PASS  window_lower  phi^11 < R_lo
PASS  window_upper  R_hi < phi^12
PASS  nearest_lower phi^21 < R_lo^2
PASS  nearest_upper R_hi^2 < phi^23
PASS  eps_lower  phi^698 < R_lo^63
PASS  eps_upper  R_hi^88 < phi^975

eps in [0.079525603, 0.079525698]
5/63  = 0.079365079   7/88  = 0.079545455
1/(4pi)          = 0.079577472
1/(4pi) - alpha^2 = 0.079524220

simplest m/n in (1/4pi-a^2, eps_lo): 87/1094 = 0.079524680
  requires phi^12121 < R_lo^1094: PASS
upper 1/2 = 0.500000000  (exp 23)
upper 1/3 = 0.333333333  (exp 34)
upper 1/4 = 0.250000000  (exp 45)
upper 1/5 = 0.200000000  (exp 56)
upper 1/6 = 0.166666667  (exp 67)
upper 1/7 = 0.142857143  (exp 78)
upper 1/8 = 0.125000000  (exp 89)
upper 1/9 = 0.111111111  (exp 100)
upper 1/10 = 0.100000000  (exp 111)
upper 1/11 = 0.090909091  (exp 122)
upper 1/12 = 0.083333333  (exp 133)
lower 1/13 = 0.076923077  (exp 144)
upper 2/25 = 0.080000000  (exp 277)
lower 3/38 = 0.078947368  (exp 421)
lower 5/63 = 0.079365079  (exp 698)
upper 7/88 = 0.079545455  (exp 975)
lower 12/151 = 0.079470199  (exp 1673)
lower 19/239 = 0.079497908  (exp 2648)
lower 26/327 = 0.079510703  (exp 3623)