A unified mathematical framework, referred to as Picard—Fuchs Hypergeometric Manifolds (PFHM), is introduced to integrate modular symmetry, coupled dynamics and energy conservation. PFHM are constructed using a synthesis of Ramanujan’s real period functions, Picard—Fuchs differential equations and Gaussian hypergeometric functions. We argue that PFHM provide an effective representation of two-dimensional coupled subsystems embedded in three-dimensionalmanifolds with dihedral symmetry. These coupled subsystems exhibit constrained energy reciprocity, making PFHM a robust tool for elucidating stable, closed and homoclinic orbits in Hamiltonian systems. An application of the proposed method is explored in the context of quantum entanglement. The intrinsic energetic reciprocity and symmetry of PFHM are shown to be analogous to the nonlocal correlations in entangled quantum systems. Modelling entangled pairs as constrained subsystems, the PFHM framework sheds new light on the energy dynamics and nonlocal correlations underpinning quantum entanglement.