# Exact-rational coverage audit -- re-confirmed 2026-10-22 01:13 UTC
# command: python3 rigorous/exact_coverage_U.py rigorous/results/U_007_1_cert.jsonl.gz

certificate: rigorous/results/U_007_1_cert.jsonl.gz
blocks: 2705
claimed range as exact rationals: [A0, A1] = [7/100 = 0.070000000000000007, 1 = 1]
interior abutments exact: 2704/2704  (gaps=0, overlaps=0)
bottom: first alpha = 0.069999999999999993  (alpha - A0 = -7.216e-18);  alpha <= A0 ? True
top:    last  beta  = 1  (beta  - A1 = +0.000e+00);  beta  >= A1 ? True
EXACT TILING of [7/100, 1]: COMPLETE
rational equal-weight extremals a_1 = 1/m with 1/m in range (m^2 equal weights):
  n=1, a_1=1/1: r = 1 = 1  in block [0.95076981966160135, 1]; certified at left endpoint alpha = 0.95076981966160135 <= r  (so H(a_1) >= H(alpha))
  n=4, a_1=1/2: r = 1/2 = 0.5  in block [0.49548456701089366, 0.50192423393707564]; certified at left endpoint alpha = 0.49548456701089366 <= r  (so H(a_1) >= H(alpha))
  n=9, a_1=1/3: r = 1/3 = 0.33333333333333331  in block [0.33212195851272308, 0.33392298656301783]; certified at left endpoint alpha = 0.33212195851272308 <= r  (so H(a_1) >= H(alpha))
  n=16, a_1=1/4: r = 1/4 = 0.25  in block [0.24940375246795807, 0.25014425975687948]; certified at left endpoint alpha = 0.24940375246795807 <= r  (so H(a_1) >= H(alpha))
  n=25, a_1=1/5: r = 1/5 = 0.20000000000000001  in block [0.19983875099199169, 0.20021431461979783]; certified at left endpoint alpha = 0.19983875099199169 <= r  (so H(a_1) >= H(alpha))
  n=36, a_1=1/6: r = 1/6 = 0.16666666666666666  in block [0.1665025163988671, 0.16671801955244581]; certified at left endpoint alpha = 0.1665025163988671 <= r  (so H(a_1) >= H(alpha))
  n=49, a_1=1/7: r = 1/7 = 0.14285714285714285  in block [0.1428404055795473, 0.14292657548207671]; certified at left endpoint alpha = 0.1428404055795473 <= r  (so H(a_1) >= H(alpha))
  n=64, a_1=1/8: r = 1/8 = 0.125  in block [0.12494871746238133, 0.12500621119789102]; certified at left endpoint alpha = 0.12494871746238133 <= r  (so H(a_1) >= H(alpha))
  n=81, a_1=1/9: r = 1/9 = 0.1111111111111111  in block [0.11107167328001899, 0.11111197182459083]; certified at left endpoint alpha = 0.11107167328001899 <= r  (so H(a_1) >= H(alpha))
  n=100, a_1=1/10  <-- 100 equal weights, tightest case on the whole range: r = 1/10 = 0.10000000000000001  in block [0.099967349784789217, 0.10000000000000001]; certified at left endpoint alpha = 0.099967349784789217 <= r  (so H(a_1) >= H(alpha))
  n=121, a_1=1/11: r = 1/11 = 0.090909090909090912  in block [0.090879107428466777, 0.090913700122468022]; certified at left endpoint alpha = 0.090879107428466777 <= r  (so H(a_1) >= H(alpha))
  n=144, a_1=1/12: r = 1/12 = 0.083333333333333329  in block [0.083330670474436347, 0.083357316006308982]; certified at left endpoint alpha = 0.083330670474436347 <= r  (so H(a_1) >= H(alpha))
  n=169, a_1=1/13: r = 1/13 = 0.076923076923076927  in block [0.076913281854583876, 0.076934218849130181]; certified at left endpoint alpha = 0.076913281854583876 <= r  (so H(a_1) >= H(alpha))
  n=196, a_1=1/14: r = 1/14 = 0.071428571428571425  in block [0.071413541082768886, 0.071430291060702866]; certified at left endpoint alpha = 0.071413541082768886 <= r  (so H(a_1) >= H(alpha))
decimal stage joins:
  0.07 (bottom): r = 7/100 = 0.070000000000000007  in block [0.069999999999999993, 0.070015772784090088]; certified at left endpoint alpha = 0.069999999999999993 <= r  (so H(a_1) >= H(alpha))
  0.08: r = 2/25 = 0.080000000000000002  in block [0.07999898142762589, 0.080000000000000002]; certified at left endpoint alpha = 0.07999898142762589 <= r  (so H(a_1) >= H(alpha))
  0.10: r = 1/10 = 0.10000000000000001  in block [0.099967349784789217, 0.10000000000000001]; certified at left endpoint alpha = 0.099967349784789217 <= r  (so H(a_1) >= H(alpha))
  0.11: r = 11/100 = 0.11  in block [0.10997549296721414, 0.11]; certified at left endpoint alpha = 0.10997549296721414 <= r  (so H(a_1) >= H(alpha))
  0.15: r = 3/20 = 0.14999999999999999  in block [0.14999999999999999, 0.15015702687582644]; certified at left endpoint alpha = 0.14999999999999999 <= r  (so H(a_1) >= H(alpha))
  1: r = 1 = 1  in block [0.95076981966160135, 1]; certified at left endpoint alpha = 0.95076981966160135 <= r  (so H(a_1) >= H(alpha))
RESULT: PASS -- exact-rational cover of [7/100,1] complete; a_1=1/10 and all stage joins covered
