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7335 bytes, as of the version from 2026-11-02 08:35 (current). Recorded here, not run.
# Numerical evidence and build handoff: T672
Run: 20261102T075604Z-build. Table: https://numberdb.org/T672, private draft.
The controller owns the subsequent independent critique, repair and offer.
No publication, review or offer action was performed by this build.
## Policy and source provenance
The coordinated skill manifest is
`agents/runs/20261102T075604Z-build-skills/manifest.json`.
Both archived files were read and byte-compared. Their SHA-256 is
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The snapshot canonical skill has the same digest. See skill-evidence.json.
The historical digest in campaign-gate.md is not substituted for this manifest.
Family issue #298 was read before work. It prescribes the q normalization,
parameter order (g,n), class notation, signed integers and companion links.
The screen.py already_here and already_asked calls, direct name searches,
integer lookups and tag-list retrieval are archived in screening.json/log.
26752 matches T255's irreducible degree, not a duplicate coefficient family.
No O’Nan coefficient family was found. The successful empty draft claim is
claim-result.json. All thirteen listed companions were read and archived.
Their actual long URLs were read from the page links, not derived from titles.
Primary mathematical source: Duncan–Mertens–Ono, arXiv:1702.03516v4,
Theorems 3.1 and 4.1, Table 5.1, Proposition 5.1 and Appendix D.
The downloaded HTML is archived as dmo.html. Appendix A/B extraction and its
source SHA-256 are preserved in source-controls.json. CTblLib 1.3.11 is
archived with its digest there, and gap-ON-excerpt.txt preserves the raw ON
ordinary character-table definition. Companion document digests and values
are in companion-controls.json.
## Computation and precision
All nine classes of order <=7, each at all positive grades n<=200 with
n congruent to 0 or 3 modulo 4: 900 exact signed integers. There are 700
distinct level-grade computations, because the class pairs 4A/4B and 7A/7B
have identical series. Separate class identities are retained.
The generator enumerates reduced positive definite integral quadratic forms
including imprimitive forms. For each form, it enumerates SL2(Z)/Gamma0(N)
using the projective line of first columns modulo N. Only transforms with
N dividing their first quadratic coefficient contribute. Dividing this coset
sum by the original PSL2 stabilizer gives the required Gamma0(N) orbit sum:
an orbit of size w_1/w_N contributes 1/w_N. The stabilizers are three for
multiples of [1,1,1], two for multiples of [1,0,1], and one otherwise.
Arb encloses the CM arguments, Dedekind eta, modular j, and weighted trace
sums at 1024 bits. Exact rational class-number corrections are propagated
in the same arithmetic. DMO's integrality theorem, not the decimal pattern,
allows each narrow enclosure to identify its unique integer. The generator
requires finite real and imaginary components, imaginary containment of zero,
and component radii below 1e-110; thus all approximate intermediate checks
exceed the requested 100-digit standard. No approximate coefficient is rounded
without a theorem and isolating interval. The integers themselves are exact.
The dry_run.py checks report 900 entries, 39 characters for the longest at
(1A,199), and a 43.8 KB entries block. Every value passes exactness().
The generator imports numberdb.sage first and names its rings.
SageMath 10.9 and numberdb 0.1.10 were used.
## Independent checks and the source discrepancy
The checks reconstruct 162 class-grade traces through n=36 from all 30
irreducible character multiplicities in DMO Appendix B. The first nine
ordinary character columns of CTblLib use the same row and class order on
this restriction. Exactly one of these 270 character values differs from
DMO's printed Appendix A: chi_22(7A) is printed as +5, but CTblLib gives -5.
Both the HTML and PDF of the DMO final preprint contain the positive sign.
The discrepancy has a mathematical diagnosis. CTblLib explicitly marks
chi_22 as the Galois conjugate of chi_21, so the rational character value -5
must be shared. Independently, an integer-valued character at an element of
prime order seven is congruent to its degree modulo seven. Here 169290 is
2 modulo seven; -5 is 2 modulo seven and +5 is not. In grade seven the printed
sign produces 24 for class 7A versus 4 for 7B and the modular trace; replacing
only this diagnosed sign restores agreement in all 162 checks. The original
printed matrix is retained unmodified; the independent GAP matrix is separate.
This is a correction to a verification source, not a choice of a different
moonshine normalization or a modification of the computed coefficients.
All 100 identity-class entries agree exactly with (t_2(n)-t_1(n))/2 from
T643. Those traces were computed independently using Hilbert class polynomials
and formal modular-form recurrences. The seven leading Hauptmodul
coefficients are checked by independent finite integer eta products (and T79
at level one), and five degree-two Faber polynomials agree with T635.
The degree-26752 and degree-143376 decompositions also agree with T255.
All 400 stated prime-power congruences, 200 alias comparisons, and 50
order-four odd-grade zeros pass. All 700 level-grade coefficients agree at
1280 bits. The latter is a precision stability check sharing the algorithm,
not a claim of an independent algorithm. The external representation and
Zagier checks provide the independent evidence; nonidentity grades above
36 are not checked against a separately published full coefficient list.
Controls run before comparison: j(i)-744=984, J^(2)(i)=536, exact isolation
of 26752, and rejection of a nonintegral real, nonreal, broad or nonfinite
ball. A deliberately changed sign is detected by the representation control.
The stored-values checker repeats the numerical identities on database data.
## Scope and follow-up
argument-selection.md justifies both free parameters as discrete_index and
reviews breadth before depth. The completeness note and generator agree.
The missing moonshine tag would serve T672, T652, T668 and T634–T636/T657;
this build retains the existing compatible tags and recommends that subject
tag for a later family review. No published table was changed.
No unbuilt Thompson T-number was invented. Its later complementary link must
be added during the family review. Higher-order classes remain extensions
of this same named quantity, not new tables.
The critic should inspect the source-sign diagnosis, independently check
numerical claims, and fingerprint generate.py plus every attached source or
verification dependency used in their review. The build does not create an
independent -arguments.json acceptance.
## Measured results
```json
{
"alias_comparisons": 200,
"character_decomposition_comparisons": 162,
"degree_comparisons": 2,
"entries": 900,
"faber_companion_checks": 5,
"hauptmodul_normalization_checks": 7,
"higher_precision_coefficient_checks": 700,
"max_characters": 39,
"maximum_1024_bit_component_radius": "8.0670106e-268",
"maximum_1280_bit_component_radius": "6.9832777e-345",
"negative_entries": 259,
"order_four_odd_grade_zeros": 50,
"prime_power_congruences": 400,
"stored": false,
"zagier_trace_comparisons": 100,
"zero_entries": 103
}
```