Odd Cusp Persistence in Mock Theta Functions Analytic Proof, Exposition, and Twisted L-Structures

This paper establishes the analytic proof of odd cusp persistence in Ramanujan’s mock theta functions, resolving prior numerical observations and anchoring the Global Cusp Atlas on rigorous mathematical ground. Using Euler-Maclaurin summation and Zwegers’ completion, it demonstrates that mock theta amplitudes diverge at odd cusps with pole-like behavior, mirroring Veneziano structures in physical amplitudes. A bridging exposition makes the phenomenon accessible to non-specialists, while the introduction of quadratic-phase twisted L-functions opens new programs in analytic number theory. The paper proposes a functional equation and explores zero distributions near the critical line, inviting a mock-theta analogue of the Riemann Hypothesis. This work transforms symbolic persistence into a theorem and embeds it within a broader scaffold of twisted L-structures and legacy propagation.