=== exact certificates for AMR-022-5039 ===
[EVEN] f=z at r=1/2: p=16, a=23/100, r=1/2 (~0.500000), h=['1'], N=160
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 1.000472064342
       exact M_p(r,f')^p                                ~ 1.000000000000
       ratio lower bound ~ 1.0004720643   CERTIFIED VIOLATION: True   (0.0s)
[FRAC] p=12.1 at r=1/2: p=121/10=12.1, a=55/73 (s=48/73), r=1/2, case B, z_-=-5/11, e(0)=121/146
       h=['1', '13/50'], R=1, eta=0.2600, N=70
       Le=257616357/2500000000 <= e(0)^p ~ 0.103046542802151  (exact check Le^Q <= e0^P passed)
       lower bound of M_p(r,g')^p ~ 1.689255618445
       upper bound of M_p(r,f')^p ~ 1.687306385936   (tail bound 4.825e-42)
       ratio lower bound ~ 1.0011552333   CERTIFIED VIOLATION: True   (0.0s)
[FRAC] p=12.01 at r=1/2: p=1201/100=12.01, a=171/221 (s=140/221), r=1/2, case B, z_-=-9/19, e(0)=361/442
       h=['1', '7/25', '-3/100', '3/500'], R=1, eta=0.3160, N=70
       Le=43965300343/500000000000 <= e(0)^p ~ 0.087930600688458  (exact check Le^Q <= e0^P passed)
       lower bound of M_p(r,g')^p ~ 1.769922176939
       upper bound of M_p(r,f')^p ~ 1.769812318901   (tail bound 8.492e-42)
       ratio lower bound ~ 1.0000620733   CERTIFIED VIOLATION: True   (0.1s)
[FRAC] p=12.0068 at r=1/2: p=120068/10000=12.0068, a=171/221 (s=140/221), r=1/2, case B, z_-=-9/19, e(0)=361/442
       h=['1', '699/2500', '-77/2500', '11/2000', '9/5000'], R=1, eta=0.3177, N=70
       Le=87987579013/1000000000000 <= e(0)^p ~ 0.087987579015328  (exact check Le^Q <= e0^P passed)
       lower bound of M_p(r,g')^p ~ 1.766255209587
       upper bound of M_p(r,f')^p ~ 1.766247228067   (tail bound 8.636e-42)
       ratio lower bound ~ 1.0000045189   CERTIFIED VIOLATION: True   (0.3s)
[EVEN] p=14 at r=0.4932: p=14, a=823/1000, r=1233/2500 (~0.493200), h=['1', '139/500'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 2.087818159704
       exact M_p(r,f')^p                                ~ 2.085296752928
       ratio lower bound ~ 1.0012091357   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=16 at r=0.4872: p=16, a=857/1000, r=609/1250 (~0.487200), h=['1', '137/500'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 2.410587761661
       exact M_p(r,f')^p                                ~ 2.407717942309
       ratio lower bound ~ 1.0011919251   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=18 at r=0.4821: p=18, a=883/1000, r=4821/10000 (~0.482100), h=['1', '67/250'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 2.749886835743
       exact M_p(r,f')^p                                ~ 2.747394171267
       ratio lower bound ~ 1.0009072832   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=20 at r=0.4779: p=20, a=449/500, r=4779/10000 (~0.477900), h=['1', '257/1000'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 3.024337106107
       exact M_p(r,f')^p                                ~ 3.021038164472
       ratio lower bound ~ 1.0010919894   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=24 at r=0.4710: p=24, a=923/1000, r=471/1000 (~0.471000), h=['1', '6/25'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 3.662098983407
       exact M_p(r,f')^p                                ~ 3.659008899600
       ratio lower bound ~ 1.0008445139   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=30 at r=0.4635: p=30, a=189/200, r=927/2000 (~0.463500), h=['1', '27/125'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 4.594164083162
       exact M_p(r,f')^p                                ~ 4.591061618238
       ratio lower bound ~ 1.0006757620   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=40 at r=0.4553: p=40, a=481/500, r=4553/10000 (~0.455300), h=['1', '23/125'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 6.100291506948
       exact M_p(r,f')^p                                ~ 6.095961466262
       ratio lower bound ~ 1.0007103130   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=50 at r=0.4498: p=50, a=973/1000, r=2249/5000 (~0.449800), h=['1', '81/500'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 7.791666701569
       exact M_p(r,f')^p                                ~ 7.787625339713
       ratio lower bound ~ 1.0005189466   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=76 at r=0.4415: p=76, a=4923/5000, r=883/2000 (~0.441500), h=['1', '1/8'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 12.539381103638
       exact M_p(r,f')^p                                ~ 12.534144592976
       ratio lower bound ~ 1.0004177797   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=100 at r=0.4372: p=100, a=989/1000, r=1093/2500 (~0.437200), h=['1', '21/200'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 17.791208145456
       exact M_p(r,f')^p                                ~ 17.784561274961
       ratio lower bound ~ 1.0003737439   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=150 at r=0.4322: p=150, a=124/125, r=2161/5000 (~0.432200), h=['1', '2/25'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 30.412501374684
       exact M_p(r,f')^p                                ~ 30.399798212841
       ratio lower bound ~ 1.0004178699   CERTIFIED VIOLATION: True   (0.0s)
[EVEN] p=200 at r=0.4294: p=200, a=2483/2500, r=2147/5000 (~0.429400), h=['1', '33/500'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 47.143901137460
       exact M_p(r,f')^p                                ~ 47.122129342979
       ratio lower bound ~ 1.0004620291   CERTIFIED VIOLATION: True   (0.1s)
[EVEN] p=300 at r=0.4263: p=300, a=4969/5000, r=4263/10000 (~0.426300), h=['1', '51/1000'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 103.268392123956
       exact M_p(r,f')^p                                ~ 103.200327393298
       ratio lower bound ~ 1.0006595399   CERTIFIED VIOLATION: True   (0.1s)
[EVEN] p=500 at r=0.4234: p=500, a=497/500, r=2117/5000 (~0.423400), h=['1', '37/1000'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 349.621552778987
       exact M_p(r,f')^p                                ~ 349.232016044559
       ratio lower bound ~ 1.0011154096   CERTIFIED VIOLATION: True   (0.2s)
[EVEN] p=1000 at r=0.4207: p=1000, a=4969/5000, r=4207/10000 (~0.420700), h=['1', '123/5000'], N=120
       lower bound of M_p(r,g')^p (partial Parseval sum) ~ 3765.758416348966
       exact M_p(r,f')^p                                ~ 3756.169903742484
       ratio lower bound ~ 1.0025527367   CERTIFIED VIOLATION: True   (0.9s)
ALL CERTIFICATES PASSED
