Enumerating Toric-Colorable Seeds of Picard Number Five via Binary Matroids

Mathieu Vall\'ee; Suyoung Choi

arxiv: 2606.29309 · v1 · pith:TCZ2OJPEnew · submitted 2026-06-28 · 🧮 math.CO · math.GT

Enumerating Toric-Colorable Seeds of Picard Number Five via Binary Matroids

Suyoung Choi , Mathieu Vall\'ee This is my paper

Pith reviewed 2026-06-30 07:40 UTC · model grok-4.3

classification 🧮 math.CO math.GT

keywords toric-colorable seedsbinary matroidsPicard numberenumerationdynamic programmingweak pseudomanifoldscontraction categoryGray code

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The pith

Binary matroids enumerate 198846 mod 2 toric-colorable seeds of dimension four with Picard number five.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors present a binary matroid framework for enumerating mod 2 toric-colorable seeds of a fixed Picard number. Matroids are grouped by contraction category and weak pseudomanifold subcomplexes are counted using dynamic programming on the mod 2 kernel of the ridge-facet incidence matrix via Gray code traversal. This yields a complete list of 198846 such seeds in dimension four and Picard number five. All listed seeds are confirmed to be toric-colorable, and the method recovers the Picard number four case more efficiently than earlier approaches.

Core claim

The central result is that there exist exactly 198846 mod 2 toric-colorable seeds of dimension four and Picard number five. The enumeration proceeds by classifying binary matroids according to their contraction categories and employing a dynamic programming algorithm to generate all qualifying weak pseudomanifold subcomplexes exactly once. Verification shows that every seed in this complete list satisfies the stronger toric-colorable condition.

What carries the argument

Binary matroids organized by contraction category, with dynamic programming on the mod 2 kernel of the ridge-facet incidence matrix to enumerate weak pseudomanifold subcomplexes.

If this is right

There are exactly 198846 mod 2 toric-colorable seeds of dimension four and Picard number five.
Every seed in the list is toric-colorable.
The same framework reproduces the known Picard number four enumeration faster than the prior method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The contraction-category grouping combined with Gray-code kernel traversal supplies a template that could be reused for enumerations at higher Picard numbers.
The resulting list supplies raw data that could feed geometric constructions or classification efforts for four-dimensional toric varieties of Picard number five.
Because the algorithm works entirely over GF(2), similar incidence-matrix techniques might apply to other problems that count pseudomanifolds or matroid realizations modulo two.

Load-bearing premise

The organization of matroids by contraction category together with the dynamic programming algorithm on the mod 2 kernel of the ridge-facet incidence matrix enumerates every mod 2 toric-colorable seed exactly once without omissions or duplicates.

What would settle it

An independent enumeration that yields a total different from 198846 or identifies either a duplicate or a mod 2 toric-colorable seed absent from the generated list.

read the original abstract

We introduce a binary matroid approach to the enumeration of mod 2 toric-colorable seeds of fixed Picard number. We organize these matroids by their contraction category and enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The main computational step uses a Gray code traversal of the mod 2 kernel of the ridge-facet incidence matrix. As the main new result, we find that there are 198,846 mod 2 toric-colorable seeds of dimension four and Picard number five. We also check that they all are toric-colorable. Finally, the same framework independently reproduces the Picard number 4 enumeration of Choi, Jang, and Vall\'{e}e much faster than their previous method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes a new count of 198846 mod-2 toric-colorable seeds for dimension 4 and Picard number 5 by organizing binary matroids into contraction categories and running Gray-code DP on the mod-2 kernel, while reproducing the Picard-4 case faster.

read the letter

The main deliverable is the explicit count of 198846 seeds together with a verification that each is toric-colorable. The binary-matroid framing and the dynamic-programming traversal over contraction categories are the concrete technical steps that make the Picard-5 case feasible.

The work reproduces the known Picard-4 enumeration more quickly than the earlier method, which shows the new organization is at least as sound as the previous one for the smaller case. The approach is direct: it enumerates weak pseudomanifold subcomplexes without fitting parameters or circular definitions.

The soft spot is that the completeness claim rests on the DP recurrence being exhaustive and duplicate-free once the matroids are partitioned by contraction category. The abstract and the stress-test note give no separate mathematical argument that the categories form a partition or that the Gray-code traversal covers every valid seed exactly once for the new Picard-5 instance. Reproduction of the Picard-4 number is reassuring but does not substitute for an independent check or a proof of the recurrence for the larger case.

The paper is aimed at people already working on the classification of toric varieties or on matroid-based enumerations in algebraic combinatorics. A reader who needs the actual list or who wants to adapt the DP technique will get usable output; a reader looking for a general theorem will not.

It is worth sending to a serious referee who can examine the code and the partitioning argument in detail.

Referee Report

1 major / 2 minor

Summary. The paper develops a binary matroid approach to enumerate mod 2 toric-colorable seeds of fixed Picard number. Matroids are organized by contraction category, and weak pseudomanifold subcomplexes are enumerated using dynamic programming on the mod 2 kernel of the ridge-facet incidence matrix via Gray code traversal. The main result is the discovery of 198,846 such seeds for dimension four and Picard number five, all verified to be toric-colorable. The framework also reproduces the Picard number four enumeration of Choi, Jang, and Vallée more rapidly.

Significance. Should the enumeration prove complete and without duplicates, this constitutes a significant expansion of the catalog of toric-colorable seeds at Picard number five. The reproduction of the prior Picard number four result provides concrete validation of the method's correctness for the lower case and suggests reliability for the new computation. The structured use of matroid contraction categories and dynamic programming represents a methodological advance over previous approaches.

major comments (1)

[Enumeration algorithm and main result] The central claim of 198,846 seeds relies on the dynamic programming algorithm enumerating each mod 2 toric-colorable seed exactly once without omissions or duplicates when organized by contraction category; however, while the method reproduces the known Picard number 4 enumeration, no explicit mathematical argument is given that the contraction categories form a partition of the relevant matroids or that the DP recurrence is exhaustive for the Picard number 5 case, which is load-bearing for the new count.

minor comments (2)

Consider adding a table summarizing the counts for both Picard numbers 4 and 5 for easy comparison.
The notation for 'mod 2 toric-colorable seeds' could be defined more explicitly in the introduction for readers unfamiliar with the term.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the methodological contribution, and the validation provided by reproducing the Picard number 4 result. We address the single major comment below.

read point-by-point responses

Referee: [Enumeration algorithm and main result] The central claim of 198,846 seeds relies on the dynamic programming algorithm enumerating each mod 2 toric-colorable seed exactly once without omissions or duplicates when organized by contraction category; however, while the method reproduces the known Picard number 4 enumeration, no explicit mathematical argument is given that the contraction categories form a partition of the relevant matroids or that the DP recurrence is exhaustive for the Picard number 5 case, which is load-bearing for the new count.

Authors: We agree that an explicit argument strengthens the paper. Contraction categories are defined by the isomorphism type of the contraction of the matroid to a fixed 5-element set (corresponding to the Picard number); because matroid contraction is unique for any given matroid, the categories are disjoint and cover every binary matroid of the relevant rank and corank. The DP recurrence enumerates all weak pseudomanifold subcomplexes by induction on the number of facets: each state transition corresponds to a valid facet addition that preserves the mod-2 kernel condition, and the Gray-code ordering of basis vectors ensures each extension is considered exactly once. The same recurrence and category partition apply verbatim to the Picard-number-5 case. We will add a short subsection containing this justification in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; direct computational enumeration of combinatorial objects

full rationale

The paper enumerates mod-2 toric-colorable seeds via binary matroids organized by contraction category, using dynamic programming on the mod-2 kernel of the ridge-facet incidence matrix with Gray-code traversal. The central output is a raw count (198846 for Picard number 5) obtained by exhaustive search, cross-checked by reproducing the independent prior Picard-number-4 enumeration. No equation, parameter fit, or self-citation reduces this count to a quantity defined by the authors' own inputs; the method is presented as a self-contained algorithm whose completeness is asserted on combinatorial grounds rather than derived from prior results by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The enumeration rests on standard facts about binary matroids and incidence matrices; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math Binary matroids correctly model the mod-2 linear dependencies arising from ridge-facet incidences in the relevant simplicial complexes.
Invoked when the kernel of the incidence matrix is used to generate the search space.
domain assumption The contraction category partitions the set of all relevant matroids without overlap or omission.
Used to organize the dynamic programming recursion.

pith-pipeline@v0.9.1-grok · 5651 in / 1335 out tokens · 39127 ms · 2026-06-30T07:40:59.025625+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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¯Dor¯de Barali´ c, Aleˇ s Vavpetiˇ c, and Aleksandar Vuˇ ci´ c,Universal complexes in toric topology, Results Math78(2023), no. 6, 218 (en)

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1972