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# T521: bounded even-moment extension
The current selection preserves the 40 odd moments described in the original
`argument-selection.md`, and adds exactly `(a,b,k)=(0,3,0),(0,3,2),
(0,4,0),(0,4,2),(1,3,0)`. The generator enumerates all 45, ordered by total
factor count, then first-kind factor count, then increasing moment power.
The original `checks.py` remains a historical checker for the 40 odd rows;
`even-checks.py` independently checks the five additions.
These are explicitly studied even moments, not a grid of factor counts.
They provide the two initial even moments for each of the three- and
four-K0 recurrences, together with the mixed I0*K0^3 zeroth moment.
The finite extension does not claim to exhaust all interesting even moments.
The unit scale and absence of absorbed factors of pi remain unchanged.
Bailey, Borwein, Broadhurst and Glasser,
[Elliptic integral evaluations of Bessel moments](https://arxiv.org/html/0801.0891v2),
Section 3.3 gives the three-K0 gamma evaluations, Sections 4.4 and 4.6 the
four-K0 even moments, and Section 4.11 the mixed zeroth moment. These are
mathematical identities and integral families; the odd-moment physical
interpretation is not asserted for the even additions.
The attached generator extends the previous direct Arb-Bessel/tanh-sinh
method. It uses `t=u^2` only for zeroth moments, so the transformed integrand
has limit zero at the lower endpoint. The previous exponential upper-tail
bound covers all five additions, from the [DLMF I0 and K0 integral
representations](https://dlmf.nist.gov/10.32). Its two truncation bounds are
below 4e-254 and 2e-318. Arb midpoints still enter numerical quadrature;
the table remains **heuristic (agreement-checked)**, not proven.
Each addition is computed at 150 and 200 working decimal digits and checked
independently at 140 and 180 digits using gamma or elliptic integrals, without
Bessel evaluations. Only 100 significant digits are stored. Source hashes,
baseline provenance, controls, and old-row reproduction are recorded in
`even-checks-evidence.json`. Existing values and manual table comments are
preserved, rather than regenerated into different decimal strings.