"The Geometric Origin of Mock Modularity: Z₂ Involutions, Twisted de Rham Cohomology, and the Bruinier--Funke Exact Sequence. " Paper XV in MEF Series.

This paper proves that the entire mathematical structure of mock modular forms—including Zwegers' non-holomorphic completion, the holomorphic shadow, the Bruinier–Funke exact sequence, and the arithmetic of their Fourier coefficients—has a single, unified geometric origin: the Z₂ involution ² = id on a torus orbifold T²/Z₂. We establish the Twisted Torus Theorem, demonstrating that the mock modular decomposition of a harmonic Maass form into holomorphic and non-holomorphic parts is identically the Z₂ eigenspace decomposition of the orbifold's state space. We construct an explicit isomorphism proving that the Bruinier–Funke differential ₖ is the analytic realisation of the Tate cohomology differential D = 1 -, establishing that BRST nilpotency descends directly from the geometric identity D N = 0. Furthermore, we identify the non-holomorphic Zwegers shadow with the twisted de Rham cohomology of the flat Z₂ local system over the orbifold. Arithmetically, we rigorously prove that the geometric Z₂ projection operator acts as the exact analytic multiplier (1 - 2^-s) on the p = 2 Euler factor of the divisor Dirichlet series, completely preserving multiplicativity over all odd primes. This pure mathematical unification resolves the geometric meaning of mock modularity, answering Ramanujan's century-old intuition: an interacting mock modular form is simply a free modular form defined on a space that has been folded in half. Keywords: mock modular forms, Bruinier–Funke exact sequence, Tate cohomology, twisted de Rham cohomology, Zwegers completion, T²/Z₂ orbifold, analytic number theory.