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3527 bytes, as of the version from 2026-11-10 22:53 (current). Recorded here, not run.
# Argument selection: minimal polynomials of Bettin–Conrey cotangent sums
The only free parameter is the integer denominator k >= 2, a discrete_index.
The quantity is the primitive polynomial P_k in Z[x], irreducible over Q,
with positive leading coefficient and P_k(c_0(1/k))=0. The real cotangent
sum is c_0(h/k) = -sum_{m=1}^{k-1} (m/k) cot(pi*m*h/k), with pi positive.
There is no scaling of the algebraic number. The polynomial variable x is
not a table parameter. Exact integer coefficients require no decimal precision.
The inspected source is Bettin and Conrey, *Period functions and cotangent
sums*, Algebra & Number Theory 7 (2013), pp. 217 and 223, definition and (12),
https://msp.org/ant/2013/7-1/ant-v7-n1-p08-p.pdf . The denominator is the order
of the root of unity in this arithmetic sum, not a rational sample of a
continuous argument. The default enumeration is every integer 3 <= k <= 24,
in increasing order. The omitted k=2 has P_2=x and is explained in Comments;
k=4 remains as the rational normalization control P_4=2*x-1.
Fixing numerator 1 is supported by the signed Galois action, rather than by
selecting famous residues. Put L=lcm(4,k), z=exp(2*pi*i/L), w=z^(L/k), and
i=z^(L/4). Then alpha_k=-(i/k) sum m*(w^m+1)/(w^m-1). For a coprime to L,
the automorphism z -> z^a sends alpha_k to (-1)^((a-1)/2)c_0(a/k).
Every reduced residue h modulo k has a lift coprime to L (choose an odd lift
if k is odd). Hence its sum is a conjugate of alpha_k or its negative; its
primitive polynomial is P_k(x) or (-1)^deg(P_k)*P_k(-x). This is an algebraic
derivation; the executable checks will also test the action throughout the
finite selection. It does not assert that all unsigned sums are conjugate.
The consecutive prefix includes odd primes, the odd prime power 9, powers of two 4,8,16, twice odd denominators,
and mixed composites such as 12,15,20,21,24. It avoids pruning to closed forms.
The cutoff 24 is a practical, noncanonical initial reference bound: degrees
are at most phi(lcm(4,k))/2, at most 22 here. It is chosen for readable exact
polynomials across these arithmetic cases, not to match T639's scalar bound 30
or a target entry count. Actual serialized sizes will be measured before
filling; the largest entry and overall block must remain readable. Extension
beyond this prefix is possible but outside this selected build. There are no
continuous couplings, probability arguments, or reference_sample groups.
One named polynomial family is stored. Ambient-field characteristic
polynomials, numerical root balls, degrees, and sign-transformed numerator
polynomials are checking artifacts, not additional tables or value columns.
The table is incomplete in the natural infinite domain: complete: no, with
the exact prefix and zero omission stated in complete-note and Programs.
Related tables (live definitions checked): T639 numerical cotangent sums;
T95 cyclotomic polynomials; T146 unweighted Gaussian period polynomials;
T640 fractional-part autocorrelation. These are context, not independent
validation. No sibling data is used as exact computational input.
Instructions: ../20261110T221640Z-build-skills/manifest.json, matching bundle
SHA-256 d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
Both complete archived versions were validated against every resource hash.
A fresh independent critic must assess this selection and bind its verdict
to the live document and generator dependencies. This is the builder's
justification, not an independent acceptance.