# Sources and reproducibility for T673

The quantity is the Harvey–Rayhaun / Griffin–Mertens weight-one-half Thompson
supertrace, not the weight-zero generalized moonshine function attached to a
Thompson element in the Monster. Classes use the first eight columns of the
Thompson character table. No library class-label permutation is involved.

Primary mathematical sources:

* J. A. Harvey and B. C. Rayhaun, *Traces of Singular Moduli and Moonshine for
  the Thompson Group*, Communications in Number Theory and Physics 10 (2016),
  23–62; https://arxiv.org/pdf/1504.08179 (v1 as served), equations (1.3)–(1.11),
  (2.10)–(2.14), Conjecture 3.1 and Tables 1, 2, 6.
* M. J. Griffin and M. H. Mertens, *A proof of the Thompson moonshine conjecture*,
  Research in the Mathematical Sciences 3 (2016), article 36,
  https://doi.org/10.1186/s40687-016-0084-7;
  https://arxiv.org/pdf/1607.03078v2. Theorem 1.2 proves module existence;
  equation (2.6), Table A.5 and Proposition 3.2 define its functions and their
  integrality. Section 5 / Table A.6 give the theta-split eta-rational identities.
  Appendix B gives the congruences. Labels in this note use the arXiv v2 numbering.
* The author's supplement:
  https://www.math.rwth-aachen.de/~Michael.Mertens/EtaCoeffs.html.
  The links lead to `EtaCoeffs/EtaCoeffs[g].m` and `EtaWt4/EtaWt[k]_[N].m`.
  `source-inputs.json` preserves the complete raw text for the selected classes
  and bases, including all zero coefficients.
* https://www.math.rwth-aachen.de/~Michael.Mertens/Identifyq.tar.gz,
  linked from the author's research page. The standalone etahalf.m link returns
  404, but this archive contains the required exact file. It also contains
  `Identifyq.m`, `Coeffs_all.m` and `MoonshineCoeffs10000.m`.

To construct each eta description, subtract the eta exponents of the auxiliary
cusp form from each basis element, then multiply by the supplied rational
coefficient. `Identifyq.m` chooses g4 for level 4, g8 for levels 8,16,32,
g12 for levels 12,36, and g20 for level 20. These are auxiliary cusp-form
names, not group classes. The resulting class levels in order are
4,8,36,12,36,16,32,20. `rebuild_eta_data.py` reproduces eta-data.json byte for
byte from source-inputs.json without executing any downloaded MAGMA code.
`generate.py` expands the Euler products with an exact integer recurrence
and combines their coefficients as Sage rationals, requiring integral output.

The source coefficient controls are an excerpt at exponents -3 through 201
of `MoonshineCoeffs10000.m`, with class order read from `Coeffs_all.m`.
MAGMA's backslash-newline continuation is removed before parsing the integer
lists. The source member's SHA-256 is retained in source-controls.json.
This is a reference comparison with data from the proof's authors, not a claim
that those data have an independent provenance from their eta descriptions.
The following checks supply distinct mathematical constructions:

* HR's Rankin–Cohen expression for the identity series, computed using divisor
  sums for E10, finite products for Delta, and the theta sum. Formal precision
  includes extra terms beyond the stored range before Laurent division.
* GM Table A.6 for all eight theta-split total functions, and its separate odd
  components for 1A,2A,3B,4A,4B. Finite factor-by-factor Euler multiplication
  supplies these eta quotients, sharing no expansion code with generate.py.
  Combining a total relation with the odd relation also determines the even
  component for these five classes. For 3A,3C,5A only the combined relation is
  asserted as this particular check; it is not represented as two independent
  scalar checks. The separate coefficient-control comparison covers every row.
* Seven selected Appendix B congruences on every admissible grade. The printed
  coefficient 53 is fifty-three, not 5 cubed; 2^7 is 128, not 27. Their baselines
  and superscripts were verified on the PDF page after initial extraction
  flattened the distinction. These were checker transcription errors, and no
  stored numbers were changed to satisfy them.
* T671's separately computed genus-character traces for every overlap at
  Hecke order 1 and negative discriminant magnitude 3; T255's Thompson degrees
  for the six decompositions at grades 0,4,5,8,9,12.
* cm_checks.py independently enumerates the reduced forms of discriminant -3D
  and encloses j at their CM roots in 1024-bit Arb arithmetic for every positive
  fundamental discriminant D in the table's range. The genus character is
  computed on a coprime represented integer, with zero on forms sharing a
  divisor with D, and with elliptic stabilizer weights. Nonfinite or wide balls
  are rejected before containment is tested. See cm-evidence.json.

Source discrepancy resolved: HR Table 6 prints a positive identity coefficient
at grade 21. HR Table 1 instead gives -2*14733025125. The exact Rankin–Cohen
formula, the author's 10,000-coefficient data, the eta-basis computation and
negative parity of the grade all give -29466050250. The table stores that
negative value and explains it in that entry's comment.

The frozen original PDFs, rendered source pages, download hashes, failed and
successful checker runs, and dependency versions remain in the build artifacts.
The authoritative attachments contain the generator, all source inputs needed
for it, checks, controls, selection rationale and final numerical evidence.

Canonical instructions: `agents/runs/20261102T085118Z-build-skills/manifest.json`.
Both archived copies have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.