--- row 1: p=16 r=1/2 a=23/100 h=['1']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=23/100 ; eta=0
    (informative) zero z_- of phi_a' outside |z|<=r: False
    RHS: exact finite sum, 1 terms ; a^p exact
    N=   30 : strict violation certified: True ; certified lower bound of ratio = 1.000472064342
    N=   60 : strict violation certified: True ; certified lower bound of ratio = 1.000472064342
    N=   90 : strict violation certified: True ; certified lower bound of ratio = 1.000472064342
    N=  120 : strict violation certified: True ; certified lower bound of ratio = 1.000472064342
    time 0.00s ; digits of largest integer ~ 555
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.0s)
--- row 2: p=121/10 r=1/2 a=55/73 h=['1', '13/50']
    case: B (|z_-| < r) ; s=sqrt(1-a^2)=48/73 ; e(0)=121/146 ; eta=13/50
    z_-=-5/11 (|z_-|=0.454545, r=0.500000)
    RHS: N_rhs=60, eta=13/50 (R=1), tail<=5.059e-36 ; Le=25761635700537751932118475687/250000000000000000000000000000
    N=   75 : strict violation certified: True ; certified lower bound of ratio = 1.001155233311
    N=  150 : strict violation certified: True ; certified lower bound of ratio = 1.001155233311
    N=  225 : strict violation certified: True ; certified lower bound of ratio = 1.001155233311
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.001155233311
    time 0.03s ; digits of largest integer ~ 3321
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.2s)
--- row 3: p=1201/100 r=1/2 a=171/221 h=['1', '7/25', '-3/100', '3/500']
    case: B (|z_-| < r) ; s=sqrt(1-a^2)=140/221 ; e(0)=361/442 ; eta=79/250
    z_-=-9/19 (|z_-|=0.473684, r=0.500000)
    RHS: N_rhs=60, eta=79/250 (R=1), tail<=8.904e-36 ; Le=87930600688458161780245914199/10^30, Le^Q<=e0^P checked exactly
    N=   75 : strict violation certified: True ; certified lower bound of ratio = 1.000062073298
    N=  150 : strict violation certified: True ; certified lower bound of ratio = 1.000062073298
    N=  225 : strict violation certified: True ; certified lower bound of ratio = 1.000062073298
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000062073298
    time 0.05s ; digits of largest integer ~ 5763
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.3s)
--- row 4: p=30017/2500 r=1/2 a=171/221 h=['1', '699/2500', '-77/2500', '11/2000', '9/5000']
    case: B (|z_-| < r) ; s=sqrt(1-a^2)=140/221 ; e(0)=361/442 ; eta=3177/10000
    z_-=-9/19 (|z_-|=0.473684, r=0.500000)
    RHS: N_rhs=60, eta=3177/10000 (R=1), tail<=9.055e-36 ; Le=21996894753832078704768621699/250000000000000000000000000000
    N=   75 : strict violation certified: True ; certified lower bound of ratio = 1.000004518941
    N=  150 : strict violation certified: True ; certified lower bound of ratio = 1.000004518941
    N=  225 : strict violation certified: True ; certified lower bound of ratio = 1.000004518941
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000004518941
    time 0.14s ; digits of largest integer ~ 7276
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.5s)
--- row 5: p=14 r=1233/2500 a=823/1000 h=['1', '139/500']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=823/1000 ; eta=139/500
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 8 terms ; a^p exact
    N=  100 : strict violation certified: True ; certified lower bound of ratio = 1.001209135713
    N=  200 : strict violation certified: True ; certified lower bound of ratio = 1.001209135713
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.001209135713
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.001209135713
    time 0.07s ; digits of largest integer ~ 5506
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 6: p=16 r=609/1250 a=857/1000 h=['1', '137/500']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=857/1000 ; eta=137/500
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 9 terms ; a^p exact
    N=  100 : strict violation certified: True ; certified lower bound of ratio = 1.001191925059
    N=  200 : strict violation certified: True ; certified lower bound of ratio = 1.001191925059
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.001191925059
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.001191925059
    time 0.07s ; digits of largest integer ~ 5522
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 7: p=18 r=4821/10000 a=883/1000 h=['1', '67/250']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=883/1000 ; eta=67/250
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 10 terms ; a^p exact
    N=  100 : strict violation certified: True ; certified lower bound of ratio = 1.000907283163
    N=  200 : strict violation certified: True ; certified lower bound of ratio = 1.000907283163
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000907283163
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.000907283163
    time 0.07s ; digits of largest integer ~ 5412
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 8: p=20 r=4779/10000 a=449/500 h=['1', '257/1000']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=449/500 ; eta=257/1000
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 11 terms ; a^p exact
    N=  100 : strict violation certified: True ; certified lower bound of ratio = 1.001091989394
    N=  200 : strict violation certified: True ; certified lower bound of ratio = 1.001091989394
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.001091989394
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.001091989394
    time 0.07s ; digits of largest integer ~ 5299
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 9: p=24 r=471/1000 a=923/1000 h=['1', '6/25']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=923/1000 ; eta=6/25
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 13 terms ; a^p exact
    N=  150 : strict violation certified: True ; certified lower bound of ratio = 1.000844513881
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000844513881
    N=  450 : strict violation certified: True ; certified lower bound of ratio = 1.000844513881
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000844513881
    time 0.16s ; digits of largest integer ~ 7640
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 10: p=30 r=927/2000 a=189/200 h=['1', '27/125']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=189/200 ; eta=27/125
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 16 terms ; a^p exact
    N=  150 : strict violation certified: True ; certified lower bound of ratio = 1.000675761987
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000675761987
    N=  450 : strict violation certified: True ; certified lower bound of ratio = 1.000675761987
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000675761987
    time 0.13s ; digits of largest integer ~ 6274
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 11: p=40 r=4553/10000 a=481/500 h=['1', '23/125']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=481/500 ; eta=23/125
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 21 terms ; a^p exact
    N=  200 : strict violation certified: True ; certified lower bound of ratio = 1.000710313001
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.000710313001
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000710313001
    N=  800 : strict violation certified: True ; certified lower bound of ratio = 1.000710313001
    time 0.35s ; digits of largest integer ~ 9669
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 12: p=50 r=2249/5000 a=973/1000 h=['1', '81/500']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=973/1000 ; eta=81/500
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 26 terms ; a^p exact
    N=  200 : strict violation certified: True ; certified lower bound of ratio = 1.000518946620
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.000518946620
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000518946620
    N=  800 : strict violation certified: True ; certified lower bound of ratio = 1.000518946620
    time 0.39s ; digits of largest integer ~ 11365
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 13: p=76 r=883/2000 a=4923/5000 h=['1', '1/8']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=4923/5000 ; eta=1/8
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 39 terms ; a^p exact
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000417779659
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000417779659
    N=  900 : strict violation certified: True ; certified lower bound of ratio = 1.000417779659
    N= 1200 : strict violation certified: True ; certified lower bound of ratio = 1.000417779659
    time 1.05s ; digits of largest integer ~ 17621
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.2s)
--- row 14: p=100 r=1093/2500 a=989/1000 h=['1', '21/200']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=989/1000 ; eta=21/200
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 51 terms ; a^p exact
    N=  300 : strict violation certified: True ; certified lower bound of ratio = 1.000373743855
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000373743855
    N=  900 : strict violation certified: True ; certified lower bound of ratio = 1.000373743855
    N= 1200 : strict violation certified: True ; certified lower bound of ratio = 1.000373743855
    time 1.01s ; digits of largest integer ~ 16782
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.2s)
--- row 15: p=150 r=2161/5000 a=124/125 h=['1', '2/25']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=124/125 ; eta=2/25
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 76 terms ; a^p exact
    N=  400 : strict violation certified: True ; certified lower bound of ratio = 1.000417869939
    N=  800 : strict violation certified: True ; certified lower bound of ratio = 1.000417869939
    N= 1200 : strict violation certified: True ; certified lower bound of ratio = 1.000417869939
    N= 1600 : strict violation certified: True ; certified lower bound of ratio = 1.000417869939
    time 1.54s ; digits of largest integer ~ 16793
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.1s)
--- row 16: p=200 r=2147/5000 a=2483/2500 h=['1', '33/500']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=2483/2500 ; eta=33/500
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 101 terms ; a^p exact
    N=  500 : strict violation certified: True ; certified lower bound of ratio = 1.000462029088
    N= 1000 : strict violation certified: True ; certified lower bound of ratio = 1.000462029088
    N= 1500 : strict violation certified: True ; certified lower bound of ratio = 1.000462029088
    N= 2000 : strict violation certified: True ; certified lower bound of ratio = 1.000462029088
    time 4.46s ; digits of largest integer ~ 31591
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.3s)
--- row 17: p=300 r=4263/10000 a=4969/5000 h=['1', '51/1000']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=4969/5000 ; eta=51/1000
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 151 terms ; a^p exact
    N=  600 : strict violation certified: True ; certified lower bound of ratio = 1.000659539871
    N= 1200 : strict violation certified: True ; certified lower bound of ratio = 1.000659539871
    N= 1800 : strict violation certified: True ; certified lower bound of ratio = 1.000659539871
    N= 2400 : strict violation certified: True ; certified lower bound of ratio = 1.000659539871
    time 8.12s ; digits of largest integer ~ 40992
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.3s)
--- row 18: p=500 r=2117/5000 a=497/500 h=['1', '37/1000']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=497/500 ; eta=37/1000
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 251 terms ; a^p exact
    N=  750 : strict violation certified: True ; certified lower bound of ratio = 1.001115409574
    N= 1500 : strict violation certified: True ; certified lower bound of ratio = 1.001115409574
    N= 2250 : strict violation certified: True ; certified lower bound of ratio = 1.001115409574
    N= 3000 : strict violation certified: True ; certified lower bound of ratio = 1.001115409574
    time 11.37s ; digits of largest integer ~ 42538
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.2s)
--- row 19: p=1000 r=4207/10000 a=4969/5000 h=['1', '123/5000']
    case: even (Lemma 7.1) ; s=sqrt(1-a^2)=None ; e(0)=4969/5000 ; eta=123/5000
    (informative) zero z_- of phi_a' outside |z|<=r: True
    RHS: exact finite sum, 501 terms ; a^p exact
    N= 1000 : strict violation certified: True ; certified lower bound of ratio = 1.002552736657
    N= 2000 : strict violation certified: True ; certified lower bound of ratio = 1.002552736657
    N= 3000 : strict violation certified: True ; certified lower bound of ratio = 1.002552736657
    N= 4000 : strict violation certified: True ; certified lower bound of ratio = 1.002552736657
    time 31.26s ; digits of largest integer ~ 72035
    two exact paths (ODE recurrence vs. recursion of Lemma 7.2) agree for n<=120: True  (0.3s)
