# Build handoff: T460

T460 is a filled, unpublished draft for the Weil-Petersson volume polynomials. It has not been offered for review. The controller must run the separate critique and repair stages before any offer. No publication or human review action was performed.

- Draft: https://numberdb.org/T460
- Canonical address returned by the API: https://numberdb.org/Weil--Petersson_volume_polynomials
- Proposal: numberdb-data family #198, “Weil--Petersson volume polynomials $V_{g,n}(L_1,\dots,L_n)$”.
- Author account: zeta3. Assisted method recorded as codex-cli.
- Range: all stable nonnegative integer pairs with 0 < 2g-2+n <= 3. The eight pairs are (0,3), (0,4), (0,5), (1,1), (1,2), (1,3), (2,0), (2,1).
- Stored quantity: exact polynomial over QQ in L1,...,Ln,p, with p representing pi formally. The parameters are g,n, in that order. There is no decimal approximation in an entry.

## Decisions and source reconciliation

The family specifies labelled boundaries and orbifold weights, with V11=(L1^2+4*pi^2)/48 and V20=43*pi^6/2160. These numerical conventions agree with Do's survey and the existing companion table T459. The family incorrectly calls the 1/48 version Mirzakhani's original convention. Do and Norbury explicitly distinguish their orbifold convention from Mirzakhani's doubled torus volume with compensating recursion factors. The draft preserves the family's numbers and accurately describes this distinction. Source: arXiv:math/0603406, page 1; Do arXiv:1103.4674, Section 3.1.

The formal variable is named p to keep it distinct from Sage's numerical pi constant. Boundary variables are L1,...,Ln. Natural initial layers by 2g-2+n count the number of pairs of pants. This gives a complete region without holes, rather than choosing unrelated genus-dependent cutoffs. The next layer contains V14, measured at 1,385 characters, and V06 requires seven variables after including p. The chosen range has 61 monomial coefficients and a longest entry of 339 characters, V13. No precision choice is needed for exact rational polynomials.

## Evidence

The attached generate.py computes pure psi intersections by DVV with string/dilaton base reductions, converts kappa_1 powers by Liu-Xu Lemma 2.2, and applies the volume coefficient formula. All divisions are exact rational operations.

The attached checks.py has a separately written geometric Mirzakhani recurrence using exact kernel moments, without the generator's psi recursion or set-partition transform. Known F1 and F3 kernel moments and printed V04 and V12 values serve as four controls. It also expands a direct transcription of all eight published polynomials from Do's Appendix B.

The final stored-value evidence is verification-stored.json:

- Generator verify(): 8/8 matched, with no differing, missing, or extra entries.
- Independent geometric recurrence: 8/8 polynomial matches, covering all 61 coefficients.
- Published polynomial transcription: 8/8 matches.
- Exactness, positivity, evenness and total degree: all 61 terms pass.
- Boundary symmetry: 12 adjacent-transposition checks pass.
- String and dilaton: five applicable pairs each pass as polynomial identities.
- Genus-zero leading multinomial formula: 20 coefficient checks pass.
- T458 leading terms: 35 coefficient comparisons pass.
- T459 zero-length values: eight comparisons pass, with 512-bit MPFI computations enclosed by the entire intervals claimed by the 100-digit decimals. The exact pair-of-pants value is checked as 1.
- dry_run.py passes. The no-write client preview reported eight additions and no replacements.
- Server audit: clean, no findings. Its initial two prose findings were repaired without changing numbers.

Checks were run on the locally computed values before filling, and repeated on values read back from the draft. Audit was run after writing. The numbers were sent once. History contains an initial claim, the filled document, a prose-only audit repair, and attached reproducibility files; no numerical repair was required.

## Remaining controller considerations

T458 and T459 are readable through the API, but ordinary public GET requests to /T458 and /T459 returned 404 during this run. They were absent from public search. Treat them as publication dependencies until the controller establishes their lifecycle state. The draft links to the actual T-number addresses; no slug was guessed. No companion table was modified.

The proposed tag “moduli spaces” does not exist. The draft uses existing volume, period and polynomial tags. A moduli-spaces tag would serve T458, T459 and T460; proposing or creating that shared tag belongs with the family/controller.

The public corpus screening used already_here, already_asked, three polynomial api/lookup queries, and tag queries. No duplicate polynomial family was found. The title claim succeeded and assigned T460; no retired number or collision bypass was used. The proposal checklist should only be marked built according to the controller's successful critique/repair workflow. It was not marked complete here and no exhausted-batch marker was written.

## Reproduction

Download generate.py, checks.py and check.py into the same directory. With numberdb installed in Sage:

    sage -python checks.py
    sage -python generate.py
    sage -python checks.py --stored

The last two commands need an authorized NUMBERDB_API_KEY while T460 is a draft. To run the repository's independent dry run, also download dry_run.py:

    sage -python dry_run.py generate.py

The generator's --fill option writes entries only; it does not offer or publish the table. Source PDFs, local range measurements, screening responses, API responses, and the working scripts are preserved in this run's artifact directory. The database and its attached files are authoritative.