check_certificates.py; data file certificates.json, sha256 ba5b98ee9b0bc2bcf33fe642341008676f06ab81911b83927e4972be5323e3cc
Python 3.13.5; parts ABCD
==============================================================================================================
N1000   (first results)   N = 1000
  parameters: alpha = 1, beta = 343447/200000000000, gamma = 1, phi = 1, b = 1/100, kappa = 13/2000000, nu1 = 13/4000000, L1 = 1, R = 1, S_T = 9906777/10000000
  Sigma* = 1/2;  nu = 13/4000000,  x0 = 1/100,  X = 10006777/10000000,  rho = 343447/100000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/100                                    sign +   [integers of 21892 and 21932 bits; sha256 of the reduced fraction P(x): 613e906173438d24034e0f9d]
       x = 197/200                                  sign -   [integers of 21904 and 21938 bits; sha256 of the reduced fraction P(x): cd7c5b992b64391178a6af4e]
       x = 1977/2000                                sign +   [integers of 21905 and 21939 bits; sha256 of the reduced fraction P(x): 1e5d20bf592efaa9f0ca9ed2]
       x = 9913/10000                               sign -   [integers of 21906 and 21939 bits; sha256 of the reduced fraction P(x): 8980fcbff7c35645af553c60]
       x = 9931/10000                               sign +   [integers of 21907 and 21940 bits; sha256 of the reduced fraction P(x): 8e42325f1b92f7f9261ee860]
       x = 10006777/10000000                        sign -   [integers of 24232 and 24265 bits; sha256 of the reduced fraction P(x): 6d79bfd18ab9f9b61ca6d1f7]
  C  enclosures (PREC = 256 bits): 939 closed intervals cover [x0, X] (910 without a zero, 29 monotone), deepest bisection 16, 1811 point evaluations, smallest interval 3.149e-07
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.983077006202698, 0.983079525626373), width 2.52e-06, P' < 0;  S in (0.973077006202698, 0.973079525626373)
       root 2 in (0.986976444196510, 0.986977074052429), width 6.30e-07, P' > 0;  S in (0.976976444196510, 0.976977074052429)
       root 3 in (0.990131707422161, 0.990132022350121), width 3.15e-07, P' < 0;  S in (0.980131707422161, 0.980132022350121)
       root 4 in (0.992433200949764, 0.992433515877724), width 3.15e-07, P' > 0;  S in (0.982433200949764, 0.982433515877724)
       root 5 in (0.993715272672653, 0.993716532384491), width 1.26e-06, P' < 0;  S in (0.983715272672653, 0.983716532384491)
  D  Descartes after Moebius maps (degree 1001, coefficients up to 21975 bits), 10 intervals:
       (1/100, 9/10): 0; (9/10, 97/100): 0; (97/100, 49/50): 0; (49/50, 197/200): 1; (197/200, 1977/2000): 1; (1977/2000, 9913/10000): 1; (9913/10000, 9931/10000): 1; (9931/10000, 199/200): 1; (199/200, 499/500): 0; (499/500, 10006777/10000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N1000: all executed parts agree with the expected values, 29.0 s]
==============================================================================================================
N2000   (first results)   N = 2000
  parameters: alpha = 1, beta = 363553/1000000000000, gamma = 1, phi = 1, b = 1/100, kappa = 7/5000000, nu1 = 7/10000000, L1 = 1, R = 1, S_T = 19804997/20000000
  Sigma* = 1/2;  nu = 7/10000000,  x0 = 1/100,  X = 20004997/20000000,  rho = 363553/500000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/100                                    sign +   [integers of 46464 and 46508 bits; sha256 of the reduced fraction P(x): d1eabbe5e97504e13e9fb67b]
       x = 1237/1250                                sign -   [integers of 46477 and 46514 bits; sha256 of the reduced fraction P(x): 5a7f8e3a74e1d6011708237e]
       x = 9929/10000                               sign +   [integers of 46478 and 46515 bits; sha256 of the reduced fraction P(x): e58ddfcdd2cef9d8a7a26b79]
       x = 4977/5000                                sign -   [integers of 46480 and 46516 bits; sha256 of the reduced fraction P(x): 16ee36b973c4f8db72967dde]
       x = 997/1000                                 sign +   [integers of 46481 and 46516 bits; sha256 of the reduced fraction P(x): 7aff75a7ec81213581e7a72b]
       x = 20004997/20000000                        sign -   [integers of 48485 and 48519 bits; sha256 of the reduced fraction P(x): c7c9820ae7a144d171753883]
  C  enclosures (PREC = 256 bits): 623 closed intervals cover [x0, X] (595 without a zero, 28 monotone), deepest bisection 14, 1180 point evaluations, smallest interval 1.259e-06
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.987821863830185, 0.987824382165782), width 2.52e-06, P' < 0;  S in (0.977821863830185, 0.977824382165782)
       root 2 in (0.991317313638433, 0.991318572806231), width 1.26e-06, P' > 0;  S in (0.981317313638433, 0.981318572806231)
       root 3 in (0.994530709859848, 0.994531969027646), width 1.26e-06, P' < 0;  S in (0.984530709859848, 0.984531969027646)
       root 4 in (0.996346429825083, 0.996347688992882), width 1.26e-06, P' > 0;  S in (0.986346429825083, 0.986347688992882)
       root 5 in (0.997711367718506, 0.997716404389699), width 5.04e-06, P' < 0;  S in (0.987711367718506, 0.987716404389699)
  D  Descartes after Moebius maps (degree 2001, coefficients up to 46553 bits), 10 intervals:
       (1/100, 9/10): 0; (9/10, 97/100): 0; (97/100, 197/200): 0; (197/200, 1237/1250): 1; (1237/1250, 9929/10000): 1; (9929/10000, 4977/5000): 1; (4977/5000, 997/1000): 1; (997/1000, 1997/2000): 1; (1997/2000, 1999/2000): 0; (1999/2000, 20004997/20000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N2000: all executed parts agree with the expected values, 227.3 s]
==============================================================================================================
N651-B   (verification run B)   N = 651
  parameters: alpha = 1, beta = 899032669057340117937713848273401/200000000000000000000000000000000000000, gamma = 1, phi = 1, b = 1/5000, kappa = 102451/6054300000, nu1 = 102451/12108600000, L1 = 1, R = 1, S_T = 500531032276116980167182393677011/500000000000000000000000000000000
  Sigma* = 1/2;  nu = 102451/12108600000,  x0 = 1/5000,  X = 500631032276116980167182393677011/500000000000000000000000000000000,  rho = 899032669057340117937713848273401/100000000000000000000000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/5000                                   sign +   [integers of 21756 and 21806 bits; sha256 of the reduced fraction P(x): 5f17f26fb9b21986a7a38825]
       x = 4925053534900169338751929153552433/5000000000000000000000000000000000 sign -   [integers of 83086 and 83225 bits; sha256 of the reduced fraction P(x): 67f9568933441a989ddf2c56]
       x = 9851439277791762446370751859307379/10000000000000000000000000000000000 sign +   [integers of 83736 and 83876 bits; sha256 of the reduced fraction P(x): 52a5031939b6d9399a3a5980]
       x = 9853261417168845942364713535616911/10000000000000000000000000000000000 sign -   [integers of 83736 and 83876 bits; sha256 of the reduced fraction P(x): 2c3ce2d25d6ee3860a74e7b7]
       x = 9854704834990181374116217174322753/10000000000000000000000000000000000 sign +   [integers of 83737 and 83876 bits; sha256 of the reduced fraction P(x): 5f88f0ea6734862d973d5c75]
       x = 500631032276116980167182393677011/500000000000000000000000000000000 sign -   [integers of 80934 and 81067 bits; sha256 of the reduced fraction P(x): 549e25a2af3d6f7960be63c1]
  C  enclosures (PREC = 256 bits): 101257 closed intervals cover [x0, X] (101234 without a zero, 23 monotone), deepest bisection 34, 202452 point evaluations, smallest interval 1.214e-12
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984972003872753, 0.984972003877608), width 4.86e-12, P' < 0;  S in (0.984772003872753, 0.984772003877608)
       root 2 in (0.985070392189187, 0.985070392190401), width 1.21e-12, P' > 0;  S in (0.984870392189187, 0.984870392190401)
       root 3 in (0.985219749793206, 0.985219749794420), width 1.21e-12, P' < 0;  S in (0.985019749793206, 0.985019749794420)
       root 4 in (0.985419251809892, 0.985419251812320), width 2.43e-12, P' > 0;  S in (0.985219251809892, 0.985219251812320)
       root 5 in (0.985505968403266, 0.985505968408122), width 4.86e-12, P' < 0;  S in (0.985305968403266, 0.985305968408122)
  D  Descartes after Moebius maps (degree 652, coefficients up to 21944 bits), 12 intervals:
       (1/5000, 4/5): 0; (4/5, 24/25): 0; (24/25, 491/500): 0; (491/500, 4923/5000): 0; (4923/5000, 197/200): 1; (197/200, 19703/20000): 1; (19703/20000, 9853/10000): 1; (9853/10000, 49273/50000): 1; (49273/50000, 24641/25000): 1; (24641/25000, 1973/2000): 0; (1973/2000, 497/500): 0; (497/500, 500631032276116980167182393677011/500000000000000000000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N651-B: all executed parts agree with the expected values, 194.1 s]
==============================================================================================================
N652-B   (verification run B)   N = 652
  parameters: alpha = 1, beta = 2240504329174974763041/500000000000000000000000000, gamma = 1, phi = 1, b = 1/2000, kappa = 17923/1062760000, nu1 = 17923/2125520000, L1 = 1, R = 1, S_T = 15636864963754464603/15625000000000000000
  Sigma* = 1/2;  nu = 17923/2125520000,  x0 = 1/2000,  X = 15644677463754464603/15625000000000000000,  rho = 2240504329174974763041/250000000000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/2000                                   sign +   [integers of 20155 and 20203 bits; sha256 of the reduced fraction P(x): 933a805d99c630136963b0a1]
       x = 9848694412313073313877/10000000000000000000000 sign -   [integers of 57141 and 57239 bits; sha256 of the reduced fraction P(x): ca1375fa45c1d8fb48786a8e]
       x = 1970207995574092915849/2000000000000000000000 sign +   [integers of 55629 and 55726 bits; sha256 of the reduced fraction P(x): 69d464151553b328c7e3b486]
       x = 9853768556003230077161/10000000000000000000000 sign -   [integers of 57141 and 57239 bits; sha256 of the reduced fraction P(x): 0281e68c7110bdaeadd04cdc]
       x = 4928379053658813722873/5000000000000000000000 sign +   [integers of 56490 and 56588 bits; sha256 of the reduced fraction P(x): aca06d98e76bce370c7d3614]
       x = 15644677463754464603/15625000000000000000 sign -   [integers of 51079 and 51166 bits; sha256 of the reduced fraction P(x): 82aaf7350d5b21755ab27ec7]
  C  enclosures (PREC = 256 bits): 43384 closed intervals cover [x0, X] (43361 without a zero, 23 monotone), deepest bisection 31, 86706 point evaluations, smallest interval 9.709e-12
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984800851183084, 0.984800851221919), width 3.88e-11, P' < 0;  S in (0.984300851183084, 0.984300851221919)
       root 2 in (0.984976544680347, 0.984976544699765), width 1.94e-11, P' > 0;  S in (0.984476544680348, 0.984476544699765)
       root 3 in (0.985243182420426, 0.985243182430135), width 9.71e-12, P' < 0;  S in (0.984743182420426, 0.984743182430135)
       root 4 in (0.985483683967780, 0.985483683987197), width 1.94e-11, P' > 0;  S in (0.984983683967780, 0.984983683987197)
       root 5 in (0.985773297620601, 0.985773297698270), width 7.77e-11, P' < 0;  S in (0.985273297620601, 0.985273297698270)
  D  Descartes after Moebius maps (degree 653, coefficients up to 20301 bits), 11 intervals:
       (1/2000, 4/5): 0; (4/5, 24/25): 0; (24/25, 491/500): 0; (491/500, 2461/2500): 0; (2461/2500, 9849/10000): 1; (9849/10000, 9851/10000): 1; (9851/10000, 4927/5000): 1; (4927/5000, 616/625): 1; (616/625, 493/500): 1; (493/500, 247/250): 0; (247/250, 15644677463754464603/15625000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N652-B: all executed parts agree with the expected values, 91.5 s]
==============================================================================================================
N655-B   (verification run B)   N = 655
  parameters: alpha = 1, beta = 8873320104916924370947/2000000000000000000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 35807/2145125000, nu1 = 35807/4290250000, L1 = 1, R = 1, S_T = 1000251064899138279703/1000000000000000000000
  Sigma* = 1/2;  nu = 35807/4290250000,  x0 = 1/1000,  X = 1001251064899138279703/1000000000000000000000,  rho = 8873320104916924370947/1000000000000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 20911 and 20959 bits; sha256 of the reduced fraction P(x): eacb4bd7b66213ae0908c164]
       x = 9845919235258108333059/10000000000000000000000 sign -   [integers of 56992 and 57090 bits; sha256 of the reduced fraction P(x): 7f36b59e2791d7faae22430f]
       x = 9850481597220069273089/10000000000000000000000 sign +   [integers of 56992 and 57090 bits; sha256 of the reduced fraction P(x): 52ea6c9ff83ef0372830decd]
       x = 4927943151056230688619/5000000000000000000000 sign -   [integers of 56338 and 56435 bits; sha256 of the reduced fraction P(x): d0b46a146573a8c69aa707c1]
       x = 986015581557359231213/1000000000000000000000 sign +   [integers of 54819 and 54914 bits; sha256 of the reduced fraction P(x): 346fac64a5f3e4aa982797f3]
       x = 1001251064899138279703/1000000000000000000000 sign -   [integers of 54826 and 54918 bits; sha256 of the reduced fraction P(x): e46213b931c1bce83c93ce39]
  C  enclosures (PREC = 256 bits): 18728 closed intervals cover [x0, X] (18703 without a zero, 25 monotone), deepest bisection 27, 37392 point evaluations, smallest interval 1.553e-10
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984458565966447, 0.984458566587484), width 6.21e-10, P' < 0;  S in (0.983458565966447, 0.983458566587484)
       root 2 in (0.984798713848218, 0.984798714003478), width 1.55e-10, P' > 0;  S in (0.983798713848218, 0.983798714003478)
       root 3 in (0.985320860834833, 0.985320860990092), width 1.55e-10, P' < 0;  S in (0.984320860834833, 0.984320860990092)
       root 4 in (0.985825225807163, 0.985825225962422), width 1.55e-10, P' > 0;  S in (0.984825225807163, 0.984825225962422)
       root 5 in (0.986135682968189, 0.986135683589227), width 6.21e-10, P' < 0;  S in (0.985135682968189, 0.985135683589227)
  D  Descartes after Moebius maps (degree 656, coefficients up to 21059 bits), 11 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 491/500): 0; (491/500, 9841/10000): 0; (9841/10000, 4923/5000): 1; (4923/5000, 197/200): 1; (197/200, 616/625): 1; (616/625, 493/500): 1; (493/500, 4933/5000): 1; (4933/5000, 99/100): 0; (99/100, 1001251064899138279703/1000000000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N655-B: all executed parts agree with the expected values, 48.0 s]
==============================================================================================================
N660-B   (verification run B)   N = 660
  parameters: alpha = 1, beta = 8718964109872981977743/2000000000000000000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 7147/435600000, nu1 = 7147/871200000, L1 = 1, R = 1, S_T = 250059242277367963817/250000000000000000000
  Sigma* = 1/2;  nu = 7147/871200000,  x0 = 1/1000,  X = 250309242277367963817/250000000000000000000,  rho = 8718964109872981977743/1000000000000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 19555 and 19601 bits; sha256 of the reduced fraction P(x): 6cca0934b7b03134b5bbd679]
       x = 9842863541815611322239/10000000000000000000000 sign -   [integers of 54806 and 54900 bits; sha256 of the reduced fraction P(x): 4837e206e637e639edcf948d]
       x = 4925225352191618348831/5000000000000000000000 sign +   [integers of 54147 and 54240 bits; sha256 of the reduced fraction P(x): b2f26410fd4df55c1747485b]
       x = 4929126954532510729091/5000000000000000000000 sign -   [integers of 54147 and 54240 bits; sha256 of the reduced fraction P(x): b5f37a62d56671080bdd2de9]
       x = 4932182990557152138029/5000000000000000000000 sign +   [integers of 54147 and 54240 bits; sha256 of the reduced fraction P(x): f9953947d78b980ecc5233f1]
       x = 250309242277367963817/250000000000000000000 sign -   [integers of 51306 and 51392 bits; sha256 of the reduced fraction P(x): 5ce0eb4a6fabb26d3f11294b]
  C  enclosures (PREC = 256 bits): 9849 closed intervals cover [x0, X] (9828 without a zero, 21 monotone), deepest bisection 25, 19638 point evaluations, smallest interval 6.210e-10
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984040690294718, 0.984040692778833), width 2.48e-09, P' < 0;  S in (0.983040690294718, 0.983040692778833)
       root 2 in (0.984701684831284, 0.984701686073342), width 1.24e-09, P' > 0;  S in (0.983701684831284, 0.983701686073342)
       root 3 in (0.985437783491365, 0.985437784112394), width 6.21e-10, P' < 0;  S in (0.984437783491365, 0.984437784112394)
       root 4 in (0.986167777437887, 0.986167778058916), width 6.21e-10, P' > 0;  S in (0.985167777437887, 0.985167778058916)
       root 5 in (0.986606493329822, 0.986606495813938), width 2.48e-09, P' < 0;  S in (0.985606493329822, 0.985606495813938)
  D  Descartes after Moebius maps (degree 661, coefficients up to 19701 bits), 11 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 981/1000): 0; (981/1000, 9837/10000): 0; (9837/10000, 2461/2500): 1; (2461/2500, 197/200): 1; (197/200, 4929/5000): 1; (4929/5000, 1233/1250): 1; (1233/1250, 9873/10000): 1; (9873/10000, 124/125): 0; (124/125, 250309242277367963817/250000000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N660-B: all executed parts agree with the expected values, 31.5 s]
==============================================================================================================
N700-B   (verification run B)   N = 700
  parameters: alpha = 1, beta = 7698967720440356451097/2000000000000000000000000000, gamma = 1, phi = 1, b = 1/100, kappa = 141/9800000, nu1 = 141/19600000, L1 = 1, R = 1, S_T = 991137518398439861899/1000000000000000000000
  Sigma* = 1/2;  nu = 141/19600000,  x0 = 1/100,  X = 1001137518398439861899/1000000000000000000000,  rho = 7698967720440356451097/1000000000000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/100                                    sign +   [integers of 16916 and 16958 bits; sha256 of the reduced fraction P(x): 6215b26294a6e6147ef6ad4a]
       x = 2459896093001758549731/2500000000000000000000 sign -   [integers of 53607 and 53695 bits; sha256 of the reduced fraction P(x): 062663a1d94ae5acdd488dfa]
       x = 9852902535805758084433/10000000000000000000000 sign +   [integers of 55005 and 55095 bits; sha256 of the reduced fraction P(x): 1b226e8d56663de2d475d8c3]
       x = 1233393510397404143039/1250000000000000000000 sign -   [integers of 52909 and 52995 bits; sha256 of the reduced fraction P(x): 85a698449148bebb2a892f83]
       x = 9883776008410726103333/10000000000000000000000 sign +   [integers of 55006 and 55095 bits; sha256 of the reduced fraction P(x): bb7b7294435ad8c5885dbb3f]
       x = 1001137518398439861899/1000000000000000000000 sign -   [integers of 52690 and 52774 bits; sha256 of the reduced fraction P(x): 02375c9cac8e27d070976b34]
  C  enclosures (PREC = 256 bits): 3404 closed intervals cover [x0, X] (3378 without a zero, 26 monotone), deepest bisection 21, 6744 point evaluations, smallest interval 9.846e-09
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.983551931423317, 0.983551970807583), width 3.94e-08, P' < 0;  S in (0.973551931423317, 0.973551970807583)
       root 2 in (0.984566450435497, 0.984566460281564), width 9.85e-09, P' > 0;  S in (0.974566450435497, 0.974566460281564)
       root 3 in (0.986036074338892, 0.986036084184959), width 9.85e-09, P' < 0;  S in (0.976036074338892, 0.976036084184959)
       root 4 in (0.987220339539566, 0.987220359231699), width 1.97e-08, P' > 0;  S in (0.977220339539566, 0.977220359231699)
       root 5 in (0.988892319804093, 0.988892477341159), width 1.58e-07, P' < 0;  S in (0.978892319804093, 0.978892477341159)
  D  Descartes after Moebius maps (degree 701, coefficients up to 17054 bits), 9 intervals:
       (1/100, 4/5): 0; (4/5, 24/25): 0; (24/25, 981/1000): 0; (981/1000, 123/125): 1; (123/125, 197/200): 1; (197/200, 4933/5000): 1; (4933/5000, 247/250): 1; (247/250, 619/625): 1; (619/625, 1001137518398439861899/1000000000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N700-B: all executed parts agree with the expected values, 18.7 s]
==============================================================================================================
N652-A   (verification run A)   N = 652
  parameters: alpha = 1, beta = 8956252108919407/2000000000000000000000, gamma = 1, phi = 1, b = 1/100000, kappa = 843117449/50000000000000, nu1 = 843117449/100000000000000, L1 = 1, R = 1, S_T = 312890330958161/312500000000000
  Sigma* = 1/2;  nu = 843117449/100000000000000,  x0 = 1/100000,  X = 312893455958161/312500000000000,  rho = 8956252108919407/1000000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/100000                                 sign +   [integers of 30260 and 30323 bits; sha256 of the reduced fraction P(x): acba0218f238a5100ac674c2]
       x = 61550049/62500000                        sign -   [integers of 30272 and 30330 bits; sha256 of the reduced fraction P(x): ea1e4b5e252f347bd31a9828]
       x = 492541509/500000000                      sign +   [integers of 30272 and 30330 bits; sha256 of the reduced fraction P(x): 609d3d000a5a0453f0bf28fb]
       x = 492715851/500000000                      sign -   [integers of 30272 and 30330 bits; sha256 of the reduced fraction P(x): e352a245656b91efadfe9508]
       x = 985702221/1000000000                     sign +   [integers of 30272 and 30330 bits; sha256 of the reduced fraction P(x): bba640c765787293596bbd9b]
       x = 312893455958161/312500000000000          sign -   [integers of 33302 and 33362 bits; sha256 of the reduced fraction P(x): 11058f3da4f92e5523a7e8eb]
  C  enclosures (PREC = 256 bits): 36491 closed intervals cover [x0, X] (36466 without a zero, 25 monotone), deepest bisection 30, 72918 point evaluations, smallest interval 1.943e-11
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984719869045673, 0.984719869123380), width 7.77e-11, P' < 0;  S in (0.984709869045673, 0.984709869123380)
       root 2 in (0.984929075160480, 0.984929075199333), width 3.89e-11, P' > 0;  S in (0.984919075160480, 0.984919075199333)
       root 3 in (0.985259333927061, 0.985259333946488), width 1.94e-11, P' < 0;  S in (0.985249333927061, 0.985249333946488)
       root 4 in (0.985581996197653, 0.985581996236507), width 3.89e-11, P' > 0;  S in (0.985571996197653, 0.985571996236507)
       root 5 in (0.985778076835882, 0.985778076913589), width 7.77e-11, P' < 0;  S in (0.985768076835882, 0.985768076913589)
  D  Descartes after Moebius maps (degree 653, coefficients up to 30409 bits), 11 intervals:
       (1/100000, 4/5): 0; (4/5, 24/25): 0; (24/25, 491/500): 0; (491/500, 2461/2500): 0; (2461/2500, 1231/1250): 1; (1231/1250, 9851/10000): 1; (9851/10000, 4927/5000): 1; (4927/5000, 9857/10000): 1; (9857/10000, 9861/10000): 1; (9861/10000, 247/250): 0; (247/250, 312893455958161/312500000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N652-A: all executed parts agree with the expected values, 81.6 s]
==============================================================================================================
N653-A   (verification run A)   N = 653
  parameters: alpha = 1, beta = 446229587927767/100000000000000000000, gamma = 1, phi = 1, b = 1/100000, kappa = 84019273/5000000000000, nu1 = 84019273/10000000000000, L1 = 1, R = 1, S_T = 3128894318443009/3125000000000000
  Sigma* = 1/2;  nu = 84019273/10000000000000,  x0 = 1/100000,  X = 3128925568443009/3125000000000000,  rho = 446229587927767/50000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/100000                                 sign +   [integers of 28140 and 28200 bits; sha256 of the reduced fraction P(x): 2815d1605514521723dbadb9]
       x = 984692359/1000000000                     sign -   [integers of 28153 and 28207 bits; sha256 of the reduced fraction P(x): db4f210665cb7250b33b6d23]
       x = 985056887/1000000000                     sign +   [integers of 28153 and 28207 bits; sha256 of the reduced fraction P(x): 95970329d5eb82d926d4d959]
       x = 123187141/125000000                      sign -   [integers of 28153 and 28207 bits; sha256 of the reduced fraction P(x): 6cfcafd493e6751c7a07d60e]
       x = 985842491/1000000000                     sign +   [integers of 28153 and 28207 bits; sha256 of the reduced fraction P(x): 36501c580bda6acbfa706b4a]
       x = 3128925568443009/3125000000000000        sign -   [integers of 34215 and 34276 bits; sha256 of the reduced fraction P(x): 2315aba8e3aaff520e75f31d]
  C  enclosures (PREC = 256 bits): 26599 closed intervals cover [x0, X] (26576 without a zero, 23 monotone), deepest bisection 28, 53136 point evaluations, smallest interval 7.771e-11
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984582625455500, 0.984582625610914), width 1.55e-10, P' < 0;  S in (0.984572625455500, 0.984572625610914)
       root 2 in (0.984855890970058, 0.984855891047765), width 7.77e-11, P' > 0;  S in (0.984845890970058, 0.984845891047765)
       root 3 in (0.985279491499306, 0.985279491577013), width 7.77e-11, P' < 0;  S in (0.985269491499306, 0.985269491577013)
       root 4 in (0.985692164802210, 0.985692164879917), width 7.77e-11, P' > 0;  S in (0.985682164802210, 0.985682164879917)
       root 5 in (0.985942489909721, 0.985942490065135), width 1.55e-10, P' < 0;  S in (0.985932489909721, 0.985932490065135)
  D  Descartes after Moebius maps (degree 654, coefficients up to 28282 bits), 11 intervals:
       (1/100000, 4/5): 0; (4/5, 24/25): 0; (24/25, 491/500): 0; (491/500, 4921/5000): 0; (4921/5000, 9847/10000): 1; (9847/10000, 197/200): 1; (197/200, 1971/2000): 1; (1971/2000, 4929/5000): 1; (4929/5000, 9863/10000): 1; (9863/10000, 989/1000): 0; (989/1000, 3128925568443009/3125000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N653-A: all executed parts agree with the expected values, 64.9 s]
==============================================================================================================
N655-A   (verification run A)   N = 655
  parameters: alpha = 1, beta = 8873328232221/2000000000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 8346141/500000000000, nu1 = 8346141/1000000000000, L1 = 1, R = 1, S_T = 5001255330114563/5000000000000000
  Sigma* = 1/2;  nu = 8346141/1000000000000,  x0 = 1/1000,  X = 5006255330114563/5000000000000000,  rho = 8873328232221/1000000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 26054 and 26111 bits; sha256 of the reduced fraction P(x): 629266a61ae015a02fc11d13]
       x = 984587209/1000000000                     sign -   [integers of 26067 and 26117 bits; sha256 of the reduced fraction P(x): 52fc38a34bc0e0ffc9164191]
       x = 197010941/200000000                      sign +   [integers of 26067 and 26117 bits; sha256 of the reduced fraction P(x): 74dd053f63a6212b16a82c1e]
       x = 985591877/1000000000                     sign -   [integers of 26067 and 26117 bits; sha256 of the reduced fraction P(x): c41ea9885341634b51da0f2c]
       x = 986010177/1000000000                     sign +   [integers of 26067 and 26117 bits; sha256 of the reduced fraction P(x): 223052061b8453ba2ee21ae6]
       x = 5006255330114563/5000000000000000        sign -   [integers of 34110 and 34170 bits; sha256 of the reduced fraction P(x): 354c2d51aeffc284f86b86c7]
  C  enclosures (PREC = 256 bits): 18739 closed intervals cover [x0, X] (18710 without a zero, 29 monotone), deepest bisection 27, 37410 point evaluations, smallest interval 1.553e-10
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984451289131736, 0.984451289752773), width 6.21e-10, P' < 0;  S in (0.983451289131736, 0.983451289752773)
       root 2 in (0.984804689179949, 0.984804689335208), width 1.55e-10, P' > 0;  S in (0.983804689179949, 0.983804689335208)
       root 3 in (0.985324548575155, 0.985324548730415), width 1.55e-10, P' < 0;  S in (0.984324548575155, 0.984324548730415)
       root 4 in (0.985827571641060, 0.985827571796320), width 1.55e-10, P' > 0;  S in (0.984827571641060, 0.984827571796320)
       root 5 in (0.986130925063863, 0.986130925684900), width 6.21e-10, P' < 0;  S in (0.985130925063863, 0.985130925684900)
  D  Descartes after Moebius maps (degree 656, coefficients up to 26181 bits), 11 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 491/500): 0; (491/500, 9841/10000): 0; (9841/10000, 4923/5000): 1; (4923/5000, 197/200): 1; (197/200, 616/625): 1; (616/625, 493/500): 1; (493/500, 4933/5000): 1; (4933/5000, 99/100): 0; (99/100, 5006255330114563/5000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N655-A: all executed parts agree with the expected values, 55.3 s]
==============================================================================================================
N660-A   (verification run A)   N = 660
  parameters: alpha = 1, beta = 4359422493891/1000000000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 6409/390625000, nu1 = 6409/781250000, L1 = 1, R = 1, S_T = 1000236952091663/1000000000000000
  Sigma* = 1/2;  nu = 6409/781250000,  x0 = 1/1000,  X = 1001236952091663/1000000000000000,  rho = 4359422493891/500000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 19451 and 19498 bits; sha256 of the reduced fraction P(x): 5c58fb015fb2eb45b5d7da5b]
       x = 1968623/2000000                          sign -   [integers of 21441 and 21484 bits; sha256 of the reduced fraction P(x): e5cea367a3ff31774b6ad542]
       x = 985010201/1000000000                     sign +   [integers of 22759 and 22804 bits; sha256 of the reduced fraction P(x): dc5bc1c0d7dbdfb7811d4d6f]
       x = 985824163/1000000000                     sign -   [integers of 22759 and 22804 bits; sha256 of the reduced fraction P(x): a595d4b8640c5ef6ff4dd903]
       x = 493224723/500000000                      sign +   [integers of 22100 and 22144 bits; sha256 of the reduced fraction P(x): f0cad76be3e6a7a5ff7c2e31]
       x = 1001236952091663/1000000000000000        sign -   [integers of 32840 and 32898 bits; sha256 of the reduced fraction P(x): afd183f09ccd17ee64704f0d]
  C  enclosures (PREC = 256 bits): 9713 closed intervals cover [x0, X] (9690 without a zero, 23 monotone), deepest bisection 25, 19364 point evaluations, smallest interval 6.210e-10
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.984104758777089, 0.984104761261204), width 2.48e-09, P' < 0;  S in (0.983104758777089, 0.983104761261204)
       root 2 in (0.984631036001211, 0.984631037243269), width 1.24e-09, P' > 0;  S in (0.983631036001211, 0.983631037243269)
       root 3 in (0.985421710041814, 0.985421710662843), width 6.21e-10, P' < 0;  S in (0.984421710041814, 0.984421710662843)
       root 4 in (0.986174236765160, 0.986174238007218), width 1.24e-09, P' > 0;  S in (0.985174236765160, 0.985174238007218)
       root 5 in (0.986623100260805, 0.986623102744921), width 2.48e-09, P' < 0;  S in (0.985623100260805, 0.985623102744921)
  D  Descartes after Moebius maps (degree 661, coefficients up to 19566 bits), 11 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 981/1000): 0; (981/1000, 9837/10000): 0; (9837/10000, 2461/2500): 1; (2461/2500, 197/200): 1; (197/200, 4929/5000): 1; (4929/5000, 1233/1250): 1; (1233/1250, 9873/10000): 1; (9873/10000, 124/125): 0; (124/125, 1001236952091663/1000000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N660-A: all executed parts agree with the expected values, 37.8 s]
==============================================================================================================
N680-A   (verification run A)   N = 680
  parameters: alpha = 1, beta = 4069150047/1000000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 7667/500000000, nu1 = 7667/1000000000, L1 = 1, R = 1, S_T = 1000183091843/1000000000000
  Sigma* = 1/2;  nu = 7667/1000000000,  x0 = 1/1000,  X = 1001183091843/1000000000000,  rho = 4069150047/500000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 20284 and 20331 bits; sha256 of the reduced fraction P(x): 922dafcd93ee2524b194e3ab]
       x = 983838721/1000000000                     sign -   [integers of 20296 and 20337 bits; sha256 of the reduced fraction P(x): 9f8f4e23a06ee1245acb93ac]
       x = 985076491/1000000000                     sign +   [integers of 20296 and 20337 bits; sha256 of the reduced fraction P(x): 5eecb070586a7c9f28cfedef]
       x = 986457957/1000000000                     sign -   [integers of 20297 and 20337 bits; sha256 of the reduced fraction P(x): 7d8ddd904ccb14aaeb26819c]
       x = 987573247/1000000000                     sign +   [integers of 20297 and 20337 bits; sha256 of the reduced fraction P(x): 7c4b034c6874ebedd3d9acfd]
       x = 1001183091843/1000000000000              sign -   [integers of 27070 and 27118 bits; sha256 of the reduced fraction P(x): 4352538d533e1dd7edf18bc1]
  C  enclosures (PREC = 256 bits): 4140 closed intervals cover [x0, X] (4111 without a zero, 29 monotone), deepest bisection 21, 8212 point evaluations, smallest interval 9.936e-09
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.983462208282507, 0.983462228154359), width 1.99e-08, P' < 0;  S in (0.982462208282507, 0.982462228154359)
       root 2 in (0.984398450747468, 0.984398460683395), width 9.94e-09, P' > 0;  S in (0.983398450747468, 0.983398460683395)
       root 3 in (0.985786956706509, 0.985786966642435), width 9.94e-09, P' < 0;  S in (0.984786956706509, 0.984786966642435)
       root 4 in (0.987018147144179, 0.987018157080105), width 9.94e-09, P' > 0;  S in (0.986018147144179, 0.986018157080105)
       root 5 in (0.987899473744389, 0.987899513488094), width 3.97e-08, P' < 0;  S in (0.986899473744389, 0.986899513488094)
  D  Descartes after Moebius maps (degree 681, coefficients up to 20389 bits), 9 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 981/1000): 0; (981/1000, 123/125): 1; (123/125, 197/200): 1; (197/200, 1233/1250): 1; (1233/1250, 79/80): 1; (79/80, 2473/2500): 1; (2473/2500, 1001183091843/1000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N680-A: all executed parts agree with the expected values, 28.6 s]
==============================================================================================================
N690-A   (verification run A)   N = 690
  parameters: alpha = 1, beta = 7869205719/2000000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 3709/250000000, nu1 = 3709/500000000, L1 = 1, R = 1, S_T = 5000788396031/5000000000000
  Sigma* = 1/2;  nu = 3709/500000000,  x0 = 1/1000,  X = 5005788396031/5000000000000,  rho = 7869205719/1000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 19894 and 19940 bits; sha256 of the reduced fraction P(x): cc4491a964c9a80976f2b127]
       x = 196743033/200000000                      sign -   [integers of 20595 and 20636 bits; sha256 of the reduced fraction P(x): 87ff2131d360aeb21749e689]
       x = 985151193/1000000000                     sign +   [integers of 20595 and 20636 bits; sha256 of the reduced fraction P(x): f13ddb9ac52679f1cdf63679]
       x = 39469711/40000000                        sign -   [integers of 20596 and 20636 bits; sha256 of the reduced fraction P(x): 04445125a5641a2b1095802d]
       x = 987982039/1000000000                     sign +   [integers of 20596 and 20637 bits; sha256 of the reduced fraction P(x): f2da221a187b4f4ebf9b9d06]
       x = 5005788396031/5000000000000              sign -   [integers of 29069 and 29119 bits; sha256 of the reduced fraction P(x): bd0eac739bbe615df41735ea]
  C  enclosures (PREC = 256 bits): 3556 closed intervals cover [x0, X] (3528 without a zero, 28 monotone), deepest bisection 21, 7046 point evaluations, smallest interval 9.936e-09
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.983277599489988, 0.983277639232684), width 3.97e-08, P' < 0;  S in (0.982277599489988, 0.982277639232684)
       root 2 in (0.984370583229153, 0.984370603100501), width 1.99e-08, P' > 0;  S in (0.983370583229153, 0.983370603100501)
       root 3 in (0.985971766750826, 0.985971776686500), width 9.94e-09, P' < 0;  S in (0.984971766750826, 0.984971776686500)
       root 4 in (0.987383973755592, 0.987383993626939), width 1.99e-08, P' > 0;  S in (0.986383973755592, 0.986383993626939)
       root 5 in (0.988335056200620, 0.988335095943316), width 3.97e-08, P' < 0;  S in (0.987335056200620, 0.987335095943316)
  D  Descartes after Moebius maps (degree 691, coefficients up to 19999 bits), 9 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 49/50): 0; (49/50, 123/125): 1; (123/125, 197/200): 1; (197/200, 987/1000): 1; (987/1000, 247/250): 1; (247/250, 4949/5000): 1; (4949/5000, 5005788396031/5000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N690-A: all executed parts agree with the expected values, 27.5 s]
==============================================================================================================
N700-A   (verification run A)   N = 700
  parameters: alpha = 1, beta = 1903047399/500000000000000, gamma = 1, phi = 1, b = 1/1000, kappa = 359/25000000, nu1 = 359/50000000, L1 = 1, R = 1, S_T = 2500332726089/2500000000000
  Sigma* = 1/2;  nu = 359/50000000,  x0 = 1/1000,  X = 2502832726089/2500000000000,  rho = 1903047399/250000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 17861 and 17903 bits; sha256 of the reduced fraction P(x): d785b693e6650b46a46ca268]
       x = 983655307/1000000000                     sign -   [integers of 20894 and 20935 bits; sha256 of the reduced fraction P(x): 39e21b84333bec4af6a5b3e6]
       x = 492618907/500000000                      sign +   [integers of 20195 and 20235 bits; sha256 of the reduced fraction P(x): 1d626cb36afc82cdb0a65897]
       x = 197392567/200000000                      sign -   [integers of 19272 and 19310 bits; sha256 of the reduced fraction P(x): 83bb062db3be3448b3a0d9c1]
       x = 197674417/200000000                      sign +   [integers of 19272 and 19310 bits; sha256 of the reduced fraction P(x): a512d51997c76d92ebf1e7e4]
       x = 2502832726089/2500000000000              sign -   [integers of 28791 and 28841 bits; sha256 of the reduced fraction P(x): f47d22f745fcfde60a70240d]
  C  enclosures (PREC = 256 bits): 2937 closed intervals cover [x0, X] (2907 without a zero, 30 monotone), deepest bisection 20, 5806 point evaluations, smallest interval 1.987e-08
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.983167051803476, 0.983167131286913), width 7.95e-08, P' < 0;  S in (0.982167051803476, 0.982167131286913)
       root 2 in (0.984379492148089, 0.984379512018948), width 1.99e-08, P' > 0;  S in (0.983379492148089, 0.983379512018948)
       root 3 in (0.986131267352958, 0.986131287223817), width 1.99e-08, P' < 0;  S in (0.985131267352958, 0.985131287223817)
       root 4 in (0.987637756671931, 0.987637776542790), width 1.99e-08, P' > 0;  S in (0.986637756671931, 0.986637776542790)
       root 5 in (0.988779854174994, 0.988779933658431), width 7.95e-08, P' < 0;  S in (0.987779854174994, 0.987779933658431)
  D  Descartes after Moebius maps (degree 701, coefficients up to 17961 bits), 9 intervals:
       (1/1000, 4/5): 0; (4/5, 24/25): 0; (24/25, 49/50): 0; (49/50, 123/125): 1; (123/125, 197/200): 1; (197/200, 987/1000): 1; (987/1000, 247/250): 1; (247/250, 1981/2000): 1; (1981/2000, 2502832726089/2500000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N700-A: all executed parts agree with the expected values, 24.0 s]
==============================================================================================================
N4000-A   (verification run A)   N = 4000
  parameters: alpha = 1, beta = 1668474131/20000000000000000, gamma = 1, phi = 1, b = 1/100, kappa = 811/2500000000, nu1 = 811/5000000000, L1 = 1, R = 1, S_T = 9900998411859/10000000000000
  Sigma* = 1/2;  nu = 811/5000000000,  x0 = 1/100,  X = 10000998411859/10000000000000,  rho = 1668474131/10000000000000000
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 6 values of S: True;  signs of the right-hand side of (1): + - + - + -   [ok]
  B  signs of P at the test points: + - + - + -   [ok]
       x = 1/100                                    sign +   [integers of 128823 and 128878 bits; sha256 of the reduced fraction P(x): 1483d0a06b35d7dec4ad79ac]
       x = 248120229/250000000                      sign -   [integers of 128837 and 128885 bits; sha256 of the reduced fraction P(x): 715d2591efb825492670712d]
       x = 249021379/250000000                      sign +   [integers of 128839 and 128886 bits; sha256 of the reduced fraction P(x): cd8d1b65ccde4c45a346b640]
       x = 997893897/1000000000                     sign -   [integers of 128841 and 128887 bits; sha256 of the reduced fraction P(x): 27bee82131045e1824db55d5]
       x = 249676357/250000000                      sign +   [integers of 128842 and 128888 bits; sha256 of the reduced fraction P(x): aa7c2e3e521316e965907cd4]
       x = 10000998411859/10000000000000            sign -   [integers of 172699 and 172754 bits; sha256 of the reduced fraction P(x): 700099855fb5ff268d7112d8]
  C  enclosures (PREC = 256 bits): 564 closed intervals cover [x0, X] (537 without a zero, 27 monotone), deepest bisection 14, 1063 point evaluations, smallest interval 1.259e-06
     number of roots of P in [x0, X]: 5, all simple; signs of P' at the roots: - + - + -   [ok]
       root 1 in (0.990808590540006, 0.990811108494110), width 2.52e-06, P' < 0;  S in (0.980808590540006, 0.980811108494110)
       root 2 in (0.994455847060488, 0.994457106037540), width 1.26e-06, P' > 0;  S in (0.984455847060487, 0.984457106037540)
       root 3 in (0.997180273401641, 0.997181532378694), width 1.26e-06, P' < 0;  S in (0.987180273401641, 0.987181532378694)
       root 4 in (0.998375042624264, 0.998376301601316), width 1.26e-06, P' > 0;  S in (0.988375042624264, 0.988376301601316)
       root 5 in (0.998901295032121, 0.998903812986225), width 2.52e-06, P' < 0;  S in (0.988901295032121, 0.988903812986225)
  D  Descartes after Moebius maps (degree 4001, coefficients up to 128937 bits), 9 intervals:
       (1/100, 4/5): 0; (4/5, 97/100): 0; (97/100, 247/250): 0; (247/250, 993/1000): 1; (993/1000, 249/250): 1; (249/250, 499/500): 1; (499/500, 4993/5000): 1; (4993/5000, 9997/10000): 1; (9997/10000, 10000998411859/10000000000000): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 5   [ok]
  [N4000-A: all executed parts agree with the expected values, 2481.2 s]
==============================================================================================================
control-N3-source-example   (Rendall and Sontag (2017), example after Theorem 1 (Theorem 3.1), with b = 0)   N = 3
  parameters: alpha = 1, beta = 1, gamma = 1, phi = 1, b = 0, kappa = 1/5000, nu1 = 1/10000, L1 = 1, R = 1, S_T = 10
  Sigma* = 1/2;  nu = 1/10000,  x0 = 0,  X = 10,  rho = 2
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 4 values of S: True;  signs of the right-hand side of (1): + - + -   [ok]
  B  signs of P at the test points: + - + -   [ok]
       x = 0                                        sign +   [integers of 14 and 40 bits; sha256 of the reduced fraction P(x): ab2a5c16553d4c58a2f84ab6]
       x = 1/10                                     sign -   [integers of 20 and 41 bits; sha256 of the reduced fraction P(x): 299dcc1b86cce3f1c9236e44]
       x = 1/2                                      sign +   [integers of 25 and 41 bits; sha256 of the reduced fraction P(x): f6d3835f176f4f86580cc009]
       x = 10                                       sign -   [integers of 34 and 50 bits; sha256 of the reduced fraction P(x): 1b16e6a2d7b57e56f1c581cf]
  C  enclosures (PREC = 256 bits): 53 closed intervals cover [x0, X] (44 without a zero, 9 monotone), deepest bisection 2, 57 point evaluations, smallest interval 5.208e-02
     number of roots of P in [x0, X]: 3, all simple; signs of P' at the roots: - + -   [ok]
       root 1 in (0.000000000000000, 0.052083333333333), width 5.21e-02, P' < 0;  S in (0.000000000000000, 0.052083333333333)
       root 2 in (0.260416666666667, 0.312500000000000), width 5.21e-02, P' > 0;  S in (0.260416666666667, 0.312500000000000)
       root 3 in (1.041666666666667, 1.250000000000000), width 2.08e-01, P' < 0;  S in (1.041666666666667, 1.250000000000000)
  D  Descartes after Moebius maps (degree 4, coefficients up to 43 bits), 5 intervals:
       (0, 1/5): 1; (1/5, 7/10): 1; (7/10, 3/2): 1; (3/2, 4): 0; (4, 10): 0
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 3   [ok]
  [control-N3-source-example: all executed parts agree with the expected values, 0.0 s]
==============================================================================================================
control-N4-three-steady-states   (Section 6 of the note: three steady states for N = 4)   N = 4
  parameters: alpha = 1, beta = 1/200, gamma = 1, phi = 1, b = 1/1000, kappa = 1/5000, nu1 = 1/10000, L1 = 1, R = 1, S_T = 1999/1000
  Sigma* = 1/2;  nu = 1/10000,  x0 = 1/1000,  X = 2,  rho = 1/100
  A  original equations (2),(3),(4) hold exactly, all C_j > 0, Sigma = Sigma* at the 4 values of S: True;  signs of the right-hand side of (1): + - + -   [ok]
  B  signs of P at the test points: + - + -   [ok]
       x = 1/1000                                   sign +   [integers of 27 and 54 bits; sha256 of the reduced fraction P(x): aff3d349b1ce8bc4b64c0228]
       x = 1/20                                     sign -   [integers of 28 and 54 bits; sha256 of the reduced fraction P(x): 9a8fb3998eacb6225166169d]
       x = 1                                        sign +   [integers of 40 and 56 bits; sha256 of the reduced fraction P(x): 3239fa836ce5961035a46486]
       x = 2                                        sign -   [integers of 43 and 59 bits; sha256 of the reduced fraction P(x): 64a657f63ed04012ef4d7189]
  C  enclosures (PREC = 256 bits): 52 closed intervals cover [x0, X] (46 without a zero, 6 monotone), deepest bisection 2, 55 point evaluations, smallest interval 1.041e-02
     number of roots of P in [x0, X]: 3, all simple; signs of P' at the roots: - + -   [ok]
       root 1 in (0.021822916666667, 0.032234375000000), width 1.04e-02, P' < 0;  S in (0.020822916666667, 0.031234375000000)
       root 2 in (0.053057291666667, 0.063468750000000), width 1.04e-02, P' > 0;  S in (0.052057291666667, 0.062468750000000)
       root 3 in (1.916708333333333, 1.958354166666667), width 4.16e-02, P' < 0;  S in (1.915708333333333, 1.957354166666667)
  D  Descartes after Moebius maps (degree 5, coefficients up to 71 bits), 3 intervals:
       (1/1000, 1/25): 1; (1/25, 1): 1; (1, 2): 1
     every interval has 0 or 1 sign changes: True; number of roots of P in (x0, X): 3   [ok]
  [control-N4-three-steady-states: all executed parts agree with the expected values, 0.0 s]
==============================================================================================================
17 parameter points checked in 3441 s:  ALL CHECKS PASSED
