Klein Bottle Degeneracy Spectrum: Zero Mode Structure of the Twisted Laplacian on Non-Orientable Surfaces

We prove that the twisted Laplacian ∆ξ acting on sections of the orientation bundle ξ→M over the Klein bottle has a unique zero mode (up to scalar): the binary step function φstep = [+1,...,+1,−1,...,−1], with dim ker(∆Kξ ) = 1. The Möbius strip has no zero mode (ker(∆Mξ ) = {0}), and the cylinder admits the constant mode. The three surfaces form an ordered degeneracy spectrum determined solely by topology: each antiperiodic identification eliminates one class of degenerate equilibrium. The proof proceeds by direct verification and a boundary-equation uniqueness argument, with a monodromy interpretation. The result is verified at three independent levels: analytically (complete algebraic proof), computationally (SymPy exact rational arithmetic, N = 8 to 128), and formally (Lean 4/Mathlib4, machine-checked, all goals closed). A Majorana correspondence establishes that the Klein bottle zero mode is the closed-surface generalisation of Majorana zero modes in topological superconducting nanowires, extending the Majorana literature from one-dimensional Möbius geometries to the full closed Klein bottle. The cultural inversion zones detected by the Concordia platform across 308 NUTS-2 regions are identified as the empirical realisation of this topologically protected zero mode. Supplementary files include the Lean 4 verification (Theorem2.lean) and SymPy verification (verify_theorem2.py).