0, z^2: 1/4, z^3: 0, z^4: 1/8, (x^2+y^2) z^0: 1/2, (x^2+y^2) z^1: 0, (x^2+y^2) z^2: 1/8, (x^2+y^2)^2: 3/8, x^2 y^2: 1/16
A5   PASS  degree 5 (all integrals are 0): rule values of the invariants z^5 -> -1/64*sqrt2; (x^2+y^2) z^3 -> 1/32*sqrt2; (x^2+y^2)^2 z -> 0; x^2 y^2 z -> -1/64*sqrt2; z^5 alone shows that the degree is 4
A6   PASS  Lemma 2.1: every weight equals 1/K_2(x_k,x_k), K_2(x,x) = sum_{|alpha|<=2} P_alpha(x)^2 (exact)
A7   PASS  the 10 quasi-orthogonal polynomials P_alpha + sum_beta h*(alpha+beta) P_beta (alpha in Lambda_3, h* = +-1/sqrt2 on (0,0,5),(2,2,1) / (4,0,1),(2,0,3),(0,4,1),(0,2,3)) vanish at all 10 nodes
A8   PASS  h* solves (*) for d = 3, n = 3 (all 108 equations E_ij(b,b'), exact); its support is {gamma in Lambda_5: gamma_1, gamma_2 even} (6 indices), as Lemma 4.1 predicts for the node set, which is invariant under x -> -x and y -> -y: True
Part B: Lemmas 3.2 and 3.3, the system (star) for d = 3, n = 4
B1   PASS  three-term relation from the moments of w (Stieltjes, exact): alpha_k = 0 (k <= 8), beta_k = 1/4 (1 <= k <= 8); hence t p_k = p_{k+1}/2 + p_{k-1}/2 and A_{k,i} = (1/2) * shift
B2a  PASS  d=2 n=4: Lemma 3.2: (S) has rank 12 on 20 unknowns, nullity 8 = |Lambda_7|, and every multi-Hankel Gamma satisfies (S); Lemma 3.3: (C) has rank 6 on the 15 entries of a symmetric G, nullity 9 = |Lambda_8|, and every multi-Hankel G satisfies (C)
B2b  PASS  d=3 n=3: Lemma 3.2: (S) has rank 39 on 60 unknowns, nullity 21 = |Lambda_5|, and every multi-Hankel Gamma satisfies (S); Lemma 3.3: (C) has rank 27 on the 55 entries of a symmetric G, nullity 28 = |Lambda_6|, and every multi-Hankel G satisfies (C)
B2c  PASS  d=3 n=4: Lemma 3.2: (S) has rank 114 on 150 unknowns, nullity 36 = |Lambda_7|, and every multi-Hankel Gamma satisfies (S); Lemma 3.3: (C) has rank 75 on the 120 entries of a symmetric G, nullity 45 = |Lambda_8|, and every multi-Hankel G satisfies (C)
B3   PASS  the system (*) for d = 3, n = 4: 36 unknowns h(gamma), gamma in Lambda_7; 135 equations E_ij(b,b') (i < j, b < b'), all nonzero: True; distinct up to sign: 117; rank of the coefficient matrix over Q: 75; E_ij(b',b) = -E_ij(b,b'), E_ij(b,b) = 0
B4   PASS  route 2: 4 (A_i G A_j^T - (A_i G A_j^T)^T)[b,b'], computed from the matrices of the three-term relation and G = Gamma Gamma^T - I, equals E_ij(b,b') for all i < j and all b, b'
B5   PASS  Lemma 3.3 in pair form: the 75 equations G(alpha,alpha') = G(alpha'',alpha''') for equal sums (45 sums in Lambda_8) have rank 75, and together with the 135 equations E_ij(b,b') still rank 75: the two systems are equivalent
Part C: Theorems 6.1-6.3 (n = 4): the direct proof of 6.1 and the three Nullstellensatz certificates
C0   PASS  Theorem 6.1, direct proof: the 15 values of G_h used there (a = h(1,2,4), b = h(1,4,2), c = h(3,2,2); support gamma_2, gamma_3 even) are as stated, and the 9 pairs compared have equal sums
C1   PASS  Theorem 6.1 (two reflections y -> -y, z -> -z): support from Lemma 4.1 = the 10 unknowns of nullstellensatz_cert_100.json: True; restricted system: 36 distinct equations, rank 23; stored equations reproduced: 36/36; sum q_k E_k = 1 exactly: True (11 nonzero multipliers, 151 terms, degree <= 2, coefficients: numerators <= 8, denominators <= 1; pairs (i,j) used: [(1, 2)])
C2   PASS  Theorem 6.2 (one reflection z -> -z): support from Lemma 4.1 = the 20 unknowns of nullstellensatz_cert_100_010.json: True; restricted system: 63 distinct equations, rank 41; stored equations reproduced: 63/63; sum q_k E_k = 1 exactly: True (63 nonzero multipliers, 1575 terms, degree <= 2, coefficients: numerators <= 6623, denominators <= 144; pairs (i,j) used: [(1, 2), (1, 3), (2, 3)])
C3   PASS  Theorem 6.3 (half-turn (x,y,z) -> (x,-y,-z)): support from Lemma 4.1 = the 16 unknowns of nullstellensatz_cert_100_111.json: True; restricted system: 63 distinct equations, rank 41; stored equations reproduced: 63/63; sum q_k E_k = 1 exactly: True (60 nonzero multipliers, 394 terms, degree <= 2, coefficients: numerators <= 89, denominators <= 39; pairs (i,j) used: [(1, 2), (1, 3), (2, 3)])
Part D: two variables (Morrow-Patterson)
D1   PASS  d = 2, n = [2, 3, 4, 5, 6, 7, 8]: h = delta_(n,n-1) and h = delta_(n-1,n) solve (*) exactly; for h = delta_(n,n-1): G(alpha,alpha') = -[alpha = alpha' = (0,n)], a function of alpha + alpha' (g = -delta_(0,2n))
D2   PASS  float, n = 2..8: the n(n+1)/2 points (cos(i pi/(n+2)), cos(j pi/(n+1))), i+j odd, are common zeros of U_a(x)U_(n-a)(y) + U_(n-a)(x)U_(a-1)(y) (max |value| 1.1e-14); with weights 1/K_(n-1)(x,x) the rule is exact on Pi_(2n-2) (max error 2.2e-16) and not on Pi_(2n-1): True; invariant under y -> -y for odd n and under x -> -x for even n (not both): True; U_n(x) - U_(n-1)(y) does not vanish on them: True
Part E: negative controls
E1   PASS  a weight of Proposition 5.1 changed by 1/1000: moment check rejects it (5 of 35 identities fail)
E2   PASS  Theorem 6.1 certificate with one multiplier coefficient changed by 1/7: sum q_k E_k != 1 (7 leftover terms)
E3   PASS  Theorem 6.3 certificate against a perturbed equation (constant term + 1): sum q_k E_k != 1 (2 leftover terms)
E4   PASS  d = 2: the wrong input h = delta_(2n-1,0) is rejected by (*) for n = 2..8
E5   PASS  h* with one sign changed is not a solution of (*) (d = 3, n = 3)
E6   PASS  the support check distinguishes the reflection z -> -z (20 unknowns) from the half-turn (16 unknowns)
SUMMARY: 27 checks, 0 failed; ALL PASS
(running time 1 s)
