Re-run of the reproducibility package of the note (version 1, release/source.zip, sha256
d1cd7b8394a2748fa9ce720d357a8eecce0fe07aae05dd5b643d03882dbe5ea7) by the third independent verification run,
15 October 2026.

The archive was unpacked into an empty scratch folder; every program was run from its own folder with
`python -I -B`, with nice -n 10: check_note.py with Python 3.13.5 (standard library), the other programs with
Python 3.13.5, numpy 2.5.2, sympy 1.14.0, mpmath 1.3.0. Each output was written to a new file and compared
with the recorded output by `cmp` and `diff`.

| program (folder) | recorded output | result |
|---|---|---|
| check_note.py | check_note.out | identical apart from the line `time:` (26.4 s recorded, 27.8 s now) |
| original/sanity_numerics.py | original/sanity_numerics.out | identical apart from the timing in line 1 (6.1 s / 6.0 s) |
| original/sanity_numerics.py | verification_run_A/claim_sanity_numerics_rerun.out | byte-identical |
| original/sanity_numerics.py | verification_run_B/rerun_claim_sanity_numerics.out | identical apart from the timing in line 1 (5.7 s / 6.0 s) |
| verification_run_A/own_checks.py | own_checks.out | byte-identical |
| verification_run_A/own_checks2.py | own_checks2.out | identical apart from the line `# time` (1.7 s / 1.8 s) |
| verification_run_A/tv_phase_check.py | own_checks2b.out | byte-identical |
| verification_run_A/band_dp.py | band_dp.out | byte-identical |
| verification_run_A/claim_sanity_numerics_minratio.py | claim_sanity_numerics_minratio.out | identical apart from the timing in line 1 (6.1 s / 6.0 s) |
| verification_run_B/own_discrepancy.py | own_discrepancy.out | byte-identical |
| verification_run_B/own_exact_identities.py | own_exact_identities.out | byte-identical |

All programs exited normally; check_note.py reports "checks: 16, TOTAL failures: 0" and "RESULT: ALL CHECKS
PASSED". Every number quoted in Section 7 of the note was compared with check_note.out: (i) 360 point sets,
60/60 rejections; (ii) 3000 point sets, 200 pairs (x, N) with N in {16, 64, 256, 1024, 4096}; (iii) mean of
N D_N / log N 5.01 (N = 2^10) and 94.12 (N = 2^20), sample means of N D_N / sqrt(N log log N) 0.716..0.860,
overall minimum 0.383, overall maximum 1.577; (iv) m <= 3000 and eight cases. Two descriptions in Section 7 are
imprecise: Lemma 2.3 is checked on {2^n x} with E_N in floating point (line 232 of check_note.py), not exactly,
and the sample means lie between 0.72 and 0.86 (the note says 0.71 and 0.87).
