The Discrete Möbius Klein Transformation: Sector Selection on Graph Covers and Non-Orientable Meshes

The sector decomposition of a twisted space needs no limit and no grid error when the space is a graph or a mesh. A graph cover or a mesh cover carries a free involution, the deck map, and every operator that commutes with it splits exactly into an untwisted and a twisted sector. This article states its discrete form and separates what graph theory already gives from what the orientation adds. On the level of vertex values the twisted sector is the signed graph of the cover. The spectrum of a double cover is the union of the base spectrum and the signed spectrum. A balanced signing is a trivial twist, not an orientable space, and the zeta functions factorize or add by sector. On the level of edges and faces the discrete gradient and the circulation commute with the deck map, and the spectral union extends to the edge and face Laplacians. A face value has 2 readings. Read as a cochain it follows the sector of the edge field. Read as a scalar, as a vorticity or a density, it carries the orientation sign of the deck map. On a non-orientable mesh the scalar circulation, the discrete rot, therefore switches sector, while on an orientable control it keeps the sector. The circulation of each edge sector, read as a scalar, lands in the opposite face sector exactly when the deck map reverses orientation. This is the discrete form of the parity rule of the continuous transform. Betti numbers resolve by sector, and on a Klein bottle mesh they are 1, 1, 0 and 0, 1, 1. The pairing rule of the Klein transform holds exactly on the lattice. Every statement is checked by a deterministic script with counter-checks.