#!/usr/bin/env python3 r"""banana-4loop-evaluate.py -- the four-loop equal-mass banana I_{11111} (a Calabi-Yau threefold: Picard-Fuchs operator AESZ #34 of the Hulek-Verrill family), d = 2-2eps, m^2 = 1. Standalone evaluator of the explicit log-Frobenius form of the complete five-master Laurent window eps^{-4}..eps^0 at any Euclidean point p^2 < 0. REQUIREMENTS: python3 with the mpmath library. Nothing else -- no network, no computer-algebra system, no reduction software, no data files: everything a run needs (the exact connection, the boundary vector, the held-out reference values) is written into this file, and a sha256 pin of that block is checked before any computation. THE MASTERS (propagators k_i^2 - 1, i = 1..4, and (k1+k2+k3+k4-p)^2 - 1; irreducible numerators D6 = (k1-p)^2 and D10 = (k1+k2)^2): m0 = I[1,1,1,1,0] = tadpole^4 = Gamma(eps)^4 (exact) m1 = I[1,1,1,1,1] = the top banana I_{11111} m2 = I[1,1,1,1,1] with D6^2 in the numerator m3 = I[1,1,1,1,1] with D10^2 in the numerator m4 = I[1,1,1,1,1] with D6^3 in the numerator Measure d^dk/(i pi^{d/2}) per loop, no e^{gamma_E eps} prefactor (the raw convention of the auxiliary-mass-flow references). A run prints the window eps^{-4}..eps^0 of all five masters; the poles of m1 vanish identically, so its finite part is the value. THE FORM. Let z = -1/p^2 (the MUM point z = 0 is p^2 -> infinity), L = log z, and M-hat = diag(z^a) e^{4 gamma_E eps} (m0..m4) with a = (0,-1,1,1,2). The masters obey an exact first-order theta-system theta M-hat = C(z,eps) M-hat (theta = z d/dz) whose integer polynomial data are written below: C = Num(z)/den(z) with den = 20 (1+z)(1+9z)(1+25z), graded over the window. Its eps^0 maximal-cut block is the Picard-Fuchs operator L4 = th^4 + z (35 th^4 + 70 th^3 + 63 th^2 + 28 th + 5) + z^2 (259 th^4 + 1036 th^3 + 1580 th^2 + 1088 th + 285) + 225 z^3 (th+1)^2 (th+2)^2, th = z d/dz, whose holomorphic solution is the Hulek-Verrill period varpi_0(z) = sum_n A_n (-z)^n, A_n = sum_{i+j+k+l+m=n} (n!/(i! j! k! l! m!))^2 = 1, 5, 45, 545, 7885, ... The system is regular at z = 0 with a nilpotent residue C0 (nilpotency index 8, checked exactly in integers at every load), so the full solution is the explicit log-Frobenius series F(z) = sum_{n>=0} z^n sum_{j=0}^{7} F_{n,j} L^j, F_0(L) = exp(C0 L) v, with every F_{n,j} generated by the exact linear recursion (n den(0) - Num(0)) F_{n,j} = [ sum_{m=1..6} (Num_m - den_m ((n-m) + d/dL)) F_{n-m} ]_j - den(0) (j+1) F_{n,j+1}. THE BOUNDARY VECTOR v (the complete MUM datum, e^{4 gamma_E eps} frame) is derived, not fit: requiring (i) the tadpole block Gamma(1+eps)^4 e^{4 gamma_E eps}, (ii) the vanishing of the top master's pole window, and (iii) the n = 0 log-tower to reproduce the classical large-momentum boundary of the equal-mass banana (Boenisch-Duhr-Fischbach-Klemm-Nega, arXiv:2108.05310; Poegel-Wang-Weinzierl, arXiv:2211.04292, the alternating Gamma-product sum over hard and soft regions) gives a rank-25 linear system through the C0 log-chains whose unique solution is m0: eps^{-4..0} = 1, 0, 2 z2, -(4/3) z3, 6 z4 (= Gamma(1+eps)^4 e^{4 gamma_E eps}) m1: 0, 0, 0, 0, 0 m2: -1, -2, -2-2 z2, -2-4 z2+(4/3) z3, -50-4 z2+(104/3) z3-6 z4 m3: -3/2, -1, -3 z2, -2 z2+2 z3, -60+(52/3) z3-9 z4 m4: 1, 3, 9/2+2 z2, 21/4+6 z2-(4/3) z3, 453/8+9 z2-52 z3+6 z4 with z_k = zeta(k). An independent extraction (backward transport of a recorded p^2 = -11 value onto the Frobenius basis, integer-relation search layer by layer, two precisions) returned the same expressions. EVALUATION. The local series is summed at z0 = 1/128 and continued to the target by eps-graded Taylor transport of the same exact connection along the Euclidean axis (the system's poles sit at p^2 in {0, 1, 9, 25} only; the physical layers are regular at p^2 = 0 and the nearest true singularity is p^2 = 25); the window is then mapped back to the raw convention by the exact e^{-4 gamma_E eps} mixing. Every truncation is certified at run time and fail-closed: the local series is accepted only when a trailing-8-term geometric tail envelope (ratio r = 25 z0, the exact distance to the nearest z-frame pole) beats 10^-(dps+12); each transport step is accepted only when its own trailing-8-term envelope (ratio |h|/R, R the exact distance to the pole set) beats the same tolerance; a miss continues the SAME recursion exactly to 8x the starting depth and then refuses (RuntimeError, nonzero exit). The accepted per-step bounds are summed into the printed certified BOUND on every value. The starting depths are seeds only, never the answer. WHAT A RUN CHECKS (every check is a RAISING gate: a miss names itself and exits 1): * structural: the embedded connection's denominator is 20(1+z)(1+9z)(1+25z) and its residue Num(0)/20 is nilpotent of index exactly 8, both checked in exact integers; * the tadpole tower m0 against the exact expansion of Gamma(eps)^4 (analytic, no stored digits involved); * the held-out references: independent auxiliary-mass-flow (AMFlow) towers of all 21 non-vanishing layers at p^2 = -2 and p^2 = -7 (about 67 digits each, capped by their own precision) and 100-digit Bessel-moment references for m1 at eps^0. None of these entered the construction of the form; * positive controls: the AESZ-34 recurrence returning the Hulek-Verrill integers 1, 5, 45, 545, 7885, 127905; |L4[varpi_0]| at z = 1/128; the n = 0 log-tower against the literature Gamma-product at L = -3; and the three-loop sibling identity int_0^inf x K_0(x)^4 dx = 7 zeta(3)/8 (the p^2 = 0 value of the three-loop K3 banana, computed by live quadrature); * --bessel (opt-in, alone or with --full): the live independent oracle I_{11111}|_{eps^0} = -16 int_0^inf x J_0(x sqrt(-p^2)) K_0(x)^5 dx, computed by quadrature at run time at a 115-digit working cap; it shares no code path with the connection or with the AMFlow references. Bessel-K quadrature at that precision is slow (many minutes per point), which is why it is not part of --full by default. MODES python3 banana-4loop-evaluate.py # QUICK default: p^2 = -2 at 30 quoted digits # (+30 working), all gates, cheap controls python3 banana-4loop-evaluate.py --full # the paper's held-out table against the # stored references: p^2 = -2 and -7 at # 100 digits, full-depth controls (minutes) python3 banana-4loop-evaluate.py --full --bessel # adds the table's live-oracle rows # (long: the 115-digit quadratures run # many minutes per point) python3 banana-4loop-evaluate.py --dps 60 --pp -5 --pp=-11/2 # any Euclidean points python3 banana-4loop-evaluate.py --check # two-precision self-agreement gate: the run # is repeated at dps+60 guard digits and the # worst nonzero layer must agree to dps+10 # (long at 100 digits) python3 banana-4loop-evaluate.py --mutate # control: one connection entry perturbed # by 1e-6; the held-out gate MUST collapse # and the run MUST exit nonzero The quick default finishes in about a minute on a laptop-class machine (mpmath on its gmpy2 backend; the pure-Python backend takes roughly half again as long) and prints its first line at once; --full takes a few minutes; --bessel and --check at 100 digits are long runs (tens of minutes). Each mode prints its own measured wall time. THE PAPER'S TABLE (the four-loop equal-mass banana section of the portfolio paper) was produced by this form at 100 working digits at the two held-out points and reads, for p^2 = -2 / -7: m1 eps^0 vs AMFlow 67 / 67 digits; worst of the 21 layers vs AMFlow 67 / 67; m1 eps^0 vs the 100-digit Bessel references 99 / 100; m1 eps^0 vs the live Bessel quadrature 116 / 116 (saturating the oracle's 115-digit cap); self-agreement at sixty guard digits 125 (p^2 = -7); mutation control 67 -> 3 at both points. --full recomputes the first three rows and gates each against the printed value (floor: printed value minus one digit); --full --bessel adds the fourth; --check and --mutate reproduce the last two. HONESTY NOTES * The AMFlow comparisons are capped at about 67 digits by the references' own precision and the Bessel-reference comparisons at 100 digits by the stored strings; only the live Bessel oracle and the two-precision self-agreement grow with dps. The quick default certifies 30 digits and typically agrees with the references to about 55. * Only Euclidean p^2 < 0 is evaluated (the transport axis is singularity-free there); the physical region p^2 > 0 is not part of this script. * The quadratures (--bessel, the K_0^4 control) are gate legs only: they never enter the value path. EXIT CODES: 0 every check passed; 1 a check failed or a certified bound could not be met (what --mutate must produce); 2 usage error (e.g. a point with p^2 >= 0); 3 the embedded data block does not match its sha256 pin (a tampered or truncated copy; refused before any computation). """ import argparse, hashlib, json, sys, time from fractions import Fraction EXIT_OK, EXIT_GATE, EXIT_USAGE, EXIT_PIN = 0, 1, 2, 3 # --------------------------------------------------------------------------- # The embedded data block: the exact theta-system in both frames (integer polynomial # coefficients as strings), the rescale exponents, the derived boundary constants # (named in the zeta-ring), and the held-out reference values. Pinned below. # --------------------------------------------------------------------------- DATA_TEXT = r"""{"pp_frame":{"den":["0","-4500","5180","-700","20"],"num":{"5,0":["-7050","-2600","50"],"5,5":["-3650","-9810","-1110","-150"],"5,10":["4050","650"],"5,15":["0","-200"],"5,20":["-400"],"6,0":["19450","1300","-750"],"6,1":["-7050","-2600","50"],"6,5":["9650","14090","5270","430"],"6,6":["-3650","-9810","-1110","-150"],"6,10":["-10850","-2150"],"6,11":["4050","650"],"6,15":["0","1200"],"6,16":["0","-200"],"6,20":["1200"],"6,21":["-400"],"7,0":["-4200","5000","1200"],"7,1":["19450","1300","-750"],"7,2":["-7050","-2600","50"],"7,5":["-600","5760","-1080","-240"],"7,6":["9650","14090","5270","430"],"7,7":["-3650","-9810","-1110","-150"],"7,10":["900","900"],"7,11":["-10850","-2150"],"7,12":["4050","650"],"7,15":["0","-1200"],"7,16":["0","1200"],"7,17":["0","-200"],"7,20":["-300"],"7,21":["1200"],"7,22":["-400"],"8,0":["-600","7400","1200"],"8,1":["-4200","5000","1200"],"8,2":["19450","1300","-750"],"8,3":["-7050","-2600","50"],"8,5":["-600","5760","-1080","-240"],"8,6":["-600","5760","-1080","-240"],"8,7":["9650","14090","5270","430"],"8,8":["-3650","-9810","-1110","-150"],"8,10":["900","900"],"8,11":["900","900"],"8,12":["-10850","-2150"],"8,13":["4050","650"],"8,15":["0","-1200"],"8,16":["0","-1200"],"8,17":["0","1200"],"8,18":["0","-200"],"8,20":["-300"],"8,21":["-300"],"8,22":["1200"],"8,23":["-400"],"9,0":["3000","9800","1200"],"9,1":["-600","7400","1200"],"9,2":["-4200","5000","1200"],"9,3":["19450","1300","-750"],"9,4":["-7050","-2600","50"],"9,5":["-600","5760","-1080","-240"],"9,6":["-600","5760","-1080","-240"],"9,7":["-600","5760","-1080","-240"],"9,8":["9650","14090","5270","430"],"9,9":["-3650","-9810","-1110","-150"],"9,10":["900","900"],"9,11":["900","900"],"9,12":["900","900"],"9,13":["-10850","-2150"],"9,14":["4050","650"],"9,15":["0","-1200"],"9,16":["0","-1200"],"9,17":["0","-1200"],"9,18":["0","1200"],"9,19":["0","-200"],"9,20":["-300"],"9,21":["-300"],"9,22":["-300"],"9,23":["1200"],"9,24":["-400"],"10,0":["-7050","-43760","-9700","-960","30"],"10,5":["-3650","-43190","-39876","-5980","-1498","-14"],"10,10":["4050","21710","4190","130"],"10,15":["0","-200","-1040","-40"],"10,20":["-400","-2080","-80"],"11,0":["41950","72040","18580","-4440","-130"],"11,1":["-7050","-43760","-9700","-960","30"],"11,5":["14150","64490","79452","26292","3966","66"],"11,6":["-3650","-43190","-39876","-5980","-1498","-14"],"11,10":["-15350","-53390","-14050","-410"],"11,11":["4050","21710","4190","130"],"11,15":["0","1200","6240","240"],"11,16":["0","-200","-1040","-40"],"11,20":["1200","6240","240"],"11,21":["-400","-2080","-80"],"12,0":["-4200","-16840","26360","7240","240"],"12,1":["41950","72040","18580","-4440","-130"],"12,2":["-7050","-43760","-9700","-960","30"],"12,5":["-600","2640","28752","-4704","-1464","-48"],"12,6":["14150","64490","79452","26292","3966","66"],"12,7":["-3650","-43190","-39876","-5980","-1498","-14"],"12,10":["900","5580","4860","180"],"12,11":["-15350","-53390","-14050","-410"],"12,12":["4050","21710","4190","130"],"12,15":["0","-1200","-6240","-240"],"12,16":["0","1200","6240","240"],"12,17":["0","-200","-1040","-40"],"12,20":["-300","-1560","-60"],"12,21":["1200","6240","240"],"12,22":["-400","-2080","-80"],"13,0":["-600","4280","39560","7720","240"],"13,1":["-4200","-16840","26360","7240","240"],"13,2":["41950","72040","18580","-4440","-130"],"13,3":["-7050","-43760","-9700","-960","30"],"13,5":["-600","2640","28752","-4704","-1464","-48"],"13,6":["-600","2640","28752","-4704","-1464","-48"],"13,7":["14150","64490","79452","26292","3966","66"],"13,8":["-3650","-43190","-39876","-5980","-1498","-14"],"13,10":["900","5580","4860","180"],"13,11":["900","5580","4860","180"],"13,12":["-15350","-53390","-14050","-410"],"13,13":["4050","21710","4190","130"],"13,15":["0","-1200","-6240","-240"],"13,16":["0","-1200","-6240","-240"],"13,17":["0","1200","6240","240"],"13,18":["0","-200","-1040","-40"],"13,20":["-300","-1560","-60"],"13,21":["-300","-1560","-60"],"13,22":["1200","6240","240"],"13,23":["-400","-2080","-80"],"14,0":["3000","25400","52760","8200","240"],"14,1":["-600","4280","39560","7720","240"],"14,2":["-4200","-16840","26360","7240","240"],"14,3":["41950","72040","18580","-4440","-130"],"14,4":["-7050","-43760","-9700","-960","30"],"14,5":["-600","2640","28752","-4704","-1464","-48"],"14,6":["-600","2640","28752","-4704","-1464","-48"],"14,7":["-600","2640","28752","-4704","-1464","-48"],"14,8":["14150","64490","79452","26292","3966","66"],"14,9":["-3650","-43190","-39876","-5980","-1498","-14"],"14,10":["900","5580","4860","180"],"14,11":["900","5580","4860","180"],"14,12":["900","5580","4860","180"],"14,13":["-15350","-53390","-14050","-410"],"14,14":["4050","21710","4190","130"],"14,15":["0","-1200","-6240","-240"],"14,16":["0","-1200","-6240","-240"],"14,17":["0","-1200","-6240","-240"],"14,18":["0","1200","6240","240"],"14,19":["0","-200","-1040","-40"],"14,20":["-300","-1560","-60"],"14,21":["-300","-1560","-60"],"14,22":["-300","-1560","-60"],"14,23":["1200","6240","240"],"14,24":["-400","-2080","-80"],"15,0":["-36675","-30520","6610","-880","25"],"15,5":["-20775","-61165","-10702","-1538","-27","-1"],"15,10":["23175","6815","85","5"],"15,15":["0","-1200","-80"],"15,20":["-2400","-160"],"16,0":["111075","18680","-970","-800","15"],"16,1":["-36675","-30520","6610","-880","25"],"16,5":["56775","91045","35302","5162","131","1"],"16,6":["-20775","-61165","-10702","-1538","-27","-1"],"16,10":["-63975","-18535","-685","-5"],"16,11":["23175","6815","85","5"],"16,15":["0","7200","480"],"16,16":["0","-1200","-80"],"16,20":["7200","480"],"16,21":["-2400","-160"],"17,0":["-25200","28320","9200","480"],"17,1":["111075","18680","-970","-800","15"],"17,2":["-36675","-30520","6610","-880","25"],"17,5":["-3600","34320","-4176","-1872","-96"],"17,6":["56775","91045","35302","5162","131","1"],"17,7":["-20775","-61165","-10702","-1538","-27","-1"],"17,10":["5400","5760","360"],"17,11":["-63975","-18535","-685","-5"],"17,12":["23175","6815","85","5"],"17,15":["0","-7200","-480"],"17,16":["0","7200","480"],"17,17":["0","-1200","-80"],"17,20":["-1800","-120"],"17,21":["7200","480"],"17,22":["-2400","-160"],"18,0":["-3600","44160","10160","480"],"18,1":["-25200","28320","9200","480"],"18,2":["111075","18680","-970","-800","15"],"18,3":["-36675","-30520","6610","-880","25"],"18,5":["-3600","34320","-4176","-1872","-96"],"18,6":["-3600","34320","-4176","-1872","-96"],"18,7":["56775","91045","35302","5162","131","1"],"18,8":["-20775","-61165","-10702","-1538","-27","-1"],"18,10":["5400","5760","360"],"18,11":["5400","5760","360"],"18,12":["-63975","-18535","-685","-5"],"18,13":["23175","6815","85","5"],"18,15":["0","-7200","-480"],"18,16":["0","-7200","-480"],"18,17":["0","7200","480"],"18,18":["0","-1200","-80"],"18,20":["-1800","-120"],"18,21":["-1800","-120"],"18,22":["7200","480"],"18,23":["-2400","-160"],"19,0":["18000","60000","11120","480"],"19,1":["-3600","44160","10160","480"],"19,2":["-25200","28320","9200","480"],"19,3":["111075","18680","-970","-800","15"],"19,4":["-36675","-30520","6610","-880","25"],"19,5":["-3600","34320","-4176","-1872","-96"],"19,6":["-3600","34320","-4176","-1872","-96"],"19,7":["-3600","34320","-4176","-1872","-96"],"19,8":["56775","91045","35302","5162","131","1"],"19,9":["-20775","-61165","-10702","-1538","-27","-1"],"19,10":["5400","5760","360"],"19,11":["5400","5760","360"],"19,12":["5400","5760","360"],"19,13":["-63975","-18535","-685","-5"],"19,14":["23175","6815","85","5"],"19,15":["0","-7200","-480"],"19,16":["0","-7200","-480"],"19,17":["0","-7200","-480"],"19,18":["0","7200","480"],"19,19":["0","-1200","-80"],"19,20":["-1800","-120"],"19,21":["-1800","-120"],"19,22":["-1800","-120"],"19,23":["7200","480"],"19,24":["-2400","-160"],"20,0":["-7050","-113390","-102340","-23420","430","10"],"20,5":["-3650","-90780","-181014","-83624","-16270","-1532","38"],"20,10":["4050","52040","58700","5640","-110"],"20,15":["0","-200","-2760","-2200","40"],"20,20":["-400","-5520","-4400","80"],"21,0":["167950","116770","229020","7380","-9290","170"],"21,1":["-7050","-113390","-102340","-23420","430","10"],"21,5":["14150","159180","311178","212584","51322","5388","-138"],"21,6":["-3650","-90780","-181014","-83624","-16270","-1532","38"],"21,10":["-10850","-160880","-138660","-22880","470"],"21,11":["4050","52040","58700","5640","-110"],"21,15":["0","1200","16560","13200","-240"],"21,16":["0","-200","-2760","-2200","40"],"21,20":["-3300","21740","12500","-220"],"21,21":["-400","-5520","-4400","80"],"22,0":["49800","-115120","32400","72160","12200","-240"],"22,1":["167950","116770","229020","7380","-9290","170"],"22,2":["-7050","-113390","-102340","-23420","430","10"],"22,5":["-600","-2520","71808","48336","-16344","-2424","48"],"22,6":["14150","159180","311178","212584","51322","5388","-138"],"22,7":["-3650","-90780","-181014","-83624","-16270","-1532","38"],"22,10":["900","13320","22320","9720","-180"],"22,11":["-10850","-160880","-138660","-22880","470"],"22,12":["4050","52040","58700","5640","-110"],"22,15":["0","-1200","-16560","-13200","240"],"22,16":["0","1200","16560","13200","-240"],"22,17":["0","-200","-2760","-2200","40"],"22,20":["-300","-4140","-3300","60"],"22,21":["-3300","21740","12500","-220"],"22,22":["-400","-5520","-4400","80"],"23,0":["53400","-63040","105120","97840","11720","-240"],"23,1":["49800","-115120","32400","72160","12200","-240"],"23,2":["167950","116770","229020","7380","-9290","170"],"23,3":["-7050","-113390","-102340","-23420","430","10"],"23,5":["-600","-2520","71808","48336","-16344","-2424","48"],"23,6":["-600","-2520","71808","48336","-16344","-2424","48"],"23,7":["14150","159180","311178","212584","51322","5388","-138"],"23,8":["-3650","-90780","-181014","-83624","-16270","-1532","38"],"23,10":["900","13320","22320","9720","-180"],"23,11":["900","13320","22320","9720","-180"],"23,12":["-10850","-160880","-138660","-22880","470"],"23,13":["4050","52040","58700","5640","-110"],"23,15":["0","-1200","-16560","-13200","240"],"23,16":["0","-1200","-16560","-13200","240"],"23,17":["0","1200","16560","13200","-240"],"23,18":["0","-200","-2760","-2200","40"],"23,20":["-300","-4140","-3300","60"],"23,21":["-300","-4140","-3300","60"],"23,22":["-3300","21740","12500","-220"],"23,23":["-400","-5520","-4400","80"],"24,0":["57000","-10960","177840","123520","11240","-240"],"24,1":["53400","-63040","105120","97840","11720","-240"],"24,2":["49800","-115120","32400","72160","12200","-240"],"24,3":["167950","116770","229020","7380","-9290","170"],"24,4":["-7050","-113390","-102340","-23420","430","10"],"24,5":["-600","-2520","71808","48336","-16344","-2424","48"],"24,6":["-600","-2520","71808","48336","-16344","-2424","48"],"24,7":["-600","-2520","71808","48336","-16344","-2424","48"],"24,8":["14150","159180","311178","212584","51322","5388","-138"],"24,9":["-3650","-90780","-181014","-83624","-16270","-1532","38"],"24,10":["900","13320","22320","9720","-180"],"24,11":["900","13320","22320","9720","-180"],"24,12":["900","13320","22320","9720","-180"],"24,13":["-10850","-160880","-138660","-22880","470"],"24,14":["4050","52040","58700","5640","-110"],"24,15":["0","-1200","-16560","-13200","240"],"24,16":["0","-1200","-16560","-13200","240"],"24,17":["0","-1200","-16560","-13200","240"],"24,18":["0","1200","16560","13200","-240"],"24,19":["0","-200","-2760","-2200","40"],"24,20":["-300","-4140","-3300","60"],"24,21":["-300","-4140","-3300","60"],"24,22":["-300","-4140","-3300","60"],"24,23":["-3300","21740","12500","-220"],"24,24":["-400","-5520","-4400","80"]}},"z_frame":{"den":["20","700","5180","4500"],"num":{"5,0":["50","2600","-7050"],"5,5":["130","-1810","4630","-8150"],"5,10":["-650","4050"],"5,15":["200"],"5,20":["-400"],"6,0":["-750","-1300","19450"],"6,1":["50","2600","-7050"],"6,5":["-430","5270","-14090","9650"],"6,6":["130","-1810","4630","-8150"],"6,10":["2150","-10850"],"6,11":["-650","4050"],"6,15":["-1200"],"6,16":["200"],"6,20":["1200"],"6,21":["-400"],"7,0":["1200","-5000","-4200"],"7,1":["-750","-1300","19450"],"7,2":["50","2600","-7050"],"7,5":["240","-1080","-5760","-600"],"7,6":["-430","5270","-14090","9650"],"7,7":["130","-1810","4630","-8150"],"7,10":["-900","900"],"7,11":["2150","-10850"],"7,12":["-650","4050"],"7,15":["1200"],"7,16":["-1200"],"7,17":["200"],"7,20":["-300"],"7,21":["1200"],"7,22":["-400"],"8,0":["1200","-7400","-600"],"8,1":["1200","-5000","-4200"],"8,2":["-750","-1300","19450"],"8,3":["50","2600","-7050"],"8,5":["240","-1080","-5760","-600"],"8,6":["240","-1080","-5760","-600"],"8,7":["-430","5270","-14090","9650"],"8,8":["130","-1810","4630","-8150"],"8,10":["-900","900"],"8,11":["-900","900"],"8,12":["2150","-10850"],"8,13":["-650","4050"],"8,15":["1200"],"8,16":["1200"],"8,17":["-1200"],"8,18":["200"],"8,20":["-300"],"8,21":["-300"],"8,22":["1200"],"8,23":["-400"],"9,0":["1200","-9800","3000"],"9,1":["1200","-7400","-600"],"9,2":["1200","-5000","-4200"],"9,3":["-750","-1300","19450"],"9,4":["50","2600","-7050"],"9,5":["240","-1080","-5760","-600"],"9,6":["240","-1080","-5760","-600"],"9,7":["240","-1080","-5760","-600"],"9,8":["-430","5270","-14090","9650"],"9,9":["130","-1810","4630","-8150"],"9,10":["-900","900"],"9,11":["-900","900"],"9,12":["-900","900"],"9,13":["2150","-10850"],"9,14":["-650","4050"],"9,15":["1200"],"9,16":["1200"],"9,17":["1200"],"9,18":["-1200"],"9,19":["200"],"9,20":["-300"],"9,21":["-300"],"9,22":["-300"],"9,23":["1200"],"9,24":["-400"],"10,0":["30","960","-9700","43760","-7050"],"10,5":["14","-1498","5980","-39876","43190","-3650"],"10,10":["-110","4890","-16530","8550"],"10,15":["40","-1040","200"],"10,20":["-80","2080","-400"],"11,0":["-130","4440","18580","-72040","41950"],"11,1":["30","960","-9700","43760","-7050"],"11,5":["-66","3966","-26292","79452","-64490","14150"],"11,6":["14","-1498","5980","-39876","43190","-3650"],"11,10":["410","-14050","53390","-15350"],"11,11":["-110","4890","-16530","8550"],"11,15":["-240","6240","-1200"],"11,16":["40","-1040","200"],"11,20":["240","-6240","1200"],"11,21":["-80","2080","-400"],"12,0":["240","-7240","26360","16840","-4200"],"12,1":["-130","4440","18580","-72040","41950"],"12,2":["30","960","-9700","43760","-7050"],"12,5":["48","-1464","4704","28752","-2640","-600"],"12,6":["-66","3966","-26292","79452","-64490","14150"],"12,7":["14","-1498","5980","-39876","43190","-3650"],"12,10":["-180","4860","-5580","900"],"12,11":["410","-14050","53390","-15350"],"12,12":["-110","4890","-16530","8550"],"12,15":["240","-6240","1200"],"12,16":["-240","6240","-1200"],"12,17":["40","-1040","200"],"12,20":["-60","1560","-300"],"12,21":["240","-6240","1200"],"12,22":["-80","2080","-400"],"13,0":["240","-7720","39560","-4280","-600"],"13,1":["240","-7240","26360","16840","-4200"],"13,2":["-130","4440","18580","-72040","41950"],"13,3":["30","960","-9700","43760","-7050"],"13,5":["48","-1464","4704","28752","-2640","-600"],"13,6":["48","-1464","4704","28752","-2640","-600"],"13,7":["-66","3966","-26292","79452","-64490","14150"],"13,8":["14","-1498","5980","-39876","43190","-3650"],"13,10":["-180","4860","-5580","900"],"13,11":["-180","4860","-5580","900"],"13,12":["410","-14050","53390","-15350"],"13,13":["-110","4890","-16530","8550"],"13,15":["240","-6240","1200"],"13,16":["240","-6240","1200"],"13,17":["-240","6240","-1200"],"13,18":["40","-1040","200"],"13,20":["-60","1560","-300"],"13,21":["-60","1560","-300"],"13,22":["240","-6240","1200"],"13,23":["-80","2080","-400"],"14,0":["240","-8200","52760","-25400","3000"],"14,1":["240","-7720","39560","-4280","-600"],"14,2":["240","-7240","26360","16840","-4200"],"14,3":["-130","4440","18580","-72040","41950"],"14,4":["30","960","-9700","43760","-7050"],"14,5":["48","-1464","4704","28752","-2640","-600"],"14,6":["48","-1464","4704","28752","-2640","-600"],"14,7":["48","-1464","4704","28752","-2640","-600"],"14,8":["-66","3966","-26292","79452","-64490","14150"],"14,9":["14","-1498","5980","-39876","43190","-3650"],"14,10":["-180","4860","-5580","900"],"14,11":["-180","4860","-5580","900"],"14,12":["-180","4860","-5580","900"],"14,13":["410","-14050","53390","-15350"],"14,14":["-110","4890","-16530","8550"],"14,15":["240","-6240","1200"],"14,16":["240","-6240","1200"],"14,17":["240","-6240","1200"],"14,18":["-240","6240","-1200"],"14,19":["40","-1040","200"],"14,20":["-60","1560","-300"],"14,21":["-60","1560","-300"],"14,22":["-60","1560","-300"],"14,23":["240","-6240","1200"],"14,24":["-80","2080","-400"],"15,0":["25","880","6610","30520","-36675"],"15,5":["1","-27","1538","-10702","61165","-20775"],"15,10":["-5","85","-6815","23175"],"15,15":["20","620","6380","4500"],"15,20":["0","160","-2400"],"16,0":["15","800","-970","-18680","111075"],"16,1":["25","880","6610","30520","-36675"],"16,5":["-1","131","-5162","35302","-91045","56775"],"16,6":["1","-27","1538","-10702","61165","-20775"],"16,10":["5","-685","18535","-63975"],"16,11":["-5","85","-6815","23175"],"16,15":["0","480","-7200"],"16,16":["20","620","6380","4500"],"16,20":["0","-480","7200"],"16,21":["0","160","-2400"],"17,0":["0","-480","9200","-28320","-25200"],"17,1":["15","800","-970","-18680","111075"],"17,2":["25","880","6610","30520","-36675"],"17,5":["0","-96","1872","-4176","-34320","-3600"],"17,6":["-1","131","-5162","35302","-91045","56775"],"17,7":["1","-27","1538","-10702","61165","-20775"],"17,10":["0","360","-5760","5400"],"17,11":["5","-685","18535","-63975"],"17,12":["-5","85","-6815","23175"],"17,15":["0","-480","7200"],"17,16":["0","480","-7200"],"17,17":["20","620","6380","4500"],"17,20":["0","120","-1800"],"17,21":["0","-480","7200"],"17,22":["0","160","-2400"],"18,0":["0","-480","10160","-44160","-3600"],"18,1":["0","-480","9200","-28320","-25200"],"18,2":["15","800","-970","-18680","111075"],"18,3":["25","880","6610","30520","-36675"],"18,5":["0","-96","1872","-4176","-34320","-3600"],"18,6":["0","-96","1872","-4176","-34320","-3600"],"18,7":["-1","131","-5162","35302","-91045","56775"],"18,8":["1","-27","1538","-10702","61165","-20775"],"18,10":["0","360","-5760","5400"],"18,11":["0","360","-5760","5400"],"18,12":["5","-685","18535","-63975"],"18,13":["-5","85","-6815","23175"],"18,15":["0","-480","7200"],"18,16":["0","-480","7200"],"18,17":["0","480","-7200"],"18,18":["20","620","6380","4500"],"18,20":["0","120","-1800"],"18,21":["0","120","-1800"],"18,22":["0","-480","7200"],"18,23":["0","160","-2400"],"19,0":["0","-480","11120","-60000","18000"],"19,1":["0","-480","10160","-44160","-3600"],"19,2":["0","-480","9200","-28320","-25200"],"19,3":["15","800","-970","-18680","111075"],"19,4":["25","880","6610","30520","-36675"],"19,5":["0","-96","1872","-4176","-34320","-3600"],"19,6":["0","-96","1872","-4176","-34320","-3600"],"19,7":["0","-96","1872","-4176","-34320","-3600"],"19,8":["-1","131","-5162","35302","-91045","56775"],"19,9":["1","-27","1538","-10702","61165","-20775"],"19,10":["0","360","-5760","5400"],"19,11":["0","360","-5760","5400"],"19,12":["0","360","-5760","5400"],"19,13":["5","-685","18535","-63975"],"19,14":["-5","85","-6815","23175"],"19,15":["0","-480","7200"],"19,16":["0","-480","7200"],"19,17":["0","-480","7200"],"19,18":["0","480","-7200"],"19,19":["20","620","6380","4500"],"19,20":["0","120","-1800"],"19,21":["0","120","-1800"],"19,22":["0","120","-1800"],"19,23":["0","-480","7200"],"19,24":["0","160","-2400"],"20,0":["-10","430","23420","-102340","113390","-7050"],"20,5":["38","1532","-16270","83624","-181014","90780","-3650"],"20,10":["-110","-5640","58700","-52040","4050"],"20,15":["40","2200","-2760","200"],"20,20":["-40","-3000","15880","8600"],"21,0":["-170","-9290","-7380","229020","-116770","167950"],"21,1":["-10","430","23420","-102340","113390","-7050"],"21,5":["-138","-5388","51322","-212584","311178","-159180","14150"],"21,6":["38","1532","-16270","83624","-181014","90780","-3650"],"21,10":["470","22880","-138660","160880","-10850"],"21,11":["-110","-5640","58700","-52040","4050"],"21,15":["-240","-13200","16560","-1200"],"21,16":["40","2200","-2760","200"],"21,20":["220","12500","-21740","-3300"],"21,21":["-40","-3000","15880","8600"],"22,0":["240","12200","-72160","32400","115120","49800"],"22,1":["-170","-9290","-7380","229020","-116770","167950"],"22,2":["-10","430","23420","-102340","113390","-7050"],"22,5":["48","2424","-16344","-48336","71808","2520","-600"],"22,6":["-138","-5388","51322","-212584","311178","-159180","14150"],"22,7":["38","1532","-16270","83624","-181014","90780","-3650"],"22,10":["-180","-9720","22320","-13320","900"],"22,11":["470","22880","-138660","160880","-10850"],"22,12":["-110","-5640","58700","-52040","4050"],"22,15":["240","13200","-16560","1200"],"22,16":["-240","-13200","16560","-1200"],"22,17":["40","2200","-2760","200"],"22,20":["-60","-3300","4140","-300"],"22,21":["220","12500","-21740","-3300"],"22,22":["-40","-3000","15880","8600"],"23,0":["240","11720","-97840","105120","63040","53400"],"23,1":["240","12200","-72160","32400","115120","49800"],"23,2":["-170","-9290","-7380","229020","-116770","167950"],"23,3":["-10","430","23420","-102340","113390","-7050"],"23,5":["48","2424","-16344","-48336","71808","2520","-600"],"23,6":["48","2424","-16344","-48336","71808","2520","-600"],"23,7":["-138","-5388","51322","-212584","311178","-159180","14150"],"23,8":["38","1532","-16270","83624","-181014","90780","-3650"],"23,10":["-180","-9720","22320","-13320","900"],"23,11":["-180","-9720","22320","-13320","900"],"23,12":["470","22880","-138660","160880","-10850"],"23,13":["-110","-5640","58700","-52040","4050"],"23,15":["240","13200","-16560","1200"],"23,16":["240","13200","-16560","1200"],"23,17":["-240","-13200","16560","-1200"],"23,18":["40","2200","-2760","200"],"23,20":["-60","-3300","4140","-300"],"23,21":["-60","-3300","4140","-300"],"23,22":["220","12500","-21740","-3300"],"23,23":["-40","-3000","15880","8600"],"24,0":["240","11240","-123520","177840","10960","57000"],"24,1":["240","11720","-97840","105120","63040","53400"],"24,2":["240","12200","-72160","32400","115120","49800"],"24,3":["-170","-9290","-7380","229020","-116770","167950"],"24,4":["-10","430","23420","-102340","113390","-7050"],"24,5":["48","2424","-16344","-48336","71808","2520","-600"],"24,6":["48","2424","-16344","-48336","71808","2520","-600"],"24,7":["48","2424","-16344","-48336","71808","2520","-600"],"24,8":["-138","-5388","51322","-212584","311178","-159180","14150"],"24,9":["38","1532","-16270","83624","-181014","90780","-3650"],"24,10":["-180","-9720","22320","-13320","900"],"24,11":["-180","-9720","22320","-13320","900"],"24,12":["-180","-9720","22320","-13320","900"],"24,13":["470","22880","-138660","160880","-10850"],"24,14":["-110","-5640","58700","-52040","4050"],"24,15":["240","13200","-16560","1200"],"24,16":["240","13200","-16560","1200"],"24,17":["240","13200","-16560","1200"],"24,18":["-240","-13200","16560","-1200"],"24,19":["40","2200","-2760","200"],"24,20":["-60","-3300","4140","-300"],"24,21":["-60","-3300","4140","-300"],"24,22":["-60","-3300","4140","-300"],"24,23":["220","12500","-21740","-3300"],"24,24":["-40","-3000","15880","8600"]}},"rescale_a":[0,-1,1,1,2],"constants_named":{"0,-4":{"1":"1"},"0,-3":{},"0,-2":{"z2":"2"},"0,-1":{"z3":"-4/3"},"0,0":{"z4":"6"},"1,-4":{},"1,-3":{},"1,-2":{},"1,-1":{},"1,0":{},"2,-4":{"1":"-1"},"2,-3":{"1":"-2"},"2,-2":{"1":"-2","z2":"-2"},"2,-1":{"1":"-2","z2":"-4","z3":"4/3"},"2,0":{"1":"-50","z2":"-4","z3":"104/3","z4":"-6"},"3,-4":{"1":"-3/2"},"3,-3":{"1":"-1"},"3,-2":{"z2":"-3"},"3,-1":{"z2":"-2","z3":"2"},"3,0":{"1":"-60","z3":"52/3","z4":"-9"},"4,-4":{"1":"1"},"4,-3":{"1":"3"},"4,-2":{"1":"9/2","z2":"2"},"4,-1":{"1":"21/4","z2":"6","z3":"-4/3"},"4,0":{"1":"453/8","z2":"9","z3":"-52","z4":"6"}},"gate_amflow_heldout":{"-2":{"0,-4":"0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999948398","0,-3":"-2.3088626596061314424260483603296097241686373437596943952230689395394709071106586837477882531669868658030280051","0,-2":"5.9552915241582022674918394241340041917132369244106348206777831627581412903724551049573639240681199551914671214","0,-1":"-11.249961739225280199971549027011689132771264106466026557278235543753520854139542226381583945090295707521082238","0,0":"20.147423583662158774752631835604440822313098315879883502473476112112100242775262906636702997599948891498563540","1,0":"-39.374358516940218859656507967746135822279959247934525023677415656424721432088800598579803881090976194342726745","2,-4":"2.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999980554116113","2,-3":"-10.926587978818394327278145080988829172505912031279083185669206818618412721331976051243364759002564756469428889","2,-2":"23.101325210899132572179711713720451471814260148270682042925625246432307499560000049764417185063723247688717697","2,-1":"-52.335600675884123900177811336252645268492190642001841373653563524135244095450310072295769066298916328186253038","2,0":"235.19839793333772799953325541924649068438043186709531909775879741729092437433414323287900757712874284882969505","3,-4":"4.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000022281251850","3,-3":"-8.3898819682275914909172176214832437587588680469186247785038102279276190819979640768650471398225079520980745778","3,-2":"29.181086539499647318861180687943799414372291472328467902603886353332693992454730604925798827349653859651809919","3,-1":"-53.876283395440276461870630295591860783224676036593710632957976198150857612675403279510716588157457479764508055","3,0":"-57.295042514924710167169475050627118708988225749550009824114737421729510814357129227934626947658918879325599955","4,-4":"21.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003678834198","4,-3":"-52.486115851728760290947015566921804207541384218953582299684447730328889049323832358703553317449608822573680457","4,-2":"136.29657264574677338703282134813252692265252478766210881512572217607885072626419193928276131605248070999712660","4,-1":"-259.68808793957595615422198398444070800338676862094848577600021694893544491267085237982951356700460996461195419","4,0":"190.55949787744978005992486535085409014649303387190524641452983310356732408664903668479742060767123432380139610"},"-7":{"0,-4":"0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999948398","0,-3":"-2.3088626596061314424260483603296097241686373437596943952230689395394709071106586837477882531669868658030280051","0,-2":"5.9552915241582022674918394241340041917132369244106348206777831627581412903724551049573639240681199551914671214","0,-1":"-11.249961739225280199971549027011689132771264106466026557278235543753520854139542226381583945090295707521082238","0,0":"20.147423583662158774752631835604440822313098315879883502473476112112100242775262906636702997599948891498563540","1,0":"-38.084997568345478989740156686915017553346761314816188201884862429888538861390042724471147663336277820491154665","2,-4":"-1.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999971122467122","2,-3":"-9.3822746807877371151479032793407805516627253124806112095538621209210581857786826325044234944061545485826231721","2,-2":"6.4134941861694356589809981963465277549344489638144518917673988280363101188043113627010687553390402459576290538","2,-1":"-42.550080625278431150977976839238144280081865916181112841809528037554343657409086672569174524813779238797091016","2,0":"102.35033564730945592722436329482365960044581914478379646527571308734636533541580216073685143247646634860879850","3,-4":"-2.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999994456247739","3,-3":"3.9265879788183943272781450809888291725059120312790831856692068186184127213319760512433647593588751275222606069","3,-2":"-3.9392865936562124751973731914131834026337987419528212763641426696560111497853892636005526886509593581252756751","3,-1":"0.72197202795831370045679028632578675399362013984831444323987456620984234152223109816209194557989211737286082524","3,0":"-56.611976947691830212718543852940270374050599555182514836277708198395945065726287113968397307769941216916016309","4,-4":"26.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051315202773","4,-3":"0.96957085024058249692274263143014717161542906224794572420020757197375641512287422255750540450629819061736340431","4,-2":"148.49695739213924096679887504737791581025728206533514722901515691980394821593365302288431541707446990634004632","4,-1":"-68.518249963231625888561574294462169658227137114746862585392123577403760501701015548927531890509975838033372127","4,0":"-5021.9535198400340669081179937281634471710423455261594273880370052390466465027799878598168745035213557609913538"}},"gate_bessel_100d":{"-2":"-39.37435851694021885965650796774613582227995924793452502367741565642542762527029746632840821495859534","-7":"-38.08499756834547898974015668691501755334676131481618820188486242988924502402851995314207725439272947"}}""" DATA_SHA256 = "8fe414140dc5c70e429ab2ce60dbfa46789b083f73b74d6db827772a22e003e5" def check_pin(): """sha256 of the embedded data block against the pin. Exit 3 on mismatch, before any computation.""" got = hashlib.sha256(DATA_TEXT.encode("ascii")).hexdigest() if got != DATA_SHA256: sys.stderr.write("REFUSED (exit %d): embedded data block sha256 %s... does not match " "the pin %s... -- this copy of the script is tampered or truncated; " "nothing was computed\n" % (EXIT_PIN, got[:16], DATA_SHA256[:16])) sys.exit(EXIT_PIN) return got DATA = json.loads(DATA_TEXT) STATE = [(i, K) for i in range(5) for K in range(-4, 1)] IDX = {s: n for n, s in enumerate(STATE)} NS = len(STATE) JMAX = 7 # certified-truncation knobs: the depth formulas below are STARTING SEEDS only; every # truncation is gated at run time by a trailing-window geometric tail bound and # escalates by exact continuation of the same recurrence, refusing at the cap. TAILWIN = 8 # trailing-term window for the geometric tail envelope TGUARD = 12 # certified tail: tol = 10^-(quoted_dps + TGUARD) ESC = 1.5 # depth escalation factor (exact continuation of the recurrence) NCAP_MULT = 8 # escalation cap = NCAP_MULT * starting depth (fail-closed) CHK_MARGIN = 10 # --check floor = dps + CHK_MARGIN CHK_ROUNDS = 3 # --check auto-deepen rounds (x2 both leg guards per round) GUARD = 30 # working guard digits on top of the quoted dps QUICK_DPS = 30 # quoted digits of the quick default FULL_DPS = 100 # quoted digits of --full (the paper's table) QUICK_PP = ["-2"] FULL_PP = ["-2", "-7"] # The paper's printed held-out table (digits of agreement at 100 working digits); --full # gates each recomputed row at (printed - 1). PAPER_TABLE = { "-2": {"amflow_m1": 67, "amflow_worst": 67, "bessel_ref": 99, "bessel_live": 116}, "-7": {"amflow_m1": 67, "amflow_worst": 67, "bessel_ref": 100, "bessel_live": 116}, } PAPER_MUTATE_TO = 3 # printed collapse: 67 -> 3 digits at both points PAPER_SELF_AGREE = 125 # printed self-agreement at sixty guard digits, p^2 = -7 # raw-mantissa arithmetic (bit-identical to the mpf operators at the same precision and # rounding; used in the two hot loops so the run stays laptop-class) from mpmath.libmp import mpf_add, mpf_mul, mpf_div, mpf_neg, from_int, fzero, fone # ---------- structural checks on the embedded data (exact integers) ---------- def structural_checks(): """den(z) = 20(1+z)(1+9z)(1+25z) and Num(0)/20 nilpotent of index exactly 8, both in exact integer arithmetic. A miss means a corrupted data block -> exit 3.""" zf = DATA["z_frame"] den = [int(c) for c in zf["den"]] if den != [20, 700, 5180, 4500]: sys.stderr.write("REFUSED (exit %d): z-frame denominator %s is not " "20(1+z)(1+9z)(1+25z)\n" % (EXIT_PIN, den)) sys.exit(EXIT_PIN) N0 = {} for k, cl in zf["num"].items(): n, m = map(int, k.split(",")) c0 = int(cl[0]) if c0: N0[(n, m)] = c0 def smul(A, B): out = {} for (i, j), a in A.items(): for (jj, k), b in B.items(): if jj == j: out[(i, k)] = out.get((i, k), 0) + a * b return {k: v for k, v in out.items() if v} P = dict(N0) idx = None for p in range(2, 10): P = smul(P, N0) if not P: idx = p break if idx != 8: sys.stderr.write("REFUSED (exit %d): residue Num(0)/20 is not nilpotent of index 8 " "(found %s)\n" % (EXIT_PIN, idx)) sys.exit(EXIT_PIN) return idx def _load(mp): def rat(s): if "/" in s: p, q = s.split("/") return mp.mpf(int(p)) / mp.mpf(int(q)) return mp.mpf(int(s)) def frame(fr): den = [rat(c) for c in fr["den"]] num = {} for k, cl in fr["num"].items(): n, m = map(int, k.split(",")) num[(n, m)] = [rat(c) for c in cl] return den, num return rat, frame # ---------- polynomial / series helpers ---------- def poly_shift(mp, c, t0): out = list(c) n = len(c) for i in range(n - 1): for j in range(n - 2, i - 1, -1): out[j] = out[j] + t0 * out[j + 1] return out def series_inv_raw(c, order, prec, rnd): inv0 = mpf_div(fone, c[0], prec, rnd) out = [inv0] + [fzero] * order for n in range(1, order + 1): s = fzero for k in range(1, min(n, len(c) - 1) + 1): s = mpf_add(s, mpf_mul(c[k], out[n - k], prec, rnd), prec, rnd) out[n] = mpf_neg(mpf_mul(inv0, s, prec, rnd)) return out def series_mul_raw(a, b, order, prec, rnd): out = [fzero] * (order + 1) for i, ai in enumerate(a[: order + 1]): if ai == fzero: continue for j in range(0, min(order - i, len(b) - 1) + 1): out[i + j] = mpf_add(out[i + j], mpf_mul(ai, b[j], prec, rnd), prec, rnd) return out # ---------- transport (p^2-frame) ---------- def transport(mp, den, num, y0, t0, t1, order0, ratio, tol): """eps-graded Taylor transport with PER-STEP CERTIFIED tail gates. order0 is a STARTING seed only. Step rule h = min(|t1-t|, ratio*R), R = exact distance to the connection's pole set {0,1,9,25} (den = 20 t (t-1)(t-9)(t-25)), so the local Taylor series has certified ratio r = |h|/R <= ratio < 1; a step is accepted only when bound = max(|a_n h^n| over the trailing TAILWIN orders, all 25 comps) * r/(1-r) beats tol. On a miss the SAME Taylor recursion is continued EXACTLY (x ESC depth) up to NCAP_MULT*order0, then RuntimeError naming step/t/h/r/bound/tol/N/cap. Accepted per-step bounds accumulate into err_sum (sup norm over the 25-layer window). Returns (y, nsteps, err_sum, worst_step_bound, max_order_used).""" prec, rnd = mp.mp._prec_rounding t = mp.mpf(t0) y = [mp.mpf(v)._mpf_ for v in y0] nst = 0 ncap = NCAP_MULT * order0 err_sum = mp.mpf(0) worst = mp.mpf(0) omax = order0 while abs(t - t1) > 0: R = min(abs(t), abs(t - 1), abs(t - 9), abs(t - 25)) h = mp.sign(mp.mpf(t1) - t) * min(abs(mp.mpf(t1) - t), ratio * R) r = abs(h) / R # certified series ratio (exact pole set; r <= ratio < 1) dsh = [c._mpf_ for c in poly_shift(mp, den, t)] csh = {key: [c._mpf_ for c in poly_shift(mp, c, t)] for key, c in num.items()} order = order0 def _mtrip(ordr): # connection Taylor coefficients at t as sparse triples; the entries fed by the # tadpole columns (cx < 5) are kept apart: the tadpole rows of the connection # vanish, so a[j] has exactly zero tadpole components for every j >= 1 and those # products are exact zeros -- skipping them changes no bit of the result dinv = series_inv_raw(dsh, ordr, prec, rnd) # Mt[k][rr] = [(cx, value), ...] in the connection's entry order; Mt_rest drops # the tadpole columns Mt = [[[] for _ in range(NS)] for _ in range(ordr + 1)] Mt_rest = [[[] for _ in range(NS)] for _ in range(ordr + 1)] for (n, m), cs in csh.items(): nc = series_mul_raw(cs, dinv, ordr, prec, rnd) for k in range(ordr + 1): if nc[k] != fzero: Mt[k][n].append((m, nc[k])) if m >= 5: Mt_rest[k][n].append((m, nc[k])) return Mt, Mt_rest Mtrip, Mrest = _mtrip(order) a = [list(y)] def _extend(upto, add=mpf_add, mul=mpf_mul, div=mpf_div): # a[j] = (sum_{k=0..j-1} M_k a[j-1-k]) / j, continued from wherever a ends; each # component accumulates its terms in k order, then in the connection's entry # order (the same summation order for every value) for j in range(len(a), upto + 1): s = [] for rr in range(NS): acc = fzero for k in range(j): ak = a[j - 1 - k] for (cx, mv) in (Mtrip[k][rr] if k == j - 1 else Mrest[k][rr]): acc = add(acc, mul(mv, ak[cx], prec, rnd), prec, rnd) s.append(acc) jr = from_int(j) a.append([div(v, jr, prec, rnd) for v in s]) _extend(order + 1) hraw = h._mpf_ while True: hp = abs(h) ** (len(a) - TAILWIN) mx = mp.mpf(0) for j in range(len(a) - TAILWIN, len(a)): aj = a[j] mj = max(abs(mp.mp.make_mpf(v)) for v in aj) * hp if mj > mx: mx = mj hp *= abs(h) bound = mx * r / (1 - r) if bound < tol: break if order >= ncap: raise RuntimeError( f"transport: certified tail bound {mp.nstr(bound, 3)} >= tol " f"{mp.nstr(tol, 2)} at step {nst} (t={mp.nstr(t, 8)}, h={mp.nstr(h, 8)}, " f"r={mp.nstr(r, 4)}), Taylor order N={order} (cap {ncap}); fail-closed") neworder = min(ncap, int(order * ESC) + 2) print(f" [transport tail-escalate: step {nst} order {order}->{neworder} " f"(bound {mp.nstr(bound, 3)} >= tol {mp.nstr(tol, 2)})]", file=sys.stderr) order = neworder Mtrip, Mrest = _mtrip(order) # exact continuation: same shifted polys, longer series _extend(order + 1) ynew = [fzero] * NS hp = fone for nn in range(len(a)): an = a[nn] for rr in range(NS): ynew[rr] = mpf_add(ynew[rr], mpf_mul(an[rr], hp, prec, rnd), prec, rnd) hp = mpf_mul(hp, hraw, prec, rnd) y = ynew t += h nst += 1 err_sum += bound if bound > worst: worst = bound if order > omax: omax = order if nst > 1000: raise RuntimeError("transport: step overflow") return [mp.mp.make_mpf(v) for v in y], nst, err_sum, worst, omax # ---------- MUM local solution (z-frame), single physical vector ---------- def local_solution(mp, den, num, v, Nloc0, z0, tol, scale): """MUM log-Frobenius local series with a CERTIFIED tail gate. Nloc0 is a STARTING seed only. The z-frame denominator factors exactly as 20(1+z)(1+9z)(1+25z), so the nearest singularity to the MUM point sits at |z| = 1/25 and r = 25*z0 (< 1, asserted) certifies the geometric envelope of the local solution at z0. The series is accepted only when bound = max(|term_n| over trailing TAILWIN n, all comps) * r/(1-r) (term_n = sum_j F_{n,j} L0^j z0^n) satisfies bound*scale < tol, scale = sup_i z0^{-a_i} covering the raw-frame rescale; on a miss the SAME Frobenius recursion is continued exactly (x ESC) to NCAP_MULT*Nloc0, then RuntimeError (fail-closed). The linear solve (n den(0) - Num(0)) X = B uses the exact finite Neumann series of the nilpotent residue (index 8, checked in integers at load). Returns (out, err_scaled, N_used).""" prec, rnd = mp.mp._prec_rounding d0 = den[0] denc = list(den) + [mp.mpf(0)] * (7 - len(den)) denr = [c._mpf_ for c in denc] d0r = d0._mpf_ # sparse integer matrices Num_k (k = 0..6) as (row, col, raw value) numk = [[] for _ in range(7)] for (n, m), c in num.items(): for k in range(min(7, len(c))): if c[k] != 0: numk[k].append((n, m, c[k]._mpf_)) N0 = numk[0] def matvec(M, x): out = [fzero] * NS for (rr, cx, mv) in M: out[rr] = mpf_add(out[rr], mpf_mul(mv, x[cx], prec, rnd), prec, rnd) return out def axpy(alpha, x, y): # y + alpha*x return [mpf_add(y[k], mpf_mul(alpha, x[k], prec, rnd), prec, rnd) for k in range(NS)] def solve_A(n, B): # (n d0 I - N0)^{-1} B = sum_{k=0}^{7} N0^k B / (n d0)^{k+1} (N0 nilpotent, index 8) nd0 = mpf_mul(from_int(n), d0r, prec, rnd) u = [mpf_div(b, nd0, prec, rnd) for b in B] x = list(u) for _ in range(7): u = [mpf_div(w, nd0, prec, rnd) for w in matvec(N0, u)] x = [mpf_add(x[k], u[k], prec, rnd) for k in range(NS)] return x # F0(L) = e^{C0 L} v, C0 = N0/d0 F = [] F0 = [] X = [mp.mpf(c)._mpf_ for c in v] fact = 1 for j in range(JMAX + 1): fr = from_int(fact) F0.append([mpf_div(x, fr, prec, rnd) for x in X]) X = [mpf_div(w, d0r, prec, rnd) for w in matvec(N0, X)] fact *= (j + 1) F.append(F0) def _add_next(): n = len(F) RHS = [[fzero] * NS for _ in range(JMAX + 1)] for m_ in range(1, min(n, 6) + 1): Fnm = F[n - m_] Nm = numk[m_] dm = denr[m_] for j in range(JMAX + 1): T = matvec(Nm, Fnm[j]) if dm != fzero: c1 = mpf_neg(mpf_mul(dm, from_int(n - m_), prec, rnd)) T = axpy(c1, Fnm[j], T) if j + 1 <= JMAX: c2 = mpf_neg(mpf_mul(dm, from_int(j + 1), prec, rnd)) T = axpy(c2, Fnm[j + 1], T) RHS[j] = [mpf_add(RHS[j][k], T[k], prec, rnd) for k in range(NS)] Fn = [None] * (JMAX + 1) for j in range(JMAX, -1, -1): B = RHS[j] if j + 1 <= JMAX: c3 = mpf_neg(mpf_mul(d0r, from_int(j + 1), prec, rnd)) B = axpy(c3, Fn[j + 1], B) Fn[j] = solve_A(n, B) F.append(Fn) for n in range(1, Nloc0 + 1): _add_next() L0 = mp.log(z0) L0r = L0._mpf_ z0r = z0._mpf_ r = 25 * z0 # certified: exact z-frame poles at z = -1, -1/9, -1/25 if not r < 1: raise RuntimeError("local series point z0 outside the certified radius 1/25") ncap = NCAP_MULT * Nloc0 def _term(n): Lp = fone acc = [fzero] * NS for j in range(JMAX + 1): acc = axpy(Lp, F[n][j], acc) Lp = mpf_mul(Lp, L0r, prec, rnd) return acc def _termmax(n): acc = _term(n) return max(abs(mp.mp.make_mpf(a)) for a in acc) * z0 ** n while True: mx = mp.mpf(0) for n in range(len(F) - TAILWIN, len(F)): mn = _termmax(n) if mn > mx: mx = mn bound = mx * r / (1 - r) if bound * scale < tol: break if len(F) - 1 >= ncap: raise RuntimeError( f"local Frobenius series: certified tail bound " f"{mp.nstr(bound * scale, 3)} >= tol {mp.nstr(tol, 2)} at z0={mp.nstr(z0, 6)} " f"(r=25*z0={mp.nstr(r, 4)}), N={len(F) - 1} (cap {ncap}); fail-closed") target = min(ncap, int((len(F) - 1) * ESC) + 2) print(f" [local-series tail-escalate: N {len(F) - 1}->{target} " f"(bound {mp.nstr(bound * scale, 3)} >= tol {mp.nstr(tol, 2)})]", file=sys.stderr) while len(F) - 1 < target: _add_next() out = [fzero] * NS zp = fone for n in range(len(F)): out = axpy(zp, _term(n), out) zp = mpf_mul(zp, z0r, prec, rnd) return [mp.mp.make_mpf(o) for o in out], bound * scale, len(F) - 1 # ---------- eps-series ---------- def exp_gamma_series(mp, nmax, c): g = c * mp.euler out = [mp.mpf(1)] for n in range(1, nmax + 1): out.append(out[-1] * g / n) return out def layer_mix(mp, vec, mixer): out = [mp.mpf(0)] * NS for n, (i, K) in enumerate(STATE): s = mp.mpf(0) for r in range(0, K + 5): if r < len(mixer): s += mixer[r] * vec[IDX[(i, K - r)]] out[n] = s return out def constants_vector(mp): zv = {"1": mp.mpf(1), "z2": mp.zeta(2), "z3": mp.zeta(3), "z4": mp.zeta(4)} rat, _ = _load(mp) v = [] for (i, K) in STATE: c = DATA["constants_named"].get(f"{i},{K}") or {} v.append(sum(rat(fr) * zv[nm] for nm, fr in c.items()) if c else mp.mpf(0)) return v def gamma4_series(mp, nmax=4): """Exact Taylor coefficients of Gamma(1+eps)^4 = exp(-4 gamma_E eps + 4 sum_{k>=2} (-1)^k zeta(k) eps^k / k) through eps^nmax; Gamma(eps)^4 = eps^-4 times this.""" l = [mp.mpf(0), -4 * mp.euler] + [4 * (-1) ** k * mp.zeta(k) / k for k in range(2, nmax + 1)] E = [mp.mpf(1)] for n in range(1, nmax + 1): E.append(sum(k * l[k] * E[n - k] for k in range(1, n + 1)) / n) return E # ---------- main computation ---------- def compute(mp, pp_targets, mutate=0.0, verbose=False, qdps=None): """returns ({pp_key: {(i,K) -> value}}, {pp_key: certificate}) in the raw convention; one shared local series + one sequential transport pass through the targets (sorted deep-to-shallow). pp_targets: dict key -> mpf value. qdps = the QUOTED digit count the run must certify (tol = 10^-(qdps+TGUARD)); defaults to mp.mp.dps - GUARD. Certificate per point: value-level |err| <= (local tail + accumulated per-step transport tails) * l1(mixer).""" rat, frame = _load(mp) denz, numz = frame(DATA["z_frame"]) denp, nump = frame(DATA["pp_frame"]) if mutate: key = (IDX[(1, 0)], IDX[(2, 0)]) # connection entry feeding m1 eps^0 from m2 eps^0 nump[key] = [c * (1 + mp.mpf(mutate)) for c in nump[key]] dps = mp.mp.dps if qdps is None: qdps = max(dps - GUARD, 1) tol = mp.mpf(10) ** (-(qdps + TGUARD)) # STARTING SEEDS only: certified/escalated at run time. Nloc = int((dps + 12) / 0.709) + 8 # near-optimal step economy: minimize steps*order^2 s.t. ratio^(order+1) <= 1e-(dps+10) order = int(1.2 * dps) + 25 ratio = mp.mpf(10) ** (-mp.mpf(dps + 10) / (order + 1)) z0 = mp.mpf(1) / 128 a = DATA["rescale_a"] scale = z0 ** (-max(a)) # sup_i |z0^{-a_i}| = 128^2, covers the raw-frame rescale v = constants_vector(mp) t0 = time.time() fh, errL, Nused = local_solution(mp, denz, numz, v, Nloc, z0, tol, scale) t_loc = time.time() - t0 # to raw-master frame at p^2 = -128 (still e^{4 gamma_E eps}-normalized) y = [fh[n] / z0 ** a[i] for n, (i, K) in enumerate(STATE)] mixer = exp_gamma_series(mp, 6, -4) l1mix = sum(abs(mixer[rr]) for rr in range(5)) # layer_mix uses orders 0..4 only out = {} cert = {} t = mp.mpf(-128) t0 = time.time() nst_tot = 0 errT = mp.mpf(0) wstep = mp.mpf(0) omax = order for key in sorted(pp_targets, key=lambda k: pp_targets[k]): tgt = pp_targets[key] y, nst, errs, ws, om = transport(mp, denp, nump, y, t, tgt, order, ratio, tol) nst_tot += nst errT += errs wstep = max(wstep, ws) omax = max(omax, om) t = tgt yr = layer_mix(mp, y, mixer) out[key] = {s: yr[n] for n, s in enumerate(STATE)} cert[key] = {"bound": (errL + errT) * l1mix, "local": errL, "transport": errT, "worst_step": wstep, "tol": tol, "Nloc": Nused, "order": omax} if verbose: print(f" [local series N={Nused}: {t_loc:.1f} s; transport {nst_tot} steps, " f"Taylor order {omax}: {time.time()-t0:.1f} s]", flush=True) return out, cert def digits(mp, a, b): """digits of agreement -log10 |a-b|/|b|, capped at the working precision (no printed count ever exceeds the digits the arithmetic carried).""" cap = float(mp.mp.dps) if b == 0: return cap if a == 0 else min(cap, float(-mp.log10(abs(a)))) d = abs(a - b) / abs(b) return min(cap, float(-mp.log10(d))) if d > 0 else cap def gate(name, measured, floor, unit="d"): """RAISING gate: print the verdict, raise (-> exit 1) below the floor.""" if not (measured >= floor): raise RuntimeError(f"{name}: {measured:.1f} {unit} < floor {floor:.1f} {unit}") print(f" [gate] {name}: {measured:.1f} {unit} >= {floor:.1f} {unit} PASS", flush=True) # ---------- positive controls ---------- def run_controls(mp, adps): """Positive controls. adps = working precision of the K_0^4 quadrature (the paper's control ran at its 90-digit cap; the quick default uses a shallower one).""" print("== positive controls ==", flush=True) # B1: the AESZ-34 recurrence returns the Hulek-Verrill integers # L4 = sum_k z^k Q_k(th): Q0=th^4, Q1=35th^4+70th^3+63th^2+28th+5, # Q2=259th^4+1036th^3+1580th^2+1088th+285, Q3=225(th+1)^2(th+2)^2 Q = [[0, 0, 0, 0, 1], [5, 28, 63, 70, 35], [285, 1088, 1580, 1036, 259], [900, 2700, 2925, 1350, 225]] A_n = [1] NN = 40 for n in range(1, NN + 1): s = 0 for m in range(1, 4): if n - m >= 0: s += sum(Q[m][j] * (n - m) ** j for j in range(5)) * A_n[n - m] * (-1) ** (m + 1) if s % n ** 4: raise RuntimeError(f"AESZ-34 recurrence: non-integer A_{n}") A_n.append(s // n ** 4) if A_n[:6] != [1, 5, 45, 545, 7885, 127905]: raise RuntimeError(f"AESZ-34 recurrence: Hulek-Verrill integers not reproduced ({A_n[:6]})") print(f" [control] AESZ-34 recurrence -> Hulek-Verrill integers {A_n[:6]} PASS", flush=True) # B2: L4 annihilates varpi_0 at z = 1/128 to truncation level zt = mp.mpf(1) / 128 l4 = mp.mpf(0) for k in range(4): for j in range(5): if Q[k][j]: sj = sum(mp.mpf(A_n[n]) * (-1) ** n * n ** j * zt ** (n + k) for n in range(NN + 1)) l4 += Q[k][j] * sj l4d = float(-mp.log10(abs(l4))) if l4 != 0 else 999.0 gate(f"[control] |L4[varpi_0]| at z=1/128, N={NN} ({mp.nstr(abs(l4), 3)}, truncation level) -log10", l4d, 20.0) # C: the n=0 log-tower against the literature Gamma-product at L = -3 rat, frame = _load(mp) denz, numz = frame(DATA["z_frame"]) N0 = mp.matrix(NS, NS) for (n, m), c in numz.items(): N0[n, m] = c[0] C0 = N0 / denz[0] v = mp.matrix(constants_vector(mp)) Lval = mp.mpf(-3) acc = mp.matrix(NS, 1) X = v.copy() fact = mp.mpf(1) for j in range(12): if j > 0: X = C0 * X fact *= j acc += X * (Lval ** j / fact) NE = 8 tot = [mp.mpf(0)] * (NE + 1) for j in range(5): lg = [mp.mpf(0)] * (NE + 1) for k in range(2, NE + 1): lg[k] = mp.zeta(k) * (-1) ** k / k * ((4 - j) + (1 + j) * (-1) ** k + j ** k - (-(j + 1)) ** k) G = [mp.mpf(1)] + [mp.mpf(0)] * NE for n in range(1, NE + 1): s = mp.mpf(0) for k in range(2, n + 1): s += k * lg[k] * G[n - k] G[n] = s / n E = [mp.mpf(1)] for n in range(1, NE + 1): E.append(E[-1] * j * Lval / n) pref = 5 * mp.binomial(4, j) * (-1) ** j for n in range(NE + 1): tot[n] += pref * sum(E[k] * G[n - k] for k in range(n + 1)) dd = digits(mp, acc[IDX[(1, 0)]], -tot[4]) gate("[control] n=0 log-tower of m1-hat vs the literature Gamma-product (eps^4 layer, L=-3)", dd, min(mp.mp.dps, 100) - 12) # A: the three-loop sibling identity int_0^inf x K0(x)^4 dx = 7 zeta(3)/8 (live quadrature) old = mp.mp.dps mp.mp.dps = adps try: t0 = time.time() f = lambda x: x * mp.besselk(0, x) ** 4 L = int(mp.mp.dps * 2.303 / 4) + 20 knots = [0, mp.mpf(1) / 2, 1, 2, 4, 8, 16, 32] knots = [k for k in knots if k < L] + [L] val = mp.quad(f, knots) ref = 7 * mp.zeta(3) / 8 da = digits(mp, val, ref) gate(f"[control] three-loop sibling int x K0^4 dx = 7 zeta(3)/8 (quadrature at {adps} dps, " f"{time.time()-t0:.1f} s)", da, adps - 20) finally: mp.mp.dps = old def bessel_oracle(mp, pp, res, qcap=115): """Live independent oracle: m1|eps^0 = -16 int_0^inf x J_0(x sqrt(-p^2)) K_0(x)^5 dx by quadrature, fail-closed refine-until-agree (double refinement must agree to 10^-(qd+10), degree escalation to a cap, refuse there). Gate leg only; the working precision is capped at qcap digits (Bessel-K evaluation slows sharply beyond).""" old = mp.mp.dps mp.mp.dps = min(old, qcap) try: s = -pp L = int((mp.mp.dps) * 2.303 / 5) + 25 f = lambda x: x * mp.besselj(0, x * mp.sqrt(s)) * mp.besselk(0, x) ** 5 t0 = time.time() knots = [0, mp.mpf(1) / 2, 1, 2, 3, 4] + list(range(5, L, 2)) + [L] qd = max(mp.mp.dps - 15, 30) md, prev = 6, None while True: vq = mp.quad(f, knots, maxdegree=md) if prev is not None and abs(vq - prev) < mp.mpf(10) ** (-(qd + 10)): MJ = vq break if md >= 12: raise RuntimeError( f"--bessel quadrature: double-refinement agreement {mp.nstr(abs(vq - prev), 3)} " f">= tol 1e-{qd + 10} at maxdegree {md} (cap 12); fail-closed") prev, md = vq, md + 2 live = -16 * MJ return digits(mp, res[(1, 0)], live), time.time() - t0, mp.mp.dps finally: mp.mp.dps = old def parse_pp(s): fr = Fraction(s) if fr >= 0: raise ValueError(s) return fr def main(): ap = argparse.ArgumentParser( description="four-loop equal-mass banana I_11111 (CY3, AESZ #34): explicit log-Frobenius " "form, all five masters eps^-4..eps^0 at Euclidean p^2 < 0.") ap.add_argument("--dps", type=int, default=None, help=f"quoted digits (default {QUICK_DPS}; {FULL_DPS} under --full); +{GUARD} working guard") ap.add_argument("--pp", action="append", default=None, help="Euclidean p^2 < 0 as an integer or rational (repeatable; write --pp=-7/2); " f"default {QUICK_PP} (quick) or {FULL_PP} (--full)") ap.add_argument("--full", action="store_true", help="the paper's held-out table against the stored references: both points at 100 " "digits, full-depth controls (minutes); add --bessel for the live-oracle rows") ap.add_argument("--check", action="store_true", help="two-precision self-agreement gate: rerun at dps+60 guard digits (long)") ap.add_argument("--bessel", action="store_true", help="live Bessel-moment oracle for m1 eps^0 by quadrature (long at high dps: many minutes " "per point at the 115-digit cap)") ap.add_argument("--control", action="store_true", help="full-depth positive controls (implied by --full)") ap.add_argument("--no-controls", action="store_true", help="skip the positive controls") ap.add_argument("--mutate", action="store_true", help="control: perturb one connection entry by 1e-6; the held-out gate must collapse " "and the run must exit nonzero") args = ap.parse_args() print("== four-loop equal-mass banana I_11111 (CY3, AESZ #34): explicit log-Frobenius form ==", flush=True) pin = check_pin() nil = structural_checks() print(f"[data] embedded connection + boundary + references sha256 {pin[:16]}... matches the pin; " f"den = 20(1+z)(1+9z)(1+25z), residue nilpotent of index {nil} (exact integers)", flush=True) import mpmath as mp qdps = args.dps if args.dps is not None else (FULL_DPS if args.full else QUICK_DPS) if qdps < 10: print(f"usage: --dps must be >= 10 (got {qdps})", file=sys.stderr) return EXIT_USAGE pps_in = args.pp if args.pp else (FULL_PP if args.full else QUICK_PP) try: pp_fr = {s: parse_pp(s) for s in pps_in} except (ValueError, ZeroDivisionError): print(f"usage: every --pp must be a rational p^2 < 0 (Euclidean); got {pps_in}", file=sys.stderr) return EXIT_USAGE mode = "FULL (the paper's held-out table)" if args.full else "QUICK" print(f"mode: {mode} -- p^2 in {pps_in}, quoted dps {qdps} (+{GUARD} working guard)" + (f"; live Bessel oracle {'ON' if args.bessel else 'off (add --bessel)'}" if args.full else f"; --full = both points at {FULL_DPS} digits against the stored references"), flush=True) mp.mp.dps = qdps + GUARD # working guard; results quoted at qdps pp_targets = {s: mp.mpf(fr.numerator) / fr.denominator for s, fr in pp_fr.items()} t_all = time.time() if not args.no_controls: run_controls(mp, adps=90 if (args.full or args.control) else 40) print("== value path ==", flush=True) t0 = time.time() allres, cert = compute(mp, pp_targets, verbose=True, qdps=qdps) print(f" [wall {time.time()-t0:.1f} s for {len(pp_targets)} point(s) at dps {qdps} (+{GUARD} guard)]", flush=True) chk = None if args.check: # ACTING two-precision gate: the self-agreement must reach qdps + CHK_MARGIN; on a # miss BOTH legs auto-deepen (x2 guards -> genuinely different internal depths, all # runtime-certified) and the check re-runs, up to CHK_ROUNDS rounds, then refuses. g1, g2 = GUARD, GUARD + 60 rnd = 0 # m1's eps^{-4..-1} vanish IDENTICALLY in the form; numerically they are truncation # noise at any precision -- treat below-threshold values as zero-consistent. zero_tol = mp.mpf(10) ** (-qdps + 5) while True: if rnd == 0: legA = allres else: mp.mp.dps = qdps + g1 legA, _ = compute(mp, pp_targets, qdps=qdps) mp.mp.dps = qdps + g2 t0 = time.time() legB, _ = compute(mp, pp_targets, qdps=qdps) tB = time.time() - t0 mp.mp.dps = qdps + GUARD worsts = {} for key in pp_targets: w = 999.0 for s in legA[key]: va, vb = legA[key][s], legB[key][s] if max(abs(va), abs(vb)) < zero_tol: continue w = min(w, digits(mp, va, vb)) worsts[key] = w overall = min(worsts.values()) if overall >= qdps + CHK_MARGIN: chk = (worsts, g2 - g1, rnd, tB) break if rnd >= CHK_ROUNDS: raise RuntimeError( f"--check: worst self-agreement {overall:.1f} d < required {qdps + CHK_MARGIN} d " f"after {rnd} auto-deepen round(s) (leg guards +{g1}/+{g2}); refusing to certify") rnd += 1 g1 *= 2 g2 *= 2 print(f" [--check auto-deepen round {rnd}: legs -> dps+{g1} vs dps+{g2} " f"(worst {overall:.1f} d < {qdps + CHK_MARGIN} d)]", flush=True) allresm = None if args.mutate: print("== --mutate control: one connection entry (m1 eps^0 <- m2 eps^0) perturbed by 1e-6 ==", flush=True) allresm, _ = compute(mp, pp_targets, mutate=1e-6, qdps=qdps, verbose=True) E4 = gamma4_series(mp, 4) floor_ref = min(qdps, 50) - 10 for key in sorted(pp_targets, key=lambda k: pp_targets[k]): res = allres[key] print(f"p^2 = {key}", flush=True) for i in range(5): for K in range(-4, 1): if i == 1 and K < 0: continue print(f" m{i} eps^{K:+d} = {mp.nstr(res[(i, K)], qdps)}") c_ = cert[key] print(f" certified BOUND: value-level |err| <= {mp.nstr(c_['bound'], 3)} (abs, sup over the " f"window; local {mp.nstr(c_['local'], 3)} + transport {mp.nstr(c_['transport'], 3)}, " f"l1-mixed; per-step tol {mp.nstr(c_['tol'], 2)}, worst step {mp.nstr(c_['worst_step'], 3)}; " f"local N={c_['Nloc']}, Taylor order {c_['order']})", flush=True) # analytic gate: the tadpole tower against the exact Gamma(eps)^4 expansion dtad = min(digits(mp, res[(0, -4 + n)], E4[n]) for n in range(5)) gate("tadpole tower m0 eps^-4..0 vs exact Gamma(eps)^4 expansion (analytic; worst layer)", dtad, qdps - 2) if chk is not None: worsts, delta, rnd, tB = chk gate(f"--check two-precision self-agreement (worst nonzero layer, legs dps+{GUARD} vs dps+{GUARD + delta}" f"{', ' + str(rnd) + ' auto-deepen round(s)' if rnd else ''}; deep leg {tB:.0f} s)", worsts[key], qdps + CHK_MARGIN) g = DATA["gate_amflow_heldout"].get(key) b = DATA["gate_bessel_100d"].get(key) if g is None: print(f" (no held-out reference at p^2 = {key}; references exist at p^2 = -2 and -7 -- " f"the certified BOUND and the tadpole gate are the checks here)", flush=True) else: worst, w_m1 = 999.0, None for kk, sval in g.items(): i, K = map(int, kk.split(",")) d = digits(mp, res[(i, K)], mp.mpf(sval)) worst = min(worst, d) if (i, K) == (1, 0): w_m1 = d db = digits(mp, res[(1, 0)], mp.mpf(b)) if b else None print(f" held-out references: m1 eps^0 vs independent AMFlow tower {w_m1:.1f} d; worst of the 21 " f"layers {worst:.1f} d" + (f"; m1 eps^0 vs 100-digit Bessel reference {db:.1f} d" if db else "") + " (AMFlow references carry about 67 digits, the Bessel strings 100)", flush=True) gate(f"m1 eps^0 vs held-out AMFlow reference (p^2 = {key})", w_m1, floor_ref) gate(f"worst of the 21 layers vs held-out AMFlow tower (p^2 = {key})", worst, floor_ref) if db is not None: gate(f"m1 eps^0 vs 100-digit Bessel reference (p^2 = {key})", db, floor_ref) if args.full and key in PAPER_TABLE and qdps >= FULL_DPS: P = PAPER_TABLE[key] gate(f"paper table row reproduced: m1 eps^0 vs AMFlow (printed {P['amflow_m1']}, p^2 = {key})", w_m1, P["amflow_m1"] - 1) gate(f"paper table row reproduced: worst layer vs AMFlow (printed {P['amflow_worst']}, p^2 = {key})", worst, P["amflow_worst"] - 1) if db is not None: gate(f"paper table row reproduced: m1 eps^0 vs 100-digit Bessel reference " f"(printed {P['bessel_ref']}, p^2 = {key})", db, P["bessel_ref"] - 1) if allresm is not None: dm = digits(mp, allresm[key][(1, 0)], mp.mpf(g["1,0"])) print(f" --mutate control: m1 eps^0 vs held-out reference {w_m1:.1f} d -> {dm:.1f} d after the " f"1e-6 connection perturbation (paper: 67 -> {PAPER_MUTATE_TO})", flush=True) gate(f"MUTATED run, m1 eps^0 vs held-out AMFlow reference (p^2 = {key}; this gate MUST fail)", dm, floor_ref) if args.bessel: dl, tq, qd = bessel_oracle(mp, pp_targets[key], res) print(f" live Bessel oracle -16 int x J0(x sqrt(-p^2)) K0(x)^5 dx at {qd} dps ({tq:.1f} s): " f"m1 eps^0 agrees to {dl:.1f} d", flush=True) gate(f"m1 eps^0 vs live independent Bessel quadrature (p^2 = {key})", dl, min(qdps, qd) - 15) if args.full and key in PAPER_TABLE and qdps >= FULL_DPS: gate(f"paper table row reproduced: live Bessel oracle (printed {PAPER_TABLE[key]['bessel_live']}, " f"p^2 = {key})", dl, PAPER_TABLE[key]["bessel_live"] - 1) if allresm is not None: # a --mutate run whose points carry no held-out reference cannot demonstrate the collapse raise RuntimeError("--mutate: no held-out reference at the requested point(s); the collapse could " "not be demonstrated -- refusing rather than exiting clean") print(f"[wall] total {time.time()-t_all:.1f} s (this run, measured)", flush=True) if not args.full: print(f"(--full reproduces the paper's held-out table against the stored references: p^2 = -2 and -7 " f"at {FULL_DPS} digits; --bessel adds the live-oracle rows; --check the two-precision " f"self-agreement gate)", flush=True) elif not args.bessel: print("(--full --bessel adds the table's live Bessel-oracle rows; --check its self-agreement row)", flush=True) print("ALL CHECKS PASSED (exit 0)", flush=True) return EXIT_OK if __name__ == "__main__": try: sys.exit(main()) except RuntimeError as e: print(f"REFUSED (exit {EXIT_GATE}): {e}", file=sys.stderr, flush=True) sys.exit(EXIT_GATE)