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# Build handoff: T505
Draft: https://numberdb.org/T505
Title: Jacobi theta₄ constants at singular nomes
Family: numberdb-data#209. Proposal: Jacobi theta₄ constants at singular nomes.
Run: 20261026T040822Z-build.
The draft is filled and remains unpublished and not offered for review.
The controller owns the separate critique, repair and offer stages. No human
review or publication action was performed. No source, pipeline, skill or
published-table edits were made, and no commits or deployment were attempted.
## Scope and convention
Exactly twelve entries, in increasing order r=1,...,12. The parameter domain
is all positive integers. The full DLMF theta-four constant is evaluated at
z=0, tau=i*sqrt(r), q=exp(-pi*sqrt(r)); the square root is positive and square
factors of r are retained. Q=q^2 is distinguished in the eta formula.
The type is R, with 100 significant digits and rigour proven.
No mathematical convention was guessed. The range and parameterisation come
from the family; implementation choices were a 96-bit guard and independent
512-bit checks with N=14 series terms and M=64 product factors.
## Screening and links
The family was read before building. Both local and live skills were read;
their range-guidance differences do not affect this natural integer family.
The screening helpers already_here and already_asked were run, the live tag
list was read, and api/lookup found no matches at the r=1,2,3 sample values
using intervals of radius 10^-30. T426 is the same function at rational nomes;
the family explicitly permits this distinct integer CM parameterisation.
T504 is the existing companion theta-three draft. Its actual slug was read
from the website's files page. Every HREF was resolved through the table API;
the Gamma link points to the actual entry key 1/4 in T9.
The theta-two and Dedekind eta siblings do not yet exist in the family queue.
Their quantities and identities are stated, but no future table address is
invented. Add sibling cross-links after those drafts exist. T504 is itself a
draft; coordinate publication so a published table does not link to an
unpublished sibling. No published neighbor was edited.
## Evidence
- dry_run.log: 12 finite enclosures; no exactness, prose or ring-import issues.
- checks-before-exact-endpoint-change.json: the original independent checks
passed before filling. checks-final-pre-fill.json reruns the generated-row
checks with exact endpoint conversion during the final build validation.
- checks-stored.json: the same checks passed on the database values, including
175 numerical comparisons, 12 wrong-nome rejection controls,
3 known-value controls, 2 serialization controls and 3 failure controls.
- fill.log: Generator.preview sent nothing; the final document and all entries
were sent once; Generator.verify(sample=None) matched 12/12 with no extras.
- audit.json: clean=true, findings=[].
- document-links-checked.json: all eight distinct internal targets resolve.
The numerical checks use explicit omitted-sum and omitted-product bounds,
Jacobi's quartic identity, the eta product identity in both q conventions,
the singular-modulus and elliptic-integral values from T295/T296, the
Ramanujan and Weber values from T297, and the theta-three values from T504.
The independent Gamma identity at r=1 is also compared with T9, and K at
r=1 is compared with T25 at m=1/2. Comparisons require finite balls of radius
less than 10^-90. Every production ball is wholly contained in its written
100-digit interval, checked using exact rational endpoints.
The generator uses 429 working bits; the minimum retained accuracy is 426
bits. The independent checks use 512 bits. Written values are all 102
characters, and the measured values-only records occupy about 1.62 KiB.
The generic dry run measures the longer diagnostic ball strings, at 147
characters maximum. Both measurements are retained.
## Tags and request tracking
The draft uses existing tags special values and special functions.
Propose complex multiplication for T504, T505, the planned theta-two and
Dedekind eta siblings, and T295, T296, T297, T299. The live tag list does
not yet contain it. The build did not invent a one-table tag or edit neighbors.
The prescribed queue command recorded T505 against the correct theta-four line
in #209. Its newly posted comments on #206 and #208 were clarified to say
that this is only an unpublished theta-four draft and theta-two remains
outstanding. Issue #208 was already closed when this build screened it. Its three-theta
request is not fully answered until theta-two exists; family #209 still
tracks that outstanding sibling. Request provenance is not mathematical
prose in the table.
## Reproduction and attachments
The draft carries generate.py, checks.py, the unchanged shared check.py,
comparison snapshots T9/T25/T295/T296/T297/T504, and the pre-fill evidence.
The stored-value evidence, version record, audit and this handoff are also
attached. The generator is read-only by default; no publication command is
embedded in it. A draft read requires the author's API key. The commands are:
sage -python checks.py
sage -python generate.py
sage -python checks.py --stored
On this builder use agents/sage.sh and pipe the key through stdin, as in the
run's scripts. No credential is present in an artifact.
## Exploratory failures
No numerical disagreement was ignored. The first check script used a
nonexistent RealBall.contains method; its serialization control failed before
any data check. It was replaced by contains_exact for rational controls and
exact rational endpoint comparisons for serialized-ball containment. The
first fill wrapper imported a private client helper absent from the builder's
installed package; it failed before reading a key or making a write. Explicit
provenance headers fixed the wrapper. Original failure logs are preserved.
The raw-download SHA-256 checks matched the locally retained generator,
independent checks, shared helper, stored-value evidence and handoff. The
HTML file-viewer links require `?raw=1` to return the downloadable bytes.
A final endpoint-conversion control found that `QQ(RealNumber)` may return a
nearby rational, rather than that binary value exactly. The serialization
containment check now uses `.exact_rational()` on both outward endpoints.
Both the pre-fill-computation and stored-value checks were rerun with this
stronger conversion. No stored value or prose changed. This is recorded in
`lesson-proposal.md`, with the control in `endpoint-conversion.log`.