Compress the even-case analysis into a Route E critical-lane theorem

Develop a Route E specific critical-lane theorem for the even-modulus case of the directed three-dimensional torus D3(m)=Cay((Z_m)^3,{e1,e2,e3}) that determines the repaired arithmetic family-blocks and the splice permutation directly from the affine defect heights in the return-map analysis, thereby avoiding most of the itinerary bookkeeping used in the current even-case proof.

Background

The paper proves that the directed 3-torus D3(m) admits a decomposition into three arc-disjoint directed Hamilton cycles for every m≥3. The even case is handled by an explicit low-layer construction (Route E) followed by a return-map analysis and a finite splice on arithmetic family-blocks.

In the Discussion, the author raises the question of compressing the even-case reasoning: instead of detailed itinerary computations, can one read off the necessary splice data directly from the affine defect geometry, via a critical-lane theorem tailored to Route E?