# Argument selection for T692

The only parameter is the integer Taylor order n >= 0, with role
`discrete_index`. It indexes coefficients of the one normalized function
pi g(1+z)=sum (-1)^n g_n z^n, rather than evaluations on a chosen continuous mesh.
The centre 1 is fixed by x -> 1/x. The order-zero cotangent sum, normalization,
and expansion centre are fixed parts of this named sequence, not free sampled
inputs. See Akatsuka–Murakami (AM), equations (1.1)-(1.3) and Section 3:
https://arxiv.org/html/2402.14216v1.

The proposed initial segment 0 <= n <= 100 is retained after reassessment.
Consecutive orders support reconstruction near x=1, reciprocal-symmetry
coefficient identities, and comparison of the signed residual g_n-1/n across
orders. Low-order coefficients alone would omit the approach to this residual
regime; selecting only particular signs or named coefficients would break the
local expansion. Both parities and every intervening order are included.
No additional free parameter family belongs to this quantity, so extending a
continuous input mesh would broaden the table into a different object.

The cutoff 100 is a practical, noncanonical reference extent, independent of
numeral base. It balances a substantial initial expansion with the severe
cancellation in AM (1.3), whose Bernoulli-weighted summands grow factorially.
The numeric evidence measures retained precision and written length for the
entire prefix. The table is incomplete as an infinite sequence; its completeness
note states this precise prefix and rationale. Generator enumeration and the
Programs example use the same indexing and normalization. Asymptotic correction
coefficients are a separate named quantity in family #308, not additional rows.

This selection has no `reference_sample` or mixed groups. No claim is made that
every order has an individually documented application. The source establishes
the family and its relationships; the finite bound is this table's reference
choice. This is build justification, not the required independent critique.

Read and compared archived local and live skills in
../20261109T150707Z-build-skills/manifest.json, both SHA-256
886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b.

The archived manifest is also attached to the draft as skill-manifest.json.