Computational companion v1.0

This archive contains exact input data, verification programs, computational certificates, dependency maps and execution records supporting the degree-four bound and the subsequent classifications in degrees one through four. It also includes the revised manuscript’s LaTeX source snapshot, an index connecting cited filenames to their locations, and detailed verification notes.

After extracting the archive, run “python3 restore_companion.py” to reconstruct and verify the complete directory tree. README.txt explains the archive’s structure, REPRODUCE.txt gives reproduction commands, and CITED_FILE_INDEX.txt locates the files cited in the paper.

This revision updates the manuscript snapshot and file mappings and includes successful restoration, integrity and exact classification-table checks. The README distinguishes fresh verification runs from preserved execution records. Not all large mathematical computations were rerun for this revision.

Exact certificates for the 800 parameter prescriptions (Section 5)

Exact certificates for the 800 parameter prescriptions (Section 5)

Companion to Classification of symmetric rank-two modular Nahm sums,
arXiv:2611.12868.

Every one of the 800 prescriptions S_i, A_i, T_i of Section 5 has an explicit
certificate. A standalone checker verifies each one in exact rational arithmetic
(Python with SymPy and python-flint; no Singular). It rebuilds the equations
from the paper's printed definitions: (4.1), the construction of Section 5.2,
and Ψ₂, Ψ₃ computed from Section 3. A separate enumeration recomputes Lemma 5.1
from the alphabet of Section 5.1 and finds exactly 450 / 319 / 31 prescriptions.
S155 and T21 are certified by the chart test of Section 5.2, independently of
the resultant slices of Lemma 5.7.

Contents of nahm_rank_two_verification_2026-11-11.zip

• parameter_case_certificates/: certificates, checker, generator programs,
  logs of the full run (every set passes), coverage.json (paper index and
  result for each prescription) and a README
• nahm_lean_pilot/: the Lean 4 pilot (42 compiled theorems), plus Lean files
  for eight unit-ideal cases of Lemma 5.2 (not yet compiled)
• section5_paragraph.tex: proposed text to follow Proposition 5.8

To check
unzip nahm_rank_two_verification_2026-11-11.zip
cd nahm_rank_two_verification_2026-11-11/parameter_case_certificates
checker/run_all.sh /anc/global_degree/replay/parameter_ideals

Here <companion> is the unpacked rank_two_classification_computations
archive. The full run takes about 45 minutes on two cores.

Not covered: the derivation of the root-condition equations (Theorem 4.7,
Section 4.4, Theorem 4.19); the real-tangency part of Lemma B.1
(anc/bilinear_repair); Lean verification of the 800 prescriptions.

SHA-256 of the zip:
06620195e87491612c37127f7373fc336c3d435152f7996dd588238d42467397