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# Argument selection for T560
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The named quantity is the Masur–Veech hypersurface volume of a connected
component of an Abelian stratum. The family issue #224 specifies the measure:
all zeros labelled, integral relative-period lattice of covolume one, area one,
phase retained, hypersurface volume equal to 2D times cone volume, D=2g+n-1.
EMZ sections 2.1–2.2 fix this convention; Table 1 says the zeros are numbered.
* `g` is a discrete_index: genus of a compact connected Riemann surface. The
domain is g >= 2; the practical extent is 2 through 4. Genus one is omitted
because a positive zero profile would be empty and the usual torus stratum
introduces a marked regular point, outside this family.
* `mu` is a discrete_index: the complete positive partitions of 2g-2. It
identifies a zero divisor type, with cone angles 2*pi*(m_i+1). Every partition
for each included genus occurs. Sorting is an indexing convention and does
not quotient out the labels of equal-order zeros. There are 2, 5, and 11
profiles in genera 2, 3, and 4, respectively.
* `component` is intrinsic: the geometric connected component determined by
the Kontsevich–Zorich classification (also EMZ section 5.2). Only valid
components occur. Hyperelliptic components are separate from even/odd spin
components. No product with all possible labels is formed. The nonhyp label
occurs only at H(3,3) in this finite selection; connected means the sole
component where no splitting occurs.
The selection has 25 components, with counts 2, 7, and 16 by genus. It covers
minimal (one zero), principal (all simple zeros), and every intermediate
stratum, rather than extending only the easy hyperelliptic formulas. All five
component labels occur. Its extent is supported by a complete published table
of exact coefficients (EMZ Table 1), with further independent formulas for
hyperelliptic and minimal components. The cutoff is a practical finite choice,
not an intrinsic boundary or an entry-count target. These are discrete
geometric families, not exploratory samples of a continuous parameter; no
reference_sample group, probability input, rational mesh, or hidden coupling
is present. No individual application is claimed for each component.
Enumeration is increasing genus, lexicographically decreasing profile, then
hyp, even, odd, nonhyp, connected when applicable. The completeness note and
default generator enumeration give precisely this selection. checks.py
independently generates partitions and the classified components, and verifies
both membership and order rather than trusting the ROWS constant.
Coverage review: retaining all profiles through genus four provides breadth
across singularity types and component phenomena at low computational cost.
A smaller famous-only selection would lose that coverage. Extending further
would require source data and independent checks for the additional genera;
there is no need to add a sparse hyperelliptic tail to meet a size target.
Sources checked: EMZ, arXiv:math/0202134v2, sections 2.1–2.2, 5.2 and Table 1;
Kontsevich–Zorich, arXiv:math/0201292, classification; AEZ,
arXiv:1212.1660v4, Theorem 1.1 and Remark 1.2; CMSZ,
arXiv:1901.01785, section 6.5. In particular, the labelled two-zero
hyperelliptic formula has prefactor 8 rather than 4. AEZ Remark 1.2 explicitly
accounts for the factor 2! for the two labels; CMSZ section 6.5 agrees.