Product Rigidity of Marked Periodic Nerves for Unequal Simplex-Strand Completion Fibers

Abstract

The paper grows out of our research on the topology of optimal-completion fibers, particularly the displacement-marked structures and higher-rank phenomena arising from shortest-completion quotients. Its central question is whether the product topology observed in periodic nerve constructions survives when the constituent simplex strands have different widths and no longer form a literal Cartesian product. The paper proves that the resulting quotient nevertheless retains the homotopy type of a product of a finite bouquet of circles with a circle, and identifies the associated fundamental-group and displacement-marking structure. It also develops a fixed six-action family exhibiting a controlled topological transition as the completion target varies. Its main novelty is a rigidity theorem demonstrating that unequal local simplex geometry does not necessarily destroy the global product homotopy type, thereby extending optimal-fiber topology beyond the equal-width and directly factorable settings.