# Argument selection for T699

Provenance: coordinated archived instructions at
`agents/runs/20261109T231810Z-build-skills/manifest.json`, copied here as
`skill-manifest.json`. Both local and live skill versions have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
This is the builder's rationale, not an independent critique acceptance.

The table contains one quantity, Lsc_(m,n)(pi*t), with the minus sign,
natural real logarithms, and powers m-1 and n-1 specified by Borwein–Straub
section 3.2, equation (10), and Charlton–Gangl–Lai–Xu–Zhao Definition 2.1.
The dummy integration variable is v, and t is the exact rational theta/pi.
The definition allows m,n>=1 and 0<t<=1. Pure-power boundaries are described
by formulas and linked to T697 rather than enumerated again.

## Discrete indices m and n

Both are discrete_index: independent orders of the sine and cosine logarithms.
The paired finite bound m,n>=2 and m+n<=10 includes every genuinely mixed
moment of total logarithmic degree at most eight. It includes both parities,
balanced and unbalanced powers. Eight is a noncanonical practical prefix,
not an assertion that every pair has a separate documented application.
The total-degree cutoff measures complexity of the same moment expansion;
it does not form a product of unrelated continuous inputs. At the midpoint
all 28 ordered pairs are retained. At the complete endpoint symmetry identifies
(m,n) with (n,m), so all 16 unordered pairs m<=n are retained. The four
application entries have (m,n)=(3,2) or (2,3), so do not deepen this bound.

## Angular parameter t: mixed, with exhaustive groups

* intrinsic: t in {1/2,1}. Reflection v -> pi-v interchanges the two
  logarithms. The midpoint therefore retains the asymmetric ordered moments,
  while the complete interval is symmetric and has a beta generating function.
  These are two structurally different slices, not a rational mesh.
  Supporting sources: BS section 3.2 equation (11); CGLXZ section 3, Step 1;
  DLMF 5.12.2 and 8.17.1 for the change of variables to beta integrals.
* documented_application: t in {1/6,1/3,5/6}. The entire group comprises
  exactly (3,2,1/6), (2,3,1/6), (3,2,1/3), (3,2,5/6).
  CGLXZ equations (3.2), (3.4), and (3.5) select these mixed moments in the
  evaluation of a harmonic-weighted central-binomial series. The reversed
  (2,3) row at pi/6 is present in (3.2) before reflection in (3.4).
  No rectangular hull of the exceptions is included.

These groups are disjoint and cover every selected t. There is no exploratory
continuous reference mesh in this proposal; no reference_sample claim is made.
The finite discrete prefix is an editorial extent of a meaningful indexed
family. Breadth was assessed before depth: complete and incomplete moments,
all ordered midpoint powers, and a separate published application slice are
represented before extending any order. Further rational angles, hyperbolic
variants, or all harmonic-weighted sums require their own selection work.

The 48 rows are enumerated by m+n, then m, then n, then increasing t. The
completeness note and generator specify these same conditional slices.
No parameter selects a different quantity or convention. No quantiles are used.

## Corpus reassessment

The authenticated catalogue, search synonyms, already_here, already_asked,
and numeric lookup were checked before claiming T699. A value pi^3/24 is
already a Favard constant, which is a theorem coincidence, not this family.
T697 and T698 are live companion drafts offered for review. T177 and T353
store beta function values, not parameter derivatives; T175 is the ordinary
Clausen boundary. The Mahler companions T681/T685 store other moment objects.
Existing tags suffice. The proposed missing tag polylogarithms would also serve
T175, T176, T697 and T698; it is not created by this run.

The independent critique must reassess these groups, all conditional pairs,
and the current generator. Its fingerprint need only list generate.py for
selection code: no imported local helper or data file controls enumeration.
Other attached scripts provide checks, not the generated selection.