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5368 bytes, as of the version from 2026-11-09 16:19 (current). Recorded here, not run.
# Build evidence for T693
Run: 20261109T154709Z-build. The table remains a private, unoffered draft.
The controller must run a separate critique and any repair before offering it.
## Instruction provenance
Read the family proposal with `python3 agents/queue.py show 308` before other
work. The archived skill manifest is
`agents/runs/20261109T154709Z-build-skills/manifest.json`.
Both archived versions, `skill-local.md` and `skill-live.md`, have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
Their equality and manifest hash are recorded in the attached
`skill-evidence.json`. The campaign gate and argument-review implementation
were read. No critique acceptance sidecar was written by the builder.
## Discovery and conventions
The authenticated catalogue included drafts. There was no table for the
asymptotic coefficient sequence; T692 was an offered Taylor-coefficient draft.
`already_here` and `already_asked` were run; the matching proposal was #308.
Live T692, T639–T642, T13, T14, T569, T570 and T209 were read for conventions,
coverage and links. The tags number theory, special values and sequence exist.
The proposed asymptotic expansions tag is absent; T693, T569 and T570 would
use it. No published table or tag was edited.
Successful api/lookup calls found no matches for C_1, C_2 and C_3 at about
19 significant digits. The exact C_0=1 has many unrelated matches. The first
handwritten HTTP attempt used the wrong record format, and an initial C_2
placeholder had incorrect digits. Those were not treated as absence evidence;
`lookups-corrected.json` in the run artifacts records valid text queries with
the values from the checked formulas.
The source is AM, arXiv:2402.14216v1, equations (1.1), (1.2), (1.6), (5.1),
(5.2), Remark 1.2 and the proof of Corollary 5.6. A source HTML copy and its
hash are retained in the work artifacts. DLMF 10.17.1 and 13.2.1 were read
with the web tool; separate direct HTTP downloads failed and are logged.
The localized constant-term typo in the statement of AM Corollary 5.6 is
resolved in `derivation.md`; equation (5.2), its proof and all printed
coefficients agree. The family’s definition and normalization are preserved.
## Results
Sage 10.9; numberdb 0.1.10. Every Sage computation ran through agents/sage.sh.
The exact polynomial generator returns 61 values, ell=0,...,60. All 61 agree
with a separate reciprocal/exponential-series construction. Six printed
polynomials agree exactly. All 60 adjacent Kummer differential identities,
61 degree/leading/constant-term tests, 62 Bernoulli ingredient checks and
1,071 Hankel ingredient checks pass. The first negative order is 50;
negative orders through 60 are exactly 50,52,54,56,58,60.
At 525 working bits the minimum retained accuracy is 517 bits. The independent
700-bit evaluations retain at least 692 bits. The maximum cancellation in
exactly assembled polynomial evaluation is about 0.4961 decimal digits.
Every main and independent enclosure lies strictly within the 100-digit
last-place interval written by the NumberDB client. C_0=1 is exact; the
remaining polynomials are nonconstant, so their values at pi^2 cannot be
rational or algebraic.
The canonical dry-run helper passes its exactness, disguised-exactness,
ring-import and prose checks: 61 raw balls, longest 176 characters. After
100-digit serialization, the longest value is 102 characters (ell=50), and
the measured entries block is 8.12 KiB. See `dry-run.log` and
`checks-output.json` for the actual results. Supporting exact polynomials
are in `coefficient-polynomials.json`, outside the value column.
Direct complex Arb U(n,0,2*pi*i), scaled by -4 Gamma(n), is compared with
truncations at L=0,1,3,5,10,20,40,60 for n=25,100,400,1600. The direct values
are computed at 1,200 bits and their accuracy is checked before comparison.
The nonzero control U(1,2,2)=1/2 must have radii below 2^-1000. With L=60,
the absolute errors after dividing by the common exponential amplitude are
about 1.3011e-23, 1.0530e-41, 4.2381e-60 and 3.6582e-78 respectively.
These are numerical tests of the asymptotic convention, not a general
certified remainder theorem or recovery of every coefficient to 100 digits.
The coefficient precision claim comes from exact formulas and ball arithmetic.
Controls also test the reciprocal series, inclusion in a written decimal
interval and rejection of a deliberately changed coefficient in Kummer’s
identity. Initial runs exposed ordinary Sage API-name differences; the fixed
checker completed successfully. These failed attempts remain in the artifacts.
## Reproduction and subsequent checks
`generate.py` computes the values and verifies the table by default.
`checks.py` recomputes all independent checks; with NUMBERDB_STORED=1 it also
reads T693, checks every stored interval and sign, and uses the stored values
in the Tricomi comparisons. Draft reads require the author’s API key.
`check.py` is the unmodified canonical build-check helper used by these checks.
The argument selection is wholly in generate.py; there are no sampled input
files or hidden source dependencies.
Post-fill verify(), stored-value checks, attachment readback and audit results
are recorded separately in `readback-evidence.md` after they have run. No
source of certification depends on those future checks being assumed to pass.