arxiv: 2604.27140 · v1 · submitted 2026-04-29 · 🧮 math.CO · math.GR

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Hamilton decompositions of the directed 5-torus for odd modulus

Sanghyun Park

Authors on Pith no claims yet

Pith reviewed 2026-05-07 09:59 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords Hamilton decompositiondirected torusCayley graphLatin tablereturn mapfive-dimensionalodd modulus

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The pith

The directed five-dimensional torus has a Hamilton decomposition for every odd integer m at least 3.

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

The paper proves that for every odd integer m at least 3, the five-dimensional directed torus graph on the group (Z_m)^5 with generators the standard basis vectors admits a decomposition into directed Hamilton cycles. The proof introduces a cyclic scheduling of the five generators across layers, where one layer uses a zero-set Latin table to choose the direction. This table is shown to form a matching through a finite exact-cover certificate, and the Hamiltonicity follows from analyzing a normalized return map that collapses to a single cycle on a chosen section. A reader might care because such decompositions help in designing efficient routing protocols and understanding the structure of multidimensional periodic networks.

Core claim

We prove that the directed five-dimensional torus D_5(m) = Cay((Z_m)^5, {e0, e1, e2, e3, e4}) has a Hamilton decomposition for every odd integer m ≥ 3. This is the first higher-dimensional case in which the return-map method requires a genuine zero-set selector rather than an odometer-type correction. The construction assigns the five outgoing generators by a cyclic layer schedule with one non-constant layer determined by a zero-set Latin table; an explicit finite exact-cover certificate proves that this layer is a matching. By cyclic symmetry, Hamiltonicity of all color classes reduces to a single normalized return map. For m ≥ 5, an explicit first-return calculation on the section p = 2 of

What carries the argument

A cyclic layer schedule of the five generators with one non-constant layer set by a zero-set Latin table (verified as a matching by finite exact-cover certificate), reduced by symmetry to a normalized return map on the p=2 section that forms a single cycle.

If this is right

The five color classes produced by the cyclic schedule are all Hamilton cycles.
For every m at least 5 the normalized return map on p=2 yields exactly one cycle whose total length is m^4.
The special case m=3 is covered by an explicit finite cycle certificate.
A Lean 4 formalization independently confirms the underlying Cayley graph statement and all finite certificates.

Where Pith is reading between the lines

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

The zero-set selector technique may apply to other dimensions where odometer corrections are insufficient.
The explicit certificates make the construction suitable for direct implementation in network routing software.
The reliance on Latin tables links the result to broader questions in combinatorial design theory for Cayley graphs.

Load-bearing premise

The zero-set Latin table defines a matching whose exact-cover certificate is finite and correct, and the normalized return map on section p=2 induces a single cycle of total length m^4.

What would settle it

An explicit computation of the return map for m=5 on the p=2 section that produces more than one cycle or excursion lengths summing to anything other than 625 would falsify the claim for that modulus.

read the original abstract

We prove that the directed five-dimensional torus $D_5(m) = \operatorname{Cay}((\mathbb{Z}_m)^5, \{e_0, e_1, e_2, e_3, e_4\})$ has a Hamilton decomposition for every odd integer $m \geq 3$. This is the first higher-dimensional case in which the return-map method requires a genuine zero-set selector rather than an odometer-type correction. The construction assigns the five outgoing generators by a cyclic layer schedule with one non-constant layer determined by a zero-set Latin table; an explicit finite exact-cover certificate proves that this layer is a matching. By cyclic symmetry, Hamiltonicity of all color classes reduces to a single normalized return map. For $m \geq 5$, an explicit first-return calculation on the section $p = 2$ gives one induced cycle whose excursion lengths sum to $m^4$. The remaining modulus $m = 3$ is settled by a printed finite cycle certificate. A companion Lean 4 formalization provides an independent machine verification of the Cayley statement and the finite certificates; source, audit scripts, and ancillary search code are available at https://github.com/aria1th/Torus-Hamilton-Decomposition-Program.

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

Park gives an explicit, Lean-verified construction for Hamilton decompositions of the directed 5-torus at odd orders.

read the letter

The main takeaway is that this paper gives an explicit construction proving the directed 5-torus has a Hamilton decomposition for every odd m at least 3, with the whole thing backed by Lean 4 verification and public code. What is new is the application to dimension 5, where the return-map method needs a genuine zero-set selector for one layer instead of a simpler odometer correction. The paper assigns generators via a cyclic layer schedule, uses a zero-set Latin table for the non-constant layer, and proves it is a matching with a finite exact-cover certificate. By symmetry, everything reduces to showing that the normalized return map on the p=2 section induces one cycle of length m^4 for m at least 5, with m=3 checked directly by a printed certificate. The paper does well on the verification side. The Lean formalization certifies the Cayley graph setup and the finite checks independently, which removes a lot of room for error in the combinatorial details. The code repository allows direct reproduction. Soft spots are limited. The general case for m greater than or equal to 5 uses a first-return calculation that is explicit but specific to this setup, and m=3 is separate. The result is only for odd m, as even cases are presumably harder or open. No load-bearing assumption seems to fail on inspection. This is for people working on decompositions of toroidal Cayley graphs or directed Hamilton cycles in abelian groups. A reader who follows the program of resolving these questions dimension by dimension will get concrete value from the construction and the machine check. It deserves a serious referee because the formalization provides unusually strong evidence for the existence result. I recommend sending it out for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the directed five-dimensional torus D_5(m) = Cay((Z_m)^5, {e_0, e_1, e_2, e_3, e_4}) admits a Hamilton decomposition into five directed Hamilton cycles for every odd integer m ≥ 3. The proof supplies an explicit construction via a cyclic layer schedule in which one layer is given by a zero-set Latin table; the matching property of this layer is established by a finite exact-cover certificate. By cyclic symmetry, the Hamiltonicity of all color classes reduces to verifying a single normalized return map on the p=2 section, which is shown to consist of one cycle of total length m^4 for m ≥ 5 by direct first-return calculation and settled for m=3 by an explicit finite cycle certificate. A companion Lean 4 formalization independently certifies the Cayley-graph statements and the finite certificates, with source code and audit scripts publicly available.

Significance. If the result holds, it constitutes the first higher-dimensional case in which the return-map method for Hamilton decompositions of directed tori requires a genuine zero-set selector rather than an odometer-type correction. The combination of an explicit combinatorial construction, finite exact-cover and cycle certificates, and machine-checked Lean verification provides strong, independently auditable evidence. This advances the program of constructing Hamilton decompositions for Cayley graphs on abelian groups and supplies a template for handling higher-dimensional tori where simpler corrections fail.

minor comments (2)

[Abstract] The abstract states that the m=3 case is settled by a 'printed finite cycle certificate'; a brief pointer to the location of this certificate (or its length) in the main text would improve readability for readers who do not immediately consult the supplementary material.
[Construction] The description of the zero-set Latin table in the construction section would benefit from an explicit small-m example (e.g., m=3 or m=5) showing the table entries and the resulting matching, to make the exact-cover certificate more immediately verifiable without external code.

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; explicit construction with independent machine verification

full rationale

The derivation supplies an explicit cyclic-layer construction whose matching property is witnessed by a finite exact-cover certificate and whose Hamiltonicity reduces to a single normalized return-map cycle whose length sum is computed directly for odd m. Both the Cayley-graph statements and the finite certificates are independently certified by a companion Lean 4 formalization, which constitutes external, machine-checked evidence rather than a self-referential fit or self-citation chain. No equation or claim reduces to its own inputs by construction, and the central result is not forced by any ansatz or uniqueness theorem imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard facts about Cayley graphs, Latin squares, and exact covers; no free parameters, invented entities, or ad-hoc axioms are introduced beyond domain-standard combinatorial assumptions.

axioms (2)

standard math Cayley graph on (Z_m)^5 with the five standard generators is well-defined and vertex-transitive
Invoked in the opening definition of D_5(m).
domain assumption A zero-set Latin table on the appropriate alphabet yields a perfect matching when the exact-cover condition holds
Used to certify the non-constant layer.

pith-pipeline@v0.9.0 · 5519 in / 1302 out tokens · 53324 ms · 2026-05-07T09:59:27.254181+00:00 · methodology

discussion (0)

Forward citations

Cited by 3 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.
Hamilton decompositions of the directed 7-torus at odd modulus via root-flat certificates and a prefix-count construction
math.CO 2026-05 unverdicted novelty 6.0 partial

Directed 7-tori D_7(m) for odd m >= 3 admit Hamilton decompositions, established by root-flat certificates for m=3,5 and a uniform prefix-count construction for m>=7, with Lean 4 verification of the boundary cases.

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages · cited by 2 Pith papers

[1]

Alspach, J.-C

B. Alspach, J.-C. Bermond, and D. Sotteau, Decomposition into cycles I: Hamilton de- compositions, inCycles and Rays(G. Hahn, G. Sabidussi, R. E. Woodrow, eds.), NATO ASI Series, vol. 301, Kluwer, 1990, pp. 9–18

1990
[2]

Aquino-Michaels, Completing Claude’s cycles, Preprint, March 2026.https: //github.com/no-way-labs/residue, accessed 29 April 2026

K. Aquino-Michaels, Completing Claude’s cycles, Preprint, March 2026.https: //github.com/no-way-labs/residue, accessed 29 April 2026

2026
[3]

S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs — a survey,Discrete Mathematics156 (1996), 1–18

1996
[4]

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
[5]

M. F. Foregger, Hamiltonian decompositions of products of cycles,Discrete Mathematics 24 (1978), 251–260. 18

1978
[6]

Keating, Multiple-ply Hamiltonian graphs and digraphs, inCycles in Graphs,Annals of Discrete Mathematics27 (1985), 81–88

K. Keating, Multiple-ply Hamiltonian graphs and digraphs, inCycles in Graphs,Annals of Discrete Mathematics27 (1985), 81–88

1985
[7]

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
[8]

Park, Hamilton decompositions of the directed 3-torus: a return-map and odometer view, arXiv:2603.24708, 2026

S. Park, Hamilton decompositions of the directed 3-torus: a return-map and odometer view, arXiv:2603.24708, 2026

work page arXiv 2026
[9]

Park, Torus Hamilton Decomposition Program, Lean 4 formalization repository, ancillary audit files, and empirical search code (including even-modulus exploration),

S. Park, Torus Hamilton Decomposition Program, Lean 4 formalization repository, ancillary audit files, and empirical search code (including even-modulus exploration),
[10]

Commitfe67dbd0cb7af889936cc28660a88e5651daed62;https://github.com/ aria1th/Torus-Hamilton-Decomposition-Program, accessed 29 April 2026

2026
[11]

W. T. Trotter and P. Erdős, When the Cartesian product of directed cycles is Hamiltonian, Journal of Graph Theory2 (1978), 137–142

1978
[12]

Witte and J

D. Witte and J. A. Gallian, A survey: Hamiltonian cycles in Cayley graphs,Discrete Mathematics51 (1984), 293–304. 19

1984