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arxiv: 2606.02513 · v1 · pith:D3MLM4VBnew · submitted 2026-06-01 · 🧮 math.CO

Neighborly Murai spheres and the Simplicial Steinitz Problem

Ivan Limonchenko , Ale\v{s} Vavpeti\v{c} This is my paper

Pith reviewed 2026-06-28 13:42 UTC · model grok-4.3

classification 🧮 math.CO
keywords neighborly spheresMurai spherespolytopal spheressimplicial Steinitz problemcombinatorial equivalenced-spheresconvex polytopes
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The pith

Classification proves all neighborly Murai spheres are polytopal.

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

The paper classifies all neighborly Murai spheres across dimensions. This classification shows each one bounds a convex polytope. It further establishes that every neighborly d-sphere on at most d+4 vertices is combinatorially equivalent to a Murai sphere. A reader would care because the result settles geometric realizability for these spheres and advances understanding of which abstract spheres arise as polytope boundaries.

Core claim

We provide a classification of neighborly Murai spheres, which implies that all of them are polytopal. Furthermore, we show that each neighborly d-sphere with no more than d+4 vertices is combinatorially equivalent to a Murai sphere for any d≥1.

What carries the argument

The exhaustive combinatorial classification of neighborly Murai spheres together with the equivalence relation that transfers polytopality.

If this is right

  • Every neighborly Murai sphere bounds a convex polytope.
  • Every neighborly d-sphere with at most d+4 vertices is polytopal.
  • The simplicial Steinitz problem has an affirmative answer for all neighborly spheres on small vertex sets.

Where Pith is reading between the lines

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

  • The same classification approach might extend to neighborly spheres on d+5 or more vertices.
  • Explicit geometric constructions for the classified cases could now be extracted from the equivalence data.
  • The result suggests testing whether non-neighborly spheres with few vertices also reduce to known polytopal families.

Load-bearing premise

The classification of neighborly Murai spheres is complete and combinatorial equivalence is enough to transfer polytopality from known examples to all others.

What would settle it

A neighborly d-sphere with d+4 vertices that is not combinatorially equivalent to any Murai sphere, or a Murai sphere from the classification that cannot be realized as a polytope.

Figures

Figures reproduced from arXiv: 2606.02513 by Ale\v{s} Vavpeti\v{c}, Ivan Limonchenko.

Figure 1
Figure 1. Figure 1: Murai polytopes in dimension 3 that are not isomorphic to Bier polytopes. Remark. The previous example shows that any join of two boundaries of simplices is a Murai sphere. On the other hand, due to the result of [24], a join of two boundaries of simplices is a Bier sphere if and only if at least one of those simplices has dimension ≤ 1. Further examples are formed by the next classification of low-dimensi… view at source ↗
read the original abstract

We provide a classification of neighborly Murai spheres, which implies that all of them are polytopal. Furthermore, we show that each neighborly $d$-sphere with no more than $d+4$ vertices is combinatorially equivalent to a Murai sphere for any $d\geq 1$.

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

0 major / 2 minor

Summary. The manuscript classifies all neighborly Murai spheres and proves they are polytopal; it further shows that every neighborly d-sphere on at most d+4 vertices is combinatorially equivalent to a Murai sphere (d ≥ 1).

Significance. If the classification is exhaustive and combinatorial equivalence correctly transfers polytopality, the result supplies an explicit positive answer to the simplicial Steinitz question for all neighborly spheres with small vertex count, reducing the problem to a finite list of known polytopal examples.

minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly state the vertex range and dimension restrictions under which the Murai-sphere classification is performed.
  2. [Introduction] Notation for the combinatorial equivalence relation and the precise definition of 'Murai sphere' should be introduced before the main theorems are stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central result is an explicit classification of neighborly Murai spheres together with a combinatorial-equivalence argument showing that all neighborly d-spheres on at most d+4 vertices are Murai. No equations, fitted parameters, or predictions appear in the abstract or described claims that reduce by construction to the inputs. The derivation consists of case-by-case enumeration and transfer of polytopality via combinatorial equivalence, which is independent of the result itself. No self-citation is load-bearing for the classification step, and no ansatz or uniqueness theorem is invoked in a self-referential manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities beyond standard domain terminology.

axioms (1)
  • domain assumption Standard definitions of simplicial spheres, the neighborly property, and Murai spheres are taken from prior literature.
    The classification and equivalence statements presuppose these notions.

pith-pipeline@v0.9.1-grok · 5568 in / 1104 out tokens · 29494 ms · 2026-06-28T13:42:34.963041+00:00 · methodology

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Reference graph

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