back to table · edit · history · where entries came from · files · download
6286 bytes, as of the version from 2026-10-27 02:01 (current). Recorded here, not run.
# Build handoff: T521, Bessel moments
Draft: https://numberdb.org/T521 (slug `Bessel_moments`). This run claimed and
filled this draft. It remains unpublished and has not been offered for review.
The controller must run an independent critique and any required repair before
it considers offering. This build supplies no argument-acceptance sidecar.
## Instructions and proposal
Read numberdb-data issue #213 first using `python3 agents/queue.py show 213`.
The linked remote batch archive returned 404; read the actual proposal in
`agents/table-ideas/BATCH-2026-10-26T1628.md` in this snapshot instead.
The required instruction manifest is
`agents/runs/20261026T164353Z-build-skills/manifest.json` (absolute path:
`/home/ubuntu/numberdb-campaign-20261026-hundred/repo-natural-arguments-v2/agents/runs/20261026T164353Z-build-skills/manifest.json`).
Its SHA-256 is `d03b756eaa1cd22ce39c8cee78fb3a44b052146786d6371925ac7bc9dec564f3`.
Both archived skill versions, the canonical local skill and the freshly fetched
live skill agree byte-for-byte, SHA-256
`b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18`.
The manifest is also cited in `numerical-evidence.json`. Read the campaign gate,
argument-review helper, and guarding-generated-tables design note.
## Selection and decisions
40 entries: all pairs `(a,b)` with `3<=a+b<=6`, `0<=a<b`, at `k=1,3,5,7`.
The parameters remain ordered `(a,b,k)`; grouping `(a,b)` preserves the
reader's order by total factor count, then a, then k. The API readback confirms
that exact order. Parameter domains describe all nonnegative a and k and b>a,
not only this finite selection.
`argument-selection.md` explains each integer parameter, cites the sources,
and fixes the equal unit scale and order-zero Bessel functions. These are
counts and moment powers of one integral, without an independently sampled
continuous coupling. The upper cutoffs are editorial prefixes of the documented
families, not intrinsically distinguished integers. The completeness note and
generator use the same selection. No regularized, equal-count or unequal-mass
integrals were introduced. The table stores the unscaled ordinary integral.
The corpus check found Bessel point-value and primitive graph-period tables,
not the IKM family. `already_here`, `already_asked`, three numerical lookups,
neighbour documents and tag searches were run. One sample value coincides
with a Dirichlet L-value and a quadratic zeta residue; this is an identity,
not a duplicate integral family. Links use slugs and entry identifiers read
from the actual corpus. No GitHub-request provenance was inserted in table
prose, and no published table was edited.
## Numerical work and evidence
* `generate.py` directly integrates every entry with tanh-sinh quadrature,
using Arb Bessel evaluations at 150 and 200 decimal working precisions.
It retains 100 significant digits. Worst relative discrepancy:
`6.06310382359e-152`. All Bessel values are checked for finite midpoints and
small relative radii before entering the quadrature.
* The nonzero controls IKM(0,1;1)=1 and IKM(0,2;1)=1/2 passed first.
* Upper-tail bounds at T=630 and 780 are less than `3.829e-254` and
`1.223e-318`. The lower endpoint has the continuous limit zero for all
selected powers. Quadrature discretization error is not certified.
* `checks.py` imports no generator computation. It checks all 14 applicable
published recurrences and 11 closed-form comparisons using zeta/L-values
and exact multinomial counts. Raw-value relative tolerance: `1e-180`.
* A separate 60-digit calculation uses mpmath's Bessel implementation,
Gauss-Legendre quadrature and the substitution t=exp(x). All 40 comparisons
passed `1e-50`; worst discrepancy `4.77261122392e-57`. It first checked
IKM(0,2;1)=1/2. This check does not independently establish all 100 digits.
* The repository's unmodified `dry_run.py` and `check.py` passed: 40 entries,
no float or disguised-exactness findings, longest value 104 characters,
entries block about 5.7 KiB.
* After filling, all 40 stored decimal strings matched the computed strings
exactly. The 25 identity checks and alternate quadrature were repeated on
the live readback. `verify(sample=None)` reported 40/40 matched, zero
differing, missing or extra. The final metadata adjustment left every
stored number unchanged.
Rigour is **heuristic (agreement-checked)**. The table never claims that
midpoint quadrature is a proven enclosure. The table’s mpmath Programs snippet was also executed at its stated 150-digit
precision and agreed with IKM(0,3;1) within a relative tolerance of 1e-140.
Software: Sage 10.9, numberdb
0.1.10, mpmath 1.4.1, Python 3.12.3. Full numerical evidence and file hashes
are in `numerical-evidence.json`; raw logs are retained alongside it.
## Audit and tags
The final site audit is clean: `audit-final.json` contains no findings.
The first audit object is retained as `audit.json`. The proposed `moments`
tag already exists but has only T36 as a published member, so the live audit
flags its coverage. It was omitted from this draft; the other five established
subject tags remain. This is a departure from the batch's suggested tag set,
not a change to the mathematical selection.
Propose **Feynman integrals** for T521, T464 and T466, covering diagrammatic
integrals and periods. Do not create it on only one table. `moments` can also
be reconsidered across T36, T521 and the batch's prospective planar-walk
moment table. No tag changes were made to published tables.
## Independent critique boundary
The next actor should independently assess every free parameter and perform
its own numerical checks. It should fingerprint the current draft with
`--generator generate.py --generator checks.py --generator argument-selection.md`.
These are all the generator's declared source/selection dependencies; the
numerical logs are evidence, not input data used by the generator. The critic
must use its own `NUMBERDB_RUN_ID` and write its own required sidecar. This
handoff is a builder's report, not an acceptance.
No publishing, offering, human-review, queue-completion, source-edit, Git
commit, push or deployment action was performed. Only the controller should
advance the draft to the next review stages.