# Argument selection for T561

This build follows the coordinated skill archived in
`agents/runs/20261027T205535Z-build-skills/manifest.json`.
Both archived versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The family proposal is numberdb-data issue 224; this is a build justification,
not the independent critique required by the campaign.

The table holds one quantity, the area Siegel–Veech constant, for connected
components of strata of holomorphic Abelian differentials. The measure is the
Masur–Veech measure restricted to unit area and normalized to probability on
each component. Cylinders are unoriented and weighted by their areas. There
are no poles or extra marked regular points.

| Parameter | Role | Justification and finite extent |
| --- | --- | --- |
| `g` | discrete_index | Genus indexes moduli spaces and the associated flat-surface dynamics. We cover every genus from 2 through 4. The cutoff is our practical choice: complete exact low-genus data and independently checkable component phenomena are available. It is not a canonical upper limit. Genus one is omitted because the shared positive-zero-profile convention would require an added regular marked point. |
| `mu` | discrete_index | Every non-increasing positive integer partition of `2*g-2` is included. These are zero orders of a holomorphic one-form, not a mesh of real-valued arguments. Including all 18 profiles covers the minimal strata, principal strata, and every intermediate profile in the chosen genera. Sorting only removes different spellings of the same order profile; zeros remain labelled in the measure convention. |
| `component` | intrinsic | Only the components allowed by the Kontsevich–Zorich classification are enumerated. The labels identify different geometric components of the same named invariant; they do not choose different quantities. `hyp`, nonhyperelliptic `even`, nonhyperelliptic `odd`, `nonhyp`, and `connected` are defined in the table. All 25 allowed components in the chosen genera are included. |

This is a discrete geometric family, with no exploratory reference_sample,
probability parameter, continuously variable coupling, or Cartesian product
of profiles with every label. All free parameters are accounted for. The
choice favors breadth across complete low-genus strata over extending only
the easy hyperelliptic closed forms. No individual-fame criterion or entry
count target was used.

The full default selection, in display order, is:

| Genus | Positive profile | Components, in order |
| --- | --- | --- |
| 2 | (2) | hyp |
| 2 | (1,1) | hyp |
| 3 | (4) | hyp, odd |
| 3 | (3,1) | connected |
| 3 | (2,2) | hyp, odd |
| 3 | (2,1,1) | connected |
| 3 | (1,1,1,1) | connected |
| 4 | (6) | hyp, even, odd |
| 4 | (5,1) | connected |
| 4 | (4,2) | even, odd |
| 4 | (4,1,1) | connected |
| 4 | (3,3) | hyp, nonhyp |
| 4 | (3,2,1) | connected |
| 4 | (3,1,1,1) | connected |
| 4 | (2,2,2) | even, odd |
| 4 | (2,2,1,1) | connected |
| 4 | (2,1,1,1,1) | connected |
| 4 | (1,1,1,1,1,1) | connected |

The generator independently enumerates all positive partitions and applies the
component classification, then checks that its generated keys equal the exact
source-data keys. Counts are 2, 7, and 16 components for genera 2, 3, and 4.
The table parameters describe the unbounded mathematical domain; the finite
extent and its rationale belong in `Data properties.complete-note`.

Supporting sources actually read:

* Kontsevich–Zorich, arXiv:math/0201292v2, Theorems 1–2; also EMZ
  arXiv:math/0202134v2, §5.2. These establish which components occur.
* Eskin–Kontsevich–Zorich, arXiv:1112.5872v6, §§1.1–1.2 and 1.6:
  positive zero profiles, unit-area probability measure, and area-weighted
  cylinder counting. Theorem 1 (2.1), Corollary 2, and the exact columns of
  Appendix A, pp. 100–101, supply all 25 entries.
* Eskin–Masur–Zorich, arXiv:math/0202134v2, §§2.1–2.2: labelled zeros,
  phase retained, relative-period lattice, and radial measure convention.

The selection-defining source is `generate.py`; it has no local imported data
or helper controlling enumeration. Attachments `checks.py` and `check.py`
provide checks, and this document and `source-evidence.md` explain the choices.
The separate critic must independently read and fingerprint the current live
draft and its attachments; this build does not create an acceptance sidecar.