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# Argument selection for T686
This build follows the coordinated skill archived in
`agents/runs/20261109T084515Z-build-skills/manifest.json`. Both `skill-local.md`
and `skill-live.md` were read and have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
The one quantity is the raw signed logarithmic moment
H_(n,k) = E[(log|1+x_1+...+x_(n-1)|)^k]. All phases are independently
Haar-uniform, every coefficient is 1, each circle has measure one, and the
logarithm is natural. There is no factorial divisor or centering. The domain
is integer n >= 2 and integer k >= 2; the selected subset is n = 3,4 and every
k = 2,...,8, enumerated in that order. No continuous argument is sampled.
## Summand count n: discrete_index
The number of summands (including the constant 1) is also the number of unit
steps in a planar uniform random walk. This is a discrete geometric family,
not an arbitrary collection of polynomials. BBSW section 4 relates its
logarithmic moments to derivatives of W_n. BSWZ sections 3 and 4 give explicit
densities for three and four steps, with different singularities. Including
both supplies breadth across the two tractable short-walk geometries.
The n=2 boundary has the same quantity and phases as the linear members of
T681 and T682; those existing tables contain orders 2 and 3. Higher orders of
that one-variable family remain with its existing owner. We do not claim that
all n=2 orders are already covered. Counts n>=5 remain part of this table's
mathematical domain, but their densities and independent high-precision
integration need further analysis. The cutoff at four is computational, not a
claim that only these walks matter.
## Moment order k: discrete_index
The logarithmic power is a genuine moment index. Every order 2 through 8 is
included for both summand counts. This initial prefix includes both parities,
negative odd moments, the second moments with known evaluations, and higher
orders beyond the short closed forms. BBSW Theorem 4.1 and Example 9.2 treat
this derivative family and report independent quadrature through order 8;
the printed values in that example are only orders 2 through 5. BSWZ equation
(70) supplies a separate four-term second-moment control. We do not present
those controls as publications of every selected higher four-term value.
Eight is a noncanonical practical extent for a reference table: the singular
endpoint contributions grow with k, and independent high-order checking is
more demanding. It is not a number-of-entries target or a famous-values-only
selection. First moments are already T283's quantity. The table has 14 compact
scalar entries; increasing k without broadening the independent checks is not
justified merely because more terms fit.
## Boundaries and corpus checks
T685 holds products of different logarithms sharing one phase, rather than a
power of the same logarithm. T522 holds radial power moments W_n(s), rather
than derivatives at zero. T681/T682 have dependent one-variable monomials;
identifying independent phases with powers of one phase changes the measure.
None of these supplies this selected two-parameter family. The authenticated
catalogue, the `already_here` and `already_asked` helpers, sample numeric
lookups, companion documents and existing tags were checked before claiming
T686. The reference count and moment order are both `discrete_index`; neither
`reference_sample` nor `mixed` applies. There are no probabilities, quantiles,
coefficient grids, arbitrary coupled inputs or quantity selectors.
Sources checked: BBSW, https://arminstraub.com/downloads/pub/logsin2.pdf,
sections 2 and 4, Theorem 4.1 and Example 9.2; BSWZ,
https://arxiv.org/html/1103.2995, equations (3), (4), (7), (26), (30), (70)
and sections 3, 4 and 6.1; family issue 304 and the live companion documents.
The independent critique must assess this selection and fingerprint the live
document and all source attachments. This build is not a critique acceptance.