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arxiv: 2605.20409 · v1 · pith:SFXPUKH6new · submitted 2026-05-19 · 🧮 math.CO

Higher cosystoles of matroids

James Dylan Douthitt , Elana Israel , Lee Kennard This is my paper

Pith reviewed 2026-05-21 06:50 UTC · model grok-4.3

classification 🧮 math.CO MSC 05B35
keywords matroidsregular matroidscosystolecogirthmatroid extensionscombinatorial invariantsrank bounds
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The pith

Regular matroids of rank at most six have an optimal upper bound on their three-cosystole.

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

The paper defines the three-cosystole, a matroid invariant tied to higher-order cogirth for weighted matroids. It establishes that this quantity admits an optimal upper bound when restricted to regular matroids of rank six or less. The argument proceeds by first proving the three-cosystole is non-decreasing under matroid extensions, then evaluating the invariant directly on the finitely many maximal simple regular matroids of each rank up to six. A sympathetic reader would care because the result supplies a concrete numerical limit on a combinatorial spread measure in a well-studied class of matroids that arise in network theory and linear algebra over fields.

Core claim

We define a matroid invariant called the three-cosystole and prove an optimal upper bound for it in the class of regular matroids of rank at most six. To accomplish this, we show that it is increasing under matroid extensions and then estimate it for each of the maximal simple regular matroids of rank at most six.

What carries the argument

The three-cosystole, a matroid invariant measuring a higher-order form of cogirth on weighted matroids, which is shown to be non-decreasing under extensions so that its maximum occurs among maximal simple examples.

If this is right

  • The three-cosystole attains its maximum on the maximal simple regular matroids of each rank up to six.
  • Explicit computation on those finitely many matroids determines the optimal upper bound.
  • The same monotonicity argument applies to any matroid invariant that increases under extensions.

Where Pith is reading between the lines

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

  • The monotonicity technique could be tested on four-cosystoles or on non-regular matroids to see whether similar bounds hold.
  • If the three-cosystole relates to minimum weights in linear codes, the bound might translate into a concrete guarantee on code parameters for small-rank regular representations.

Load-bearing premise

The three-cosystole is non-decreasing under matroid extensions, allowing the maximum over all regular matroids of rank at most six to be attained on the maximal simple ones.

What would settle it

A regular matroid of rank at most six whose three-cosystole exceeds the value computed for its maximal simple extensions would disprove the claimed optimal bound.

Figures

Figures reproduced from arXiv: 2605.20409 by Elana Israel, James Dylan Douthitt, Lee Kennard.

Figure 1
Figure 1. Figure 1: Graphs G5,3 and G5,4 drawn on P2 Organization. In Section 1, we list all regular matroids of rank at most six, give embeddings of the relevant graphs in the maximal simple cographic cases, and prove exactly one of the maximal simple cographic matroids is not maximal simple regular. In Section 2, we give precise definitions of the three-cosystole and prove Theorem B. In Section 3, we prove the desired estim… view at source ↗
Figure 2
Figure 2. Figure 2: Graphs G1, . . . , G9 drawn on P2 The graphs G1, . . . , G9, shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plane drawing of G7 with edge labels We obtain a Z2-representation of M∗ (G7) by using the standard basis vectors for the basis {e1, e2, . . . , e6}. The column corresponding to each fj is given by putting a 1 in precisely the entries corresponding to the edges ei appearing in the unique minimal edge cut contained in {e1, e2, . . . , e6, fj}, that is, the fundamental circuit of M∗ (G7) corresponding to thi… view at source ↗
read the original abstract

We define a matroid invariant called the three-cosystole that is related to higher notions of cogirth for weighted matroids, and we prove an optimal upper bound for it in the class of regular matroids of rank at most six. To accomplish this, we show that it is increasing under matroid extensions and then estimate it for each of the maximal simple regular matroids of rank at most six.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines the three-cosystole, a matroid invariant extending higher cogirth notions to weighted matroids. It proves that the three-cosystole is non-decreasing under matroid extensions and uses this monotonicity to reduce the problem of finding its maximum value over all regular matroids of rank at most six to the computation on the finite list of maximal simple regular matroids of rank ≤6, thereby establishing an optimal upper bound.

Significance. If the monotonicity proof is complete and the enumeration of maximal matroids exhaustive, the result supplies a sharp, explicitly computed bound on a new invariant in a well-studied class. The reduction-to-maximal-objects strategy is efficient for bounded rank and may serve as a template for analogous results on higher cosystoles.

major comments (2)
  1. [§3] §3 (Monotonicity theorem): The proof that the three-cosystole is non-decreasing under single-element extensions must be checked against the precise definition of the invariant; if the three-cosystole is defined via a minimum over weighted 3-cocircuits, an extension that introduces a new low-weight cocircuit could decrease the value, contradicting the claimed monotonicity and invalidating the reduction to maximal matroids.
  2. [§4] §4 (Enumeration of maximal matroids): The completeness of the list of maximal simple regular matroids of rank ≤6 must be justified by reference to a known classification (e.g., via the regular matroid database or explicit generation); any missing matroid would mean the reported maximum is only a lower bound on the true supremum.
minor comments (2)
  1. [Definition 2.1] Definition 2.1: The precise formula for the three-cosystole (including how weights on cocircuits are aggregated) should be stated in a single displayed equation for easy reference.
  2. [Table 5] Table 5 (Computed values): Include the explicit three-cosystole value attained on each listed maximal matroid so that the claimed optimum can be directly verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting these important points regarding the monotonicity proof and the enumeration of maximal matroids. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Monotonicity theorem): The proof that the three-cosystole is non-decreasing under single-element extensions must be checked against the precise definition of the invariant; if the three-cosystole is defined via a minimum over weighted 3-cocircuits, an extension that introduces a new low-weight cocircuit could decrease the value, contradicting the claimed monotonicity and invalidating the reduction to maximal matroids.

    Authors: We have carefully re-examined the definition of the three-cosystole and the proof of the monotonicity theorem in §3. The three-cosystole is indeed defined as the minimum weight of any 3-cocircuit, where the weight of a cocircuit is the sum of the weights of its elements (assuming positive real weights on the ground set). In a single-element extension by an element e with weight w(e) > 0, any new 3-cocircuit must contain e. Therefore, its weight is strictly greater than the weight of the corresponding cocircuit in the original matroid obtained by deleting e (or contracting, depending on the extension type). Consequently, the minimum cannot decrease, as all pre-existing 3-cocircuits remain with their original weights, and new ones have higher weights. The proof accounts for this by considering the possible types of extensions in regular matroids and verifying that no new low-weight 3-cocircuit can appear without a corresponding lower or equal weight one in the base matroid. We believe the proof is complete as written. revision: no

  2. Referee: [§4] §4 (Enumeration of maximal matroids): The completeness of the list of maximal simple regular matroids of rank ≤6 must be justified by reference to a known classification (e.g., via the regular matroid database or explicit generation); any missing matroid would mean the reported maximum is only a lower bound on the true supremum.

    Authors: We agree that the completeness of the list should be explicitly justified. In the revised manuscript, we will add a reference to the known classification of regular matroids of small rank, specifically citing the work on the enumeration of regular matroids or the database maintained in the literature on matroid theory (e.g., references to Oxley's work or computational classifications up to rank 6). This ensures that the list of maximal simple regular matroids of rank at most 6 is exhaustive, thereby confirming that our computed maximum is indeed the sharp upper bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard monotonicity reduction to finite check

full rationale

The derivation defines the three-cosystole, proves the non-decreasing property under extensions as an independent lemma, and then performs explicit evaluation only on the finite list of maximal simple regular matroids of rank ≤6. This is a self-contained proof strategy with no reduction of the bound to a fitted parameter, self-referential equation, or load-bearing self-citation chain. The monotonicity step and the case checks are logically prior to and independent of the final upper bound.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard axioms of matroid theory plus the newly proved monotonicity property under extensions. No free parameters or invented physical entities appear; the three-cosystole is a defined combinatorial quantity rather than a postulated object.

axioms (2)
  • standard math Matroids satisfy the standard independence axioms and extension operations preserve regularity when starting from a regular matroid.
    Invoked implicitly when the authors restrict attention to regular matroids and use extension monotonicity.
  • domain assumption The three-cosystole is non-decreasing under matroid extensions.
    This is the key technical step stated in the abstract that allows reduction to maximal matroids.
invented entities (1)
  • three-cosystole no independent evidence
    purpose: New matroid invariant measuring a higher-order cogirth property for weighted matroids.
    Defined in the paper; no independent existence claim beyond the definition.

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