# Argument selection for T698: Inverse binomial sums

Instruction provenance: `skill-manifest.json`, copied from
`agents/runs/20261109T224707Z-build-skills/manifest.json`. The archived local
and live canonical skills were read and match. Each has SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
This is the builder's justification, not the required independent critique.

The quantity is F_s(u) = sum_{j>=1} u^j/(j^s binomial(2j,j)). Its general
parameters are an integer s>=2 and a rational -4<=u<=4. The stored values
retain the sign of u^j. In particular F_s(-1)=-A(s), where BBK's Apéry sum
uses (-1)^(j+1). The hypergeometric argument in DK is u/4, not u.
There is one named quantity and no harmonic-number insertion or quantity selector.

## Denominator power s: discrete_index

The denominator power is the established weight index of these unweighted
inverse central binomial sums. BBK defines S(k) and A(k) in equations (1)
and (2), and DK's equation (1.2) treats every integer c>=2. We select the
noncanonical practical prefix 2<=s<=10. It includes both parities, the low-order
arcsine and zeta controls, and higher arithmetic periods at all five slices.
The point is a small reference across weights, not only the famous closed
forms. Ten is a computational and coverage choice, not a mathematical endpoint.
The generator's MAX_ORDER=10 and completeness note state the same cutoff.

## Series argument u: mixed

The two disjoint groups cover all selected arguments exactly once:

* **intrinsic**, values `1`, `-1`, `4`: u=1 is the unweighted central binomial
  sum and u=-1 its alternating counterpart, both explicitly defined in BBK's
  introduction. The coefficients' convergence radius is 4. The point u=4 is
  the positive convergence boundary and the endpoint theta=pi of DK's angle
  correspondence. It needs a distinct evaluation method because the defining
  series has only algebraic convergence there. DK also documents the u=1
  single-scale application; it is classified here by its simpler intrinsic role.
* **documented_application**, values `2`, `3`: DK's introduction, immediately
  before equation (1.2), distinguishes the even basis at u=2 and odd basis at
  u=3. Their corresponding angles are pi/2 and 2*pi/3. The same paper's section
  4 places inverse binomial sums in massive Feynman-diagram expansions. The
  claim is support for these argument families, not a separate published
  application of every weight through ten.

This selection is not an exploratory grid: it contains no denominator-bounded
rational mesh. It is independent of decimal notation. The four positive
arguments correspond to theta/pi=1/3,1/2,2/3,1 under u=4 sin(theta/2)^2;
this functional relation explains the match to T697's main angles.

Breadth was reviewed before increasing depth. The selection includes both
signed and unsigned sums, both documented arithmetic bases, and the positive
convergence endpoint. The negative endpoint -4 is in the family domain but
is not added to this finite selection: the alternating unit slice has an
established Apéry family, while +4 additionally reaches the singular positive
boundary of the generating function and the log-sine half-period endpoint.
The trivial zero argument is described by F_s(0)=0 and is not repeated nine
times. Other rational inputs, harmonic-weighted sums, and imaginary-angle
log-sinh companions would need their own argument-selection work. The count
of 45 is a consequence of this selection, not its reason.

Enumeration is increasing s, then u=1,-1,2,3,4, keeping the unit-signed pair
adjacent. Parameters describe the general family; only completeness and this
attachment describe the finite selection. No additional slice is generated.

## Sources checked

* Davydychev and Kalmykov, *Massive Feynman diagrams and inverse binomial sums*,
  https://arxiv.org/html/hep-th/0303162 (v4), introduction, equations
  (1.1)-(1.4), and the paragraph separating u=1,2,3. The u/4 normalization,
  log-sine leading minus sign, and positive angle correspondence were read.
* Borwein, Broadhurst and Kamnitzer, *Central Binomial Sums, Multiple Clausen
  Values and Zeta Values*, https://arxiv.org/html/hep-th/0004153 (v1), equations
  (1),(2), Lemma 1, Theorem 1, and section 6. The alternating sign convention
  and proven zeta/Clausen controls were read; no conjectural identity is used.
* Borwein and Straub, *Special Values of Generalized Log-sine Integrals*,
  https://arxiv.org/html/1103.4298 (v1), definition (1).
* The live family issue #310 and the live T697 document were read. T697 is an
  offered draft, not a published numerical authority. T14 and T175 provide
  independent published zeta and Clausen controls. T176, T177, T353, T681 and
  T685 were inspected for the family boundary; none stores this series family.

Discovery checked the authenticated catalogue including drafts, the supplied
already_here and already_asked functions, numerical lookup samples, and existing
tags. The only issue title match was #310 and no duplicate table was found.
The suggested tag `polylogarithms` is absent. It would connect T175, T176,
T697 and T698; this is a proposal for a person, not a new singleton tag or an
edit to any published table.