# Build handoff: T507

Built **Jacobi theta₂ constants at singular nomes**, https://numberdb.org/T507,
as an unpublished draft. It has not been offered for review. The separate
critique and repair stages remain for the controller.

The table contains the full positive theta-two constant at
`tau = i*sqrt(r)`, `q = exp(-pi*sqrt(r))`, for every integer `1 <= r <= 12`,
ordered increasingly. Its parameter domain is all positive integers. Square
factors of r are retained. These conventions and the initial range are those
of family #209; no unresolved mathematical convention was guessed.

All twelve entries carry 100 significant decimal digits with proven ball
arithmetic. The generator uses 429 working bits and retains at least 423 bits.
Each stored decimal's one-unit-last-place interval contains the entire
computed ball. The longest written value has 102 characters. The numbers-only
measurement is approximately 1.62 KB; the dry run including the square-point
annotation measures approximately 2.4 KB in the working ball representation.

Checks before filling, and again after reading back every stored entry:

- Independent 512-bit real-ball theta series through n=12, with explicit
  geometric tail bounds.
- Independent theta-two and eta products through 80 factors, with explicit
  omitted-product bounds.
- Jacobi's quartic identity and the theta/eta product identity, also using the
  stored companion values from T504, T505 and T506.
- Elliptic modulus and complete elliptic integral relations using T295, T296
  and an independent Arb elliptic-integral evaluation.
- Ramanujan and Weber relations using T297.
- The square-lattice Gamma identity, the equality with T505 at r=1, and the
  T25 m=1/2 / T296 r=1 equality.
- Five known-value controls and three required-failure controls for nonfinite,
  excessively wide and deliberately shifted enclosures.
- `verify(sample=None)`: 12/12 matched; zero differing, missing or extra.
- The site audit returned `clean: true`, with no findings.

The generator, independent checks, shared check utilities and pre-fill
numerical evidence are attached to the draft. Post-fill evidence and the audit
are also attached. Local evidence and source-page snapshots are in this work
directory. Sage 10.9 and numberdb 0.1.10 were used.

Live screening used `already_here`, `already_asked`, text searches and three
number lookups. The r=1 lookup found T505's equal value; r=2 and r=3 had no
matches. No theta-two CM family existed. The actual sibling/link slugs were
read and resolved against the database. Existing published tables were not
edited. Request provenance remains in the issue queue, outside the table prose.

The existing `special values` and `special functions` tags are used.
**Tag proposal:** `complex multiplication` is still absent. It would group
T504, T505, T506, T507, T295, T296, T297 and T299. No new tag was created and
no published table was retagged.

The table was claimed empty, then its prose and numbers were sent together
in one fill. Attachments share that fill revision. No publication or human
review action was performed.