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4328 bytes, as of the version from 2026-10-26 23:52 (current). Recorded here, not run.
# Numerical evidence for T532, Airy derivative zeta values
Build run: 20261026T232323Z-build. This is builder evidence, not an independent
critique, an argument-selection acceptance, or a publication decision.
Instruction manifest:
`/home/ubuntu/numberdb-campaign-20261026-hundred/repo-natural-arguments-v2/agents/runs/20261026T232323Z-build-skills/manifest.json`.
Both archived skill versions were read and were identical to the canonical
local file and a fresh live /skill read. Their SHA-256 is
`b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18`.
The manifest is attached as skill-manifest.json; skill-read-evidence.json
records the fresh hashes. The stricter combined instructions were followed.
The sole argument n is the inverse-operator power, n >= 2. All orders 2..20
are included, in increasing order. This is an editorial initial segment of a
discrete spectral family; no continuous parameter or adjustable scale is
sampled. See argument-selection.md and Voros, Section 3, equations (28),
(29), (43.2), (43.3), https://arxiv.org/pdf/math-ph/9811001.
The generator represents every Z_N(n) as an exact polynomial in
Gamma(1/3)/(3^(1/3) Gamma(2/3)), using rational arithmetic. Its final Arb
substitution uses 429 bits (100 digits plus a 96-bit guard). Across the 18
nonrational entries the minimum relative accuracy was 421 bits and the
largest relative radius was 7.336470736913143e-128. Z_N(3)=1 remains an
exact integer by the Airy equation and Voros (43.3). Every entry was
recomputed at 628 bits. SageMath 10.9 and numberdb 0.1.10 were used.
The independent check computes Ai(z)/Ai'(z) directly with Arb at 600 bits
on 512 points of |z|=1/2, then extracts coefficients by discrete Fourier
sums. It does not use the generator's recurrence, polynomial coefficients
or Gamma initialization. An explicit alias error of 2/(2^512-1), less than
1.5e-154, is included in every enclosure. The derivation and the proof that
the nearest pole has modulus greater than 1 are in checks.py. Before the
Airy computation the same coefficient extractor recovered all 19 requested
coefficients of 3+2z+z^5, including its nonzero coefficients. Every ball's
real and imaginary components were checked for finiteness and small radius.
Results:
* Shared dry_run.py: 19 entries passed exactness, ring-import and prose
checks, with no exact value disguised as a decimal.
* All 19 values agreed with the higher-precision rerun and independent
Cauchy enclosures. The Cauchy enclosures fit inside each written 100-digit
interval; the exact entry was checked by its symbolic identity and by
inclusion of 1 in the independent enclosure.
* Low-order identities, the full recurrence, positivity, strict decrease
and strict log convexity passed. The reflection-form Gamma expression
supplied a separately written initial-constant check.
* Values read back through numberdb.table('T532') passed those identities
and independent-enclosure containment checks again. The exact stored
spelling at n=3 is `1`.
* The Programs section read from the live document was executed and agreed
with the independent computation at every selected order.
* Generator.verify(sample=None): 19/19 matched, 0 differing, 0 missing,
0 extra. The read-back check took about 0.89 seconds after Sage startup.
* The stored values have maximum length 102 characters; the shared
measurement of their flat block is about 2.47 KiB before the one entry
comment. The larger dry-run size (3.3 KiB) measures full working balls.
* The authenticated site audit returned clean=true and no findings.
Full values, independent enclosures, measurements and read-back results are
in numerical-results.json. The raw dry-run and check logs are retained in the
run work directory, including the two checker interface errors corrected
before its first successful control. No failed comparison between two
numerical methods was discarded.
The independent critique must still assess the mathematical derivation,
source interpretation and argument selection. The Cauchy computation is
independent in algorithm and initialization, but uses the same Arb library;
it is not an independent software-library check. No finite root sum was
presented as a 100-digit infinite-sum computation. The remote audit does
not perform the optional outward-link check.