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# Argument selection for T579
This is the builder's justification, not the required independent critique.
The coordinated instructions are archived at
`agents/runs/20261028T110020Z-build-skills/manifest.json`.
Both `skill-local.md` and `skill-live.md` have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
## Object, convention and free parameter
There is one free parameter, `matroid`, of type `Symbolic`, displayed as M.
It names the object whose single invariant P_M(t) is stored. The domain is
finite simple matroids, not a finite list of keys. The invariant is the
Elias–Proudfoot–Wakefield Kazhdan–Lusztig polynomial, with rank-zero base 1,
2 deg(P_M) < rank(M) in positive rank, and the characteristic-polynomial
recursion in their Theorem 2.2. Restrictions and contractions have their usual
orientations. Non-simple contractions retain their actual flat intervals.
The variable t has the standard KL grading; no t-to-t^2 substitution is made.
All outputs are exact expanded polynomials in ZZ[t].
The parameter has a mixed selection, with the following exhaustive disjoint
groups. There are no other inputs, couplings, generated duals or direct sums.
## Uniform matroids: discrete_index
Membership: 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).
U(r,n) means rank r on n elements, with independence determined by cardinality
at most r. Rank and corank c=n-r are genuine discrete structural indices.
The extent 5 <= r <= 8 and 1 <= c <= 4 gives four low-corank slices: circuits
at c=1 and progressively more dependence at c=2,3,4. It tests rank variation
through both quadratic and cubic KL polynomials without devoting the table
only to one corank sequence. The finite cutoffs are practical, noncanonical
reference choices. The family is studied in EPW section 2.4 and Appendix A.1;
the paper denotes this object M_(c,r), reversing neither rank nor corank.
The Sage Uniform documentation independently specifies the rank-first input.
## Braid matroids: discrete_index
Membership: K(6), K(7), K(8).
K(m) means the cycle matroid M(K_m), of rank m-1 on binomial(m,2) edges.
These are the type-A braid arrangement matroids and their flats form the
partition lattice, as defined in EPW section 2.5. This is a second discrete
family with geometry different from the uniform one. The selected vertex
counts give ranks 5,6,7. Full independent enumeration reaches the 4,140
partitions of eight vertices; extending the symmetry-reduced generator is
cheap, but comparing against full lattices grows quickly. The bound is a
practical reference extent, not an intrinsic stopping point or an entry target.
EPW Appendix A.2 independently tabulates every selected braid polynomial.
## Named examples: reference_sample
Membership: R10, TicTacToe, K5dual.
Purpose: provide bounded exploratory recognition and comparison outside the
uniform and braid families. R10 supplies a regular rank-five example;
TicTacToe is the documented rank-five paving catalogue object on nine
elements; K5dual is the cographic matroid M*(K_5), of rank six on ten elements.
The identities mean exactly the Sage catalogue constructors of those names,
not every matroid with the same rank, characteristic polynomial or Tutte
polynomial. These keys preserve the spelling in T396.
Bounds: exactly those three objects, with no additional extensions, duals,
direct sums or search across catalogue entries. Each has at most ten elements,
so the checker can independently enumerate every subset and test flatness.
Coverage rationale: retain regular, paving, and cographic examples from the
family's small reference pool, while keeping the KL-specific rank-five lower
bound. This small group broadens the uniform/braid coverage. It is not a
classification of small matroids, a claim of a documented application for each
value, or a choice based solely on the availability of a library constructor.
It is deliberately noncanonical and invariant under numeral base or ground-set
relabelling. The official Sage catalogue and object documentation were checked
for the names, ranks and constructions; no publication using each selected
polynomial is claimed.
Reference-policy flags: `noncanonical_acknowledged: true`,
`notation_independent: true`, `sources_checked: true`.
## Breadth and exclusions
The 37-object family pool includes lower ranks. For this particular invariant,
ranks at most four give constant or linear polynomials, so the 22-object
rank-at-least-five part is retained. Four lower-rank cases are independent
controls, not stored rows. This selects the first ranks in which quadratic
coefficients can occur while preserving all three kinds of input, rather than
pruning to famous objects or deepening only a circuit sequence. No numerical
mesh, probability parameter or excluded quantile family is involved.
The measured largest polynomial is 31 characters; the numbers are readable.
The practical cutoff is a small-rank reference and independent-checking choice,
not a storage constraint. All 22 results are distinct, but distinctness was
not used to remove contexts. Entries are ordered by rank, ground-set size and
key, exactly as the generator and table completeness note specify. Symbolic
labels describe matroids, not the resulting polynomials.
## Checked sources
- Elias, Proudfoot and Wakefield, *The Kazhdan–Lusztig polynomial of a matroid*,
arXiv:1412.7408v3, Theorem 2.2, sections 2.4–2.5, Appendix A.1–A.2.
https://arxiv.org/abs/1412.7408
- Proudfoot, Xu and Young, *The Z-polynomial of a matroid*, arXiv:1706.05575,
Definition 2.1, Proposition 2.3, Corollary 2.5 and Remark 2.6.
https://arxiv.org/abs/1706.05575
- SageMath matroid catalogue and object documentation: Uniform, CompleteGraphic,
R10, TicTacToe, K5dual.
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 documentation has a copied noun “NonDesargues” in R10's explanatory
paragraph. Its constructor, heading, examples and source all specify R10;
we use that constructor and do not adopt the copied noun as an alias.