Output of verify_margin.py (X = 1e12), run by the author, 29 October 2026:
prime powers up to 1.000000e+12: 37607992088   A = 1999997.3576058184  (error <= 2.1e-09)
  N in [1000, 1e+12]:  0.042686 <= M(N) <= 0.137816   (min near N=105606345001, max near N=319439)
  N in [1e+06, 1e+12]:  0.042686 <= M(N) <= 0.133242   (min near N=105606345001, max near N=3511779809)
  N in [1e+08, 1e+12]:  0.042686 <= M(N) <= 0.133242   (min near N=105606345001, max near N=3511779809)
  N in [1e+10, 1e+12]:  0.042686 <= M(N) <= 0.131211   (min near N=105606345001, max near N=742864907663)
  N in [1e+11, 1e+12]:  0.042686 <= M(N) <= 0.131211   (min near N=105606345001, max near N=742864907663)
RESULT: 0 < M(N) < 0.18484 on the whole range
Check: 37607992088 = pi(1e12) + 80070 = 37607912018 + 80070.