# Argument selection for T580

Policy: the coordinated skill archived at
`agents/runs/20261028T114049Z-build-skills/manifest.json`. Both local and live
versions were read and verified with SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attached skill-evidence.json records the manifest and both digests.
This is the builder's rationale, not an independent critique acceptance.

## Quantity and parameter

One quantity, Z_M(t), is indexed by one Symbolic parameter `matroid`, a finite
simple matroid. Each key identifies the object, with U(r,n) giving rank first
and size second, and K(m) the complete-graph cycle matroid of rank m-1.
Catalogue names are exactly Sage's and preserve T396/T579's shared spellings.
The polynomial variable t is grading, not a sampled parameter. Cohomological
degree 2i contributes t^i. No inverse CDF or statistical thresholds occur.

## Mixed selection: exhaustive disjoint groups

1. **discrete_index: uniform matroids.** The 24 keys 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 3 through 8 and corank 1 through 4 are four low-corank slices of a
   genuine combinatorial family, including the circuit/Narayana case c=1.
   They compare generic dependencies at several ranks and coranks. Ranks below
   3 have Z polynomials of degree at most two, determined by their flat counts;
   they are controls instead of separate rows. The maximum rank and corank are
   practical reference bounds, not canonical endpoints. Symmetry compression
   keeps all four slices inexpensive and their polynomials short.
2. **discrete_index: braid matroids.** Exactly K(4), K(5), K(6), K(7), K(8).
   These cycle matroids of complete graphs are the type-A arrangement family,
   with partition lattices of flats, providing structure different from uniform
   dependence. Vertex counts 4 through 8 give ranks 3 through 7. Partition
   growth makes further depth less valuable before comparing other families.
   These bounds are practical, not an assertion that larger ranks are uninteresting.
3. **reference_sample: named catalogue objects.** Exactly Fano, NonFano,
   Pappus, NonPappus, Vamos, R10, TicTacToe, K5dual. These give bounded
   exploratory comparisons across finite geometry, representability contrasts,
   regular, paving and cographic examples, including nonrepresentable matroids.
   Each is the documented catalogue object, without freely sampled matrices.
   No paper using each resulting Z polynomial is claimed.

The named group's reference policy is:
- purpose: identification and comparison of cohomological/combinatorial
  polynomials outside the uniform and braid families;
- bounds: exactly the eight named objects, each with at most ten elements,
  without automatically adding duals, direct sums, or catalogue extensions;
- coverage rationale: several distinct small structures, rather than every
  object under a runtime threshold, broaden coverage before deepening a single
  family; the choice is not a quota or a target number of entries;
- noncanonical_acknowledged: true;
- notation_independent: true (independent of ground-set labels and numeral base);
- sources_checked: true (the sources support the objects and conventions,
  not a supposed application for each exploratory value).

## Sources actually checked

Proudfoot, Xu and Young, *The Z-polynomial of a matroid*, Definition 2.1,
Proposition 2.3, Corollary 2.5, Section 4 and Proposition 4.9;
https://doi.org/10.37236/7105, author copy https://pages.uoregon.edu/njp/z.pdf.
Section 4 explicitly treats uniform and braid families, their flat counts, and
the Narayana specialization. In their U_(m,d), d is rank and m is corank;
our U(r,n) uses rank and total size instead.

SageMath's catalogue and individual object documentation:
https://doc.sagemath.org/html/en/reference/matroids/sage/matroids/matroids_catalog.html
https://doc.sagemath.org/html/en/reference/matroids/sage/matroids/database_matroids.html
The named constructions, ranks and ground-set sizes are checked at runtime.

## Completeness and ordering

All 37 declared objects are included. Distinct objects remain distinct even if
polynomials coincide. Catalogue K4 is an alias of K(4), not an extra row.
Order by rank, ground-set size, then exact key. The parameter domain remains
all finite simple matroids; `complete: no` describes these finite subsets and
why they were chosen, not completeness at any rank or ground-set size.

This selection preserves the proposal's breadth after reassessment under the
current skill. Entry length and exactness are measured by dry_run.py; numerical
controls and independent full-coefficient calculations are in checks.py.