# command: python3 pn_families.py --xc 61 --xmax 100 --lam0 16
# started: 2026-10-27 02:03:10 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, 61 <= x <= 100, sigma < 3.2074 (regimes N, C, Sm, Ss, P); arb prec = 96
[N] tau <= 8, mu <= 1/15.967, n <= 20 m^2/E(tau): 256 cells, 0 failed  6s
[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 <= 3.207, nu >= 15.967: 40 cells, 0 failed  21s
[Ss] sigma <= 0.5, tau >= 8: 2 cells, 0 failed  0s
    N : min pl = 0.7209, max Lu = 1.1544
    C : min pl = 0.7763, max Lu = 1.1851
    Sm: min pl = 0.4560, max Lu = 1.4353
    Ss: min pl = 0.9349, max Lu = 1.0304
[E] sigma <= 0.5: N/P <= 0.1882 (Lemma 9.20 at lam = 16) < (min pl / max Lu) sqrt(1 - loss) over N, C, Ss = 0.7209 / 1.1851 * 0.96697 = 0.5882: verified   (loss <= 0.06497)
[E] 0.5 <= sigma <= 3.207: N/P <= 0.00416 < (min pl / max Lu) sqrt(1 - loss) = 0.4560 / 1.4353 * 0.96697 = 0.3072: verified
ALL VERIFIED (31s)

real	0m30.700s
user	0m30.181s
sys	0m0.037s
# finished: 2026-10-27 02:03:41 UTC