arxiv: 2605.00660 · v1 · submitted 2026-05-01 · 🧮 math.CO · cs.DM

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Hamilton decompositions of the directed 7-torus at odd modulus via root-flat certificates and a prefix-count construction

Sanghyun Park

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Pith reviewed 2026-05-09 18:44 UTC · model grok-4.3

classification 🧮 math.CO cs.DM

keywords directed Hamilton decompositiontorus Cayley graphroot-flat certificateprefix-count constructionodd modulusformal verificationseven-dimensional lattice

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The pith

The directed seven-dimensional torus has a Hamilton decomposition for every odd modulus at least three.

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

The paper shows that the directed Cayley graph on the seven-dimensional integer lattice modulo m, generated by the seven unit vectors, can be decomposed into directed Hamilton cycles whenever m is odd and at least three. This result builds on earlier work for five dimensions by formalizing a verification method that checks only local properties on one root flat. For most odd m the proof uses an explicit construction based on prefix counts and count matrices that generate all seven factors at once. The two smallest cases are confirmed by direct computation inside a formal proof assistant. Such decompositions are useful in designing efficient routing schemes on toroidal networks and in combinatorial scheduling problems.

Core claim

We prove that the directed seven-dimensional equal-side torus D_7(m) = Cay((Z/mZ)^7, {e_0, ..., e_6}) admits a directed Hamilton decomposition for every odd integer m >= 3. The proof introduces the root-flat certificate, a framework where three local conditions on a single root flat suffice to guarantee the global decomposition. For odd m >=7 a uniform prefix-coordinate construction with one-layer prefix maps and explicit 7x7 count matrices produces the seven Hamilton factors. The cases m=3 and m=5 are handled by finite certificates verified in Lean 4.

What carries the argument

The root-flat certificate, defined by the three local conditions of row Latinness, layer bijectivity, and primitive return maps on a single root flat, which together imply the existence of a directed Hamilton decomposition of the torus Cayley graph.

Load-bearing premise

The three local conditions on the root flat are sufficient to guarantee that the corresponding factors form a global directed Hamilton decomposition of the entire graph.

What would settle it

Finding an odd m where the given prefix maps fail to satisfy row Latinness, layer bijectivity, or primitive return maps would falsify the construction for that m.

Figures

Figures reproduced from arXiv: 2605.00660 by Sanghyun Park.

**Figure 1.** Figure 1: Logical structure of the proof. The symbolic view at source ↗

**Figure 2.** Figure 2: The root-flat/prefix-coordinate dictionary. The bijection view at source ↗

read the original abstract

We prove that the directed seven-dimensional equal-side torus D_7(m) = Cay((Z/mZ)^7, {e_0, e_1, ..., e_6}) admits a directed Hamilton decomposition for every odd integer m >= 3. The proof has two main contributions. First, we introduce the root-flat certificate: a named verification framework in which a Hamilton decomposition of D_n(m) follows from three local conditions on a single root flat -- row Latinness, layer bijectivity, and primitive return maps. This abstraction was used informally in the earlier odd D_5(m) construction; here it appears as a definition and a theorem, providing a common verification interface for prime-dimensional base cases. Second, for every odd m >= 7, we give a uniform prefix-coordinate construction: one-layer prefix maps, a symbol-count criterion, and explicit 7x7 count matrices produce all seven Hamilton factors without a finite search. The remaining moduli m = 3 and m = 5 are exactly the boundary where the prefix-count method provably cannot work; they are handled by finite root-flat certificates whose validity is checked in Lean 4. A Lean 4 formalization verifies the Cayley statement, with the symbolic branch and the finite boundary certificates checked in the same development.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit prefix constructions for Hamilton decompositions of the directed 7-torus at odd m >=7 plus Lean checks for m=3 and 5, but the lifting from local root-flat conditions to the global decomposition is the part that carries the load.

read the letter

The paper shows that the directed seven-dimensional torus has a Hamilton decomposition for all odd m at least three. They do this with a new formal framework called the root-flat certificate and a uniform construction that works for m seven and larger. They define the certificate around three local conditions on one flat: row Latinness, layer bijectivity, and primitive return maps. If those hold, the theorem says you get the full directed Hamilton decomposition of the Cayley graph. For the larger odd moduli they give explicit prefix maps and count matrices that satisfy the conditions without needing to search. The cases m three and five fall outside that method, so they verify the certificates directly in Lean. This approach improves on the earlier dimension five results by making the certificate a reusable interface and by providing a search-free construction for most m. The Lean development covers both the symbolic part and the finite checks, which is a plus for reproducibility. The constructions are given in enough detail that one could in principle check the three conditions by hand or with a small program for a given m. That makes the main claims more concrete than many existence proofs in this area. The main thing to watch is the sufficiency of the three conditions for the global result. The paper states this as a theorem that lifts the earlier informal usage, but the step from local properties on the root flat to a complete edge decomposition across all directions is the one that needs the most careful reading. Any gap there would make the rest of the work irrelevant. This paper is for specialists in graph decompositions and Cayley graphs on tori. Readers working on similar problems in higher dimensions or looking for explicit factors might find the prefix-count method worth adapting. It has enough new structure and verification that it should go through peer review rather than being rejected outright. I would recommend sending it to referees, with particular attention to the lifting argument from the root flat.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the directed 7-torus D_7(m) = Cay((Z/mZ)^7, {e_0,...,e_6}) admits a directed Hamilton decomposition for every odd integer m >= 3. It defines a root-flat certificate consisting of three local conditions on a single root flat (row Latinness, layer bijectivity, primitive return maps) and proves that these suffice for the global decomposition. For odd m >= 7 an explicit uniform prefix-coordinate construction is given via one-layer prefix maps, a symbol-count criterion, and 7x7 count matrices that produce the seven Hamilton factors without search. The boundary cases m=3 and m=5 are handled by finite root-flat certificates whose validity is verified in Lean 4, with the full Cayley statement also formalized in the same development.

Significance. If the lifting theorem from the three root-flat conditions to the global edge decomposition holds, the result supplies both a uniform explicit construction for all sufficiently large odd m and machine-checked verification for the small cases, extending the earlier informal D_5(m) work to a formal interface in dimension 7. The combination of parameter-free count matrices for m >= 7 and Lean formalization of the boundary certificates constitutes a reproducible and verifiable contribution to the study of Hamilton decompositions in high-dimensional directed Cayley graphs on tori.

major comments (2)

[Theorem 2.4 (root-flat certificate theorem)] The central lifting theorem (that row Latinness, layer bijectivity and primitive return maps on one root flat together imply a directed Hamilton decomposition of the full D_7(m)) is load-bearing for every claim in the paper. The propagation argument showing that these local properties guarantee that each Hamilton factor covers every generator direction exactly once must be inspected in detail; any gap here would render both the prefix-count matrices and the Lean-checked certificates irrelevant to the global statement.
[Section 4 (prefix-count construction)] Section 4 states that the explicit 7x7 count matrices for odd m >= 7 satisfy the symbol-count criterion and produce all seven factors. Because the construction is claimed to be uniform and search-free, the paper should include at least one fully worked example (for a concrete m >= 7) showing that the generated prefix maps meet all three root-flat conditions; without this the reader cannot directly verify the method on a representative instance.

minor comments (2)

[Section 3] The notation for 'one-layer prefix maps' and the precise definition of the symbol-count criterion would benefit from an additional small illustrative diagram or table for m=9, for example.
[Section 5] The Lean formalization is cited as verifying both the Cayley statement and the boundary certificates, but no repository link or key lemma names are provided in the text; adding these would improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their positive evaluation of our manuscript and for the specific suggestions that will help improve its clarity. Below we address each major comment in turn.

read point-by-point responses

Referee: [Theorem 2.4 (root-flat certificate theorem)] The central lifting theorem (that row Latinness, layer bijectivity and primitive return maps on one root flat together imply a directed Hamilton decomposition of the full D_7(m)) is load-bearing for every claim in the paper. The propagation argument showing that these local properties guarantee that each Hamilton factor covers every generator direction exactly once must be inspected in detail; any gap here would render both the prefix-count matrices and the Lean-checked certificates irrelevant to the global statement.

Authors: We thank the referee for emphasizing the centrality of Theorem 2.4. The lifting theorem and its propagation argument are proved in complete detail in Section 3 of the manuscript. The argument proceeds by showing that the three conditions on the root flat propagate to ensure each of the seven factors is a single directed cycle that traverses every vertex and uses each generator direction exactly once per factor. This is established through a combination of combinatorial counting and cycle decomposition techniques. We are confident there are no gaps in this reasoning, which underpins both the general construction and the verified cases. Therefore, we do not plan to revise this section unless a specific issue is identified. revision: no
Referee: [Section 4 (prefix-count construction)] Section 4 states that the explicit 7x7 count matrices for odd m >= 7 satisfy the symbol-count criterion and produce all seven factors. Because the construction is claimed to be uniform and search-free, the paper should include at least one fully worked example (for a concrete m >= 7) showing that the generated prefix maps meet all three root-flat conditions; without this the reader cannot directly verify the method on a representative instance.

Authors: We agree with the referee that providing a worked example would enhance the reader's ability to verify the prefix-count construction. Accordingly, we will add to Section 4 a complete example for m = 7. This will include the explicit 7x7 count matrices, the computation of the one-layer prefix maps, verification that they satisfy the symbol-count criterion, and direct checking that the resulting root flat meets row Latinness, layer bijectivity, and primitive return maps. This revision will be made without changing the theoretical claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; sufficiency theorem and explicit constructions are independent of prior informal usage.

full rationale

The paper formalizes the root-flat certificate as a new definition and theorem in this work, with explicit prefix-count constructions for m >= 7 that are verified to satisfy the three local conditions via count matrices. Boundary cases m=3,5 rely on Lean formalization of the certificates rather than fitting or self-referential arguments. The self-reference to informal D_5(m) usage is present but does not bear the load of the central sufficiency claim or the global decomposition proof, which is developed here with independent verification steps. No self-definitional, fitted-prediction, or ansatz-smuggling reductions appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proof rests on standard facts about Cayley graphs on abelian groups and on the newly introduced root-flat abstraction; no numerical parameters are fitted and no new physical entities are postulated.

axioms (2)

standard math Cayley-graph definition and basic properties of directed tori over Z/mZ
The object D_7(m) is introduced via the standard Cayley-graph construction on the abelian group (Z/mZ)^7.
domain assumption Root-flat certificate theorem: the three local conditions imply a global Hamilton decomposition
The paper introduces this as a named theorem that reduces the global claim to local checks on a single flat.

invented entities (1)

root-flat certificate no independent evidence
purpose: Named verification interface that reduces Hamilton decomposition to three local conditions on one flat
Newly defined and stated as a theorem in the paper; no independent existence proof outside the construction is supplied.

pith-pipeline@v0.9.0 · 5536 in / 1517 out tokens · 53068 ms · 2026-05-09T18:44:23.445792+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Hamilton decompositions of all directed tori at odd modulus
math.CO 2026-05 accept novelty 8.0 full

For every d >= 2 and odd m >= 3 the directed Cayley graph D_d(m) admits a decomposition of its arcs into d directed Hamilton cycles.
Hamilton decompositions of all directed tori at odd modulus
math.CO 2026-05 unverdicted novelty 6.0 partial

Directed tori D_d(m) have directed Hamilton decompositions for all d ≥ 2 and odd m ≥ 3.

Reference graph

Works this paper leans on

17 extracted references · 3 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Alspach, J.-C

B. Alspach, J.-C. Bermond, and D. Sotteau,Decomposition into cycles I: Hamilton de- compositions, in Cycles and Rays, NATO ASI Series C, vol. 301, Kluwer, 1990, pp. 9–18

1990
[2]

Aquino-Michaels, Completing Claude’s cycles: multi-agent structured exploration on an open combinatorial problem, Version v1.0.0, Zenodo, 2026

K. Aquino-Michaels,Completing Claude’s cycles: Multi-agent structured exploration on an open combinatorial problem, version v1.0.0, Zenodo, doi:10.5281/zenodo.19737970, 2026

work page doi:10.5281/zenodo.19737970 2026
[3]

Z. R. Bogdanowicz,On decomposition of the Cartesian product of directed cycles into cycles of equal lengths, Discrete Applied Mathematics 229 (2017), 148–150

2017
[4]

Z. R. Bogdanowicz,Identifying Hamilton cycles in the Cartesian product of directed cycles, AKCE International Journal of Graphs and Combinatorics 17 (2020), no. 1, 534–538

2020
[5]

S. J. Curran and D. Witte,Hamilton paths in Cartesian products of directed cycles, in Cycles in Graphs, Annals of Discrete Mathematics 27 (1985), 35–74

1985
[6]

S. J. Curran and J. A. Gallian,Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey, Discrete Mathematics 156 (1996), 1–18. 12

1996
[7]

Darijani, B

I. Darijani, B. Miraftab, and D. Witte Morris,Arc-disjoint Hamiltonian paths in Cartesian products of directed cycles, Ars Mathematica Contemporanea 25 (2025), no. 2, P2.10

2025
[8]

M. F. Foregger,Hamiltonian decompositions of products of cycles, Discrete Mathematics 24 (1978), 251–260

1978
[9]

Keating,Multiple-ply Hamiltonian graphs and digraphs, in Cycles in Graphs, Annals of Discrete Mathematics 27 (1985), 81–88

K. Keating,Multiple-ply Hamiltonian graphs and digraphs, in Cycles in Graphs, Annals of Discrete Mathematics 27 (1985), 81–88

1985
[10]

Kőnig,Theorie der endlichen und unendlichen Graphen, Akademische Verlagsge- sellschaft, Leipzig, 1936

D. Kőnig,Theorie der endlichen und unendlichen Graphen, Akademische Verlagsge- sellschaft, Leipzig, 1936

1936
[11]

D. E. Knuth,Claude’s cycles, preprint, March 2026,https://www-cs-faculty. stanford.edu/~knuth/papers/claude-cycles.pdf(accessed 29 April 2026)

2026
[12]

Sanghyun Park,A diagonal Cartesian-power lift for Hamilton decompositions of directed tori, preprint, 2026

2026
[13]

SanghyunPark,Hamilton decompositions of the directed 3-torus: a return-map and odome- ter view, arXiv:2603.24708, 2026

work page arXiv 2026
[14]

Sanghyun Park,Hamilton decompositions of the directed 5-torus for odd modulus, arXiv:2604.27140, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[15]

Sanghyun Park,Torus Hamilton Decomposition Program, Lean 4 formalization, release0.0.2-d7, commit793467a8f92afcb19feaffc192ea6418123b117c, 2026, https://github.com/aria1th/Torus-Hamilton-Decomposition-Program/releases/ tag/0.0.2-d7

2026
[16]

W. T. Trotter, Jr. and P. Erdős,When the Cartesian product of directed cycles is Hamil- tonian, Journal of Graph Theory 2 (1978), no. 2, 137–142

1978
[17]

Witte and J

D. Witte and J. A. Gallian,A survey: Hamiltonian cycles in Cayley graphs, Discrete Mathematics 51 (1984), no. 3, 293–304. 13

1984