# Selection for the dual Möbius approximation errors

The sole free parameter is N, type Z, with natural domain N >= 1. It is a
`discrete_index`: the truncation level of the dual approximation in Báez-Duarte,
Arithmetical Aspects (math/0011254v1), §1.2, equation (1.19), checked directly.
Here the paper's F_N is named D_N; it is not the untapered F_N of
math/0202141v2. Ordinary fractional parts, M(y)=sum_{n<=floor(y)} mu(n),
mu(1)=1, and the squared norm ||chi+D_N|| in L²((0,infinity),dx) fix the family.

Select every N=1,...,30, in increasing order. Thirty is a noncanonical practical
prefix, not an intrinsic threshold. It supports comparison with all currently
available B_N errors (T707) and the 30-index family context, and keeps rational
Gram ratios within the live T640 range and independently integrated residuals
small. The same-index T641 minimum is not asserted as a lower bound: its
integer dilation space differs. This remains one construction, not a selector
for four norms. There is no quantile or conventional critical value.

N/k is derived from the prescribed sum, not a free rational dilation or a
sampled continuous coupling. Every summation k=1,...,N is required by (1.19),
including zero weights; no decimal or dyadic grid has been selected. There are
no omitted continuous parameters, reference_sample groups or root choices.
Retain nonsquarefree N because the dual dilation geometry changes there.
Breadth across successive indices comes before extending beyond the shared
30-index comparison prefix. No index is selected by attractive numerical values.
The infinite natural family is not claimed complete; default enumeration,
Programs and complete-note must all say every N=1,...,30.

The 30 values use about 100 significant digits, except the proved exact E_1=1;
size is measured by dry_run.py. The generator can compute any larger positive N.
Independent coverage judgment remains for a fresh critic. Instruction bundle:
skill-manifest.json, bundle SHA-256
d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab;
original archive agents/runs/20261110T211830Z-build-skills/manifest.json.