Acyclonesto-Cosmohedra and Oriented Building Sets

• Acyclonesto-Cosmohedra are convex polytopes defined from acyclic oriented building sets that combine nested nestings with cosmological amplitude data.
• They feature a rich, non-simple boundary structure whose facets factorize into poset-associahedra and smaller acyclonesto-cosmohedra.
• Their construction generalizes classical associahedra by encoding partial Chan–Paton ordering and offering new insights into scattering-amplitude-like physical interpretations.

Searching arXiv for the most relevant papers on acyclonesto-cosmohedra, cosmohedra, and cosmological polytopes. Acyclonesto-cosmohedra are convex polytopes attached to an oriented building set $(S,\mathcal{B},\mathcal{C})$, where $\mathcal{B}$ is a building set on a finite ground set $S$ and $(S,\mathcal{C})$ is an acyclic realizable oriented matroid on the same ground set. They arise as truncations of acyclonestohedra and have face posets given by nested nestings: an acyclic nesting $\tau\subseteq\mathcal{B}$ together with a further nesting $\mathcal{N}$ on the Hasse diagram of $(\tau,\subseteq)$. Their canonical forms are intended to encode scattering-amplitude-like objects and generalized cosmological wavefunction coefficients with partial Chan–Paton ordering, extending associahedra, graph associahedra, nestohedra, Galashin’s poset associahedra, the cosmohedron, and graph cosmohedra (Forcey et al., 13 Jul 2025).

1. Definition and ambient combinatorial data

The foundational datum is an oriented building set $(S,\mathcal{B},\mathcal{C})$. A building set $\mathcal{B}$ on a finite set $S$ is a collection of nonempty subsets such that every singleton belongs to $\mathcal{B}$0, and whenever $\mathcal{B}$1 intersect nontrivially, the union $\mathcal{B}$2 also belongs to $\mathcal{B}$3. Its connected components are the inclusion-maximal elements $\mathcal{B}$4. The additional structure is an oriented matroid $\mathcal{B}$5, whose signed circuits encode oriented dependence data; the relevant case is acyclic and realizable, meaning that no signed circuit has all entries positive and that the matroid comes from actual vectors in a real vector space (Forcey et al., 13 Jul 2025).

A nesting $\mathcal{B}$6 of $\mathcal{B}$7 is a collection with $\mathcal{B}$8 such that any two elements are nested or disjoint, and any finite family of pairwise disjoint elements in $\mathcal{B}$9 has union not in $S$0. A nesting is acyclic if, for each $S$1, the oriented matroid obtained by restricting to $S$2 and contracting all strictly smaller nests is acyclic. The acyclonestohedron is the polytope whose faces are labeled by acyclic nestings, ordered by reverse inclusion. Facets correspond to singleton acyclic nestings $S$3 for which both $S$4 and $S$5 are acyclic, while vertices correspond to maximal acyclic nestings (Forcey et al., 13 Jul 2025).

The acyclonesto-cosmohedron is defined from this acyclonestohedral data by passing from a nesting to a nested nesting. If $S$6 is an acyclic nesting, then $S$7 is partially ordered by inclusion, its Hasse diagram $S$8 is a forest, and one considers the graph associahedral nesting data associated with the line graph $S$9. A nested nesting is a pair $(S,\mathcal{C})$0, with $(S,\mathcal{C})$1 an acyclic nesting and $(S,\mathcal{C})$2 a nesting on the building set corresponding to $(S,\mathcal{C})$3. The face poset of the acyclonesto-cosmohedron is the set of such pairs, ordered by two elementary operations: collapsing a minimal nest in $(S,\mathcal{C})$4, and discarding a non-maximal nest in $(S,\mathcal{C})$5 (Forcey et al., 13 Jul 2025).

This definition distinguishes the 2025 notion from an earlier heuristic use of the phrase. In work on cosmological polytopes, “Acyclonesto-Cosmohedra” was proposed only roughly as cosmological polytopes for acyclic graphs, viewed in analogy with graph-associahedra or nestohedra. That earlier usage concerned trees and forests inside the cosmological-polytope framework, not oriented building sets and nested nestings in the later formal sense (Juhnke-Kubitzke et al., 2023).

2. Face structure, factorization, and simplicity properties

Acyclonestohedra are simple polytopes: each vertex lies on exactly $(S,\mathcal{C})$6 facets. Their boundary structure is recursive. For a facet labeled by $(S,\mathcal{C})$7, one has the factorization

$(S,\mathcal{C})$8

where

$(S,\mathcal{C})$9

This iterates to higher-codimension faces and is the precise sense in which acyclonestohedra are described as physics-like positive geometries (Forcey et al., 13 Jul 2025).

Acyclonesto-cosmohedra inherit a richer, generally non-simple boundary structure. Their faces are labeled by nested nestings $\tau\subseteq\mathcal{B}$0, and the codimension is the number of nests in $\tau\subseteq\mathcal{B}$1, including the improper one. The factorization rule for a facet labeled by a nesting $\tau\subseteq\mathcal{B}$2 is a product of a poset associahedron $\tau\subseteq\mathcal{B}$3 with smaller acyclonesto-cosmohedra attached to the restricted-and-contracted oriented building sets associated to the members of $\tau\subseteq\mathcal{B}$4. In words, the boundary separates into a static combinatorial factor recording how channels are organized and recursive factors carrying the smaller cosmological data (Forcey et al., 13 Jul 2025).

Non-simplicity is not incidental. In dimension $\tau\subseteq\mathcal{B}$5, acyclonesto-cosmohedra are generically non-simple, and some vertices have degree strictly larger than $\tau\subseteq\mathcal{B}$6. For connected building sets, a maximal nesting has $\tau\subseteq\mathcal{B}$7 elements, the Hasse diagram is a tree, and the combinatorics of minimal nests imply that some vertices can have degree up to $\tau\subseteq\mathcal{B}$8. This places acyclonesto-cosmohedra closer to the original cosmohedron than to ordinary nestohedra or graph associahedra, which are simple (Forcey et al., 13 Jul 2025).

The same non-simple behavior is already present in the original cosmohedron. There, equalities among the shaving parameters force vertices where more than $\tau\subseteq\mathcal{B}$9 facets meet, and those equalities are essential to retaining the Russian-doll combinatorics rather than producing a fully resolved permuto-cosmohedron (Arkani-Hamed et al., 2024). Later work recast the face lattice of the cosmohedron in purely combinatorial terms: the face lattice of the $\mathcal{N}$0-dimensional cosmohedron is anti-isomorphic to the poset of Matryoshkas of an $\mathcal{N}$1-gon, ordered by containment (Ardila-Mantilla et al., 3 Mar 2026). Acyclonesto-cosmohedra generalize that type of nested boundary data from polygonal and graph-theoretic input to oriented building sets.

3. Geometric realizations and canonical forms

For realizable oriented building sets, acyclonestohedra admit an ABHY-like realization. If $\mathcal{N}$2 is represented by vectors $\mathcal{N}$3, then for each $\mathcal{N}$4 with acyclic restriction and contraction one defines affine functions $\mathcal{N}$5 with cut parameters $\mathcal{N}$6, taking $\mathcal{N}$7 when $\mathcal{N}$8 and $\mathcal{N}$9 when $(\tau,\subseteq)$0. The acyclonestohedron is then the region

$(\tau,\subseteq)$1

together with the linear relations induced by dependencies among the $(\tau,\subseteq)$2. Hierarchies such as $(\tau,\subseteq)$3 for $(\tau,\subseteq)$4 ensure convexity and the correct combinatorics (Forcey et al., 13 Jul 2025).

Its canonical form has the usual positive-geometry structure. Writing the wedge over an independent set of $(\tau,\subseteq)$5-variables, one obtains a rational differential form whose scalar part is the amplitube

$(\tau,\subseteq)$6

The poles occur only on facets corresponding to acyclic nestings, and residues factorize into products of lower-dimensional amplitubes on restricted and contracted data (Forcey et al., 13 Jul 2025).

The acyclonesto-cosmohedron is realized by introducing one variable $(\tau,\subseteq)$7 for each acyclic nesting $(\tau,\subseteq)$8: $(\tau,\subseteq)$9 where the $(S,\mathcal{B},\mathcal{C})$0-parameters are positive cuts attached to the regions $(S,\mathcal{B},\mathcal{C})$1. The polytope is then cut out by

$(S,\mathcal{B},\mathcal{C})$2

subject to the same linear relations among the $(S,\mathcal{B},\mathcal{C})$3. Appropriate hierarchies,

$(S,\mathcal{B},\mathcal{C})$4

are used to realize all combinatorial faces (Forcey et al., 13 Jul 2025).

The corresponding canonical form is most conveniently organized through variables $(S,\mathcal{B},\mathcal{C})$5 attached to the nests $(S,\mathcal{B},\mathcal{C})$6 in each nested nesting $(S,\mathcal{B},\mathcal{C})$7. The resulting scalar object is the cosmological amplitube, defined algebraically as a sum over vertices of products of $(S,\mathcal{B},\mathcal{C})$8. Its residues on boundaries corresponding to a nesting $(S,\mathcal{B},\mathcal{C})$9 factorize into a poset-associahedron contribution times a product of smaller cosmological amplitubes, which is the recursive cosmological analogue of locality and unitarity-like factorization (Forcey et al., 13 Jul 2025).

4. Examples and special families

The Stasheff associahedron appears when the oriented matroid is trivial and the building set is the interval building set of a totally ordered set. For the path-graph model with covers $\mathcal{B}$0,

$\mathcal{B}$1

the acyclonestohedron reduces to the usual graph associahedron for the path, hence the associahedron. At five points, the resulting amplitube is the standard five-point biadjoint amplitude in one color ordering, and the corresponding three-dimensional cosmohedral truncation reproduces the known $\mathcal{B}$2 wavefunction coefficient (Forcey et al., 13 Jul 2025).

At the opposite extreme, the trivial oriented matroid with building set $\mathcal{B}$3 yields a simplex, while the claw poset with all nonempty subsets in the building set yields the permutohedron. In the latter case, the amplitube is

$\mathcal{B}$4

and the associated acyclonesto-cosmohedron is described as a permutoassociahedron-like object whose facets are associahedra (Forcey et al., 13 Jul 2025).

The first genuinely oriented example in the 2025 framework is the diamond poset. Here the ground set is $\mathcal{B}$5, the oriented matroid is

$\mathcal{B}$6

and the building set is

$\mathcal{B}$7

There are 13 acyclic nestings, the acyclonestohedron is a hexagon, and the acyclonesto-cosmohedron is a dodecagon. The amplitube has six channels, while certain potential poles, such as $\mathcal{B}$8, are excluded by acyclicity; the cosmological amplitube has 12 terms corresponding to nested nestings (Forcey et al., 13 Jul 2025).

Further examples reinforce the same pattern. The bowtie poset yields an acyclonestohedron that is an octagon and a cosmohedron that is a 16-gon. The $\mathcal{B}$9 poset gives a three-dimensional acyclonestohedron with three octagonal facets and a cosmohedron with three 16-gon facets; every maximal nesting is totally nested, so facets of the cosmohedron are pentagons, i.e. two-dimensional associahedra (Forcey et al., 13 Jul 2025). These examples show that the oriented matroid data acts as a filter on the faces of an underlying nestohedral or graph-associahedral structure rather than merely decorating an unchanged polytope.

5. Relation to cosmohedra, graph cosmohedra, and acyclic cosmological polytopes

The term sits at the intersection of several polytope families. The following summary organizes the lineage.

Family	Input data	Face labels
Associahedron	Path graph / total order	Tubings or partial triangulations
Graph associahedron	Graph $S$0	Tubings of $S$1
Acyclonestohedron	Oriented building set $S$2	Acyclic nestings
Cosmohedron	Polygonal / associahedral data	Russian dolls or Matryoshkas
Graph cosmohedron	Graph $S$3	Regional tubings
Acyclonesto-cosmohedron	Oriented building set $S$4	Nested nestings

The original cosmohedron was introduced as a polytope underlying the cosmological wavefunction for $S$5 theory. It can be obtained from the associahedron by blowing up faces and shaving with inequalities indexed by partial triangulations, and its faces are labeled by Russian-doll configurations of subpolygons (Arkani-Hamed et al., 2024). The later combinatorial treatment proved that these faces are exactly Matryoshkas and described the cosmohedron as an $S$6 in $S$7 polytope obtained by chiseling bracket-associahedra at the vertices of an associahedron (Ardila-Mantilla et al., 3 Mar 2026).

Graph cosmohedra generalize this from paths to arbitrary graphs by replacing polygonal substructures with regions and regional tubings. They can be obtained by consistently blowing up all boundaries of the corresponding graph associahedron to codimension one, and they come with cosmological amplitubes defined as sums over vertices of products of region variables (Glew et al., 24 Feb 2025). This provides the immediate geometric background for acyclonesto-cosmohedra: the 2025 paper argues that acyclonesto-cosmohedra should be regarded as oriented-building-set analogues of graph cosmohedra rather than as direct descendants only of the polygonal cosmohedron (Forcey et al., 13 Jul 2025).

A distinct but related precursor came from the study of cosmological polytopes of graphs. In that setting, one attaches to any connected undirected graph $S$8 a polytope $S$9 whose canonical form computes contributions to cosmological wavefunctions. Earlier work suggested an “Acyclonesto-Cosmohedron” program for the acyclic case, meaning cosmological polytopes of trees or forests viewed in analogy with graph-associahedra and nestohedra. That work established that every cosmological polytope admits a regular unimodular triangulation via a Gröbner basis with squarefree initial ideal, gave explicit facet characterizations for paths and trees, and computed

$\mathcal{B}$00

for paths and cycles (Juhnke-Kubitzke et al., 2023).

The later Ehrhart-theoretic analysis completed that story. For any cosmological polytope $\mathcal{B}$01, the $\mathcal{B}$02-polynomial is a specialization of the Tutte polynomial of the defining graph; for a simple tree with $\mathcal{B}$03 edges,

$\mathcal{B}$04

and in general

$\mathcal{B}$05

so the volume is $\mathcal{B}$06 times the number of acyclic edge subsets (Benjes et al., 17 Mar 2025). These results remain highly relevant as an acyclic graph-theoretic parallel, but they do not define the later acyclonesto-cosmohedron itself. A common misconception is therefore that acyclonesto-cosmohedra are simply “cosmological polytopes of trees.” In the strict 2025 usage, they are truncations of acyclonestohedra built from acyclic realizable oriented matroids on building sets (Forcey et al., 13 Jul 2025).

6. Physical interpretation, sections of graph cosmohedra, and outlook

The physical motivation is the extension of color-ordering and cosmological wavefunction geometry beyond the totally ordered setting of the open string and the path associahedron. In the acyclonestohedral framework, Chan–Paton-like data is only partially ordered. The building set $\mathcal{B}$07 encodes which subsets can interact, while the oriented matroid $\mathcal{B}$08 encodes oriented incidence constraints; acyclicity excludes nestings that would force a directed cycle. The resulting amplitube $\mathcal{B}$09 is a scattering-amplitude-like object with poles on allowed channels $\mathcal{B}$10, and the cosmological amplitube of the acyclonesto-cosmohedron is the corresponding wavefunction-like object with additional nested-time-ordering data carried by nested nestings (Forcey et al., 13 Jul 2025).

This interpretation places acyclonesto-cosmohedra inside the broader positive-geometry program. They extend the hierarchy

$\mathcal{B}$11

to

$\mathcal{B}$12

Correlator geometries sharpen the same picture from another direction: the correlatron is a one-higher-dimensional polytope sandwiched between cosmohedron and associahedron facets, and graph correlahedra encode graph-by-graph correlator contributions without the power-of-two weights present in earlier formulations (Figueiredo et al., 24 Jun 2025). A plausible implication is that acyclonesto-cosmohedra should admit analogous correlator refinements once the oriented-building-set version of the graph-correlahedral story is formulated.

A central conjectural relation concerns sections of graph cosmohedra. It was already known that acyclonestohedra can be realized as linear sections of graph associahedra. The 2025 work provides evidence that the same holds for their cosmological truncations: for any poset, and plausibly for all acyclonesto-cosmohedra, the polytope should arise as a section of the graph cosmohedron of the line graph of the Hasse diagram. The strongest explicit evidence comes from low-dimensional examples: the diamond and bowtie posets both have Hasse diagrams whose line graph is the 4-cycle $\mathcal{B}$13, and their cosmohedra, a 12-gon and a 16-gon, appear as different linear sections of the graph cosmohedron for $\mathcal{B}$14 (Forcey et al., 13 Jul 2025).

The current outlook is therefore twofold. On the combinatorial side, acyclonesto-cosmohedra supply a common framework for nested-set, graph-associahedral, and oriented-matroid constraints. On the physical side, they furnish positive geometries whose boundaries recursively factor into the same class, with poset-associahedral factors controlling channel organization and smaller acyclonesto-cosmohedra controlling generalized cosmological dynamics (Forcey et al., 13 Jul 2025). What remains open is a fully systematic theory of their sections, their relation to graph cosmohedra and correlatrons beyond the poset case, and a more direct worldsheet or string-theoretic interpretation of the partial-order and oriented-matroid data.