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# Argument selection for T582
This is the build rationale, not an independent critique acceptance.
The sole free parameter is `matroid`, a finite simple matroid. Each key names
an object whose augmented Chow polynomial is taken. There is no quantity,
normalization or continuous coupling parameter. The overall assessment is
`mixed`, exhausted by the three disjoint groups listed here.
## Conventions and sources
The augmented ring, algebraic grading, and flat-lattice orientations are those
of [Ferroni, Matherne, Stevens and Vecchi (2024)](https://doi.org/10.1016/j.aim.2024.109733),
Definition 3.2, Propositions 3.6 and 3.7 and Theorem 3.14. Uniform matroids and
their polynomials are treated in Section 3.5 and Theorem 3.21; braid matroids
and partition lattices in Section 5.2. The
[Sage catalogue object documentation](https://doc.sagemath.org/html/en/reference/matroids/sage/matroids/database_matroids.html)
was checked at the eight named constructors. Those sources establish the
objects and conventions; they do not make this finite reference selection canonical.
`U(r,n)` has rank first and ground-set size second. `K(m)` denotes M(K_m),
not its dual. `K5dual` denotes M*(K_5). The eight other keys keep Sage's exact
spellings, shared with T396. Ground-set relabeling changes no selected object.
The same keys and display labels as T580 and T581 are retained.
## Uniform group: discrete_index
All U(r,r+c) with 3 <= r <= 8 and 1 <= c <= 4 are included. Rank and corank
are genuine matroid indices. Four low-corank slices compare generic dependence,
including circuits (c=1), across six ranks. Rank three starts beyond the
simplest low-rank cases and allows a cubic Hilbert polynomial; those lower
ranks remain controls. Rank eight and corank four are practical bounds, not
mathematical endpoints. Symmetry makes this range inexpensive while preserving
several independent coefficient patterns. Extending rank alone would privilege
one easy family over the geometric and catalogue comparisons.
## Braid group: discrete_index
All M(K_m) for 4 <= m <= 8 are included. Complete graphs and their type-A
arrangements give a separate geometric sequence. These five sizes cover
ranks three through seven, with partition lattices rather than uniform flat
lattices. The cutoff anticipates growth in explicit partition checks: Pi_8
already has 4140 flats. This is a practical reference bound, not a claim that
m=8 is intrinsically distinguished. K4 and K5 aliases are not extra rows.
## Named group: reference_sample
Purpose: make augmented Chow outputs identifiable outside the uniform and
braid sequences, and compare small matroids with different geometric and
representability structures.
Bounds: exactly Fano, NonFano, Pappus, NonPappus, Vamos, R10, TicTacToe and
K5dual, with no automatically generated extensions, duals or direct sums.
Coverage rationale: Fano/NonFano and Pappus/NonPappus preserve two familiar
representability contrasts. Vamos supplies a nonrepresentable example, R10 a
regular example, TicTacToe a paving example, and K5dual a cographic example.
This retains structural breadth instead of taking every catalogue object below
a runtime threshold. A publication using every individual polynomial is not
claimed or required. The selection and all cutoffs are explicitly noncanonical.
They depend on named isomorphism types, independently of numeral base or labels.
Reference-policy attestations for this build rationale:
`noncanonical_acknowledged: true`; `notation_independent: true`;
`sources_checked: true`. An independent critic must make its own assessment.
## Measured extent and limits
The 37 values have maximum written length 95 characters, at U(8,12).
The standard dry run reports about 3.4 KB for its complete entry records.
The eight named examples have 16, 18, 29, 31, 79, 162, 225 and 314 flats,
respectively in the named order given in Bounds. Exact independent checks
complete in under one second after imports on this builder. None of these
measurements is a reason to fill the available storage with more objects.
The selected coverage is a practical reference across three useful groups.
Distinct objects with coincident values would remain distinct contexts.
The generator orders all entries by rank, ground-set size, then key. The
table's completeness note describes exactly these three groups and bounds;
it does not claim an exhaustive list of all matroids of any rank or size.
## Exhaustive group membership
### Uniform (discrete_index)
`U(3,4)`, `U(3,5)`, `U(3,6)`, `U(3,7)`, `U(4,5)`, `U(4,6)`, `U(4,7)`, `U(4,8)`, `U(5,6)`, `U(5,7)`, `U(5,8)`, `U(5,9)`, `U(6,7)`, `U(6,8)`, `U(6,9)`, `U(6,10)`, `U(7,8)`, `U(7,9)`, `U(7,10)`, `U(7,11)`, `U(8,9)`, `U(8,10)`, `U(8,11)`, `U(8,12)`
### Braid (discrete_index)
`K(4)`, `K(5)`, `K(6)`, `K(7)`, `K(8)`
### Named (reference_sample)
`Fano`, `NonFano`, `NonPappus`, `Pappus`, `Vamos`, `TicTacToe`, `R10`, `K5dual`
## Display order
`U(3,4)`, `U(3,5)`, `K(4)`, `U(3,6)`, `Fano`, `NonFano`, `U(3,7)`, `NonPappus`, `Pappus`, `U(4,5)`, `U(4,6)`, `U(4,7)`, `U(4,8)`, `Vamos`, `K(5)`, `U(5,6)`, `U(5,7)`, `U(5,8)`, `TicTacToe`, `U(5,9)`, `R10`, `K(6)`, `U(6,7)`, `U(6,8)`, `U(6,9)`, `K5dual`, `U(6,10)`, `K(7)`, `U(7,8)`, `U(7,9)`, `U(7,10)`, `U(7,11)`, `K(8)`, `U(8,9)`, `U(8,10)`, `U(8,11)`, `U(8,12)`