[S0.1] PASS: for 1<=i<=6, 1<=n<=300 every n has exactly one i-binomial representation (exhaustive search), and the greedy one is valid
[S0.2] PASS: pseudo-power values 2^<2>=2, 3^<1>=6, 6^<2>=10, 5^<2>=7, C(10,4)^<4>=C(11,5), 0^<3>=0; Kruskal-Katona 6^(2)=4, 3^(1)=3
[S0.3] PASS: for 1<=n<=i<=60: n = C(i,i)+C(i-1,i-1)+...+C(i-n+1,i-n+1) and n^<i> = n (the case n = k of the proof of Lemma D)
[S0.4] PASS: C(v-1+i,i)^<i> = C(v+i,i+1) for v<=11, i<=14 (polynomial rings attain Macaulay's bound)
[S1.1] PASS: h=(1,4,6,4,1) of the 4-cross-polytope: classical reading True, printed reading False (fails at i=1: g_2=2 > 3^<1>_printed=1; classical 3^<1>=6)
[S1.2] PASS: Theorem 7 of the source on 135 h-vectors of simplicial polytopes (cyclic C(n,d), 2<=d<=10, d+1<=n<=d+14, and cross-polytopes): violations with the classical pseudo-power 0, with the printed operator 91, with the Kruskal-Katona operator 91
[S1.3] PASS: cyclic polytope C(8,5), h=(1,3,6,6,3,1), g=(1,2,3): classical holds, Kruskal-Katona reading fails (2^(1)=1 < 3)
[S1.4] PASS: g=(1,5,3,2,1) is an M-sequence; h=(1,6,9,11,12,11,9,6,1) satisfies a g-theorem; its condition at i=3 uses 2^<3> = 2 with 2 < 3
[S1.5] PASS: (1+2t)(1+4t) = 1+6t+8t^2, g=(1,5,2): classical holds, the printed reading fails (5^<1>_printed=1 < 2)
[S2.1] PASS: C(d,i)^2/(C(d,i-1)C(d,i+1)) = (1+1/i)(1+1/(d-i)) for 2<=d<=200, 1<=i<=d-1
[S2.2] PASS: (1,2,2) and (1,4,8) satisfy (ULC) with equality at i=1, but 1+2t+2t^2 and 1+4t+8t^2 have negative discriminant
[S3.1] PASS: Lemma D, ((k+1)! a^<k>)^k >= (k! a)^(k+1), for all 1<=k<=30, 1<=a<=2500: 75000 pairs, violations 0; (F1), (F2) and the case n=k checked at every pair: errors 0
[S3.2] PASS: Lemma D on the blocks [C(n,k),C(n+1,k)), 2<=k<=12, k<=n<=k+30 (whole blocks of size <= 2000, otherwise 140 points each): 67992 pairs, violations 0
[S3.3] PASS: Lemma D on 2000 random pairs with k<=100, a<10^45: violations 0
[S3.4] PASS: gamma_k = q^k((k-1)/(k+1))^(k-1) = (k+1)(k-1)^(k-1)/k^k, gamma_2 = 3/4 and gamma_k < 3/4 (3<=k<=400); (k+1)^k >= k*k! (2<=k<=300); hockey stick sum_j C(n-k+j,j) = C(n,k-1) (1<=k<=n<=80); P = k!C(n,k), (k+1)!C(n+1,k+1) = (n+1)P, beta_max = k!C(n,k-1) = kP/(n-k+1), P' = (n+1)P/(n-k+1) = (n+1)n...(n-k+2) (2<=k<=40, k<=n<=k+80)
[S3.5] PASS: Step 2, (9): with beta = k s^(k-1), (k+1)(beta/k)^(k/(k-1)) - c beta = G(s) = (k+1)s^k - cks^(k-1); G(s*) = -c^k((k-1)/(k+1))^(k-1) at s* = c(k-1)/(k+1), i.e. beta* = c^(k-1)k((k-1)/(k+1))^(k-1), and G >= G(s*) on a grid of [0, 3s*] (2<=k<=30, five values of c), exactly
[S3.6] PASS: (10) (n+1) - P^(1/k) >= gamma_k (n+1)/(n-k+1) exactly for 2<=k<=30, k+1<=n<=k+150 (4350 cases), with the Step 4 inequalities P^(1/k) <= n-(k-1)/2, (n+1)/(n-k+1) <= (k+2)/2, 3(k+2)/8 <= (k+1)/2; smallest ratio LHS/RHS = 1.033674 at (k,n) = (2,3) (numerical), where (10) reads 4 - sqrt(6) >= 3/2, i.e. 25/4 >= 6
[S3.7] PASS: (8) (n+1)P + (k+1)(beta/k)^(k/(k-1)) >= (P+beta)^((k+1)/k) on 41-point grids of [0, beta_max], 2<=k<=12, k<=n<=k+30 (numerical, 60 digits): min LHS/RHS - 1 = 2.246e-02 at (k,n) = (2,32)
[S4.1] PASS: Proposition C, (k+1)! g_(k+1) <= (k! g_k)^((k+1)/k), with (5), (6), (7), exactly on 6000 random rational (ULC) sequences (lambda non-increasing; constant in a third of them): 33044 indices k with h_(k-1) < h_k, 31937 with g_(k+1) > 0; failures {'(5)': 0, '(6)': 0, '(7)': 0, 'PropC': 0}
[S4.2] PASS: (1,2,4) is log-concave but not (ULC), and g_2 = 2 > Phi_1(g_1) = 1/2: the hypothesis (ULC) cannot be weakened to log-concavity (and (1,2,4) does not satisfy a g-theorem: 1^<1> = 1 < 2)
[S5.1] PASS: Theorem A on every integer (ULC) sequence with h_0 = 1, 1<=h_i<=30, 2<=d<=10: 370612 sequences (largest first peak 4); failures {'shape': 0, 'g-theorem': 0, 'plateau': 0} (shape = Lemma 2.4, including at most one plateau)
[S6.0] PASS: exact Sturm-sequence test of real-rootedness on 11 known cases (with repeated roots)
[S6.1] PASS: all h = (1,h_1,...,h_d) with only real roots (exact Sturm) in the boxes d=2, h_i<=60; d=3, h_i<=40; d=4, h_i<=25: 3008 + 7538 + 542 polynomials; 11088 polynomials (max degree 4, max first peak 3, 136 palindromic); failures: Theorem B 0, Cor. 5.5 0, Cor. 5.2 0
[S6.2] PASS: (1+t)^n (n<=200), Eulerian and Narayana polynomials (n<=22, real roots certified by Sturm): 242 polynomials (max degree 200, max first peak 100, 242 palindromic); failures: Theorem B 0, Cor. 5.5 0, Cor. 5.2 0
[S6.3] PASS: 3000 random products of real-rooted factors (1+bt, 1+ct+t^2 with c>=2, 1+3t+t^2, 1+5t+5t^2, 1+7t+11t^2), degree 2..41: 3000 polynomials (max degree 41, max first peak 34, 1014 palindromic); failures: Theorem B 0, Cor. 5.5 0, Cor. 5.2 0
[S6.4] PASS: h-vectors of sd(Delta) from 400 random M-sequences h(Delta), d<=7 (formula of Brenti-Welker): nonnegative, real-rooted (Sturm) failures 0; Corollary 5.3 (source's Theorem 9): 400 polynomials (max degree 7, max first peak 4, 22 palindromic); failures: Theorem B 0, Cor. 5.5 0, Cor. 5.2 0; middle peak and g-vector up to floor(d/2) an M-sequence (Remark 5.4): failures 0
[S7.1] PASS: (1,4,8) is (ULC) and satisfies a g-theorem, but g=(1,3,4) is not an f-vector of a simplicial complex (3^(1) = C(3,2) = 3 < 4)
[S7.2] PASS: Phi_k(C(n,k)) > C(n,k+1) = C(n,k)^(k), exactly, for 1<=k<=30, k<=n<=k+60
[S7.3] PASS: Corollary 5.5: partial sums of every M-sequence (1,g_1,...,g_s), s<=4, g_1<=4 are an M-sequence: 1321 sequences, failures 0
[S8.1] PASS: Lemma D is asymptotically sharp for every k: at a = C(n,k), a^<k>/Phi_k(a) = (n+1)/P^(1/k) and n(ratio - 1) -> (k+1)/2 [k=1,n=1000000: n(ratio-1)=1.0000; k=2,n=1000000: n(ratio-1)=1.5000; k=3,n=1000000: n(ratio-1)=2.0000; k=5,n=1000000: n(ratio-1)=3.0000; k=10,n=1000000: n(ratio-1)=5.5000]; k=2, a = C(n,2)+n/2, n=10^6: n(ratio-1) = 0.7500 (-> 3/4)
[S8.2] PASS: for a <= 2000 the smallest value of a^<k>/Phi_k(a) is (a+1)/a at k=1, a=2000 (smallest values for k=1..4: 1.000500, 1.012032, 1.061621, 1.134535)
[S8.3] PASS: Proposition C is asymptotically sharp for every k: for h_i = L^i/i! the ratio g_(k+1)/Phi_k(g_k) < 1 tends to 1 and L^2(1 - ratio) -> (k+1)/2 [k=1,L=1000000: L^2(1-ratio)=1.000; k=2,L=1000000: L^2(1-ratio)=1.500; k=3,L=1000000: L^2(1-ratio)=2.000; k=4,L=1000000: L^2(1-ratio)=2.500]
ALL CHECKS PASSED
