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# Argument selection for T648
This build uses the coordinated canonical NumberDB skill archived at
`agents/runs/20261101T095610Z-build-skills/manifest.json` (a copy is attached
as `skill-manifest.json`). The local and live copies both have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The source-snapshot canonical file has the same hash.
The sole free parameter is the total number of Bessel factors, integer `n`.
Its selected identities, in presentation order, are `5`, `6`, `8`.
The mathematical object is the vacuum-containing Broadhurst–Mellit determinant
with the specific ordered rows and columns in the cited constructions. This
is a finite source-defined catalogue, not a formula for every integer n.
Category: documented_application. Assess its two constructions separately:
* `n=5,6`: Zhou, arXiv:1711.01829v2, Theorem 1.3, equations (1.9) and (1.10),
specifies the determinants with rows a=0,2 and columns r=1,3. These combine
vacuum and two-I0 moments and evaluate the Mahler measures of five- and
six-term sums of monomials, respectively. Broadhurst, arXiv:1604.03057,
Sections 7.2 and 7.3, studies these in noncritical modular L-values.
* `n=8`: Broadhurst, Section 7.8, Conjecture 6, equation (175), specifies
the same rows but the second column is r=3 minus twice r=5. This exact
polynomial column, not arbitrary moment columns, is the published
vacuum-containing construction for a weight-six noncritical L-value.
Both groups have the same documented-application role. No exploratory group,
continuous mesh, independent Bessel order or scale, or arbitrary minor is
included. The unit scale, cylindrical order zero, unscaled moments and signed
row/column order are fixed conventions, agreeing with issue #284 and T647.
The a=0 row comes before a=2; the r=1 column comes before the specified
polynomial column. The eight-factor row is not an absolute value.
Coverage review: all three explicitly defined constructions in the selected
proposal are included. Keeping only n=5,6 would omit a different, documented
column construction; inserting n=7 or extending to n>8 would require a new
matrix definition rather than a larger compute budget. Scalar vacuum moments
at n=3,4 already belong to T521 and are not additional vacuum determinants
of this catalogue. Full-period determinants belong to T647. Off-shell masses,
arbitrary minors and higher-factor scalar moments are separate work.
The practical extent is therefore the complete three-member catalogue selected
from these sources; `complete: no` refers to coverage of the broader research
subject and does not assert a definition at other factor counts. The note,
program and generator all enumerate exactly 5,6,8. The three-value size is
justified by the source-defined constructions, not by a minimum or maximum
entry quota. Values target 100 significant digits with agreement-checked
quadrature, without using conjectural L-value identities as certificates.
Selection-controlling source dependency: `generate.py` only. The standalone
`checks.py` does not determine enumeration or supply generator data.
This build records its assessment; independent critique is still required.