[OK] pi > 3.14159 implies sqrt(2/pi) < 79789/10^5   (needs pi > 3.1415498214)
[OK] pi > 314159/100000 (Arb enclosure of pi)

== Hand cases q = 3/8, 2/5, 5/12 (exact) ==
[OK] q=3/8: tau^2, theta, kappa, kappa' as tabulated
[OK] q=3/8: a_B = kappa/(1-kappa) = 5/7
[OK] q=3/8: p_q = 400 * cubic, coprime integer coefficients, positive multiple
[OK] q=3/8: p_q(-1,0,s-,s+,2) = ('-476', '9', '-2473/10000', '-729/80', '649')
[OK] q=3/8: sign pattern -,+,-,-,+  => roots in (-1,0),(0,s-),(s+,2); p_q<0 on [s-,s+]
[OK] q=3/8: [s(a_A), s(a_B)] in [s-, s+]   (s(a_A)^2=9/91, s(a_B)^2=25/24)
[OK] q=3/8: (11) with a_A=3/10: 79789e-5 + a_A C_q <= 2tau  (margin >= 0.00575)
[OK] q=3/8: Remark 9.8(a) Case A margin 2tau - 0.7979 - a_A C_q = 0.005744 >= 0.0049
[OK] q=2/5: tau^2, theta, kappa, kappa' as tabulated
[OK] q=2/5: a_B = kappa/(1-kappa) = 3/5
[OK] q=2/5: p_q = 900 * cubic, coprime integer coefficients, positive multiple
[OK] q=2/5: p_q(-1,0,s-,s+,2) = ('-959', '25', '-1531/200', '-119', '769')
[OK] q=2/5: sign pattern -,+,-,-,+  => roots in (-1,0),(0,s-),(s+,2); p_q<0 on [s-,s+]
[OK] q=2/5: [s(a_A), s(a_B)] in [s-, s+]   (s(a_A)^2=1/8, s(a_B)^2=9/16)
[OK] q=2/5: (11) with a_A=1/3: 79789e-5 + a_A C_q <= 2tau  (margin >= 0.00500)
[OK] q=2/5: Remark 9.8(a) Case A margin 2tau - 0.7979 - a_A C_q = 0.004988 >= 0.0049
[OK] q=5/12: tau^2, theta, kappa, kappa' as tabulated
[OK] q=5/12: a_B = kappa/(1-kappa) = 7/13
[OK] q=5/12: p_q = 7056 * cubic, coprime integer coefficients, positive multiple
[OK] q=5/12: p_q(-1,0,s-,s+,2) = ('-6887', '225', '-1195767/15625', '-1343', '2521')
[OK] q=5/12: sign pattern -,+,-,-,+  => roots in (-1,0),(0,s-),(s+,2); p_q<0 on [s-,s+]
[OK] q=5/12: [s(a_A), s(a_B)] in [s-, s+]   (s(a_A)^2=49/351, s(a_B)^2=49/120)
[OK] q=5/12: (11) with a_A=7/20: 79789e-5 + a_A C_q <= 2tau  (margin >= 0.00571)
[OK] q=5/12: Remark 9.8(a) Case A margin 2tau - 0.7979 - a_A C_q = 0.005701 >= 0.0049

   displayed decimal bounds of the Case-A paragraph:
[OK] q=3/8: C_q = 17/(8 sqrt15) < 5487/10000
[OK] q=3/8: a_A C_q < 1647/10000
[OK] q=3/8: 0.7979 + 1647/10000 <= 4813/5000
[OK] q=3/8: 2tau > 4841/5000
[OK] q=3/8: 4813/5000 < 4841/5000
[OK] q=2/5: C_q = 13/(10 sqrt6) < 1327/2500
[OK] q=2/5: a_A C_q < 177/1000
[OK] q=2/5: 0.7979 + 177/1000 <= 9749/10000
[OK] q=2/5: 2tau > 9797/10000
[OK] q=2/5: 9749/10000 < 9797/10000
[OK] q=5/12: C_q = 37/(12 sqrt35) < 1303/2500
[OK] q=5/12: a_A C_q < 73/400
[OK] q=5/12: 0.7979 + 73/400 <= 2451/2500
[OK] q=5/12: 2tau > 493/500
[OK] q=5/12: 2451/2500 < 493/500
[OK] sqrt(2/pi) < 0.7979 (pi > 3.14159 => 2/pi < 0.7979^2)
[OK] s(3/10)^2 = 9/91, s(3/10) > 31/100
[OK] s(5/7)^2 = 25/24, s(5/7) < 21/20
[OK] s(1/3)^2 = 1/8, s(1/3) > 7/20
[OK] s(3/5)^2 = 9/16, s(3/5) = 3/4
[OK] s(7/20)^2 = 49/351, s(7/20) > 9/25
[OK] s(7/13)^2 = 49/120, s(7/13) < 1

   psi margins of Remark 9.8(a) (exact Sturm check of the cubic on [s(a_A), s(a_B)]):
[OK] q=3/8: max psi on [a_A, a_B] < 1987/2000
[OK] q=2/5: max psi on [a_A, a_B] < 604/625
[OK] q=5/12: max psi on [a_A, a_B] < 9539/10000

== General q in [3/8,1/2]: BF3 scheme re-implemented (64 intervals, a_A in 10^-4 Z) ==
[OK] a_A = 194/625 for j = 0
[OK] a_A = 4041/10^4 for j = 63
[OK] Case C empty (a_A >= a_B(q0)) exactly for j >= 48  (found j in 48..63, count 16)
[OK] q0 = 15/32 at j = 48
[OK] bisection mode 'quad': P < 0 certified on all 48 boxes; leaves = 2784, max depth = 6
[OK] bisection mode 's-only': P < 0 certified on all 48 boxes; leaves = 249, max depth = 6
[OK] bisection mode 'alternate': P < 0 certified on all 48 boxes; leaves = 1407, max depth = 11
   (informative) largest P at grid endpoints: -0.022288

ALL CHECKS OK
