Higher-dimensional directed tori: nested return maps, odometer models, and parity behavior

Investigate whether higher-dimensional directed tori Dd(m)=Cay((Z_m)^d,{e1,...,ed}) for d≥4 admit nested return-map analyses that yield finite-defect odometer models comparable to those used for D3(m), and determine whether odd and even ambient dimensions exhibit genuinely different parity behavior in such constructions.

Background

Beyond the 3-dimensional case, the author envisions a broader program where low-layer defects are organized by successive return sections until a finite-defect odometer emerges.

This raises two related questions: whether such nested-return/odometer models extend to composite higher-dimensional products of directed cycles, and whether there are systematic parity distinctions between odd and even ambient dimensions.