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# Argument selection for T581
The sole parameter `matroid`, of type `Symbolic`, identifies a finite simple
matroid. It never selects a second quantity. Values are the ordinary Chow
polynomial for the maximal building set, with a divisor in degree one.
The domain is not restricted to the finite selection computed here.
The governing archived skill manifest is
`agents/runs/20261028T121807Z-build-skills/manifest.json`.
Both `skill-local.md` and `skill-live.md` were read and matched SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
See the attached skills-evidence.json for the full artifact location.
This is the builder's justification, not independent critique approval.
## Exhaustive mixed selection
### Uniform matroids: discrete_index
The 24 identities are:
`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)`.
Rank comes first. These are four low-corank slices of a genuine discrete family,
U_(r,r+c), 3 <= r <= 8, 1 <= c <= 4. They include circuits (c=1) and vary
both grading length and generic dependence. Rank three starts with a nontrivial
middle coefficient; ranks one and two have only 1 and 1+t and are controls.
The Boolean case c=0 is checked against permutation descent counts but is not
added to the selected dependence slices. The upper rank and corank cutoffs are
practical, noncanonical choices, not a classification statement. Symmetry
permits computation without enumerating thousands of uniform flats. FMSV
Theorem 3.21 and Table 1 explicitly study this family and grading convention.
### Braid matroids: discrete_index
The five identities are `K(4)`, `K(5)`, `K(6)`, `K(7)`, `K(8)`.
K(m) is the cycle matroid of the complete graph on m vertices, of rank m-1
and size binomial(m,2). It is not a Coxeter group or a polynomial of a graph.
These type-A arrangement examples provide partition lattices, contrasting
with truncated Boolean lattices in the uniform group. The five vertex counts
cover ranks three through seven. The largest full independent check enumerates
4140 partitions; partition growth makes further depth less attractive than
including the contrasting catalogue objects. The cutoff is practical, not
intrinsic. FMSV Section 5.2 explicitly discusses this family. The generator
counts flats by block number, and the independent checker enumerates actual
set partitions and their block sizes.
### Named objects: reference_sample
The eight identities and (rank, size) are:
| Identity | Rank | Size | Role in reference coverage |
| --- | --- | --- | --- |
| Fano | 3 | 7 | Finite geometry and binary representability |
| NonFano | 3 | 7 | Representability contrast to Fano |
| Pappus | 3 | 9 | Classical incidence geometry |
| NonPappus | 3 | 9 | Nonrepresentable contrast to Pappus |
| Vamos | 4 | 8 | A nonrepresentable paving example |
| R10 | 5 | 10 | A regular matroid |
| TicTacToe | 5 | 9 | A higher-rank paving example |
| K5dual | 6 | 10 | The cographic matroid dual to M(K_5) |
Every identity is exactly the Sage catalogue constructor, preserving T396's
spelling. K4 is not added as an alias for K(4), and K5 is not added beside K(5).
K5dual is matroid duality; it does not mean reversing the flat lattice.
The runtime check verifies simplicity, rank and size for each object.
The largest named lattice has 314 flats, so complete independent subset checks
are inexpensive. Distinct matroids remain distinct rows even when their
polynomials coincide: Fano and K(4) both have 1+8t+t^2.
Reference policy:
- **Purpose:** identify ordinary Chow-ring Hilbert polynomials beyond the
uniform and braid families, including examples with different incidence and
representability structures.
- **Bounds:** exactly the eight named objects, with no generated matrices,
further duals, direct sums, or automatic catalogue extensions.
- **Coverage rationale:** compare several mathematical structures at small
size, instead of selecting every catalogue object below a runtime threshold.
Breadth is assessed before adding depth to uniform or braid sequences.
- **Noncanonical acknowledged:** true. This is an exploratory reference choice,
not a claim of a published application for every polynomial.
- **Notation independent:** true. Ground-set relabelling and numeral base do
not change the selected objects.
- **Sources checked:** true. The Sage catalogue and the individual constructor
documentation identify these objects; FMSV fixes the invariant and convention.
## Coverage and reproducibility
The groups are disjoint and contain all 37 identities. Enumeration orders them
by rank, ground-set size, then key, as specified in family issue 229. The table
has `complete: no` with the same groups, bounds and reasons. No quota or desired
entry count selected these bounds. The longest polynomial has 79 characters;
readability does not require pruning any group. No decimal mesh is involved.
The three groups are a `mixed` assessment of the single parameter, with leaf
categories `discrete_index`, `discrete_index`, and `reference_sample`.
## Checked sources
- Ferroni, Matherne, Stevens and Vecchi, *Hilbert–Poincaré series of matroid Chow
rings and intersection cohomology*, arXiv:2212.03190v3,
https://arxiv.org/html/2212.03190v3 . Section 3.1 and Definition 3.1 specify the
ordinary maximal-building-set ring. Proposition 3.5 gives weighted chains,
including the empty product and the top flat; Theorem 3.12 gives the distinct
characteristic convolution. Theorem 3.21 treats uniforms; Section 5.2 treats
braid matroids. Section 2.3 fixes derangement and Eulerian normalization.
- SageMath, *Catalog of matroids*,
https://doc.sagemath.org/html/en/reference/matroids/sage/matroids/matroids_catalog.html .
The Uniform, CompleteGraphic and individual catalogue constructors distinguish
the parameterized families from named examples.
- SageMath, *Database of matroids*,
https://doc.sagemath.org/html/en/reference/matroids/sage/matroids/database_matroids.html .
The sections named Fano, NonFano, Pappus, NonPappus, Vamos, R10, TicTacToe and
K5dual fix the eight reference objects. The R10 prose has an apparent copied
name; its constructor, regular-field description and example identify R10.