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4546 bytes, as of the version from 2026-11-09 09:52 (current). Recorded here, not run.
# Argument selection for T685
This build follows the archived coordinated skill recorded in
`agents/runs/20261109T080742Z-build-skills/manifest.json` (attached as
`skill-manifest.json`). Both archived local and live versions have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
The family proposal is numberdb-data issue 304, read before this build.
This document is the builder's justification; independent critique remains
required and must supply its own argument-review evidence.
## Quantity and conventions
For integers n >= 3 and k >= 1, set
P_(n,j) = 1 + x + sum_(ell=1)^(n-2) y_(j,ell). The quantity M_(n,k) is the
expectation of product_(j=1)^k log|P_(n,j)|. The unit-circle variable x is shared
by all factors, and all y_(j,ell) are mutually independent and independent of x.
Every circle has normalized Haar measure, all coefficients are one, and log is
natural. There is no factorial divisor, absolute value around the logarithm,
centering, or exponentiation. Zeros are treated by convergent improper integrals.
The ordered parameter identities are (n,k); n counts the constant summand too.
The two slices have y_j for n=3 and y_j,z_j for n=4.
## Parameter n: discrete_index
The selection contains both n=3 and n=4, the three- and four-term sharing
patterns of Borwein–Straub (BS), sections 2, 3.1 and 3.2, equations (11) and (41).
It is a count of summands in one general family of correlated logarithmic
moments, not a selector between different numerical quantities. Including both
slices covers two distinct conditional expectations. For n=3, Jensen's formula
clips the conditional mean at zero. For n=4, averaging two private phases yields
a positive smooth conditional mean with a convergent cotangent-derived series.
These reductions provide computationally practical certified integration for
both slices. Extending n beyond four would require another reduction or a
certified higher-dimensional integration method; this run does not establish
one. The definition retains n >= 3, while the completeness note states the two
computed slices. The source-defined small-summand cases justify the breadth;
no continuous coefficient, angular, scale, or probability parameter is sampled.
## Parameter k: discrete_index
The selection includes every integer 1 <= k <= 12 at each selected n. The index
counts logarithmic factors. Its initial prefix includes the first-order Mahler
controls, both parities, low-order formulas and higher orders beyond those
closed forms. Twelve is a noncanonical practical cutoff, not a special order
selected by an identity and not a target entry count. It supplies a short joint
moment reference sequence for each sharing pattern, while allowing all values
to be checked independently at substantially more than 100 significant digits.
The source defines the families for every positive k; individual orders need
not have separate application papers.
Breadth was assessed before depth: both source-defined sharing patterns are
included instead of extending only the three-term case. All 24 values remain
compact scalars. The dry run measured a longest enclosure string of 152
characters and a 4.6 KB entries block before 100-digit serialization. There is
no arbitrary decimal mesh or product of independent continuous sample grids.
Neither parameter needs a reference_sample or mixed classification.
## Boundaries and existing coverage
Live catalogue, draft catalogue, text searches, issue search and value lookups
found no existing table of this sharing pattern. T283 contains its k=1 boundary,
which is recomputed and cross-linked. T681 and T682 concern repeated factors
built from one variable, with different phase dependencies. T522 holds radial
Mellin moments, rather than these correlated products. The proposed diagonal
higher measures are not built or assigned an invented T-number here. Further
sharing patterns in BS section 3.3 are excluded from this table.
## Checked source and synchronization
Jonathan M. Borwein and Armin Straub, *Log-sine evaluations of Mahler measures*,
Journal of the Australian Mathematical Society 92 (2012), 15–36,
DOI 10.1017/S1446788712000067. Author PDF:
https://arminstraub.com/downloads/pub/logsin1.pdf
Definition in section 1; equations (11)–(13), (41)–(42); Theorems 2.1 and 3.3;
Examples 3.1 and 3.4; distinction from a further sharing pattern in section 3.3.
The attached generator enumerates increasing n, then k, exactly as the
completeness note declares. There are no additional selection source files.