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# Reproducing the checks for T509
The table stores 360 signed values of kei_nu(x), for nu=0,1,2 and reduced positive rationals x=a/b with b<=4 and x<=20, at 100 significant digits. DLMF 10.61.2 fixes the principal-branch convention and the order-dependent phase.
Install numberdb in SageMath, download generate.py, checks.py, and check.py together, and run:
sage -python checks.py
sage -python generate.py
sage -python checks.py --stored
A draft requires NUMBERDB_API_KEY for the last two commands. All three commands are read-only. generate.py also documents its explicit draft-fill mode; that mode writes numbers but cannot publish a draft.
The generation uses certified Sage complex balls at 397 bits. All 360 balls fit their stored decimal intervals. The largest absolute radius is less than 6.40e-110 and the largest relative radius is less than 2.25e-102.
Independent verification uses 480-bit rigorous quadrature of an elementary integrand, with an analytic tail bound below 8.19e-307. All 360 written intervals contain those independent enclosures. The largest integral radius is less than 3.55e-133. Controls are integral(t,0,1)=1/2, integral(sin(t),0,pi/2)=1, and the elementary half-order Kelvin value at x=1. Deliberately displaced answers fail the comparisons.
The Hankel identity and integer negative-order reflection were checked for all 360 entries. The recurrence was checked at all stored orders and arguments, using T493’s actual stored ker values and supplementary neighboring orders where needed. The origin limit was derived from DLMF 10.65.4 and supplemented with three small-argument numerical checks.
computed-checks.jsonl and stored-checks.jsonl give per-entry numerical evidence and summaries. verify.log records 360/360 matched, with no differing, missing, or extra entries. dry-run.log records exactness and size checks. The longest written value is 108 characters; the measured written-entry block is about 52.9 KiB. The program snippets both ran successfully. audit.json records a clean site audit.
This is build-stage validation. Separate critique and any repair precede a person’s decision to publish. This run did not publish the draft, review it, or offer it for review.