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4769 bytes, as of the version from 2026-10-26 21:52 (current). Recorded here, not run.
# T529 build handoff
Built the private draft **Moments of the positive-part Brownian bridge area**,
https://numberdb.org/T529, with twelve raw moments indexed by n = 1,...,12.
No offer, publication, or human review action was taken. The controller must
run the separate critique and repair stages, including independent argument
selection review, before considering an offer.
The model is the centered unit-duration Brownian bridge with covariance
min(s,t)-st, zero endpoints, variance-t underlying Brownian motion and positive
part applied before integration. The proposal's convention is unambiguous in
Janson, arXiv:0704.2289v3, Section 24, equations (236)–(245). There is only
one free parameter, positive integer moment order. Its selection justification
is attached as argument-selection.md and agrees with default enumeration and
the completeness note. Twelve is an editorial truncation of the discrete
sequence of Laplace-transform coefficients, not a canonical cutoff.
Six even moments are exact rationals. Six odd moments have 100 significant
decimal digits enclosed using Arb at 513 working bits. Minimum measured
relative accuracy is 509 bits. The longest written entry is 104 characters;
the measured number block is approximately 1.13 KiB. The first two values are
sqrt(pi/2)/8 and 1/20.
The dry run passed exactness, named-import and size checks. All orders passed
an independent exact polynomial solution of the Brownian Feynman–Kac
resolvent problem, an independent factorial-series quotient, and a 768-bit
Gamma-formula check. Controls were M0=1, the explicit first resolvent
polynomial, and the Gaussian-marginal beta integral for the mean. The first
four positive moments match Janson's Table 6. Every written decimal interval
contains both computed balls; exact rows compare as rationals. Strict moment
log-convexity holds through order 11. The one-value program matches all twelve
rows. On database readback, Generator.verify(sample=None) reported 12/12
matched with no differing, missing or extra entries; the independent checks
and recovered recurrence passed again. See readback-evidence.json.
The live API audit is clean after removing the unused moments tag. Although
that tag exists in the tag index, it grouped no other table. The table uses
probability theory and integral. Brownian motion remains an absent subject
tag that would connect T523, T524, T525, T526, T527 and T529; it is proposed
here, not created. T524 is linked by its stable T-number, read through the
API, because its private draft has no public page yet. It must be public when
a future public T529 links to it; the current draft audit allows draft links.
The Gamma-function link was read and its public address checked.
Screening used already_here, already_asked, authenticated and unauthenticated
text searches, numeric lookup, and the tag index. The final sample lookups
for orders 1 and 3 had no matches; the exact second moment matched unrelated
families. Related Brownian tables were read through the API. No existing table
for the positive-part bridge area was found, and the exact-title draft claim
succeeded as T529. The proposal remains for the controller to mark built when
its subsequent stages succeed. No batch-exhausted marker was made.
Instruction provenance is
agents/runs/20261026T213949Z-build-skills/manifest.json,
SHA-256 d03b756eaa1cd22ce39c8cee78fb3a44b052146786d6371925ac7bc9dec564f3.
Both archived skill versions, the canonical local file, and a fresh live fetch
have SHA-256 b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
The manifest and individual read hashes are attached, and the manifest is cited
in the numerical evidence. The source PDF and its hash remain in the run
artifacts. The web reader could not read /skill's markdown MIME type, so the
live source was downloaded directly and compared byte for byte.
The generator has no local imported computational or selection dependencies:
generate.py contains both enumeration and computation. checks.py imports the
attached reusable check.py and generate.py for comparisons; its independent
resolvent and quotient implementations share no computational code with the
generator. A critic should fingerprint the live generate.py (and may also bind
checks.py and check.py) and the current selection with agents/argument_review.py.
The build has not written an independent acceptance sidecar.
Portable findings are recorded in lessons-20261026T213949Z-build.md.
Checker interface errors and their logs have been retained. No failed checker
result was treated as agreement, and no incorrect values were uploaded.
All work is in this run's artifact directory and the website attachments;
no commits, pipeline source changes, pushes, or deployments were performed.