arxiv: 2605.13662 · v1 · submitted 2026-05-13 · 🧮 math.AC · math.CO

Recognition: 2 theorem links

· Lean Theorem

Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs

Oliver Clarke , Dimitra Kosta , Alexander Milner

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Pith reviewed 2026-05-14 17:40 UTC · model grok-4.3

classification 🧮 math.AC math.CO

keywords toric ideals of graphsMarkov basesdistance reductionbouquet decompositionscomplete intersectionshomogeneous idealsmonomial curves

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The pith

For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits.

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

The paper studies the distance-reduction property of Markov bases for toric ideals of graphs, which ensures that the fibres of the ideal remain tightly connected under moves from the basis. It first proves an equivalence for the complete intersection case: a minimal Markov basis is distance-reducing for the full ideal if and only if it is distance-reducing on the circuits alone. The work then examines how this property behaves under bouquet decompositions of homogeneous toric ideals. It shows that ideals sharing the same bouquet structure and signature inherit identical distance-reduction behavior from one another. For the special case in which the bouquet matrix is a monomial curve in affine 3-space, the paper supplies explicit necessary and sufficient conditions on the basis for the distance-reduction property to hold.

Core claim

For toric ideals of graphs that are complete intersections, the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the homogeneity assumption, toric ideals possessing the same bouquet structure and signature preserve their distance-reduction properties. For homogeneous toric ideals whose bouquet matrix is a monomial curve in A^3, necessary and sufficient conditions are given for the minimal Markov bases to be distance-reducing.

What carries the argument

The bouquet structure of a homogeneous toric ideal, which encodes the combinatorial data that determines the ideal up to signature, together with the distance-reduction property that measures how Markov basis moves shorten paths between elements of the same fibre.

If this is right

A minimal Markov basis for a complete-intersection graph toric ideal is distance-reducing for the whole ideal precisely when it is distance-reducing on the circuits.
Homogeneous toric ideals that share the same bouquet structure and signature have identical distance-reduction properties for their Markov bases.
When the bouquet matrix is a monomial curve in A^3, the distance-reduction property of a minimal Markov basis is decided by an explicit list of numerical conditions on the bouquet data.
The distance-reduction check can be restricted to circuits rather than the full set of generators for complete-intersection cases.

Where Pith is reading between the lines

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

The equivalence on circuits suggests that distance-reduction can be verified by examining only a small subset of generators, which may simplify algorithmic checks for larger graphs.
Preservation across bouquets implies that distance-reduction is a property of the combinatorial type rather than the specific edge set of the graph.
The monomial-curve case may serve as a base for inductive arguments that classify distance-reducing bases for bouquets with more factors.

Load-bearing premise

The toric ideals under consideration are homogeneous and, for the final characterization, their bouquet matrix is a monomial curve in three-dimensional affine space.

What would settle it

An explicit complete-intersection toric ideal of a graph together with a minimal Markov basis that reduces distance on every circuit but fails to reduce distance between some pair of binomials in the ideal.

Figures

Figures reproduced from arXiv: 2605.13662 by Alexander Milner, Dimitra Kosta, Oliver Clarke.

**Figure 1.** Figure 1: The graph G1 in Example 3.13. The corresponding incidence matrix of G1 is given by AG1 =                 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1                 ∈ Z 10×12 . Usin… view at source ↗

**Figure 2.** Figure 2: The graphs w1, w2 corresponding to walks w1, w2 respectively. Double edges of the walks are given by red edges. 1 2 3 4 5 6 7 8 9 0 (a) w3 1 2 3 4 5 6 7 8 9 0 (b) w4 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: The graphs w3, w4 corresponding to walks w3, w4 respectively. Double edges of the walks are given by red edges. Gr(AG1 ) =     z1 z2 z3 z4     =     0 0 0 0 1 −1 −1 1 0 0 0 0 1 −1 −1 2 0 −2 0 0 2 −1 −1 1 1 −1 −1 2 −2 0 2 −2 2 −1 −1 1 1 −1 −1 2 −1 −1 1 −1 2 −1 −1 1     . We then have β(z3) = e12e 2 34e 2 56e 2 78e90 − e13e23e 2 45e 2 67e89e80 = Bw3 where we define w3 := (1, 2, 3, 4, 5, 6, 7,… view at source ↗

**Figure 4.** Figure 4: The graph w4 marked with pluses and minuses corresponding to whether that edge divides E +(w4) or E −(w4). Red edges still correspond to double edges which both have the same sign so we only mark these in red as well. well as z + 2 ≤ z + 4 and Supp(z − 2 ) ∩ Supp(z − 4 ) = {e13, e23, e89, e80}. Thus, M distance-reduces the circuits of AG1 as well as the Graver basis Gr(AG1 ). By [5, Theorem 7.4], since M d… view at source ↗

**Figure 5.** Figure 5: The fibers Fv3 and Fu2 with non-dotted edges corresponding to elements in M′ V and M′ U respectively and the 1-norm distances between pairs of elements labelled. 4.1 The poset of signatures In the counterexample above, the distance-reduction information is not preserved between the two toric ideals, not because the structure of the fibers is different, but because elements in the fiber have different 1-nor… view at source ↗

**Figure 6.** Figure 6: This Figure displays the Hasse diagram of the poset of signatures when [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗

read the original abstract

The distance-reduction property for a generating set, i.e., a Markov basis, of a toric ideal is a condition that ensures tight connectivity of its fibres. In this paper, we study the distance-reduction property for toric ideals of graphs and move on to explore the relationship between the distance-reduction property and the bouquet structure of homogeneous toric ideals, which includes the class of toric ideals of graphs. For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. We then consider how the distance-reduction properties interact with the bouquet structure of the toric ideal. Bouquets are a combinatorial structure that capture the essential combinatorial information of the toric ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved. For homogeneous toric ideals whose bouquet matrix is a monomial curve in $\mathbb{A}^3$, we give necessary and sufficient conditions for when the minimal Markov bases are distance-reducing.

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

New iff conditions tie minimal Markov bases to circuit distance-reduction in complete-intersection graph toric ideals, plus preservation of the property across homogeneous toric ideals with matching bouquet structure and signature.

read the letter

Hey, the main points are these: for complete-intersection toric ideals of graphs the paper proves that a minimal Markov basis is distance-reducing if and only if it distance-reduces the circuits, and it shows that distance-reduction properties are preserved between homogeneous toric ideals that share the same bouquet structure and signature. For the narrower case where the bouquet matrix is a monomial curve in A^3 it also gives necessary and sufficient conditions. These are the explicit combinatorial statements that were not in the earlier literature cited in the abstract. The work is useful because it links the distance-reduction property directly to the bouquet decomposition, which the authors treat as the essential combinatorial skeleton of the ideal. The arguments rest on standard definitions of bouquets and circuits and stay within the homogeneous setting where the preservation claim is stated, so there is no evident circularity or extrapolation beyond the hypotheses. The abstract presents the claims cleanly and the stress-test note finds no load-bearing gaps. The soft spots are modest and already flagged in the paper. The iff result requires the complete-intersection hypothesis, and the preservation result requires homogeneity; without those the statements do not hold, but the authors do not claim they do. Full proofs are not visible here, so the combinatorial details still need checking, yet nothing in the stated theorems contradicts standard toric-algebra techniques. This is a paper for people already working on Markov bases, toric ideals of graphs, and bouquet decompositions. A reader who knows the background will pick up the new criteria quickly and can test them on examples. It is narrow but the results are sharp enough that it deserves a serious referee rather than a desk reject. I would send it out for review with reviewers who know the toric-ideal literature.

Referee Report

2 major / 2 minor

Summary. The paper examines the distance-reduction property of Markov bases for toric ideals of graphs, focusing on bouquet decompositions. For complete-intersection toric ideals of graphs, it proves that minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits. Under homogeneity, it shows that distance-reduction properties are preserved for toric ideals sharing the same bouquet structure and signature. For homogeneous toric ideals whose bouquet matrix is a monomial curve in A^3, it provides necessary and sufficient conditions for minimal Markov bases to be distance-reducing.

Significance. If the claims hold, the work supplies explicit combinatorial criteria linking bouquet signatures to fiber connectivity in toric ideals, strengthening the interface between graph theory, toric algebra, and algebraic statistics. The iff result for complete intersections and the preservation theorem under homogeneity are particularly useful for classifying Markov bases without exhaustive computation.

major comments (2)

[§3, Theorem 3.4] §3, Theorem 3.4: the iff statement for complete-intersection toric ideals of graphs relies on the circuits generating the ideal; the proof sketch does not explicitly address whether the distance-reduction map on circuits extends to the full minimal Markov basis when the graph contains multiple bouquet components of the same type.
[§4, Proposition 4.7] §4, Proposition 4.7: the preservation of distance-reduction under identical bouquet structure and signature assumes homogeneity throughout; without an explicit counter-example or reduction step, it is unclear whether the argument survives when the grading is relaxed to multi-grading.

minor comments (2)

[§2] Notation for the bouquet matrix and signature is introduced in §2 but used without re-statement in later statements; a short table summarizing the invariants would improve readability.
[§5] The monomial-curve case in §5 cites an external reference for the classification of monomial curves in A^3; adding a one-sentence reminder of the relevant numerical conditions would make the necessary-and-sufficient statement self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments. We address each major comment below.

read point-by-point responses

Referee: [§3, Theorem 3.4]: the iff statement for complete-intersection toric ideals of graphs relies on the circuits generating the ideal; the proof sketch does not explicitly address whether the distance-reduction map on circuits extends to the full minimal Markov basis when the graph contains multiple bouquet components of the same type.

Authors: In the complete-intersection case for toric ideals of graphs, the circuits themselves constitute a minimal Markov basis. The distance-reduction property is therefore defined directly on this basis. When multiple bouquet components of the same type appear, the toric ideal decomposes as a direct sum of the component ideals. Fiber distances add across components, so distance reduction on each component's circuits extends to the global basis. We will insert a clarifying sentence in the proof of Theorem 3.4 to record this decomposition explicitly. revision: partial
Referee: [§4, Proposition 4.7]: the preservation of distance-reduction under identical bouquet structure and signature assumes homogeneity throughout; without an explicit counter-example or reduction step, it is unclear whether the argument survives when the grading is relaxed to multi-grading.

Authors: Proposition 4.7 is stated and proved under the standing homogeneity hypothesis; the bouquet structure and signature are defined only in that setting, and the proof relies on uniform degree control in the fibers. The manuscript makes no claim for multi-graded relaxations, which would require a separate argument. We therefore leave the statement scoped as written. revision: no

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes combinatorial theorems relating distance-reduction properties of minimal Markov bases to circuits in complete-intersection toric ideals of graphs, and preservation of these properties across homogeneous toric ideals sharing bouquet structure and signature. These results follow directly from explicit definitions of bouquets, signatures, circuits, and homogeneity conditions, without any reduction of claims to fitted parameters, self-referential definitions, or load-bearing self-citations that collapse the argument to its own inputs. The iff statements and necessary/sufficient conditions are presented as consequences of the combinatorial bouquet decomposition, remaining independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard definitions of toric ideals, Markov bases, and bouquet decompositions from prior literature; no new free parameters or invented entities are introduced.

axioms (2)

standard math Toric ideals of graphs admit Markov bases that generate the kernel of the incidence map
Invoked to define distance-reduction and circuits.
domain assumption Homogeneous toric ideals admit bouquet decompositions that capture generator combinatorics
Central to the preservation theorem.

pith-pipeline@v0.9.0 · 5492 in / 1226 out tokens · 39165 ms · 2026-05-14T17:40:32.680397+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 2.5. Let I_A be a homogeneous toric ideal... z is distance-reduced by ˆz iff ˆz+ ≤ z+ and Supp(ˆz−) ∩ Supp(z−) is non-empty

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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