Odd Cusp Persistence in Ramanujan's Mock Theta Functions - Analytic Proofs, Twisted L-Functions, and Resonance Analogies

This paper establishes the analytic law of odd cusp persistence in Ramanujan’s mock theta functions, proving that divergences at odd roots of unity survive with pole-type behavior. Using Euler-Maclaurin summation and Zwegers’ non-holomorphic completion, the work elevates prior numerical observations into rigorous theorem. It introduces a twisted L-function with quadratic-phase coefficients, proves its analytic continuation, and derives a functional equation linking ss and 1−s1 - s. The persistence phenomenon is reframed as a global analytic structure, aligning mock theta behavior with automorphic L-functions and Langlands-type symmetry. A resonance analogy with Veneziano amplitudes in physics highlights interdisciplinary potential, suggesting that mock theta divergences mirror scattering poles in string theory. This paper anchors the Global Cusp Atlas and opens new programs in zero distribution, spectral interpretation, and analytic number theory.