# Sources consulted for the build

Family #231 was read before screening or computation with
`python3 agents/queue.py show 231`. It specifies the principal-branch kei_0
normalization, the strictly positive root index, and proposed indices 1..20.

The archived canonical instructions are identified by
`agents/runs/20261028T182642Z-build-skills/manifest.json`.
Both versions and the snapshot's canonical file agree at SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.

Official DLMF pages were read through the web tool:
- 10.61.2 fixes G(x)=K_0(x exp(i pi/4)); 10.61.3 supplies the radial ODE.
- 10.65.2 and 10.65.5 supply the independent real and harmonic-number series.
- 10.68.2 and 10.68.4 identify the continuous phase representation.
- 10.67.1, .2, .5 and .6 give the decay used for the boundary term at infinity.
- 10.32.9 gives the absolutely convergent integral bounding |G'| by K_1.
- 10.75(xii) lists a twenty-root reference in Zhang and Jin (1996), p. 323.
  The book page itself was not consulted and supplies no numerical controls.

URLs are https://dlmf.nist.gov/10.61, /10.65, /10.68, /10.67, /10.32, /10.75.
Direct urllib downloads of these pages received HTTP errors; the web tool
returned their text and equations. Those failures are preserved in sources.json.

The image at https://personal.math.ubc.ca/~cbm/aands/page_381.jpg was downloaded
and visually inspected. Its kei column reads 3.91467, 8.34422, 12.78256,
17.22314, 21.66464. It is saved as page_381.jpg, SHA-256
162792fde5470ff2a8e446d4e44df7559aa2ea55598360b950d25e3feb116789.
The comparison tolerance is one last-place unit, 1e-5. In particular, the second
printed root is not the nearest-rounded five-decimal version of 8.3442250629...;
the independent high-precision series certifies the new digits. This historical
check does not certify the 100-digit enclosures.

Corpus screening used both screen.already_here and screen.already_asked, a
Kelvin/kei/Bessel search, three api/lookup intervals, and both tag-list pages.
Live neighboring documents T490, T491, T493, T509, T20, T21 and the modified
Bessel K value table were saved. Links use actual search-result addresses.
No duplicate root family was found. The existing zero and special functions
tags were reused. Kelvin functions remains a proposed tag; it could connect
T490, T491, T493, T509 and T587, plus the other pending Kelvin root families.