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# Argument selection for T639
The quantity is the signed Bettin–Conrey sum
\(c_0(h/k)=-\sum_{m=1}^{k-1}(m/k)\cot(\pi m h/k)\).
The ordered parameters are `(k, h)`. The domain is every pair of integers
\(k\ge2\), \(1\le h<k\), \(\gcd(h,k)=1\); the current computation takes
all of these pairs with \(k\le30\), ordered by denominator and then numerator.
There is no order parameter: this table fixes the documented order zero.
## Both free parameters
- **k: arithmetic_structure.** The denominator is the modulus of the finite
cotangent sum, not a continuous input discretized for evaluation. Complete
residue systems are the arithmetic objects in Bettin–Conrey, §4, pp. 11–13
of arXiv:1111.0931v2. Moduli 2 through 30 include primes, prime powers and
composite moduli with several prime factors. Thirty is a practical,
noncanonical initial limit shared with the family's finite approximation
problem; it is not a special mathematical threshold or a size target.
- **h: arithmetic_structure.** At each modulus the entire group of reduced
residues is present. Multiplication and inversion permute this group;
inversion is needed in the exact conversion to the Vasyunin sum. Reflection
changes the sign, and both signs are retained. Restricting to h=1 or a few
surds would omit the residue dependence central to this family. No Cartesian
product of independent continuous arguments is being sampled.
## Application and breadth
For the first 30 integer dilates in the Nyman–Beurling approximation problem,
reduced ratios j/k have numerator and denominator at most 30. The Vasyunin
sums in the Gram formula use those ratios and their reciprocals; reduction
modulo the denominator and modular inversion place every nontrivial argument
in this selection. Conversely each selected h/k occurs among those dilation
pairs. Denominator 1 contributes an empty sum, zero, and is not a free omitted
family here. This follows from Bettin–Conrey §4, equation (13) and the paragraph
on the approximation distances following it, not from a claim that every
individual fraction has a separate famous application.
All residue classes are included before extending the depth in the modulus.
The finite cutoff preserves complete modular inversion/reflection orbits and
agrees with the companion proposals. It does not claim completeness over all
rationals. Exact minimal polynomials, orders other than zero, autocorrelation
values and approximation errors are separate quantities, outside this table.
There is no exploratory reference_sample component and no probability input.
## Sources and conventions checked
- S. Bettin and B. Conrey, *Period functions and cotangent sums*,
arXiv:1111.0931v2, p. 3 and §4 pp. 11–13, equations (12)–(14): the minus sign,
modular-inverse Vasyunin conversion, and the finite approximation context.
- L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias,
arXiv:math/0306251, §4: the Vassiounine spelling and cotangent-sum context.
- DLMF 5.5.4: the digamma reflection formula used for the independent check.
The live family proposal is issue 281, archived as family-281.md. Its other
three proposed tables were absent from the live text searches on this run.
Existing T226, T145 and T622 are linked using their actual live slugs; they
contain digamma ingredients, character-weighted sums and cotangent products,
respectively, not this quantity.
## Skill provenance
The coordinated manifest is
`agents/runs/20261031T200151Z-build-skills/manifest.json`.
Both archived skill-local.md and skill-live.md have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
This is the build justification; the separate critique actor must independently
assess it and bind its assessment to the live selection and generator sources.