Re-runs made for this package on 2026-10-14 (macOS, Python 3.13.5; mpmath 1.3.0, SymPy 1.14.0), each from a fresh copy
of this reproducibility folder in a scratch directory, so that no recorded output was overwritten.  "PY" is a Python
with mpmath and SymPy.  Comparisons with the recorded outputs: byte for byte ("IDENTICAL"), or after removing running
times ("SAME modulo times") for outputs that print running times.

1. Quick part: sh run_quick.sh PY   (8.5 s)
   IDENTICAL  check1_k5_A.txt, check1_k6_A.txt, check1_k6_B.txt, check1_k7_A.txt, check1_k8_d2_s6_A.txt,
              check1_k9_d2_s7_A.txt                                   (original/outputs/)
   IDENTICAL  check2_k5_A.txt, check2_k6_A.txt, check2_k6_B.txt, check2_k7_A.txt, check2_k8_d2_s6_A.txt,
              check2_k9_d2_s7_A.txt                                   (original/outputs/)
   IDENTICAL  identity_k5_A.txt, identity_k6_A.txt, identity_k6_B.txt, identity_k7_A.txt, identity_k8_d2_s6_A.txt,
              identity_k9_d2_s7_A.txt   (show_identity.py with the argument 22; original/outputs/)
   IDENTICAL  real_structure_k5_A.txt, check_k7_exact.txt, exact_symmetric_k7-31.txt      (original/outputs/)
   IDENTICAL  check_note.txt, rejection_tests_k7_exact.txt                                 (outputs/)
   23 of 23 identical.  check_note.py: ALL CHECKS PASSED, 4.4 s.
   outputs/rejection_tests_k7_exact.txt is also byte-identical to original/outputs/rejection_tests_k7_exact.txt
   (SHA-256 1bccc8819a7f6ef57c4f9be87c15c4b4d3b61254dfe1b3e7e83b434dd0f9a225).

2. Hashes: in original/, shasum -a 256 -c outputs/sha256.txt   ->  14 x OK (six certificates, eight programs).

3. Slow part of original/ (total of item 3 to 5: 186 s of CPU time, one process at a time)
   python3 -I rejection_tests.py certificates/<cert>.json PY  (six certificates, about 3 s each)
       SAME  rejection_tests_k5_A.txt, _k6_A, _k6_B, _k7_A, _k8_d2_s6_A, _k9_d2_s7_A: 36 corruptions, all rejected by both checkers
   checker 1 with P = 256 and checker 2 with 250 bits, six certificates
       SAME  precision_variation.txt: 12 x PASS
   PY -I exact_symmetric.py 201   (35 s)
       SAME  exact_symmetric_k7-201.txt: (C1)-(C3) hold for all odd k in [7, 201]; k = 7 and k = 9 identities verified

4. Verification run A (run in a copy of verification_run_A/)
       SAME  vA_k7_exact.out          (all coefficients reduce to 0; RESULT: PASS)
       SAME  vA_k7_controls.out       (five corruptions: FAIL)
       SAME  vA_k7_numeric.out        (24 root choices, worst relative residual 7.4e-92)
       SAME  vA_cert_check_all.out    (six certificates PASS)
       SAME  vA_cert_controls.out     (21 corrupted certificates rejected)
       SAME  vA_k5_realstructure.out
       SAME  vA_oddk_numeric.out      (k = 7, ..., 101)
       SAME  vA_oddk_201.out          (98/98, modular and exact; 52 s)

5. Verification run B (run in a copy of verification_run_B/)
       SAME  vb_k7_exact.out.txt      (identity verified; five corruptions rejected)
       SAME  vb_k7_numeric.out.txt    (48 root choices, error 5.4e-109)
       SAME  vb_kantorovich_cert_<cert>.out.txt for the six certificates, with all eight variants each
       SAME  vb_kantorovich_summary.txt (6 x exit 0, 42 x exit 1)
       SAME  vb_orbit_invariants_k5.out.txt
       SAME  vb_oddk.out.txt          (OVERALL: PASS; 29 s)
       SAME  vb_oddk_extend.out.txt   (odd k in [7, 201] and [203, 1001]: no failure; 27 s for the second range)
   The searches (vb_search.py, vb_refine_cert.py) and the search programs of original/search/ were not re-run; they
   are not part of any proof.

The programs were run with "nice -n 10", one at a time.
