files / problems / riemann-hypothesis / compute / q2 / certificate / analytic / window.txt
--- A: exact target and elementary enclosures [PASS] A1 exact y ceiling [PASS] A2 target domain [PASS] A3 exact y0 square bracket 0.187271994702891975 < y0 < 0.187271994702891976 [PASS] A4 exact ymax square bracket 0.823103881657716397 < ymax < 0.823103881657716398 [PASS] A5 exact criterion-radius bracket 0.982308098307246981 < sqrt(1-y0^2) < 0.982308098307246982 [PASS] A6 rational Machin enclosure is ordered [PASS] A7 rational pi enclosure is narrow width < 1e-100 [PASS] A8 independent interval pi overlaps exact Machin bracket [PASS] A9 independent interval square roots fit exact brackets --- B: exact site and half-open window coverage [PASS] B1 first q_N is positive [PASS] B2 paper-domain denominators are positive [PASS] B3 target start is strictly right of x_N0 margin > 5377393.987856431175 [PASS] B4 target start is strictly left of x_(N0+1) margin > 11989041.174663815586 [PASS] B5 target start belongs to the closed-left/open-right N0 window [PASS] B6 symbolic consecutive-window identity q_(N+1)-q_N = 2N+1 > 0 for every N>=690988 [PASS] B7 finite row count is exact [PASS] B8 finite leg endpoints are consecutive combined with B6: union of half-open windows is [x*,x_(Nmid+1)) [PASS] B9 right edge of the finite union is well ordered x_(Nmid+1) > 185298771040833.378711 [PASS] B10 independent interval site margins are strict left >= [5377393.987856431175136911633099149776126187727656572931700057951232141261099799182947049268286019895591627289060738620228035, 5377393.987856431175136911633099149776126187727656572931700057951232141261099799182947049268286019895591627289060738620228035], right >= [11989041.17466381558610664242647868045345041857250426953691390606014297551374098583024085346700127178144343297556071481021988, 11989041.17466381558610664242647868045345041857250426953691390606014297551374098583024085346700127178144343297556071481021988] --- C: universal x-monotonicity and freeze directions [PASS] C1 gamma derivative hypotheses [PASS] C2 kappa derivative hypotheses [PASS] C3 positive-part atom has nonnegative coefficient [PASS] C4 log part of sigma is strictly increasing [PASS] C5 correction x^-2 max(0,h) is nonincreasing h_x=-8y(1+y)/x^3 <= 0 [PASS] C6 frozen directions G(x)<=G_N, K(x)<=K_N, Sigma(x)>=Sigma_N on every W_N [PASS] C7 boundary convention is fail-closed W_N=[x_N,x_(N+1)); x_(N+1) belongs to W_(N+1), where all three frozen bounds reset in the safer direction [PASS] C8 interval gamma derivative is strictly negative upper=[-0.000000000000000000001930226366128654794640712049833361247186855633261085953388146900220904274511034645976665298124379292731783516809839758622785, -0.000000000000000000001930226366128654794640712049833361247186855633261085953388146900220904274511034645976665298124379292731783516809839758622785] [PASS] C9 interval kappa derivative is strictly negative upper=[-0.0000000000000000000000000000004397415329915446034522144403270445352853953956126115148457536856931647705549068230081993293926264821105146376530345671259061, -0.0000000000000000000000000000004397415329915446034522144403270445352853953956126115148457536856931647705549068230081993293926264821105146376530345671259061] [PASS] C10 interval sigma derivative lower bound is strictly positive lower=[0.0000000000000002175540602539481099274928562447763753277053346890384099291191116406017473202423488797357455854834822001258297026476904917802, 0.0000000000000002175540602539481099274928562447763753277053346890384099291191116406017473202423488797357455854834822001258297026476904917802] [PASS] C11 first frozen corner is numerically regular G_N0=[0.08092027208085031433269964767525708205622001442886977828449103755857678506948407599522939937699358960331217398122318541179305, 0.08092027208085031433269964767525708205622001442886977828449103755857678506948407599522939937699358960331217398122318541247076] K_N0=[0.0000000000000025164696062256376651233246474987410759720491757814734824770485947965010942202669261531061906814746427406931532140401819058, 0.000000000000002516469606225637665123324647498741075972049175781473482477048594796501094220266926153106190681474642740693153214040181915431] Sigma_N0=[1.677709889853772829047842425377833293899018723165824745255299351772631839957065629033163473909112318226247845908805644471385, 1.677709889853772829047842425377833293899018723165824745255299351772631839957065629033163473909112318226247845908805644474483] TOTAL CHECKS RUN: 30 RESULT: ALL PASS CERTIFIED WINDOW-FREEZE CONCLUSION: at t0=129/800, for every integer N in [690988,3840000] and every x in [x_N(t0),x_(N+1)(t0)), the producer's frozen gamma and kappa values are upper bounds and its frozen sigma value is a lower bound, uniformly for y in [sqrt(87677/2500000),sqrt(271/400)]. The criterion start X+sqrt(1-y0^2) lies strictly inside W_N0, so the finite windows cover from that closed start through x_(Nmid+1) with only the final right endpoint excluded.