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3580 bytes, as of the version from 2026-10-29 05:38 (current). Recorded here, not run.
# Argument selection for Landau constants G_n (T601)
This is the builder's justification, not the independent critique or its
acceptance sidecar. The governing archived instructions are recorded in
`agents/runs/20261029T051809Z-build-skills/manifest.json`; both local and live
copies have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The attached `skill-manifest.json` and `skill-evidence.json` preserve that record.
## Quantity and convention
G_n is the sharp bound on the absolute value of the sum of the coefficients
of degrees 0 through n of a function analytic in the open unit disk whose
supremum norm is at most one. Coefficients may be complex. There is no
condition f(0)=0 or requirement of univalence. The coefficient sum has n+1
terms and does not require a boundary value f(1). Li, Liu, Xu and Zhao,
section 1, equation (1.1), explicitly use this indexing and state sharpness:
https://arxiv.org/abs/1309.4564 . Their pointwise condition |f(z)|<1 gives the
same best constant as the closed unit ball used here: the maximum-modulus
principle reduces the additional cases to constant functions of modulus one,
which have coefficient sum of modulus one.
The central-binomial expression is also given in Chen and Choi, section 1,
equation (1.1): https://rgmia.org/papers/v17/v17a09.pdf . The family proposal
#238 agrees with these conventions. Both sources define rational values, so
these are stored exactly rather than as 100-digit approximations.
## Every free parameter
- Parameter: `n`.
- Domain: nonnegative integers; type `Z`.
- Category: `discrete_index`.
- Meaning: largest retained Taylor degree, not a continuous input encoded as
an integer and not a selector between different quantities.
- Selected set: every integer from 0 through 100, in ascending order.
Each degree defines a different sharp coefficient-sum problem. Consecutive
coverage supplies the constant-term case and all low and moderate truncation
degrees, so it is not restricted to the few initial values with short names
or fractions. The theoretical family is infinite. The upper bound 100 is a
practical reference cutoff, not a theorem or an assertion that every degree
has a separately documented application.
## Breadth, finite extent, and cost
There are no other free parameters or independent families of inputs to
sample. The analytic functions in the definition are quantified over, not
selected as rows. Thus the complete initial segment covers the available
breadth for this single named quantity. Fourier projection norms and
Landau–Kolmogorov derivative inequalities are different quantities; the
existing tables T499 and T500 explain those relations without adding a
second parameter or quantity to this table.
The dry run gives 101 exact entries, a maximum written length of 240
characters at n=100, and an estimated entry block of 15.174 KiB. This makes a
consecutive moderate-degree reference inexpensive to reproduce and readable,
with room for later extension. Larger n is not required merely because the
arithmetic is cheap. No rational mesh, decimal grid, Cartesian product,
exploratory continuous-argument group, or campaign-excluded quantile is used.
The generator's default enumeration, parameter declaration, Programs snippet,
and completeness note all specify this same initial segment; the declaration
itself retains the unbounded mathematical domain n>=0. A separate critique
must check this argument, the stored enumeration and values, and fingerprint
the live attachments. This build provides no independent acceptance.