# command: python3 pn_families.py --xc 101
# started: 2026-10-27 02:02:26 UTC
# Python 3.13.17, python-flint 0.9.0, numpy 2.5.3, sympy 1.14.0
# machine: 2 cores, 7 GB RAM, Linux x86_64
# sha256 of the scripts in the directory:
#   62ba14c054023c20a139428938aa4561365d93366172e919afde228d09b40d4e  band_check.py
#   e3b29b32dee33a3771e46eee7fd3c7feb9d5281a0f880fb0ea2a9dc9bde9ea99  bergeron_certificate.py
#   a86969c6ce89c5a3aba23d4d30aa55d8fb9924613d794ab271728fe3f2035110  circle_check.py
#   0ad5f6f485561df541b4e83a0e780275e5077b2e3a6baa864ca0419d01ef15af  corner_check.py
#   85f217ba1cc63c5421bb2df726223642e8289baab3f46dc55e026417f27cbb21  cww_certificate.py
#   1b487046af3818f1b6e61f9903bb4dd3cbdb45e7d451c23cdce74faad8cf4eb0  pn_certificate.py
#   ea6a0e64e29ea4599808a3cb8023f9a18bfb4469f6b282d051f24034a5903108  pn_families.py
#   33684b0c7134b1b7da3e1e5fd944fe28c067fcc4255b832d327d3aea4043f665  row_check.py
#   0dd9195b2ba9a6ca36055c39fd90cd069ab530072d01786042400059e129e650  row_sweep.py
#   b077df5d9e0b9b7bf8be56cf2ab6763e5d8167a7408763c6ba54667837f077b3  rq_certificate.py
#
sources: pn_families.py ea6a0e64e29ea459, pn_certificate.py 1b487046af3818f1, rq_certificate.py b077df5d9e0b9b7b, cww_certificate.py 85f217ba1cc63c54
Theorem 9.24 (ii): R(x, A) for lam = A/x^2 >= 16, x >= 101 (regimes N, C, Sm, Ss, P); arb prec = 96
[N] tau <= 8, mu <= 1/15.980, n <= 20 m^2/E(tau): 256 cells, 0 failed  7s
[C] tau <= 8, U(rho) >= 20 m^2: 64 cells, 0 failed  3s
    N/C split: every C cell ends no later than the N cells it meets
[Sm] 0.5 <= sigma <= 6, nu >= 15.980: 54 cells, 0 failed  33s
[Ss] sigma <= 0.5, tau >= 8: 2 cells, 0 failed  0s
    N : min pl = 0.8358, max Lu = 1.0806
    C : min pl = 0.8579, max Lu = 1.1015
    Sm: min pl = 0.2155, max Lu = 1.4499
    Ss: min pl = 0.9646, max Lu = 1.0197
[E] sigma <= 0.5: N/P <= 0.1868 (Lemma 9.20 at lam = 16) < (min pl / max Lu) sqrt(1 - loss) over N, C, Ss = 0.8358 / 1.1015 * 0.97119 = 0.7370: verified   (loss <= 0.05678)
[E] 0.5 <= sigma <= 6: N/P <= 0.00412 < (min pl / max Lu) sqrt(1 - loss) = 0.2155 / 1.4499 * 0.97119 = 0.1444: verified
[P] sigma >= 6: N/P <= 4.14e-38 < 0.8888: verified  0s
ALL VERIFIED (44s)

real	0m44.204s
user	0m43.681s
sys	0m0.026s
# finished: 2026-10-27 02:03:10 UTC