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3010 bytes, as of the version from 2026-10-31 22:58 (current). Recorded here, not run.
# Argument selection for T642
The only free parameter is **N**, an integer truncation index with domain
N >= 2. The table holds one quantity: the squared error E_N of the specified
Selberg approximation. Neither a choice of approximant nor a normalization is
an additional parameter.
**Assessment: discrete_index.** Báez-Duarte, *A strengthening of the
Nyman–Beurling criterion*, §1 (arXiv:math/0202141v2), explicitly defines the
Selberg approximation S_N as the sum over 1 <= n <= N of
mu(n)(1-log(n)/log(N)) rho_n in L2((0,infinity),dx). Bettin–Conrey–Farmer,
*An optimal choice of Dirichlet polynomials for the Nyman–Beurling criterion*,
§1, defines the same logarithmically tapered coefficients in V_N(s). These
sources establish the index family, norm and sign convention. N=1 is excluded
because log(N) vanishes. The complete prefix N=2,...,30 retains every defined
small truncation, including prime, composite and nonsquarefree endpoints.
An endpoint with mu(N)=0 is still a distinct approximant: its log(N) changes
all the earlier weights. No interior truncation is omitted on that basis.
Thirty is a practical, noncanonical initial bound, not a mathematical
threshold. It permits comparison with every matching optimum in T641 and
uses the same Gram problem whose distinct rational ratios occur in T640.
The independent checker recomputes all 465 symmetric Gram entries and 29
least-squares minima; retaining a modest dense matrix makes the evidence
inspectable. This common bound is tied to one actual approximation problem,
not a mesh imposed on unrelated continuous functions or an entry-count target.
The numbers themselves are short: each stored value has 100 significant digits;
the dry run measured 29 internal ball strings with longest length 155 and a
5.5 KB block. The finite extent does not restrict the parameter's domain.
A breadth review finds no other independent input family within this named
quantity: the n values are summation indices, the Möbius coefficients and
logarithmic taper are fixed by the published definition, and the measure dx
on the full positive half-line is fixed by family #281. Other mollifiers,
anchored functions, constrained coefficients and L2(0,1) errors would change
the quantity; they are not omitted slices of E_N. Optimal distances remain
the separate companion T641. No exploratory reference group, probability
parameter or conventional statistical threshold is present.
The default enumeration, table completeness note and Programs example all use
this convention; the example evaluates N=10 within the declared prefix.
All 29 generated identities and their increasing order are checked against
the stored draft. This justification is supplied for the independent critique;
it is not an independent acceptance or an argument-review sidecar.
Policy evidence: `agents/runs/20261031T225043Z-build-skills/manifest.json`.
Archived local and live skills both have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.