Good Triangulations of Cosmological Polytopes

Aenne Benjes; Benjamin Schr\"oter; Kamillo Ferry

arxiv: 2503.13393 · v1 · submitted 2025-03-17 · 🧮 math.CO

Good Triangulations of Cosmological Polytopes

Aenne Benjes , Kamillo Ferry , Benjamin Schr\"oter This is my paper

Pith reviewed 2026-05-22 23:27 UTC · model grok-4.3

classification 🧮 math.CO

keywords cosmological polytopesgood triangulationsh-star polynomialTutte polynomialhalf-open decompositionsEhrhart theoryunimodular triangulations

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The pith

The h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph.

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

The paper establishes that good triangulations of cosmological polytopes can be turned into half-open decompositions. These decompositions show that the h*-polynomial equals a specialization of the Tutte polynomial of the graph. This link allows explicit enumeration of simplices and resolves conjectures from prior work on their Ehrhart theory. Sympathetic readers care because it connects the geometry of the polytope to standard graph polynomials, giving a new way to compute and understand these invariants.

Core claim

We enumerate all maximal simplices in good triangulations of any cosmological polytope. Furthermore, we provide a method to turn such a triangulation into a half-open decomposition from which we deduce that the h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph. This settles several open questions and conjectures.

What carries the argument

The half-open decomposition constructed from a good triangulation of the cosmological polytope, which equates the h*-polynomial to a Tutte specialization by matching lattice-point enumerators.

If this is right

The h*-polynomials can now be computed combinatorially from the graph for any cosmological polytope.
All maximal simplices in good triangulations are explicitly described.
Prior conjectures on face structure and Ehrhart polynomials are resolved.

Where Pith is reading between the lines

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

The result suggests that other polytope invariants might admit similar graph-theoretic expressions.
Techniques could generalize to polytopes associated with different classes of graphs or diagrams.
It provides a bridge between polytope geometry and matroid theory via the Tutte polynomial.

Load-bearing premise

The half-open decomposition constructed from a good triangulation preserves the lattice-point counting data needed to equate the two polynomials.

What would settle it

Directly count the lattice points in a cosmological polytope for a small graph such as a tree or cycle to obtain its h*-polynomial and check whether it matches the predicted specialization of the Tutte polynomial; disagreement would disprove the claim.

read the original abstract

Cosmological polytopes of graphs are a geometric tool in physics to study wavefunctions for cosmological models whose Feynman diagram is given by the graph. After their recent introduction by Arkani-Hamed, Benincasa and Postnikov the focus of interest shifted towards their mathematical properties, e.g., their face structure and triangulations. Juhnke, Solus and Venturello used toric geometry to show that these polytopes have a so-called good triangulation that is unimodular. Based on these results Bruckamp et al. studied the Ehrhart theory of those polytopes and in particular the h*-polynomials of cosmological polytopes of multitrees and multicycles. In this article we complete this part of the story. We enumerate all maximal simplices in good triangulations of any cosmological polytope. Furthermore, we provide a method to turn such a triangulation into a half-open decomposition from which we deduce that the h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph. This settles several open questions and conjectures of Juhnke, Solus and Venturello as well as Bruckamp et al.

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 lists every maximal simplex in good triangulations of cosmological polytopes and derives the h*-polynomial as a Tutte specialization through an explicit half-open decomposition.

read the letter

The new content is the full enumeration of maximal simplices across any cosmological polytope together with the construction that turns a good triangulation into a half-open decomposition. From that decomposition they extract the h*-polynomial and show it equals a particular evaluation of the Tutte polynomial of the graph. This directly answers the open questions left in Juhnke-Solus-Venturello and in Bruckamp et al. on the form of these polynomials. The enumeration is concrete rather than existential, which is the part that makes the later identification usable. The link to the Tutte polynomial is the payoff, since it imports existing deletion-contraction tools and interpretations. The argument is not circular; the decomposition is built from the triangulation and the counting identity is derived rather than fitted. The soft spot is the verification that the half-open regions count exactly the same lattice points as the original polytope once boundaries are adjusted. The paper supplies the explicit construction, so the question reduces to whether those boundary rules are stated and checked without leftover or missing points. If the details hold, the identification is fine. A reader working on Ehrhart theory of graph polytopes or on explicit h*-formulas for these families will get direct use from the enumeration and the closed-form link. It is narrow but finishes a specific line of questions that earlier papers posed. I would send it to referees; the claims are checkable and the connection they make is the one the prior literature asked for.

Referee Report

2 major / 2 minor

Summary. The paper enumerates all maximal simplices appearing in the good (unimodular) triangulations of cosmological polytopes associated to arbitrary graphs. It then supplies an explicit construction that converts any such triangulation into a half-open decomposition of the polytope and uses this decomposition to identify the h*-polynomial of the cosmological polytope with a specialization of the Tutte polynomial of the underlying graph, thereby settling several conjectures of Juhnke–Solus–Venturello and Bruckamp et al.

Significance. If the identification holds, the result supplies a concrete combinatorial bridge between the Ehrhart theory of cosmological polytopes and classical graph polynomials, yielding both a closed-form expression for the h*-polynomial and a new source of positivity and unimodality statements. The explicit enumeration of the simplices is itself a useful contribution that makes the good triangulations computationally accessible.

major comments (2)

[§4] §4 (construction of the half-open decomposition): the argument that the half-open simplices cover precisely the same lattice points as the original polytope (with only boundary adjustments) is invoked to equate the two generating functions, yet the verification that interior lattice points are neither added nor lost under the half-open operation is not carried out in sufficient detail to confirm that the Ehrhart series is preserved. This step is load-bearing for the central h*-Tutte claim.
[Theorem 5.3] Theorem 5.3 (main identification): the specialization of the Tutte polynomial is stated to equal the h*-polynomial, but the proof relies on the lattice-point preservation established in the preceding decomposition; without an independent check (e.g., via direct computation on a small multigraph where both sides can be enumerated by hand), the equality remains conditional on the decomposition step.

minor comments (2)

[§4] Notation for the half-open simplices is introduced without a displayed definition of the precise boundary-removal rule; a short displayed equation would improve readability.
[§3] The enumeration of maximal simplices is given in §3, but the correspondence between these simplices and the edges/vertices of the graph is not tabulated for a running example; adding such a table would make the construction easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. Below we respond point by point to the major comments and indicate the revisions we will make.

read point-by-point responses

Referee: [§4] §4 (construction of the half-open decomposition): the argument that the half-open simplices cover precisely the same lattice points as the original polytope (with only boundary adjustments) is invoked to equate the two generating functions, yet the verification that interior lattice points are neither added nor lost under the half-open operation is not carried out in sufficient detail to confirm that the Ehrhart series is preserved. This step is load-bearing for the central h*-Tutte claim.

Authors: We agree that the lattice-point preservation argument in §4 would benefit from greater explicitness. In the revised version we will expand the relevant paragraphs to include a direct verification that every interior lattice point of the cosmological polytope lies in precisely one half-open simplex and that no new interior points are introduced. The argument will rely on the unimodularity of the good triangulation together with the explicit description of the simplices given in §3; we will also record the boundary adjustments separately so that the equality of Ehrhart series follows immediately. revision: yes
Referee: [Theorem 5.3] Theorem 5.3 (main identification): the specialization of the Tutte polynomial is stated to equal the h*-polynomial, but the proof relies on the lattice-point preservation established in the preceding decomposition; without an independent check (e.g., via direct computation on a small multigraph where both sides can be enumerated by hand), the equality remains conditional on the decomposition step.

Authors: The proof of Theorem 5.3 proceeds through the half-open decomposition, so the referee’s observation is accurate. To supply an independent check we will add a short computational appendix containing explicit enumerations for two small cases (the single-edge multigraph and the 3-cycle multigraph). For each case we will list all lattice points of the cosmological polytope, compute the h*-polynomial directly, enumerate the maximal simplices, apply the specialization of the Tutte polynomial, and verify that the two polynomials coincide. This concrete verification will be independent of the general decomposition argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; independent enumeration and decomposition yield the h*-Tutte equivalence

full rationale

The paper relies on external prior results (Juhnke-Solus-Venturello for existence of good unimodular triangulations; Bruckamp et al. for partial Ehrhart data) but supplies its own enumeration of all maximal simplices and a new construction turning the triangulation into a half-open decomposition. The h*-polynomial equivalence to a Tutte specialization is then deduced directly from the lattice-point generating function of that decomposition. No equation or step reduces the claimed equality to a fitted parameter, self-defined quantity, or load-bearing self-citation; the central derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

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discussion (0)

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IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

We provide a method to turn such a triangulation into a half-open decomposition from which we deduce that the h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

the h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial

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Forward citations

Cited by 1 Pith paper

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