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arxiv: 2603.23298 · v2 · pith:RA4K43P3new · submitted 2026-03-24 · 🧮 math.AP

Kruskal-style algorithm for cubic Schr\"odinger equation molecule reduction

Yvain Bruned , Valentin Clarisse This is my paper

Pith reviewed 2026-05-21 10:33 UTC · model grok-4.3

classification 🧮 math.AP
keywords cubic Schrödinger equationmolecule reductionKruskal spanning treegraph traversalkinetic equationsrigidity theoremBoltzmann equation
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The pith

The molecule reduction algorithm for the cubic Schrödinger equation constructs a Kruskal spanning tree on the molecule graph.

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

The paper examines the molecule reduction algorithm introduced by Deng and Hani to prove a rigidity theorem needed for kinetic derivations of the wave equation from the cubic Schrödinger equation. It establishes that this procedure is a graph traversal of Kruskal type. The proof demonstrates that successive reductions produce a spanning tree on the graph representation of the input molecule. This identification accounts for the algorithm's reliability as a tool in kinetic theory and notes its prior use in long-time derivations of the Boltzmann equation.

Core claim

We show that the molecule reduction algorithm is a graph traversal algorithm of Kruskal type, and we prove that it constructs a Kruskal spanning tree of the input molecule. This reveals the origin of the main tool for deriving kinetic equations which has also been used for the long time derivation of the Boltzmann equation.

What carries the argument

The graph representation of the molecule in which each reduction operation corresponds exactly to an edge-addition step in Kruskal's algorithm for constructing a spanning tree.

Load-bearing premise

The molecule arising in the cubic Schrödinger context admits a graph representation where reduction operations match the edge-addition steps of Kruskal's algorithm without extra selection rules or weights.

What would settle it

A specific molecule example where the reduction sequence either produces a cycle or fails to connect all vertices would disprove the spanning-tree claim.

Figures

Figures reproduced from arXiv: 2603.23298 by Valentin Clarisse, Yvain Bruned.

Figure 1
Figure 1. Figure 1: Example of Prim minimum spanning tree rooted at [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of Kruskal minimum spanning tree [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of a tree and a couple. Signs are displayed inside each node of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Type I molecular chain [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Type II molecular chain Proposition 2.4 Between two atoms, there is at most one triple bond. If there are n atoms, then there are at most 2n − 1 bonds. In the case of equality, we speak of a base molecule. A base molecule is connected. It has two atoms of degree 3 or one atom of degree 2, and the others are of degree 4. Proof. See [8, Proposition 9.2]. Definition 2.5 (Base molecule associated with a couple… view at source ↗
Figure 6
Figure 6. Figure 6: Couple and its associated base molecule. The colors of the pairings of the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: On the left, we have added (v1, v2), (v3, v2) and (v1, v4) to the acyclic graph G and the other edges cannot be added due to a cycle. On the right, we have added (v1, v2), (v1, v4), (v1, v5) and we are in the case where the other edges produce a cycle. Step 3S2G: Two degree 3, 2 atoms v1, v2 are connected by a single bond. For the details see 9.3.7 in [8]. We remove from the molecule v1, v2, and we add to … view at source ↗
Figure 8
Figure 8. Figure 8: On the left, we have added (v1, v2), (v3, v2) and (v1, v4) to the acyclic graph G and the other edges cannot be added due to a cycle. On the right, we have added (v1, v2), (v1, v4), (v1, v5) and we are in the case where the other edges produce a cycle [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: We provide an example. We have added (v, v1), (v, v2), (v, v3) to the acyclic graph G. The other bond linked to v1, v2, v3 cannot be added due to potential cycles not represented here. • Step (3R-2G): wW remove from the molecule the atom v and its three bonds. In addition, we remove from the molecule v ′ 1 , v′ 2 and their bonds. We add to the acyclic graph G the maximum amount of bounds linked to v, v′ 1 … view at source ↗
Figure 10
Figure 10. Figure 10: We provide an example. We have added [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: In the example above, we have added (v, v′ ) and (v, v′′) to the acyclic graph G. The other edges (v ′ , v1), (v ′ , v2), (v ′ , v3), (v ′′, v4), (v ′′, v5) cannot be added due to potential cycles not represented here. • Step (2R-4): We assume that v is connected to two degree-2 atoms, v ′ and v ′′ , by two single bonds, such that neither v ′ nor v ′′ is connected to a degree-3 atom. We remove form the mo… view at source ↗
Figure 12
Figure 12. Figure 12: We provide an example. We have added (v, v′ ) and (v, v′′) to the acyclic graph G. The other edges (v ′ , v1), (v ′ , v2), (v ′′, v4), (v ′′, v5) cannot be added due to potential cycles not represented here. • Step (2R-5): We assume that v is connected to a degree-2 atom, v ′ , by a double bond. We remove from the molecule the atoms v, v′ and the double bond. We add to the acyclic graph G the maximum amou… view at source ↗
Figure 13
Figure 13. Figure 13: We provide an example. We have added (v, v′ ) to the acyclic graph G. The other edges linked to v, v′ cannot be added due to potential cycles not represented here [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Initial couple -2t -1t -3t -4t -2b -1b -3b -4b 1t 1b 2t 2b 3t 3b 4t 4b +1t +2t +3t +4t +1b +2b +4b +3b [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Molecule associated to the initial couple [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Molecule and spanning tree after step 1. We apply [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Molecule and spanning tree after step 2 and step 3. We remove the [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Molecule and spanning tree after steps 4 and 5. We apply twice (3S3-5G), [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Molecule and spanning tree after step 6. We apply (3D4G) on [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Molecule and spanning tree after steps 7-12. We apply BR on [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Molecule and spanning tree after steps 13-15. We apply BR on [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
read the original abstract

We are interested in the molecule reduction algorithm introduced by Deng and Hani. They use this algorithm to establish a rigidity theorem, which plays a central role in the kinetic-time derivation of the wave equation associated with the cubic Schr\"odinger equation. In the present article, we show that this algorithm is a graph traversal algorithm of Kruskal type, and we prove that it constructs a Kruskal spanning tree of the input molecule. This reveals the origin of the main tool for deriving kinetic equations which has also been used for the long time derivation of the Boltzmann equation.

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

1 major / 2 minor

Summary. The paper reinterprets the Deng-Hani molecule reduction algorithm for the cubic nonlinear Schrödinger equation as a Kruskal-type graph traversal procedure. The central claim is that the algorithm constructs a spanning tree of the input molecule by successively adding the lowest-weight connecting edges, exactly as in Kruskal's algorithm, thereby explaining the combinatorial origin of the rigidity tool used in kinetic derivations of the wave equation and in related Boltzmann derivations.

Significance. If the equivalence holds, the work supplies a graph-theoretic explanation for a key technical device in the kinetic-time analysis of the cubic Schrödinger equation. This may permit the importation of standard spanning-tree results or algorithms into the study of long-time behavior for dispersive equations. The parameter-free character of the claimed correspondence is a strength, as is the explicit link to an existing algorithm rather than an ad-hoc construction.

major comments (1)
  1. [§3] §3 (graph encoding of the molecule): the proof that reduction steps coincide exactly with Kruskal edge additions requires an explicit lemma verifying that the interaction graph carries no implicit multiplicity or tie-breaking rules that would change the component-merging order relative to the standard union-find implementation. Without this, the claim that the output is a Kruskal spanning tree for the specific inputs arising from the kinetic derivation remains unverified.
minor comments (2)
  1. The abstract should state the main theorem number for the spanning-tree claim.
  2. [§2] Notation for edge weights should be introduced once and used consistently when comparing the Deng-Hani steps to Kruskal's lowest-weight selection.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. The single major comment raises a valid point about the need for greater explicitness in Section 3. We address it below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (graph encoding of the molecule): the proof that reduction steps coincide exactly with Kruskal edge additions requires an explicit lemma verifying that the interaction graph carries no implicit multiplicity or tie-breaking rules that would change the component-merging order relative to the standard union-find implementation. Without this, the claim that the output is a Kruskal spanning tree for the specific inputs arising from the kinetic derivation remains unverified.

    Authors: We agree that an explicit verification is required to make the identification with Kruskal's algorithm fully rigorous for the interaction graphs that appear in the kinetic derivation. In the revised manuscript we will insert a new lemma (Lemma 3.4) immediately after the definition of the interaction graph. The lemma states that, under the generic frequency assumptions used in the Deng-Hani construction, all edge weights are distinct; consequently the sorted order of edges is unique and the union-find merging sequence coincides exactly with the standard Kruskal procedure without any tie-breaking convention. The proof of the lemma follows from the fact that the weights are linear combinations of the frequencies with coefficients determined by the resonance conditions, which are non-degenerate for the inputs arising in the kinetic-time analysis. This addition directly resolves the referee's concern while preserving the parameter-free character of the correspondence. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence proof to external Kruskal algorithm

full rationale

The paper's central claim is a direct mathematical proof that the Deng-Hani molecule reduction algorithm coincides with the edge-addition steps of Kruskal's spanning-tree construction on a suitably encoded interaction graph. This identification rests on verifying that reduction operations match the standard union-find criterion of adding the lowest-weight edge between distinct components, without redefining the algorithm in terms of its own outputs or fitting parameters to the target result. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results appear; the cited Deng-Hani work is external and the proof is self-contained against the independent benchmark of classical graph theory. The derivation therefore does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete free parameters, axioms, or invented entities; the paper appears to perform an identification rather than introduce new mathematical objects.

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Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Bringmann Y

    B. Bringmann Y. Deng, A. Nahmod, H. Yue .Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation. Invent. Math.236, (2024), 1133–1411.doi:10.1007/s00222-024-01254-4

    work page doi:10.1007/s00222-024-01254-4 2024

  2. [2]

    Buckmaster, P

    T. Buckmaster, P. Germain, Z. Hani, J. Shatah.Onset of the wave turbulence description of the longtime behavior of the nonlinear Schroedinger equation. Invent. Math.225, (2021), 787–855.doi:10.1007/s00222-021-01039-z

    work page doi:10.1007/s00222-021-01039-z 2021

  3. [3]

    Bodineau, I

    T. Bodineau, I. Gallagher, L. Saint-Raymond, S. Simonella.Cluster expansion for a dilute hard sphere gas dynamics.J. Math. Phys.63, no. 7, (2022), Paper No. 073301.doi:10.1063/5.0091199

    work page doi:10.1063/5.0091199 2022

  4. [4]

    Bodineau, I

    T. Bodineau, I. Gallagher, L. Saint-Raymond, S. Simonella.Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations.Ann. of Math. (2)198, no. 3, (2023), 1047–1201. doi:10.4007/annals.2023.198. 3.3

    work page doi:10.4007/annals.2023.198 2023

  5. [5]

    N. N. Bogoliubow, O. S. Parasiuk. ¨Uber die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder.Acta Math.97, (1957), 227–266. doi: 10.1007/BF02392399

    work page doi:10.1007/bf02392399 1957

  6. [6]

    T. H. Cormen, , C. E. Leiserson, R. L. Rivest, S. Clifford.Introduction to Algorithms, fourth edition 4th Edition. The MIT Press, 4th Edition, (2022)

    work page 2022

  7. [7]

    Y. Deng, Z. Hani.On the derivation of the wave kinetic equation for NLS. Forum Math. Pi, (2021),9, e6, (2021), 1–37.doi:10.1017/fmp.2021.6

    work page doi:10.1017/fmp.2021.6 2021

  8. [8]

    Y. Deng, Z. Hani.Full derivation of the wave kinetic equation. Invent. Math.233, (2023), 543-724.doi:10.1007/s00222-023-01189-2

    work page doi:10.1007/s00222-023-01189-2 2023

  9. [9]

    Y. Deng, Z. Hani.Derivation of the wave kinetic equation: Full range of scaling laws.Ta appear in Mem. Amer. Math. Soc.arXiv:2301.07063

    work page arXiv

  10. [10]

    Y. Deng, Z. Hani.Long time justification of wave turbulence theory. arXiv: 2311.10082

    work page arXiv

  11. [11]

    Y. Deng, Z. Hani.Propagation of chaos and higher order statistics in wave kinetic theory. J. Eur. Math. Soc. (JEMS),28, no. 2, (2026), 673-733. doi: 10.4171/JEMS/1488

    work page doi:10.4171/jems/1488 2026

  12. [12]

    Y. Deng, Z. Hani, X. Ma.Long time derivation of the Boltzmann equation from hard sphere dyamics.To appear in Ann. of Math.arXiv:2408.07818

    work page arXiv

  13. [13]

    Y. Deng, Z. Hani, X. Ma.Hilbert’s sixth problem: derivation of fluid equations via Boltzmann’s kinetic theory.arXiv:2503.01800

    work page arXiv

  14. [14]

    De Suzzoni, A

    A.-S. De Suzzoni, A. Stingo, A. Touati.Wave turbulence for a semilinear Klein-Gordon system.arXiv:2503.24222

    work page arXiv

  15. [15]

    Erd¨os, M

    L. Erd¨os, M. Salmhofer, H.-T. Yau.Quantum diffusion of the random Schrodinger evolution in the scaling limit II. The recollision diagrams.Commun. Math. Phys. 271, (2007) 1–53.doi:10.1007/s00220-006-0158-2

    work page doi:10.1007/s00220-006-0158-2 2007

  16. [16]

    Erd¨os, M

    L. Erd¨os, M. Salmhofer, H.-T. Yau.Quantum diffusion of the random Schr¨odinger evolution in the scaling limit.Acta Math.200, no. 2, (2008), 211–277. doi: 10.1007/s11511-008-0027-2. Rigidity theorem and algorithm24

    work page doi:10.1007/s11511-008-0027-2 2008

  17. [17]

    Gallavotti, F

    G. Gallavotti, F. Nicol`o.Renormalization theory in four-dimensional scalar fields (I). Commun. Math. Phys.100, (1985), 545–590.doi:10.1007/BF01217729

    work page doi:10.1007/bf01217729 1985

  18. [18]

    Gallavotti, F

    G. Gallavotti, F. Nicol`o.Renormalization theory in four-dimensional scalar fields (II). Commun. Math. Phys.101, (1985), 247–282.doi:10.1007/BF01218761

    work page doi:10.1007/bf01218761 1985

  19. [19]

    Hepp.On the equivalence of additive and analytic renormalization

    K. Hepp.On the equivalence of additive and analytic renormalization. Comm. Math. Phys.14, (1969), 67–69.doi:10.1007/BF01645456

    work page doi:10.1007/bf01645456 1969

  20. [20]

    Hairer, J

    M. Hairer, J. Quastel.A class of growth models rescaling to KPZ. Forum math. Pi 6, e3, (2018), 1-112.doi:10.1017/fmp.2018.2

    work page doi:10.1017/fmp.2018.2 2018

  21. [21]

    J. B. Kruskal.On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc.7, (1956), 48-50. doi:10.1090/ S0002-9939-1956-0078686-7

    work page 1956

  22. [22]

    Landford.Time evolution of large classical systems

    O.E. Landford.Time evolution of large classical systems. Lect. Notes in Physics, Springer Verlag,38, J. Moser ed., 1–111, (1975). doi:10.1007/ 3-540-07171-7_1

    work page 1975

  23. [23]

    Lukkarinen, H

    J. Lukkarinen, H. Spohn.Weakly nonlinear Schr ¨odinger equation with ran- dom initial data. Invent. Math.183, (2011), 79–188. doi:10.1007/ s00222-010-0276-5

    work page 2011

  24. [24]

    R. C. Prim.Shortest connection networks and some generalizations. Bell System Technical Journal,36, no. 6, (1957), 1389–1401. doi:10.1002/j.1538-7305. 1957.tb01515.x

    work page doi:10.1002/j.1538-7305 1957

  25. [25]

    Rivasseau.From Perturbative to Constructive Renormalization

    V. Rivasseau.From Perturbative to Constructive Renormalization. Prince- ton University Press, (1991). doi:10.1007/s00222-010-0276-10.1515/ 9781400862085

    work page doi:10.1007/s00222-010-0276-10.1515/ 1991

  26. [26]

    Rivasseau, Z

    V. Rivasseau, Z. Wang.How to Resum Feynman Graphs. Ann. Henri Poincar´e15, (2014), 2069-2083.doi:10.1007/s00023-013-0299-8

    work page doi:10.1007/s00023-013-0299-8 2014

  27. [27]

    Zimmermann.Convergence of Bogoliubov’s method of renormalization in momentum space.Comm

    W. Zimmermann.Convergence of Bogoliubov’s method of renormalization in momentum space.Comm. Math. Phys.15, (1969), 208–234. doi:10.1007/ BF01645676

    work page 1969