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Ternary relations and their polytopes

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Pith's one-line read The paper establishes that cosmological polytopes, introduced for computing the wavefunction of the Universe, are a special case of Graev's ternary-relation polytopes, and uses this to transfer extreme-metric techniques to root-system polyt

desk verdict A clean unification of Graev and cosmological polytopes with a genuinely useful marking/cocycle theorem; the new D_n facet family rests on a proof gap that should be fixed. read the letter →

arxiv 2509.06811 v1 pith:PTOGQJBC submitted 2025-09-08 math.CO math.DG

Aleksei Lavrov This is my paper · ORCID
classification math.COmath.DG MSC 52B1205E4553C25
keywords ternaryrelationspolytopescosmologicalGraevrootsystemsextrememetricsmetricconesimplicialposets
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that the cosmological polytope construction, invented for computing the wavefunction of the Universe, is a special case of a polytope construction due to Graev that arose in the study of left-invariant Einstein metrics. The bridge is an isomorphism of ternary relations: for every graph G, the ternary relation built from the cone over G is isomorphic to the ternary relation built from G itself, so the two polytopes coincide. The author then inserts 2-dimensional simplicial posets between graphs and root systems, showing that the Graev polytopes for A_{n-1}, D_n, and B_n are ternary polytopes of such posets. Using the marking/cutset technique and a generalization of Avis' extreme-metric criterion, the paper produces new families of extreme metrics and hence facets for these polytopes. If correct, the two theories—one from cosmology, one from homogeneous-space geometry—are one combinatorial subject.

What carries the argument

The central object is the symmetric ternary relation T=(Σ,R) and its ternary polytope P(T)=Conv(e_i+e_j−e_k, e_i−e_j+e_k, −e_i+e_j+e_k : [i,j,k]∈R). The load-bearing identity is Lemma 2.2: for any graph G seen as a 1-dimensional simplicial poset, T(C_G)≃T(G), which forces P(C_G)=P(G). The cone C_G adds an apex vertex; its edges encode exactly the original edges and vertices of G, and its triangles encode the relation triples [v_1,e,v_2]. For root systems, the 2-dimensional simplicial posets K_n (doubled complete graph) and (C K_n)^(2) play the analogous role. The facet machinery in Section 4 is carried by 'bypassing subgraphs' and '1-ic-colorable' graphs: a subgraph whose every missing edge

What would settle it

Compute the ternary relation T(C_G) for a single-edge graph G and compare its polytope with the triangle P(G); if the vertex sets or facet counts differ, Lemma 2.2 and the unification fail. To target Theorem 4.10 directly, take n=5 and a Hamiltonian graph H containing the diagonal chord e_i^{s^2(i)} used in the proof, then check whether the 4-cycle C_i is isometric in G; the proof's 'similar argument' is exactly the point that would break.

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Extended reading notes

Core claim

The paper claims that the cosmological polytope of a graph G is the ternary polytope P(T(G)) from Definition 1.1. Lemma 2.2 proves the stronger isomorphism T(C_G)≃T(G), where C_G is the cone over G viewed as a 1-dimensional simplicial poset, so P(C_G)=P(G); this makes the cosmological construction a special case of Graev's. The same dictionary gives P(A_{n−1})=P(K_n), P(D_n)=P(K_n), and P(B_n)=P((C K_n)^(2)). On facets, Theorem 3.2 identifies minimal one-corner markings with minimal 1-cocycles, yielding a cutset description when H^1(P,Z2)=0; Theorem 4.9 shows a bypassing, 1-ic-colorable subgraph induces an extreme graph metric; Theorem 4.10 builds such subgraphs in K_n for n≥5 with n−1 not d

Load-bearing premise

The load-bearing premise of the new D_n facet family is that certain walks in the doubled complete graph always contract to straight edges and that the specific 4-cycles C_i are isometric; the second assertion is only sketched and stops applying if the chosen Hamiltonian graph already contains the diagonal edge used in the argument.

Editorial extensions

If this is right

  • The known facet theorem for cosmological polytopes (facets correspond to connected subgraphs of G) transfers to the corresponding Graev polytope, giving a direct geometric interpretation of cut-metric facets for type A.
  • Because P(A_{n−1})^∨ is the metric cone M_n, the marking/cutset description of Corollary 3.3 reconstructs the classic cut-metric facets of M_n and isolates where higher A_n needs new extreme rays.
  • Theorem 4.9 gives a general sufficient condition—bypassing plus 1-ic-colorable—for a graph metric on an arbitrary 2-dimensional simplicial poset to be extreme, extending Avis' criterion beyond the complete graph case.
  • Theorem 4.10 supplies, for every n≥5 with n−1 not divisible by 3, a family of spanning subgraphs of the doubled complete graph K_n whose induced graph metrics are extreme; under Lemma 2.4 these are facets of P(D_n).
  • The cone identifications put the B_n Graev polytope into the same simplicial-poset framework, so the same extreme-metric machinery is available for it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: since the isomorphism T(C_G)≃T(G) holds at the level of ternary relations, constructions on cosmological polytopes that depend only on the relation—triangulations, Ehrhart data, facet complexes—should transfer verbatim to the corresponding Graev polytopes.
  • Extension: applying Theorem 4.9 to the B_n simplicial poset (C K_n)^(2) is the natural next step; the paper sets up the definitions but does not carry out the search for bypassing 1-ic-colorable subgraphs there.
  • Extension: the number-theoretic condition 'n−1 not divisible by 3' in Theorem 4.10 comes from iterating a 3-step shift around the Hamiltonian cycle; other cycle structures or auxiliary edges might produce analogous facet families when n−1 is divisible by 3.
  • Extension: the metric-on-poset definition is new; testing it on other finite configurations, such as other root systems, could yield new extreme metrics on M_n for n=8, where the complete classification is not known.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a common framework for three a priori unrelated constructions: Graev's ternary-relation polytopes (which bound the number of Einstein metrics on homogeneous spaces), cosmological polytopes of Arkani-Hamed, Benincasa and Postnikov, and the metric cone of finite metric spaces. Section 2 introduces ternary relations attached to 2-dimensional simplicial posets and proves exact isomorphisms showing that any cosmological polytope is a ternary polytope (Lemma 2.2), that the Graev polytope P(A_{n-1}) is P(K_n) (Lemma 2.3), and that D_n and B_n polytopes are represented by the simplicial posets K_n and (C K_n)^{(2)} (Lemmas 2.4 and 2.5). Section 3 extends the marking technique from cosmological polytopes to simplicial posets, proving a bijection between one-corner markings and Z_2 1-cocycles (Theorem 3.2). Section 4 defines metrics on simplicial posets, generalizes Avis's criterion for extremality (Theorem 4.9), and claims an infinite family of facets for the D_n polytope for n ≥ 5 with 3 ∤ (n−1) (Theorem 4.10).

Significance. If the results are correct, the paper gives a clean conceptual unification: the cosmological polytope construction is literally a special case of Graev's construction, and the metric cone is also recovered as the dual of a Graev polytope. The cocycle reformulation in Theorem 3.2 is elegant and is proved from first definitions. The explicit isomorphisms in Section 2 are exact and are the strongest parts of the paper. The proposed D_n facet family in Theorem 4.10 would be a genuinely new contribution to the metric-cone literature, but the proof as written relies on unproved contraction assertions; until those are supplied, this part of the paper should be viewed as conditional.

major comments (2)
  1. [Section 4.2, proof of Theorem 4.10] The assertion 'One can show that any walk in K^+_n connecting v_i and v_j which contains odd number of edges can be contracted to the edge e_j^i along K_n' is not proved and, as stated, is false. For example, in K_4 the odd walk e_1^2, e_2^1, e_1^3 (a backtrack along the same positive edge) cannot be contracted, since no triangle contains a repeated edge. This matters because the bypassing argument for edges in E(K^+_{n-1})\E(H) uses exactly this statement. The lemma that is actually needed is the simple-path version: a simple path in K^+_n of odd length from v_i to v_j contracts to e_i^j. This is plausible and can be proved by induction on length using triangles of the form \Delta_b^{ac} and \Delta_d^{ac}, but it must be stated and proved explicitly. As written, Theorem 4.10 is not established.
  2. [Section 4.2, proof of Theorem 4.10] The isometricity of the 4-cycles C_i is asserted with 'One can show', and the analogous claim for the cycles used for E(H)\E(C) is asserted with 'the similar argument holds'. These are load-bearing for the 1-ic-colorability of G. The arguments are likely repairable: for the pair (v_{s(i)}, v_n) one should use the ordinary edge e_{s(i),n}, which is not in E(G), and exhibit the two length-2 contractable walks in G. Please provide these details explicitly, including the corresponding B_G witnesses, rather than leaving them as unchecked assertions.
minor comments (5)
  1. [Section 4.2, proof of Theorem 4.7] The last line says 'd'(e_1)+\cdots+d'(e_n)=d'(e_n)'; this should read '=d'(e)'.
  2. [Section 4.2, proof of Theorem 4.9] The proof assumes there is a multiplier \lambda\in R_{>0} with d'=\lambda d_G on E(G). If d' vanishes on E(G), one must allow \lambda=0. This is harmless (then induction gives d'=0), but it should be stated so the proof is formally correct.
  3. [Section 2.2, after Definition 2.1] The notation K_n for the boolean 2-skeleton and K_n for its extension is hard to distinguish in plain text. A different letter or font (e.g. K'_n) would improve readability.
  4. [Section 3, before Lemma 3.1] The statement 'one can show that markings from Mmin_f(P) are in one-to-one correspondence with extreme rays of P(P)^∨' is standard but is not proved or referenced; a short justification would be helpful.
  5. [Throughout] There are several typographical errors: 'inlcusion', 'simlicial', 'partial case' for 'special case', and the reference title 'metic cone' in [5]. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main equivalences are proved directly from the definitions, and there is no fitted input, self-citation chain, or prediction that reduces to its own premise.

full rationale

The paper's central claims are explicit isomorphisms of ternary relations (Lemmas 2.2–2.5), each proved by direct bijections between the relevant vertex/edge/triangle sets. For example, Lemma 2.2 constructs the bijection Σ_{C_G} → Σ_G and checks that triangles of C_G correspond exactly to triples of T(G); the equality P(C_G)=P(G) then follows from the functoriality of the polytope construction in Definition 1.1, not from assuming the conclusion. The identification of cosmological polytopes as Graev polytopes is established by first defining T(G) and then exhibiting T(C_G)≃T(G), so it is a genuine structural result rather than a renaming. There is no parameter fitting anywhere, and no quantity called a prediction is derived from the data that would force it. The metric-cone duality P(A_{n-1})^∨≃M_n is cited as known background and is not used to prove the main new facet theorems; Theorem 4.9 gives a self-contained proof of the extremality criterion, with Avis’s result mentioned only as a comparison after the fact. Theorem 4.10 does contain assertions introduced by “one can show” and “the similar argument holds,” and those are unproved contraction claims that constitute a proof gap or correctness risk, not circularity: they do not assume the conclusion of the theorem. There is also no load-bearing self-citation chain, and no uniqueness theorem is imported from the author’s own prior work. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

No free parameters are fitted. The paper introduces new combinatorial objects (K_n' and (C K_n')^{(2)}) and a generalized metric notion; these are defined, not derived, and serve as the scaffolding for the facet theorems. The main proofs rest on standard convex duality and Z2 cohomology facts, plus an unproven geometric assumption about walk contractions in Section 4.

assumptions (3)
  • standard math Facets of a convex polytope whose affine hull does not contain the origin correspond to extreme rays of its dual cone; minimal feasible markings correspond to extreme rays of P(P)^∨.
    Used in Section 3 as the bridge from markings to facets. This is standard convex duality, stated but not proven.
  • standard math The Z2 chain and cochain groups of a 2D simplicial poset satisfy the usual boundary/coboundary identities, including ker(δ^1) ≅ im(δ^0) when H^1(P, Z2) = 0.
    Invoked in Theorem 3.2 and Corollary 3.3 to identify minimal markings with cutsets.
  • domain assumption The contraction geometry of walks in a 2D simplicial poset is well-defined: an elementary contraction replaces two consecutive edges of a triangle by the third, and sequences of such contractions preserve endpoints and behave like geodesic shortcuts.
    Section 4.2's proofs of Theorems 4.7 and 4.9 depend on this geometric interpretation. The paper does not prove confluence or that shortest contractable walks have the path-length property used in the induction.
invented entities (1)
  • Simplicial posets K_n' and (C K_n')^{(2)}
    purpose: To encode the root systems D_n and B_n as 2D simplicial posets so that their ternary polytopes coincide with Graev polytopes P(D_n) and P(B_n) (Lemmas 2.4 and 2.5).
    These are new combinatorial objects constructed by the author. They are proven internally to satisfy the poset axioms and give the desired ternary relations, but they have no external falsifiable handle beyond the paper's own isomorphisms.

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Pith. "Pith review of Ternary relations and their polytopes." pith.science (2026). https://pith.science/paper/PTOGQJBC

@misc{pith2026250906811,
  author       = {Pith},
  title        = {Pith review of: Ternary relations and their polytopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTOGQJBC}},
  note         = {Machine review of arXiv:2509.06811}
}
read the original abstract

Graev introduced the construction of a convex polytope associated with a symmetric ternary relation. He showed that the number of left-invariant Einstein metrics on a homogeneous space under some conditions is no more than the normalized volume of certain polytope of such form. It happens that the construction of a cosmological polytope introduced by Arkani-Hamed, Benincasa and Postnikov for computation of the wave function of the Universe is the special case of the Graev construction. The paper is devoted to unification of these two theories from combinatorial perspective.

Figures

Figures reproduced from arXiv: 2509.06811 by the authors.

Figure 1
Figure 1. Examples of ternary polytopes: 𝑃(𝐴2), 𝑃(𝐵2) and 𝑃(𝐴3) (from left to right). pair (ΣΩ, 𝑅Ω) such that ΣΩ = {1, ..., 𝑛} is the set of indices of Ω, while 𝑅Ω contains all triples [𝑖, 𝑗, 𝑘] for which the equality ±𝛼𝑖 ± 𝛼𝑗 ± 𝛼𝑘 = 0 holds for some distribution of signs. Note that considering different enumerations of vectors of Ω we obtain isomorphic ternary relations so we do not distinguish them. Various combinatorial pr… view at source ↗
Figure 2
Figure 2. Cones over some graphs. Note that the set of edges of the cone is the union of edges and vertices of the initial graph. The apex of each cone is denoted by 𝑣0. as 3-element sets Δ𝑖 𝑗𝑘 := {𝑖, 𝑗, 𝑘} for any 1 ≤ 𝑖 ≠ 𝑗 ≠ 𝑘 ≤ 𝑛. Note that the notations 𝑒𝑖 𝑗 and Δ𝑖 𝑗𝑘 are invariant under permutations of indicies. The triples in the ternary relation T (K𝑛) have the form [𝑒𝑖 𝑗, 𝑒 𝑗𝑘, 𝑒𝑖𝑘] for 1 ≤ 𝑖 < 𝑗 < 𝑘 ≤ 𝑛. On the other… view at source ↗
Figure 3
Figure 3. Possible locations of marks in the triangle. Note that these three markings are the only minimal feasible markings of this poset. the other hand, the maximal marking containing all corners of P corresponds to the interior set of the cone 𝑃(P)∨ . There are markings 𝑀 for which the subsets 𝑃(P)∨ 𝑀 are empty. We will call a marking 𝑀 feasible if it defines non-empty subset 𝑃(P)∨ 𝑀 and infeasible, otherwise. One can eas… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Not locally feasible markings [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: All minimal feasible markings of the cone over the line graph with 2 subsequent edges. The corresponding connected subgraphs of the initial graph are highlighted by dashed line according to the theory of cosmological polytopes. the cosmological case, i. e. for simplici…
Figure 6
Figure 6. Figure 6: Cut-metrics on three points. of 𝐾𝑛 satisfying the triangle inequalities. For any partition 𝑉(𝐾𝑛) = 𝑆 ⊔ 𝑆 we can con￾sider the corresponding cutset 𝐻({𝑆, 𝑆) ⊂ 𝐸(𝐾𝑛) as it was discussed in the previous section. This cut-set defines the metric 𝑑{𝑆,𝑆} which is equal to 1 o…
Figure 7
Figure 7. Figure 7: The extreme metric on 5 points associated with the subgraph 𝐾3,2 of 𝐾5. The subgraph 𝐾3,2 is highlighted by solid line. if it has no repetitions in vertices. A cycle is a path with the first and the last vertices coincided, i. e. 𝑣1 = 𝑣𝑛+1. For any two walks 𝑝 = 𝑒1..𝑒𝑛…
Figure 8
Figure 8. Figure 8: The graph metric on the cone over 3-cycle. This poset contains only 3 triangles. The subgraph of the 1-skeleton which induces the metric is highlighted by solid line. Note that the bottom edge has the contractable bypassing walk with repetitions. Also note that this me…

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