Return-section/odometer framework for the directed 4-torus

Ascertain whether the directed four-dimensional torus D4(m)=Cay((Z_m)^4,{e1,e2,e3,e4}) admits a nested return-section analysis that isolates a finite-defect clock-and-carry (odometer) core analogous to the mechanism established for D3(m), thereby providing a comparable structural framework in dimension four.

Background

The proof for D3(m) uses a return-section (Poincaré section) reduction to expose a clock-and-carry odometer mechanism on P0 and, in the even case, a finite-defect repair via Route E.

The author proposes extending this methodology to higher dimensions, with the directed 4-torus as the first test case, to see if the same finite-defect odometer core emerges under a nested return-section analysis.

References

Several structural questions remain. Second, the proof suggests a higher-dimensional program in which low-layer defect arrangements are analyzed by nested return sections until a finite-defect odometer emerges. The first natural test case is the directed $4$-torus, where one can ask whether the same return-section viewpoint still isolates a finite-defect clock-and-carry core.

— Hamilton decompositions of the directed 3-torus: a return-map and odometer view  (2603.24708 - Park, 25 Mar 2026) in Section 6, Discussion and outlook