# Argument selection for T562

Policy read: the matching archived local and live canonical skills recorded in
`agents/runs/20261027T213908Z-build-skills/manifest.json`. Both SHA-256 values are
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attached `skill-evidence.json` records the manifest hash as well.
Family proposal: numberdb-data issue 224, read before screening or construction.
This build's justification is not an independent critique acceptance.

## Quantity and fixed conventions

One named quantity is stored: the sum of the g nonnegative Lyapunov exponents
of the real Hodge bundle for the Teichmüller flow diag(exp(t),exp(-t)), with
lambda_1=1. It is evaluated for the Masur–Veech probability measure on each
area-one connected component, not for a particular Teichmüller curve. The
surfaces are compact and connected; the differentials are holomorphic, with
positive zero orders, no poles and no added regular marked points. All zeros
are labelled. The relative-period integral lattice has covolume one; the
phase is retained. The radial measure convention and component probability
normalization agree with T560 and T561. Exact rational sums need no decimal
precision or error estimate. EKZ Section 1, Theorem 1 and Corollaries 1–2
specify these dynamical conventions; EMZ Sections 2.1–2.2 specify the measure.

## Every parameter

* `g` is a discrete_index: the topological genus of the Riemann surface.
  The domain is every integer g>=2. The cutoff g<=4 is a practical finite
  choice, not a claim that higher genera lack mathematical interest.
* `mu` is a discrete_index: the non-increasing positive partition of 2g-2
  giving the zero orders. All such partitions are included for each selected
  genus, not a sampling of numerical coordinates. KZ Section 1.1 defines
  these strata, and Theorems 1–2 classify their connected components.
* `component` is intrinsic: the actual connected component specified by that
  classification. It is dependent on the zero profile, not freely combined
  with every profile. `hyp` is a hyperelliptic component, `even` and `odd`
  exclude hyperelliptic components, `nonhyp` is the other component of
  H(g-1,g-1) for even g>=4, and `connected` means the sole component of an
  otherwise connected stratum. Genus-two strata use `hyp`.

There is no continuous reference sampling, arithmetic mesh, or quantity selector.
No parameter needs a reference_sample or mixed category. The choice of measure
is fixed rather than silently varying over rows. No excluded quantiles occur.

## Complete finite selection and breadth

| Genus | Profiles | Allowed components |
|---|---|---|
| 2 | (2), (1,1) | hyp |
| 3 | (4), (2,2) | hyp, odd |
| 3 | (3,1), (2,1,1), (1,1,1,1) | connected |
| 4 | (6) | hyp, even, odd |
| 4 | (4,2), (2,2,2) | even, odd |
| 4 | (3,3) | hyp, nonhyp |
| 4 | (5,1), (4,1,1), (3,2,1), (3,1,1,1), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1) | connected |

These are all 18 profiles and 25 components in genera 2–4. This scope covers
minimal, principal and intermediate strata and all component phenomena that
occur in the range. EKZ supplies complete exact low-genus data, so this is a
coherent initial reference for dynamical invariants of classified geometric
objects. Each row need not have an individually famous application. Extending
only the easy hyperelliptic formulas would sacrifice this breadth. Genus one
would require a regular marked point, outside the positive-zero convention.

The largest rational occupies nine characters. Thus the cutoff is a source
coverage and verification choice, not a storage limit or entry-count target.
The table is globally incomplete (`complete: no`) and its completeness note
states this entire finite selection and rationale. Default enumeration is by
increasing genus, decreasing lexicographic profile, then hyp, even, odd,
nonhyp, connected as applicable. The checker independently generates partitions
and applies KZ Theorems 1–2, checking both the exact selected set and order.

## Sources actually checked

* Kontsevich–Zorich, arXiv:math/0201292v2, Section 1.1 and Theorems 1–2.
* Eskin–Kontsevich–Zorich, DOI 10.1007/s10240-013-0060-3, Sections 1.2–1.4,
  Theorem 1, Corollaries 1–2 and Appendix A, published p.326, exact column.
* Eskin–Masur–Zorich, arXiv:math/0202134v2, Sections 2.1–2.2 and Appendix B.2.
* Chen–Möller, arXiv:1104.3932v2, Props.4.5,4.8 and Sections 5–6, for geometric
  checks; its inconsistent numerical/formula statements are explicitly
  reconciled in source-notes.md, not silently accepted.

`generate.py` contains all enumeration and value data; it imports no local
selection helper. `checks.py`, `source-column.json`, `T561.json`, and the
standard `check.py` supply checking inputs. The separate critique must inspect
and fingerprint the live generator and all selection-relevant attachments;
this build supplies no critique sidecar and does not offer the draft.