# Argument selection for T641

The sole free parameter is the integer truncation index N >= 1. Its role is
`discrete_index`: it selects the nested spaces spanned by the first N integer
dilates rho_n(x) = {1/(n x)} in L^2((0,infinity), dx). Neither continuous
dilations nor coefficient constraints are being sampled. The quantity is the
minimum squared distance to chi = 1_(0,1), not the distance itself or a
specified approximation's error.

We include every N from 1 to 30, in increasing order. This full prefix preserves
successive enlargements of the approximation space, including all composite
and prime indices and the one-dimensional base case. It supports comparison
of the minima and their optimal coefficient vectors without omitting ordinary
members in favor of isolated famous indices. The only extent decision is 30.
It is a practical, noncanonical initial cutoff for certifying dense systems
and keeping the attached coefficient vectors and Gram data inspectable. It is
not a mathematical threshold, interpolation mesh, or target entry count.

Breadth review: the unanchored integer-dilate problem has one discrete family,
so a contiguous prefix supplies its relevant breadth. Anchored functions,
restriction to (0,1), normalized basis functions, a constraint sum c_n/n = 0,
continuous dilation samples, and a fixed Selberg approximation change the
problem; they are not missing slices of this one-quantity table. Larger N is
a possible extension of the same family. The associated A(j/k) inputs are
exactly the reduced ratios needed by these spaces, not an independently
chosen rational grid. They are recomputed and enclosed internally, not added
as another indexed quantity here. No exploratory reference_sample component
or quantile parameter is present.

The definition, generator default enumeration, Programs example, and completeness
note agree on this selection. Measured costs and precision are in
numerical-evidence.json. A separate campaign critique must independently assess
this justification; this build does not supply or accept a critique sidecar.

Checked sources:

- L. Báez-Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann
  Hypothesis, arXiv:math/0202141v2, section 1 and Theorem 1.1. The exact Hilbert
  space and integer dilation family are stated there.
  https://arxiv.org/html/math/0202141v2
- S. Bettin and B. Conrey, Period functions and cotangent sums,
  arXiv:1111.0931v2, pp. 12–13, equation (13) and the definition of d_N^2.
  https://arxiv.org/pdf/1111.0931
- S. Bettin, J. B. Conrey and D. W. Farmer, An optimal choice of Dirichlet
  polynomials for the Nyman–Beurling criterion, arXiv:1211.5191v1, section 1,
  p. 1. This uses Dirichlet-polynomial coefficients a_n = -c_n.
  https://arxiv.org/pdf/1211.5191
- L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias,
  arXiv:math/0306251, sections 8–9, Propositions 85, 87 and 89.
  https://arxiv.org/pdf/math/0306251

Policy evidence: ../20261031T205515Z-build-skills/manifest.json, archived local
and live skills both SHA-256
a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.
The attached skill-evidence.json records the absolute manifest path and hashes.