# Argument selection for Bessel moments (T521)

The table stores IKM(a,b;k) = integral from zero to infinity of
`t^k I_0(t)^a K_0(t)^b dt`, with the standard positive-real cylindrical
modified Bessel functions. The ordered parameters are `(a,b,k)`.

## Evidence and normalization

Bailey, Borwein, Broadhurst and Glasser, *Elliptic integral evaluations of
Bessel moments*, arXiv:0801.0891v2, equations (1), (2), (5), (8)–(13),
and Sections 3–6, develop these equal-scale moments. Equation (1) gives
vacuum diagrams; equation (2) gives sunrise diagrams. The differential
operator immediately following equation (2) increases the power of t by two.
Equations (12) and (13) define the one-I and two-I moment families. Section 5
relates two-I moments to lattice Green functions. Zhou, *Hilbert Transforms
and Sum Rules of Bessel Moments*, arXiv:1706.01068v2, equation (1.2), names
and defines the unified IKM family and describes its quantum-field application.

* `a` is a nonnegative integer count of I_0 factors, not a sampled order or
  continuous coupling. For each total N=3,4,5,6 all counts 0 <= a < N/2
  are included. Thus a=0,1 for N=3,4 and a=0,1,2 for N=5,6.
* `b=N-a` is the integer count of K_0 factors. The restriction b>a chooses
  the exponentially convergent equal-scale family. The factor counts N=3
  through 6 are the nontrivial low-factor families developed in Sections 3–6
  of BBBG. N=1,2 have elementary gamma quotients (equation (7)); they are
  useful controls but are not additional stored rows. N=6 is an editorial
  cutoff of the discrete family, not a distinguished boundary.
* `k` is the nonnegative integer power of the integration variable. The stored
  powers 1,3,5,7 are the first four odd moments. The base measure t dt occurs
  in the two-dimensional diagrams, and the mass differentiation operator
  produces the increments by two. Four moments permit checks of the four-term
  five- and six-factor recurrences (10)–(13). The upper cutoff is a short
  reference prefix, not a claim that seventh order alone is special.

Order zero and equal unit scales are fixed by the cited problem. No masses,
ratios, orders, angular inputs, external momenta or couplings are freely
sampled. No loop measure, factorial or power of pi is absorbed into IKM.
Only ordinary convergent improper integrals are included. The definition's
integer domain is larger than the current selection; no equal-count or
regularized moments are claimed.

## Enumeration and completeness

For N in 3,4,5,6, then a in 0..floor((N-1)/2), then k in 1,3,5,7,
yield `(a,N-a,k)`. This gives 10 factor pairs and 40 entries. The generator
and completeness note use this exact order and selection. The parameters
index one integral; they do not choose different named quantities.

This document is the builder's justification. It is not independent argument
review or acceptance. A separate critique must assess every parameter and
fingerprint generate.py, checks.py and this attachment against the live draft.

Instruction provenance: ../20261026T164353Z-build-skills/manifest.json in the
source snapshot. Both archived versions, the canonical file and a fresh live
GET have SHA-256 b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.