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# Sources and normalization for T669
The family issue #296 was read first and archived as family-296.md in the run.
The governing skill manifest is agents/runs/20261102T045202Z-build-skills/manifest.json.
Its local/live hashes agree, as recorded in skill-evidence.json.
1. Li, Long and Tu, Computing Special L-Values of Certain Modular Forms with
Complex Multiplication, SIGMA 14 (2018), 090, §2.1, equation (2.1).
https://sigma-journal.com/2018/090/sigma18-090.pdf
DOI 10.3842/SIGMA.2018.090; arXiv 1803.06072.
Read through the web tool, including page 4's displayed definition and
the explicit fields Q(i) and Q(sqrt(-3)). Their n is our w_D and their
fundamental discriminant -d is our D. The leading Gamma(1/2) is sqrt(pi).
Direct local downloading returned HTTPError; the web copy was readable.
2. Barquero-Sanchez, Cadwallader, Cannon, Genao and Masri,
Faltings heights of CM elliptic curves and special Gamma values,
Research in Number Theory 3 (2017), 13.
https://link.springer.com/article/10.1007/s40993-017-0077-7
DOI 10.1007/s40993-017-0077-7.
Read §1 (the metric and Theorem 1.3), §3 (F), §4 (CM points), and
Theorem 5.1. At conductor one, their class-product formula gives
log b_D = log(2*pi) + log(d)/4 + mean(log F(tau)).
The forms in checks.py use tau=(-b+i*sqrt(d))/(2a). Changing the sign
of b conjugates ideal classes and leaves this real class product unchanged.
The local HTML snapshot's digest is recorded in sources.json.
3. NIST DLMF 23.15 and 23.17.8.
https://dlmf.nist.gov/23.15 and https://dlmf.nist.gov/23.17#E8
Read the nome convention q=exp(pi*i*tau) and the eta product. Our Q=q^2,
hence eta(tau)=exp(pi*i*tau/12)*product(1-Q^n). Direct local fetches
failed, but the web copies were readable. Equation 23.17.3 independently
supplies the square-lattice Gamma value.
The tail estimate in checks.py is derived directly: for r=|Q|<1,
|log(1-Q^n)| <= sum(k>=1,r^(n*k)/k) <= r^n/(1-r^n).
Summing over n>M gives at most r^(M+1)/((1-r)*(1-r^(M+1))).
Since log F contains twice log|eta|, its error allowance is twice this bound.
The 80-term evaluation uses this absolute bound in an Arb ball, not two
unbounded truncations whose agreement could be accidental.
Additional relations are deductions checked in this run. The exact character
sum is sum(a*chi_D(a))=-2*d*h_D/w_D. Substituting it into T230's
imaginary-field formula gives gamma_K=2*EulerGamma+log(2*pi^2)-2*log(b_D).
Theorem 1.3 at conductor one gives h_st(D)=log(d)/4-log(b_D) in the
Silverman/Faltings metric (i/2) integral(omega wedge conjugate(omega)).
The square-lattice identities follow from Gamma reflection, T296's K(1/2),
T506's eta(i), and T177's B(1/4,1/2); all were checked numerically.
The live tables T9, T129, T211, T230, T296, T506, T665 and T177 were read.
Their addresses were resolved from live search results or page links. T665
was an unpublished draft at the time of this build; its contextual link
states no equality of its marked complex periods with this positive constant.
The proposed order-indexed height companion did not yet exist in live search.
No future identifier is invented. The absent tag 'complex multiplication'
would connect T669 with T129, T296, T506 and the proposed height companion;
the draft uses the four existing tags instead.