# T710 source and normalization record

Checked primary source: S. Bettin and B. Conrey, *Period functions and cotangent
sums*, Algebra & Number Theory 7 (2013), 215–242, DOI 10.2140/ant.2013.7.215,
https://msp.org/ant/2013/7-1/ant-v7-n1-p08-p.pdf . Pages 217 and 223 define
c_0(h/k) with the negative sign and m/k weights. Page 223, equation (12),
gives D(0,h/k)=1/4+(i/2)c_0(h/k); p.224 defines D via the divisor series.
Page 225 defines V(h/k)=-c_0(hbar/k), with the modular inverse bar verified
visually rather than trusting PDF text extraction. Source pages 223 and 225
are preserved as source-page-9.png and source-page-11.png.

DLMF 25.11.13, https://dlmf.nist.gov/25.11.E13 , checked through the web tool,
gives zeta(0,u)=1/2-u. Grouping both divisor variables modulo k yields
D(s,h/k)=k^(-2s) sum_{r,s=1}^k exp(2*pi*i*h*r*s/k) zeta(s,r/k)zeta(s,s/k),
where the second summation index can be renamed to avoid the complex-variable
s. Evaluation at zero gives the exact double sum used by the checker.
Direct attempts to download DLMF TeX/PNG returned HTTP errors; the checked
rendered web formula is recorded here, not claimed as a locally downloaded source.

Derived, not quoted: put L=lcm(4,k), z=exp(2*pi*i/L), w=z^(L/k), i=z^(L/4).
The elementary exponential formula for cotangent puts alpha_k in Q(z) and
proves the signed Galois action in argument-selection.md. The primitive
integer polynomial normalization fixes both scalar content and leading sign.
The paper's neighboring field-containment sentence on p.223 appears to name
k-th roots of unity; that sentence cannot be used literally for c_0(1/3),
which is 1/(3*sqrt(3)). Our field is explicitly Q(zeta_lcm(4,k)), justified
by the displayed algebraic identity, so the sum's definition is unambiguous.
No general degree equality is assumed. The finite checks find degree at most
22; the ambient characteristic polynomial has multiplicity two in this prefix.

Live discovery: catalogue.json contains the single complete authenticated
planning-pass fetch (706 records, four offered drafts). screen.json records
already_here, already_asked and synonym searches; plausible candidate metadata
was refreshed immediately before claiming. T639 is the scalar companion, not
a polynomial duplicate; T95 and T146 are distinct cyclotomic and Gaussian-period
families; T640 records the analytic autocorrelation context. Three typed
api/lookup polynomial probes (27*x^2-1, 2*x-1, 27*x^2-49) found no tables.
The existing tags number theory, polynomial and algebraic were checked on the
catalogue and both tag pages. The absent minimal polynomial tag is not created.

Issue #273 explicitly requests assessment of minimal polynomials alongside
cotangent values. Family proposal #314 maps this selection to contributes to
#273; both URLs are in initial-message.txt. It is partial progress, not issue
closure. No sibling is an independent correctness oracle.

Checked computation: offline-checks.json reports full exact Estermann/orbit
agreement for 22 polynomials and all 178 reduced numerators, with native integer
coefficients; 512-bit Arb radii <7.50e-152 and accuracy >=505 bits. Exact
irreducibility, annihilation, local root uniqueness, known-answer and deliberately
wrong controls passed. The checking method differs from the generator but shares
Sage arithmetic/factorization. A separate critic must still issue the independent
scientific and argument-selection verdicts; none is issued by this builder.

Artifacts and hashes are listed in evidence-manifest.json. skill-manifest.json
is the run's archived manifest: both complete bundles have SHA-256
 d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
The controller's original is ../20261110T221640Z-build-skills/manifest.json.
No mathematical alternatives remain unsettled for this normalization.