files: direct moments m_0..m_44; model moments m_0..m_306; second run cumulants k_2..k_560; main.tex found
== A  Lemma 2.2: first vectors and moments from the four operators (5)
ok     b Omega = 2*1_1
ok     b^2 Omega = 4 Omega + 4*1_2
ok     b^3 Omega = 12*1_1 + 8*1_3
ok     m_2 = 4, m_4 = 24, m_6 = 464/3 (as ||b^k Omega||^2)
ok     m_6/m_4 = 58/9 > 6
ok     m_8, m_10, m_12 from the same computation equal the recorded files
ok     vacuum components of b^2 Omega, b^4 Omega, b^6 Omega are m_2, m_4, m_6
== B  Table 1 and the integers n! m_2n
ok     direct moments m_0..m_44 equal the model moments
ok     n! m_2n for n = 0..10 is the table (3) of the source (direct moments)
ok     Table 1, rows n = 1..11 (m_2n, k_2n, n! m_2n, n! k_2n as printed)
ok     n! m_2n for n = 11..16 as displayed after Theorem 1.3 (model and direct moments)
ok     n! m_2n is an integer for all n <= 153, n! k_2n for 1 <= n <= 153
== C  Boolean cumulants
ok     k_2..k_306 recomputed from the moments by m_N = sum_j k_j m_(N-j) equal the recorded cumulants
ok     cumulants of the second run (k_2..k_560) equal those of the note on k_2..k_306
ok     k_2 = 4, k_4 = m_4 - k_2^2 = 8, k_6 = m_6 - 2 k_2 k_4 - k_2^3 = 80/3
ok     all k_2n > 0 (n <= 280)
ok     k_10/k_8 = 382/95 > 4
ok     k_306/k_304 > 2.1019907^2
ok     k_306/k_304 < 2.1019908^2 (so the square root is 2.1019907...)
ok     k_(2n+2)/k_(2n) is non-decreasing for n = 1..279 and below 6
== D  the quintic E(Y) of Proposition 5.3(b)
ok     E(Y) of (12) equals phi((Y^2-1)/4) - int_1^Y phi((Y^2-y^2)/4) dy at 8 rational points (degree 5: identity)
ok     E'(Y) = -(2/5)(Y-2)(Y^3-Y^2-5Y-2)
ok     Y^3-Y^2-5Y-2 = Y(Y^2-5) - (Y^2+2)
ok     E(2) = 1/50, E(1) = 8/5
ok     E >= 1/50 at 1251 rational points of [1, 9/4] (sanity; 81/16 > 5)
ok     float: condition (C) for Certificate B at s = 3/4 has slack sqrt(gamma)/50
== E  Theorem 1.2(d) and Table 2: enclosures recomputed with the bounds of Lemma 9.1, kappa_+ = 49/20
ok     kappa_+^2 = 2401/400 > 6
ok     the cumulants k_2..k_44 from the direct moments equal those of the model
ok     first line of (d): 2.58579 < c < 2.58604 and 0.25001 < w < 0.25313 from the direct moments m_0..m_44 (J = 20)
ok       Table 2, row 1: (2.585794744, 2.586036944) and (0.250011133, 0.253121332)
ok       2w > 1/2 from the direct moments alone: w > 0.2500111, margin over 1/4 between 1.1e-5 and 1.2e-5
ok       with m_0..m_42 only: lower bound for w is 0.2493... (not sufficient)
ok       with kappa_+ = 2.582 (Certificate A) and m_0..m_44: only 0.0717 < w < 0.6145
ok       first run (m_0..m_160, J = 78): 2.585826606048203 < c < 2.585826606048213, 0.25133165463828 < w < 0.25133165463861; widths 8.8e-15, 3.2e-13
ok       m_0..m_200 (J = 98): widths 2.2e-18 and 9.7e-17
ok     second line of (d): the two 27-digit intervals from m_0..m_306 (J = 151); widths 6.4e-28 and 4.2e-26
ok       nested: the enclosure from m_0..m_306 lies in that from m_0..m_200, which lies in those from m_0..m_160 and m_0..m_44
ok     abstract and Figure 1: c = 2.585826606048204448..., w = 0.251331654638419314..., 1 - 2w = 0.4973..., 2w > 1/2
== F  the enclosure from the cumulants k_2..k_560 of the second run (J = 278), and the displayed digits
ok     the end points recorded by the second run satisfy F_up(z_-) < 0 < F_lo(z_+); widths 9.0e-51 and 1.0e-48
ok     c = 2.5858266060482044486315152361168415078092428079979... and w = 0.25133165463841931453050567432858126056719804...
ok       this enclosure lies in the one from m_0..m_306
== G  Jacobi parameters
ok     beta_1..beta_8 = 4, 2, 4/3, 7/6, 11/10, 414/385, 16868/15939, 102259729/97766928
ok     beta_1 = k_2 and beta_1 beta_2 = k_4
ok     Table 4: beta_10, beta_20, beta_50 and k(beta_k - 1) (exact values, rounded)
ok     Table 4: beta_100, beta_150 from the recorded output of the note (25 digits)
ok     Table 4: beta_200, beta_240, beta_280 from the recorded output of the second run (30 digits)
== H  the example of Section 1.3
ok     Jacobi parameters (theta,1,1,...): zero of z - theta*G_sc(z) at theta/sqrt(theta-1), mass (theta-2)/(2(theta-1)), for theta = 13/4, 5, 29/4, 10, 53/4
ok     float: theta = 4 gives atoms at 4/sqrt(3) = 2.309... of mass 1/3
ok     sixth moment of 2 x arcsine (variance 1): 2^6 * C(6,3)/2^3 = 160, different from m_6 = 464/3
ok     m_a(u): arcsine moments C(a,a/2)(u/2)^(a/2): variance u (a = 2), and m_6(1) = 5/2
TOTAL failures: 0
