# Build handoff: T461

Built draft [T461](https://numberdb.org/T461), **Orbifold Euler characteristics of the moduli spaces of smooth curves**, for proposal 4 in numberdb-data issue #198. The draft is filled, unpublished, and has not been offered for review. No publication, offer, or human-review action was requested or attempted. The controller owns the subsequent critique, repair, and offer stages.

## Quantity and decisions

The stored quantity is the exact rational orbifold Euler characteristic of the smooth complex moduli stack M_{g,n}, with distinct ordered markings and full automorphism weights. Parameter order is `g, n`. The domain is nonnegative integral g,n with 2g-2+n>0.

The issue's shared overview says all six tables concern compactified moduli spaces, but its linked screening report explicitly specifies smooth curves for this proposal. The proposal's item-specific convention is confirmed by Giacchetto–Lewański–Norbury, Theorem 1.1, and Bini–Harer, Theorem 3.1. The final title says “smooth” to remove this ambiguity. The original proposal title was successfully claimed first; no title collision was bypassed.

The stored range is every stable pair 1 <= g <= 12, 0 <= n <= 12: **155 entries**, not 156, because (1,0) is unstable. This follows the proposal's compact reference range. Genus-zero signed factorials are stated in a formula instead of enumerated, as the proposal recommended. Genus zero remains in the mathematical parameter domain. The genus-one orbifold value is -1/12; removing the generic elliptic involution would double it to -1/6. No n! factor is removed.

The exact rationals use `rigour: exact`, the skill's current designation, rather than the family's older `proven` wording. No decimal precision is claimed or needed. Longest value: 21 characters, at (12,12). The dry-run entries block, including the single entry comment, is about 7.5 KiB.

## Checks performed

- `already_here` and `already_asked`, targeted orbifold/moduli searches, and lookup of -1/240, 1/1008, -1/12 and -7601 found no duplicate family. The initial malformed raw lookup requests returned errors; corrected client lookups were performed and preserved in `screening-extra.json`.
- Read T13, T14, T458, T459 and T460. Used actual search-result slugs for T13/T14 and verified T-number links for the companion tables; no slug was inferred from a title.
- The repository dry-run helper passed on all 155 entries: exact values, named imports, acceptable entry prose, no decimals disguising exact values.
- Independent Python Fraction calculation, using a Stirling-number sum for Bernoulli numbers and the closed formula: **155/155 exact matches**.
- Arb evaluation from positive even zeta values and the zeta functional equation, at 512 bits: **155/155 contain the exact rational**, all finite; the minimum checked relative radius bound was 2^-507. Known genus-one and zeta(2) controls passed; deliberate wrong-sign and float controls were rejected.
- All **143** adjacent pairs satisfy the forgetful-map recurrence. All genus-one, one-marked, and unmarked formulas pass in range. The displayed genus-zero formula was checked for n=3..12, including its recurrence.
- The displayed standalone Sage program agrees at all **165** stable pairs 0 <= g <= 12, 0 <= n <= 12.
- All 12 one-marked values agree with T14's negative odd zeta values and with T13's Bernoulli numbers.
- `verify(sample=None)`: **155/155 matched, 0 differing, 0 missing, 0 extra**.
- The independent exact, zeta, and recurrence checks were repeated on values read back from the database, with the same results.
- Site audit: initially one missing Bernoulli cross-reference in `rigour details`; corrected without changing any entry. Final audit is **clean**.

The generator, independent checks, shared check helper, before/after numerical evidence, program and corpus checks, and audit report are attached to the website draft. Files in this run directory preserve failed checker diagnostics as well as successful results. The checker failures were type/API issues, not disputed numbers: Fraction versus Sage Rational comparison, and unavailable RealBall convenience methods. Their causes and fixes are recorded in `lessons.md`.

## For the separate critique

The mathematical convention is smooth rather than compactified. Please assess the selection through genus/marking 12 and the decision to leave genus-zero factorials in the formulas. Independently check numerical claims rather than treating the generator's verification as a proof.

The proposed **moduli spaces** tag is absent. Existing `algebra` and `combinatorics` tags are used. A new tag would serve this table and at least T458 (psi intersections), T459 (Weil-Petersson volumes), and T460 (volume polynomials). No tag was invented on this single table.

T458 and T459 are readable with the bot key but their anonymous page requests return 404. They appear to remain drafts. T461 links to them as companion drafts, which the audit permits; publication ordering must be checked by the controller so a published table will not point at an unpublished draft.

The full-document endpoint requires entries, so the completed prose and all 155 values were sent together once after the empty claim. The audit fix is a separate prose revision; file attachments use one shared run identifier. No numerical correction was needed. Nothing was committed, pushed, deployed, or written into source files. The proposal queue remains claimed for the controller's later lifecycle handling; no request issue was closed for an unpublished table.

## Sources

- [Giacchetto, Lewański and Norbury, An intersection-theoretic proof of the Harer–Zagier formula (2023)](https://content.algebraicgeometry.nl/2023-2/2023-2-004.pdf), Theorem 1.1 and the forgetful recurrence in the proof of Corollary 1.3; DOI 10.14231/AG-2023-004.
- [Harer and Zagier, The Euler characteristic of the moduli space of curves (1986)](https://www-users.cse.umn.edu/~reiner/Classes/Math8680Fall2014Papers/HarerZagier.pdf), opening theorem and the unmarked relation.
- [Bini and Harer, Euler Characteristics of Moduli Spaces of Curves](https://arxiv.org/abs/math/0506083), Section 3 distinguishes smooth and compactified characteristics.