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arxiv: 2606.18984 · v2 · pith:XECAUSCVnew · submitted 2026-06-17 · 💻 cs.IT · math.IT

Structure of kissing arrangements in {mathbb R}¹² and a place for the 841st sphere

Rustem Takhanov , Zhenisbek Assylbekov , Stanislav Yun This is my paper

Pith reviewed 2026-06-26 19:07 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords kissing arrangementssphere packingR^12kissing numberRiesz energy1-factorizationbridge vectors
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The pith

Kissing arrangements of 840 spheres in R^12 admit positive-dimensional families of 48-systems, yielding infinitely many non-isometric examples and a numerical 841-sphere arrangement.

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

The paper examines kissing arrangements in twelve-dimensional space built from two 60-vector blocks in orthogonal six-dimensional subspaces together with 720 fixed bridge vectors from the 1-factorization of the complete graph K6. It establishes that each 60-vector block fixes twelve vectors as signed coordinate directions while the remaining forty-eight vectors can vary continuously inside families called 48-systems, all while preserving the required minimum distances to the bridges. This continuous freedom immediately produces infinitely many pairwise non-isometric kissing arrangements of size 840. The same families supply structured initial points for logarithmic Riesz-energy minimization, from which a configuration of 841 spheres is obtained numerically.

Core claim

Kissing arrangements of size 840 in R^12 with fixed bridge vectors admit 48-systems: positive-dimensional families of configurations for the 48 varying vectors in each 60-point block. Consequently there exist infinitely many pairwise non-isometric kissing arrangements of size 840. This block flexibility supplies initial configurations for logarithmic Riesz-energy optimization that numerically realize a kissing arrangement of size 841.

What carries the argument

48-systems: positive-dimensional families of 48-vector configurations that complete each 60-point block while keeping all inner products with the fixed bridge vectors at or below the kissing threshold.

If this is right

  • Continuous families of 840-kissing arrangements exist in R^12.
  • Infinitely many pairwise non-isometric 840-kissing arrangements exist in R^12.
  • Structurally informed initialization reaches a kissing arrangement of size 841 by Riesz-energy minimization.
  • The local geometry around known 840-arrangements contains positive-dimensional freedom inside the 60-point blocks.

Where Pith is reading between the lines

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

  • Similar block decompositions with variable subsystems might produce larger kissing arrangements in other dimensions once analogous bridge structures are identified.
  • The numerical 841 example indicates that the maximum kissing number in R^12 is at least 841, but proving local optimality would still be required.
  • Alternative choices of bridge vectors or different 1-factorizations could admit even larger variable blocks and therefore still bigger arrangements.

Load-bearing premise

The 48-systems remain valid non-overlapping configurations across a positive-dimensional parameter range and the numerical Riesz-energy optimizer produces a configuration whose minimum distance meets the kissing requirement.

What would settle it

An explicit choice of parameters inside a claimed 48-system for which at least one vector overlaps a bridge vector, or a verification that the reported 841-sphere configuration contains at least one pair of spheres with inner product strictly larger than the kissing bound.

Figures

Figures reproduced from arXiv: 2606.18984 by Rustem Takhanov, Stanislav Yun, Zhenisbek Assylbekov.

Figure 1
Figure 1. Figure 1: 1-factorization of K6. The bridge construction is based on the unique 1-factorization of the complete graph K6 (see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic view of the 48-system in R6 : the equatorial layer x1 = 0 and the two lifted layers x1 = ± 1 2 . 2) The + 1 2 floor vectors, i.e.  1 2  × {b1, · · · , b16}. 3) The − 1 2 floor vectors, i.e.  − 1 2  × {b1, · · · , b16}, where vectors {bi} are the same for + 1 2 and − 1 2 floors. A key observation from numerically obtained examples is that the set of pairs (xi , xj ) with an obtuse angle betwee… view at source ↗
read the original abstract

Most currently known kissing arrangements of size $840$ in $\mathbb R^{12}$ share a common structure. They consist of $60$ vectors supported on $\mathbb R^6\times\{\mathbf 0\}$, another $60$ vectors supported on $\{\mathbf 0\}\times\mathbb R^6$, and $720$ additional \emph{bridge vectors}. The bridge vectors encode the interaction between the two six-dimensional factors and are constructed from the unique $1$-factorization of the complete graph $K_6$. In this paper we investigate kissing arrangements of this type while keeping the bridge vectors fixed. We show that each $60$-point block admits substantial flexibility: $12$ of its vectors may be chosen as the signed coordinate vectors $\pm e_i$, while the remaining $48$ vectors may vary within a positive-dimensional family of configurations, which we call $48$-systems. As a consequence, we obtain infinitely many pairwise non-isometric kissing arrangements of size $840$ in $\mathbb R^{12}$. The geometric freedom revealed by these constructions provides new insight into the local structure of extremal configurations. Exploiting this structure, we develop a specialized initialization scheme for logarithmic Riesz energy optimization. Starting from such structurally informed initial configurations, we numerically construct a kissing arrangement of size $841$ in $\mathbb R^{12}$.

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

2 major / 2 minor

Summary. The paper examines kissing arrangements of 840 unit spheres in R^12 that decompose into two 60-point sets supported on orthogonal 6-dimensional subspaces together with 720 bridge vectors constructed from the unique 1-factorization of K_6. Keeping the bridges fixed, it asserts that each 60-point block admits a 12-dimensional signed-basis subset together with a positive-dimensional family of configurations for the remaining 48 vectors (termed 48-systems). This flexibility is claimed to produce infinitely many pairwise non-isometric 840-arrangements. The same structural freedom is then used to generate initial configurations for a logarithmic Riesz-energy optimizer, which numerically yields a kissing arrangement of size 841.

Significance. If the geometric claims are established, the construction supplies an explicit positive-dimensional family of 840-kissing arrangements and thereby new information on the local geometry of extremal packings in R^12. The numerical production of a 841-configuration from structurally informed initials is a concrete computational advance that could, if rigorously validated, improve the known lower bound. The paper receives credit for grounding the bridges in the 1-factorization and for the reproducible initialization scheme, both of which are concrete strengths.

major comments (2)
  1. [Abstract / 48-systems construction] Abstract and the section introducing 48-systems: the claim that the 48-vector families remain valid kissing configurations (inner products with the fixed bridge vectors and among themselves at least 1/2) for a positive-dimensional open set of parameters is asserted geometrically but not accompanied by explicit lower bounds or an interval of validity. Continuity of the distance functions alone does not guarantee that the feasible set has positive dimension in the interior.
  2. [Numerical optimization section] Numerical construction of the 841-arrangement: the manuscript reports a Riesz-energy optimizer output but supplies neither the achieved minimum inner product, error bars on the distances, nor convergence diagnostics confirming that all pairwise distances meet or exceed the kissing threshold. This leaves the 841 claim dependent on an unverified numerical minimum.
minor comments (2)
  1. [Introduction] Notation for the bridge vectors and the 1-factorization should be introduced with an explicit reference or short recap of the K_6 factorization used, to make the fixed-bridge assumption self-contained.
  2. [48-systems] The manuscript would benefit from a short table listing the dimension of the parameter space for a single 48-system and the number of independent constraints imposed by the bridge inner-product conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. Below we respond point by point to the major comments, indicating how we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / 48-systems construction] Abstract and the section introducing 48-systems: the claim that the 48-vector families remain valid kissing configurations (inner products with the fixed bridge vectors and among themselves at least 1/2) for a positive-dimensional open set of parameters is asserted geometrically but not accompanied by explicit lower bounds or an interval of validity. Continuity of the distance functions alone does not guarantee that the feasible set has positive dimension in the interior.

    Authors: We agree that providing explicit lower bounds would make the argument more transparent. The geometric construction of the 48-systems is based on a specific parameterization that allows us to derive explicit bounds on the inner products. In the revised version, we will include these bounds and specify an interval of validity for the parameters, thereby confirming that the set of valid configurations has positive dimension. revision: yes

  2. Referee: [Numerical optimization section] Numerical construction of the 841-arrangement: the manuscript reports a Riesz-energy optimizer output but supplies neither the achieved minimum inner product, error bars on the distances, nor convergence diagnostics confirming that all pairwise distances meet or exceed the kissing threshold. This leaves the 841 claim dependent on an unverified numerical minimum.

    Authors: We acknowledge the need for more rigorous reporting of the numerical results. In the revised manuscript, we will include the achieved minimum inner product value, error estimates on the distances, and convergence diagnostics from the optimizer to confirm that the configuration satisfies the kissing condition. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit geometric parameterization from external 1-factorization yields independent infinite family; numerical 841 is optimization, not derivation

full rationale

The paper constructs 48-systems by fixing 12 signed basis vectors and deforming the remaining 48 while holding bridge vectors constant (taken from the known unique 1-factorization of K_6, an external combinatorial fact). The infinite non-isometric family follows directly from the positive-dimensional parameter space of these deformations, with no equation reducing a claimed output to a fitted input by construction. The 841-sphere claim is obtained via numerical Riesz-energy minimization initialized from these configurations, not a closed-form prediction. No self-citations are load-bearing for the central claims, and no ansatz or uniqueness theorem is smuggled in. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated. The construction implicitly relies on the existence of the 1-factorization of K_6 and on the validity of the 48-systems as kissing configurations.

pith-pipeline@v0.9.1-grok · 5785 in / 1235 out tokens · 19699 ms · 2026-06-26T19:07:21.176817+00:00 · methodology

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