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REVIEW 4 major objections 4 minor 23 references

Finding Kissing Numbers with Game-theoretic Reinforcement Learning

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A two-player reinforcement learning game over cosine matrices reports new kissing-number lower bounds in dimensions 25–31, a rational 13-dimensional configuration, and new generalized kissing records.

desk verdict A genuinely new RL search formulation for kissing configurations with potentially significant new bounds, but the paper withholds the Gram matrices and certificates needed to verify the central claims. read the letter →

arxiv 2511.13391 v4 pith:6W3E4VIW submitted 2025-11-17 cs.LG cs.AImath.COmath.MG

Chengdong Ma , Théo Tao Zhaowei , Pengyu Li , Minghao Liu , Haojun Chen , Zihao Mao , Bo Li , Yuan Cheng
show 2 more authors
Yuan Qi Yaodong Yang
This is my paper · ORCID
classification cs.LGcs.AImath.COmath.MG MSC 52C17
keywords kissingnumbersphericalcodesreinforcementlearningGrammatrixcompletionLeechlatticerationalconfigurationsgeneralized
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 argues that the kissing number problem can be attacked by treating a candidate configuration as a Gram matrix and letting two cooperative agents play a completion game: one agent fills in pairwise cosines, the other deletes entries that lead to dead ends. This reformulation, the paper claims, makes high-dimensional search tractable enough for reinforcement learning to find configurations beyond human construction. At stake are long-standing lower bounds: PackingStar reports new kissing-number records from dimension 25 through 31, the first improved rational 13-dimensional structure since 1971, and new records for generalized kissing configurations with tighter angle constraints. If these configurations are valid, they give mathematicians new concrete objects to study and, in several cases, possible routes to optimality.

What carries the argument

The central object is the Gram matrix of the sphere centers, with entries bounded by 1/2 for non-adjacent spheres. PackingStar models construction as a sequential two-player Markov game: Player 1 (Filler) extends the matrix by sampling candidate entries from a cosine set, subject to positive semidefiniteness and rank constraints; Player 2 (Corrector) later removes a learned subset of entries to free the construction from local dead ends. The Cholesky factorization of the top-left n×n block and the relation g^(2)=G'(G^(1:n,1:n))^+ g^(1) reduce the candidate action set to a small, parallelizable search; the shared reward is the final matrix size, i.e., the number of spheres.

What would settle it

Check each reported configuration for a pair of unit vectors whose inner product exceeds 1/2, or a Gram matrix that is not positive semidefinite; either condition would invalidate the corresponding lower bound. Also investigate whether the fixed cosine sets C1/C2 and the Leech-lattice structural constraint (Eqs. 9–11) used in dimensions 25–31 artificially exclude larger configurations than the ones reported.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the cooperative two-player matrix-completion formulation lets an RL system, PackingStar, discover new maximal-size spherical codes where coordinate-based methods stall. Concretely, it claims K_new(25)=197056, K_new(26)=198550, K_new(27)=200044, K_new(28)=204520, K_new(29)=209496, K_new(30)=220440, K_new(31)=238350, surpassing the previous records by up to 5476 spheres; a rational configuration with 1146 spheres in dimension 13, above the previously known rational count of 1130; and generalized kissing records K_new(12,1/4)=81, K_new(14,1/3)=252, and K_new(17,1/3)=578. It further claims that the new 25-dimensional configuration sits naturall

Load-bearing premise

The load-bearing premise is that the reported configurations are genuine spherical codes—every Gram matrix is positive semidefinite and every pairwise inner product is at most 1/2—because if any claimed configuration violates these constraints, the corresponding new lower bound no longer exists.

Editorial extensions

If this is right

  • If the reported configurations are valid, the known lower bounds for K(25) through K(31) rise by 8, 38, 68, 152, 1224, 456, and 5476 spheres, respectively.
  • The 13-dimensional rational construction provides a fully rational Gram matrix, enabling exact algebraic study without numerical approximation, and improves the rational kissing number from 1130 to 1146.
  • The generalized kissing records K(12,1/4)=81, K(14,1/3)=252, and K(17,1/3)=578 would update the standard bounds for spherical codes under those angular constraints.
  • The decomposition of the 17-dimensional 1/3-configuration as 288×2+2 suggests near-optimality and may guide proof attempts.
  • The paper's claim of over 6000 newly discovered 14-dimensional configurations provides a pool of structures for further investigation.

Reading between the lines

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

  • Editorial: The manuscript reports these configurations without including full Gram matrices or validation certificates in the text, so an independent check of positive semidefiniteness and the 1/2 cosine cutoff is the first step before the records become settled bounds.
  • Editorial: If the 25-dimensional configuration truly reflects Leech-lattice substructure, a natural next step is an optimality proof via linear programming or Fourier-analysis bounds, which would make K(25) exact.
  • Editorial: The same filler-corrector game could be transferred to other constrained spherical-code problems—for example other values of alpha in K(n,alpha), or codes with prescribed antipodal structure.
  • Editorial: The fully rational 13D configuration is a natural testbed for algebraic proof techniques that require exact inner products rather than floating-point coordinates.
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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

4 major / 4 minor

Summary. The paper introduces PackingStar, a two-player reinforcement-learning system that searches for kissing configurations by treating the problem as a cooperative matrix-completion game over Gram matrices. Player 1 fills candidate cosine entries, Player 2 deletes suboptimal ones, and the shared reward is the final matrix size. The authors report new lower bounds for kissing numbers in dimensions 25–31 (e.g., K_new(25)=197056, K_new(31)=238350), a rational 13-dimensional configuration with 1146 spheres (improving the previous rational bound 1130), and new generalized kissing records for K(12,1/4)=81, K(14,1/3)=252, and K(17,1/3)=578. The method first infers candidate cosine sets from simulation, then searches over Gram matrices subject to those sets and optional structural constraints. The paper argues that this Gram-domain reformulation is more scalable and geometrically interpretable than coordinate-based optimization.

Significance. If the reported configurations are valid unit spherical codes, the results are significant: they would improve long-standing lower bounds in several dimensions, produce the first rational 13-dimensional construction beyond the 1971 record, and demonstrate that reinforcement learning can discover nontrivial geometric structures in high dimensions. The reformulation of kissing-number search as a two-player matrix completion game is conceptually interesting and, in principle, reproducible. The explicit construction forms in Table 2 and the cosine sets in Table 3 allow partial arithmetic checking, and the reported increments are internally consistent. However, the manuscript currently ships no Gram matrices, coordinate files, validation certificates, or code, and it explicitly defers implementation details to a future version. The central claims therefore cannot be verified from the text as it stands; the significance is conditional on the missing evidence.

major comments (4)
  1. [Results: New records from dimensions 25 to 31; Table 2; Table 3] The load-bearing premise is that each reported configuration is a valid unit spherical code: the Gram matrix must be positive semidefinite and every off-diagonal entry must be at most 1/2 (or the stated alpha). The paper provides only cosine sets, a repository URL, and construction formulas. A cosine set is not a certificate: it does not by itself show that the claimed number of unit vectors with those pairwise inner products exists. In particular, K_new(25) relies on a 496-element subset S_i of Leech minimal vectors with no explicit S_i or Gram submatrix supplied, and Table 3 for K_new(31) refers the reader to the repository for the full cosine set. The Methods section states: 'More implementation details will be provided in the supplementary materials of a future version.' If any claimed configuration violates PSD or the inner-product bound, the corresponding record collapses. The manu
  2. [Methods, Eqs. (9)–(11)] The search is restricted to hand-chosen cosine sets C1, C2 and, in dimensions 25–31, the structural constraint C* requiring the configuration to match the Leech-lattice minimal-vector set. The paper asserts that Step 1 'converges' to the cosine set, but no argument or proof is given that the simulated cosine frequencies exhaust all cosine values that could appear in an extremal configuration. Consequently, the reported 'new lower bounds' are lower bounds within a restricted family, not general kissing-number lower bounds. This does not invalidate the claimed improvements over previous constructions, but it does undermine the paper's stronger language that the new form 'may be optimal' or is 'optimal within this construction pathway'; those statements require either a proof of exhaustiveness or a clear restriction of the claim.
  3. [Abstract; Conclusion] The abstract claims that 'several of them provably optimal under natural inner products,' but the Conclusion states: 'A current limitation is that PackingStar cannot provide optimality proofs for these new configurations.' No optimality proof appears anywhere in the manuscript. This is an internal inconsistency in a headline claim. The abstract must be corrected to match the actual scope, or the proofs must be supplied.
  4. [Results: Data availability] No code, training hyperparameters, compute budgets, or validation scripts are included, and the statement that results are 'independently validated and are now included in the standard databases' is uncited. For a machine-learning-driven discovery paper, this is a major reproducibility gap: a reader cannot reconstruct the search, check the reported arithmetic on the full matrices, or trust the claimed inclusion in external databases. At minimum, the repository must contain the exact Gram matrices or coordinates for every reported configuration, a validation script that verifies PSD and inner-product bounds, and a description of the training setup sufficient for replication.
minor comments (4)
  1. [Abstract] The abstract mentions 'improve 15 strong bounds held for decades,' but the Results section lists 7 new bounds in dimensions 25–31, 1 rational bound in 13D, and 3 generalized bounds, totaling 11 distinct new values. The count of 15 should be clarified or justified.
  2. [Methods, Eq. (15)] The reward is written as ||diag(G^{(M)})||_F^2. Since diag(G) is all ones for a valid Gram matrix, this quantity is simply the number of spheres M. The notation is unnecessarily indirect and could be simplified to 'the number of rows M'.
  3. [Table 2] The formula for dimension 27 contains the sum '2|S1|+2|S2|+∑_{i=3}^5 |Si|' but the preceding term 'K(3)+K(24)' is not grouped consistently; consider adding parentheses or clarifying the intended combination. Also, the table's caption says 'new construction form' but does not specify which forms correspond to which previous forms [19].
  4. [Results: Diverse constructions in different dimensions] The claim of 'over 6000 new configurations in 14 dimensions' is not accompanied by any enumeration criterion, validation summary, or availability statement beyond the repository. It would help to state how a 'new' configuration is defined and how validity was checked.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor, non-load-bearing circularity: the reported cosine-set 'structures' for K_r(13) are the same sets imposed as search constraints; the kissing-number bounds themselves are search outputs, not fitted to targets.

  1. other [Results, 'Improved rational configurations in 13 dimensions'; Methods, 'Finding Kissing Numbers in two-player matrix completion game'; Table 3.]
    "guided by the simulations in Step 1 of our method, we identify two feasible cosine sets for 13-dimensional rational arrangements: C1=C2={−1,0,±1/4,±1/2} or {−1,−3/4,0,±1/4,±1/2}... Based on these cosine features, PackingStar discovers several new rational configuration with kissing number K_r(13)=1146"

    Methods states the pipeline is seeded by the Step-1 cosine set ('The cosine set from Step 1 populates an initial matrix') and restricts every candidate entry to predefined C1/C2 (Eqs. 9-10). Table 3 then lists for K_r(13) exactly the same sets, {±1/4,±1/2} and variants with −3/4. The cosine-set 'structure' is therefore an input constraint by construction, not an independently derived output feature. This affects the structural-description claim, not the 1146 lower bound, which is the result of maximizing matrix size rather than of fitting any target value.

full rationale

The main derivation chain is not circular. PackingStar's reward is the number of rows of the completed Gram matrix (Eq. 15), and the reported lower bounds K_new(25..31), K_r(13), K_new(12,1/4), etc. are outputs of a search under explicit constraints; no target kissing number appears as a fitted parameter. The 25-31 results are obtained inside the previously published Leech-lattice subset template (refs [18,19]), which is a strong prior but is acknowledged, and the counts |S_i|=496, the assembly templates of Table 2, and the generalized codes are not forced by the prior to equal the claimed values. No self-citation chain or imported uniqueness theorem is load-bearing. The one partial circularity is the cosine-set feature: C1/C2 are extracted in Step 1 and then re-imposed as constraints, while Table 3 reports the same sets as the configurations' cosine sets; this is an input echoed as an output, but it is not the source of the numerical records. Separately, the manuscript has serious verification gaps that are not circularity: no full Gram matrices or certificates are supplied ('More implementation details will be provided in the supplementary materials of a future version'), the 'independently validated... standard databases' claim is uncited, and the Conclusion's 'Trained from scratch without any human-designed priors' is contradicted by Methods' use of simulation-derived cosine sets and the Leech C* constraint. These are correctness/reproducibility risks, not equivalence-to-input circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central results are obtained by RL search, not by derivation, so the main ledger entries are the hand-set cosine sets and the Leech-lattice structural constraint, plus the assumptions that the Step-1 simulation and the two-player game produce valid, near-optimal completions. No new physical or mathematical entities are postulated.

free parameters (4)
  • Cosine set C1=C2 for 25–31D = {-1, 0, ±1/4}
    Chosen to match Leech-lattice minimal-vector inner products; imposed as the allowed cosine values in Eqs. (9)-(11).
  • Cosine sets for 13D rational configurations = {-1,0,±1/4,±1/2} or {-1,-3/4,0,±1/4,±1/2}
    Identified by Step-1 simulation and selected by hand; not proven exhaustive.
  • Structural constraint C* for 25–31D = Leech lattice minimal-vector set
    Applied as an extra action constraint via Eq. (11); restricts all 25–31D searches to the Leech-lattice template.
  • RL hyperparameters (MCTS constant c, network architecture, training budgets)
    Not specified; deferred to a future supplement. Required to reproduce the experiments.
assumptions (4)
  • domain assumption The first n vectors of each partial configuration have full rank, so Eq. (8) forces the new-table entries g^(2) from g^(1) via the pseudo-inverse.
    The candidate action set A^(m) in Eq. (9) is correct only when the top-left n×n block is invertible and the Cholesky/pseudo-inverse step is well-defined; singular cases are not discussed.
  • ad hoc to paper The Step-1 simulation's converged cosine frequencies produce an exhaustive set of cosine values appearing in extremal configurations.
    The matrix completion is restricted to C1/C2; any optimal configuration using other cosine values cannot be found by the system.
  • ad hoc to paper The two-agent MCTS/policy-gradient dynamics converges to valid, near-maximal Gram matrices.
    No convergence or optimality proof is given; validity of the final matrices is asserted through 'independent validation' rather than demonstrated.
  • domain assumption Known kissing values K(24) and K(k) for k=2..7 can be embedded as geometric components in higher-dimensional constructions.
    Table 2 combines these known components into the new configurations; the embeddings must preserve the pairwise cosine constraints.

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Cite this review

Pith. "Pith review of Finding Kissing Numbers with Game-theoretic Reinforcement Learning." pith.science (2026). https://pith.science/paper/6W3E4VIW

@misc{pith2026251113391,
  author       = {Pith},
  title        = {Pith review of: Finding Kissing Numbers with Game-theoretic Reinforcement Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6W3E4VIW}},
  note         = {Machine review of arXiv:2511.13391}
}
read the original abstract

Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a defining challenge in discrete geometry. As the local analogue of Hilbert's 18th problem, it has profound implications across geometry, number theory and information theory. Although lattices and codes have achieved significant progress, the field is confined to isolated extremal configurations, leaving underlying geometric principles obscured. Here we shift the object to the broader extremal configuration space, thereby opening a new path for the Kissing Number Problem. Accordingly, we recast this problem as a cooperative matrix-completion game, and train a reinforcement learning system, PackingStar, to solve it. One player fills cosine entries while the other corrects suboptimal ones, making explosive geometric complexity tractable. Working within extremal configuration spaces, PackingStar discovers new interpretable geometric structures that improve 15 strong bounds held for decades in kissing numbers and their generalizations, several of them provably optimal under natural inner products. These findings reveal the first explicit spherical-code realization of the Fischer group Fi22, extend the classical Euclidean representation of subgroup structure, and directly inspire subsequent breakthroughs by mathematicians. Overall, the work provides an early example of AI-driven progress on a Hilbert-calibre problem, showing how reinforcement learning advances mathematical discovery by unlocking more expressive objects.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 4 linked inside Pith

  1. [1]

    OUP Oxford, 2000

    Jeremy Gray.The Hilbert Challenge. OUP Oxford, 2000

    2000

  2. [2]

    Improved kissing numbers in seventeen through twenty-one dimen- sions.arXiv preprint arXiv:2411.04916, 2024

    Henry Cohn and Anqi Li. Improved kissing numbers in seventeen through twenty-one dimen- sions.arXiv preprint arXiv:2411.04916, 2024

    arXiv 2024

  3. [3]

    A survey on the kissing numbers

    Peter Boyvalenkov, Stefan Dodunekov, and Oleg R Musin. A survey on the kissing numbers. arXiv preprint arXiv:1507.03631, 2015

    arXiv 2015

  4. [4]

    Mathematical programming bounds for kissing numbers

    Leo Liberti. Mathematical programming bounds for kissing numbers. InInternational Con- ference on Optimization and Decision Science, pages 213–222. Springer, 2017

    2017

  5. [5]

    Kissing number of codes: A survey.Coding Theory and applications (tentative title), 2023

    Yi Liu, Wei Cheng, Olivier Rioul, Sylvain Guilley, and Patrick Sol´e. Kissing number of codes: A survey.Coding Theory and applications (tentative title), 2023

    2023

  6. [6]

    Table of kissing number bounds.., 2024

    Henry Cohn. Table of kissing number bounds.., 2024

    2024

  7. [7]

    Das problem der dreizehn kugeln.Math- ematische Annalen, 125(1):325–334, 1952

    Kurt Sch¨ utte and Bartel Leendert van der Waerden. Das problem der dreizehn kugeln.Math- ematische Annalen, 125(1):325–334, 1952

    1952

  8. [8]

    The kissing number in four dimensions.Annals of Mathematics, pages 1–32, 2008

    Oleg R Musin. The kissing number in four dimensions.Annals of Mathematics, pages 1–32, 2008

    2008

Show all 23 references
  1. [9]

    Sur les formes quadratiques.Mathematische Annalen, 6(3):366–389, 1873

    Aleksandr Korkine and G Zolotareff. Sur les formes quadratiques.Mathematische Annalen, 6(3):366–389, 1873

  2. [10]

    Notes on sphere packings.Canadian Journal of Mathematics, 19:251–267, 1967

    John Leech. Notes on sphere packings.Canadian Journal of Mathematics, 19:251–267, 1967

    1967

  3. [11]

    Some sphere packings in higher space.Canadian Journal of Mathematics, 16:657–682, 1964

    John Leech. Some sphere packings in higher space.Canadian Journal of Mathematics, 16:657–682, 1964

    1964

  4. [12]

    The sphere packing problem in dimension 8.Annals of mathematics, pages 991–1015, 2017

    Maryna S Viazovska. The sphere packing problem in dimension 8.Annals of mathematics, pages 991–1015, 2017. 16

    2017

  5. [13]

    The sphere packing problem in dimension 24.Annals of mathematics, 185(3):1017–1033, 2017

    Henry Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko, and Maryna Viazovska. The sphere packing problem in dimension 24.Annals of mathematics, 185(3):1017–1033, 2017

    2017

  6. [14]

    Springer Science & Business Media, 2013

    John Horton Conway and Neil James Alexander Sloane.Sphere packings, lattices and groups, volume 290. Springer Science & Business Media, 2013

    2013

  7. [15]

    Sphere packings and error-correcting codes.Canadian Journal of Mathematics, 23(4):718–745, 1971

    John Leech and NJA Sloane. Sphere packings and error-correcting codes.Canadian Journal of Mathematics, 23(4):718–745, 1971

    1971

  8. [16]

    Highly symmetric lines.arXiv preprint arXiv:2207.08266, 2022

    Mikhail Ganzhinov. Highly symmetric lines.arXiv preprint arXiv:2207.08266, 2022

    2022 arXiv

  9. [17]

    New lower bounds for contact numbers in small dimensions.Problemy Peredachi Informatsii, 35(4):3–11, 1999

    Victor Alexandrovich Zinov’ev and Thomas Ericson. New lower bounds for contact numbers in small dimensions.Problemy Peredachi Informatsii, 35(4):3–11, 1999

    1999

  10. [18]

    Improved lower bounds for kissing numbers in dimensions 25 through 31.SIAM Journal on Discrete Mathematics, 31(3):1895–1908, 2017

    Kenz Kallal, Tomoka Kan, and Eric Wang. Improved lower bounds for kissing numbers in dimensions 25 through 31.SIAM Journal on Discrete Mathematics, 31(3):1895–1908, 2017

    1908

  11. [19]

    Rigidity of spherical codes

    Henry Cohn, Yang Jiao, Abhinav Kumar, and Salvatore Torquato. Rigidity of spherical codes. Geometry & Topology, 15(4):2235–2273, 2011

    2011

  12. [20]

    Alphaevolve: A coding agent for scientific and algorithmic discovery.arXiv preprint arXiv:2506.13131, 2025

    Alexander Novikov, Ng ˆan V˜ u, Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco JR Ruiz, Abbas Mehra- bian, et al. Alphaevolve: A coding agent for scientific and algorithmic discovery.arXiv preprint arXiv:2...

    2025 arXiv

  13. [21]

    Spherical codes with prescribed signed permu- tation automorphisms inside shells of low-dimensional integer lattices.IEEE Transactions on Information Theory, 70(12):8669–8674, 2024

    Mikhail Ganzhinov and Patric RJ ¨Osterg˚ard. Spherical codes with prescribed signed permu- tation automorphisms inside shells of low-dimensional integer lattices.IEEE Transactions on Information Theory, 70(12):8669–8674, 2024

    2024

  14. [22]

    Elsevier, 2001

    Thomas Ericson and Victor Zinoviev.Codes on Euclidean spheres, volume 63. Elsevier, 2001

    2001

  15. [23]

    A note on five dimensional kissing arrangements.arXiv preprint arXiv:2301.08272, 2023

    Ferenc Sz ¨oll˝osi. A note on five dimensional kissing arrangements.arXiv preprint arXiv:2301.08272, 2023. 17 Acknowledgments We are grateful to Professor Henry Cohn for his influential work on the Kissing Number Problem, which has been a source of inspiration for this researc...

    2023 arXiv

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