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# Argument selection for Generalized log-sine integrals (T697)
This build follows the coordinated skill archived at
`agents/runs/20261109T220207Z-build-skills/manifest.json`, attached as
`skill-manifest.json`. Both archived versions have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
The family proposal is numberdb-data issue 310. Its enumeration was reassessed
against that skill, not accepted as a required grid.
The quantity is Ls_n^(k)(pi t) = -integral_0^(pi t)
v^k log^(n-1-k)|2 sin(v/2)| dv. Logarithms are natural, angles are in radians,
and neither signs nor factorials nor powers of pi are absorbed into the value.
The parameters describe n>=1, 0<=k<n, rational 0<t<=1. The zero-logarithm
boundary k=n-1 is elementary and described by a formula rather than rows.
## Every free parameter
* `n`, discrete_index: total order. The noncanonical prefix 2 through 10
covers logarithmic powers from one through nine and the low-order controls,
while keeping the full triangular moment family inexpensive to certify.
* `k`, discrete_index: angular moment order. At each n include every
0<=k<=n-2. This includes the basic integrals k=0 and all positive angular
moments with a nonzero logarithmic power, rather than extending only the
already familiar k=0 and k=1 slices. All these are values of one indexed
function, not different quantities under a selector.
* `t`, mixed, with disjoint exhaustive groups:
* intrinsic: `1/3`, `1`. At pi/3 the logarithm vanishes and its sign changes;
pi is the half-period endpoint with a Gamma-ratio moment-generating
function for k=0. Both occur for every selected (n,k).
* documented_application: `1/2`, `2/3`, `5/6`. Borwein–Straub explicitly
identify pi/2 and 2pi/3 among the important special angles in their
introduction; Davydychev–Kalmykov connect these to the even/odd arithmetic
bases of inverse binomial sums at u=2,3. These two occur throughout the
triangular order family. The angle 5pi/6 occurs **only** at (n,k)=(4,0),
the Ls_4 term in Charlton–Gangl–Lai–Xu–Zhao equation (3.5). No rectangular
hull of this exceptional row is generated.
This is 4*(1+2+...+9)+1=181 entries, ordered by n, then k, then increasing t.
The application classification concerns the angle families; it does not assert
that every order pair has a separately documented physical application.
There is no reference_sample mesh in this selection. Rational keys encode the
specified angle families exactly; small denominators alone are not the reason.
## Breadth and practical extent
The coverage combines an intrinsic transition, a complete half-period
endpoint, two distinct arithmetic/application angles, and a single additional
published weight-four term. Across each main angle it retains all angular
moments and both logarithmic parities through order ten. This breadth precedes
any deep extension at one angle. Ten is a reference cutoff chosen for a usable
low-order prefix, not an intrinsic bound or a row-count target. Measurements
are in numerical-evidence.json. Values remain compact at 100 significant
digits. Arbitrary rational angles, imaginary arguments, mixed log-cosine powers,
and harmonic insertions belong to different selection plans or quantities.
## Sources actually checked
* Borwein and Straub, Special Values of Generalized Log-sine Integrals,
arXiv:1103.4298v1, definition (1), introduction, equation (8), and section 5.
https://arxiv.org/html/1103.4298
* Davydychev and Kalmykov, Massive Feynman diagrams and inverse binomial sums,
arXiv:hep-th/0303162v4, equations (1.2)–(1.4) and the introduction's
discussion of u=1,2,3. The normalization is u=4 sin^2(theta/2), not u/4.
https://arxiv.org/html/hep-th/0303162
* Charlton, Gangl, Lai, Xu and Zhao, On two conjectures of Sun concerning
Apéry-like series, arXiv:2210.14704v1, Definition 2.1 and equation (3.5).
https://arxiv.org/html/2210.14704
The live corpus and draft catalogue were checked, including the definitions
and coverage of T14, T175, T176, T177, T353, T681 and T685. Only n=2,k=0
is the ordinary Clausen boundary. Higher logarithmic powers and angular
moments are not higher-order Clausen functions. T681 and T685 provide
normalized moments or differences, not this family. Their links identify the
specific relations. Unbuilt proposals have no invented T-numbers.
This is the builder's justification. The separate critique must independently
assess it and fingerprint generate.py, the sole enumeration/computation source.
No independent acceptance sidecar is supplied by the builder.