check_note.py -- OWR-15951-003 (Fillman's question; two couplings; Fricke-Vogt invariant)
== part_A
PASS A1 det(p+qJ+a s3+b s1) = p^2+q^2-a^2-b^2
PASS A2 tr(AB)/2 = yz - qA qB + Re(conj(zA) zB)
PASS A3 G(x,y,z) - [|qA zB - qB zA|^2 - (Im conj(zA) zB)^2] = (detA-y^2)(detB-z^2)-(1-y^2)(1-z^2)  [so the identity holds when detA = detB = 1]
PASS A4 Lemma 5.1 exactly on 200 random pairs in SL(2,Q)  [mismatches 0]
PASS A5 control: the wrong sign (+) fails on random pairs  [mismatches 200 of 200]
== part_B
PASS B1 X <-> (alpha, zeta) is a bijection on 100 random rational matrices
PASS B2 tr X / 2 = Re alpha and det X = |alpha|^2 - |zeta|^2
PASS B3 product rule (6) on 100 random pairs
PASS B4 X acts on C = R^2 as z -> alpha z + zeta conj(z)
PASS B5 operator norm = |alpha| + |zeta| (200 random real matrices)  [max diff 1.8e-15]
PASS B6 J <-> (-i, 0) and n <-> (i/2, i/2)
PASS B7 R(phi) <-> (e^{-i phi}, 0)
PASS B8 alpha(R(k)X) = e^{-ik} alpha(X), zeta(R(k)X) = e^{-ik} zeta(X)
PASS B9 D_k [[0,1],[g-k^2,0]] D_k^{-1} = kJ + (g/k) n
PASS B10 N(sigma) = R(-k sigma) n R(k sigma) <-> (i/2, (i/2) e^{2ik sigma})
PASS B11 zeta(N(s1)N(s2)) = (e^{2iks1} - e^{2iks2})/4, alpha = -1/4 + e^{2ik(s1-s2)}/4
PASS B12 R(k(1-s1)) n R(k(s1-s2)) n R(k s2) = R(k) N(s1) N(s2)
PASS B13 formula (9) for zeta(W_2): double integral = single integral (quadrature)  [max diff 6.7e-08 at grid 1/1200]
== part_C
PASS C1 (i/2k)(1/(2ik)) = 1/(4k^2)
PASS C2 |h1 e^{2it} - h0|^2 = (h1-h0)^2 + 4 h0 h1 sin^2 t (exact, 1125 cases)
PASS C3 two-case lower bound |h1 e^{2it} - h0|^2 >= sin^2 t (h0+h1)^2
PASS C4 h0 + h1 = 0: sin^2 t |h1 e^{2it} - h0|^2 = h1^2 sin^2(2t)
PASS C5 h constant = mu: L(t) = mu^2 sin^4 t / (4 t^4)
PASS C6 |I_(3.11) - L| t^5 bounded (no growth up to t = 254, size <= max(1,|mu|^3))  [mu=1: 0.0798, mu=-0.7: 0.0273, mu=2.5: 1.25, mu=12: 154]
PASS C7 logarithmic derivative of (3.11), last term -2t/(t^2-mu) (DFG Sec. 4.5 print 2t/sqrt(t^2-lambda)); central differences  [max rel. diff 8.3e-10]
== part_D
PASS D1 Lemma E constants: g < pi/2 < L, 0 < a < A <= L <= B = pi (3042 exact cases)
PASS D2 interval model of Lemma A (both families, two perturbations, N = 3): no gap in [E*, 6.16e+06]  [last gap ends at E* = 140.1]
PASS D3 first family alone: gap pi+alpha+beta > length pi-alpha-beta  [gap 4.242, length 2.042]
PASS D4 reflection identity Pi(2 pi(2n+1) - s) = Pi(s)  [max 4.7e-14]
== part_E
PASS E1 ||R(z)|| = e^{|Im z|} for complex z  [max rel. diff 2.2e-16]
== part_F
   pair (f_a, f_b) = (0, 1):
      t=20.42 I=1.4404e-06 (I-L)t^5=+0.0092 | t=23.04 I=4.8743e-07 (I-L)t^5=-0.0763
      t=32.99 I=2.1129e-07 (I-L)t^5=+0.0057 | t=35.60 I=8.6142e-08 (I-L)t^5=-0.0780
      t=58.12 I=2.1915e-08 (I-L)t^5=+0.0032 | t=60.74 I=1.0237e-08 (I-L)t^5=-0.0793
   pair (f_a, f_b) = (0, 1+x):
      t=20.42 I=3.2436e-06 (I-L)t^5=+0.0307 | t=23.04 I=1.1239e-06 (I-L)t^5=-0.2651
      t=32.99 I=4.7556e-07 (I-L)t^5=+0.0190 | t=35.60 I=1.9943e-07 (I-L)t^5=-0.2716
      t=58.12 I=4.9315e-08 (I-L)t^5=+0.0108 | t=60.74 I=2.3776e-08 (I-L)t^5=-0.2767
   pair (f_a, f_b) = (cos 2 pi x, x^2):
      t=20.42 I=3.9833e-07 (I-L)t^5=+0.1381 | t=23.04 I=1.6420e-07 (I-L)t^5=-0.0142
      t=32.99 I=5.4905e-08 (I-L)t^5=+0.0827 | t=35.60 I=2.8498e-08 (I-L)t^5=-0.0383
      t=58.12 I=5.5475e-09 (I-L)t^5=+0.0463 | t=60.74 I=3.3765e-09 (I-L)t^5=-0.0561
   pair (f_a, f_b) = (chi[0,1/2), chi[0,1/2)+x):
      t=20.42 I=3.6496e-07 (I-L)t^5=+0.0196 | t=23.04 I=1.5385e-07 (I-L)t^5=-0.0814
      t=32.99 I=5.3249e-08 (I-L)t^5=+0.0181 | t=35.60 I=2.7727e-08 (I-L)t^5=-0.0825
      t=58.12 I=5.5033e-09 (I-L)t^5=+0.0170 | t=60.74 I=3.3436e-09 (I-L)t^5=-0.0833
PASS F1 I by traces = I by formula (7) (relative)  [max rel. diff 2.0e-06]
PASS F2 |I - L| t^5 small for t in [20, 62] (illustration of Theorem C)  [max 0.277]
checks: 32, failed: 0
ALL CHECKS PASSED
running time 0.4 s
