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# Sources checked for T634
The family issue #254 was read first with `python3 agents/queue.py show 254`.
The family coverage map and the current public documents T79, T80, T83, T96,
T129, T215, T217, T255, T297, T309 and T506 were read, not another actor's
transcript. The current corpus search finds no duplicate quantity. Actual
slugs for all linked companions were obtained from live search results.
The empty title claim succeeded as T634.
- Matsusaka, *The Fourier coefficients of the McKay–Thompson series and the
traces of CM values*, Research in Number Theory 3 (2017), article 23,
Introduction, the two prime-level formulas preceding Theorem 1.1:
https://doi.org/10.1007/s40993-017-0090-x . Checked the pA/pB assignment,
integer exponents, constant shifts, Fricke coefficient, and q convention.
The same introduction defines CM points from binary quadratic forms and
describes the graded Monster traces. The source explicitly includes all
five primes 2,3,5,7,13. Local HTML and its SHA-256 are archived.
- Tomiyama, *Generalized modular equations and singular values of Hauptmoduln*,
version 2 (2025), https://arxiv.org/html/2505.05135v2 , Corollary 3.11:
checked algebraic integrality of all McKay–Thompson CM values. This supports
the algebraic tag and the deduction that every rational value is an integer.
It does not select our finite bound. Local HTML and SHA-256 are archived.
- DLMF 23.17.8, https://dlmf.nist.gov/23.17 : checked eta's product, with the
DLMF nome exp(pi i tau) distinguished from this table's exp(2 pi i tau).
Equations 23.17.2 and 23.17.5 give j(i)=1728 and j=1728 J.
- DLMF 20.2.2 and 20.2.3, https://dlmf.nist.gov/20.2 : checked the theta-two
and theta-three Fourier series used by the independent checker.
- DLMF 23.15.6, https://dlmf.nist.gov/23.15 : checked
lambda=(theta2/theta3)^4. DLMF 23.19.2,
https://dlmf.nist.gov/23.19#E2 , gives
J=(4/27)(1-lambda+lambda^2)^3/(lambda^2(1-lambda)^2), and j=1728 J.
The theta-to-j computation thus uses the required normalization.
- DLMF 23.18.5, https://dlmf.nist.gov/23.18 : consulted eta's modular
transformation convention. No additional Fricke identity is advertised
without a numerical test in the table.
- T309's actual definition uses m_p=C_p/R_p on these five levels. The exact
five canonical equations were read from its Numbers entries and archived
in algebraic-check-data.json. Its PARI/GP reference was consulted for the
canonical equation convention. This is a transformed relation to the
present values, not a duplicated table or a repeats declaration.
- T129's class polynomials at D=-4r, r=1,...,30, were read as exact integer
polynomials and archived in algebraic-check-data.json. T83 identifies the
rational j-values at r=1,2,3,4,7. No class polynomial of a McKay–Thompson
value was guessed from numerical digits.
The DLMF pages and Sage numerical API documentation were read with the web
tool. Attempts to download DLMF .tex URLs using urllib were refused; that
failure is preserved in sources-manifest.json. It did not prevent reading
and checking the equations on the HTML pages. The initial exploratory Sage
probe used nonexisting method names eta(), contains() and rel_accuracy_bits();
the working Sage names are modular_eta(), membership (`n in ball`) and
accuracy(). ComplexBall finiteness is checked on both real components.
All successful evidence was produced after these local failures were fixed.
This source review and numerical checking were performed by the builder.
They are not a substitute for the separate independent critique session.