Energy–Entropy–Time Coupling in the Timeless Energy Principle: Mathematical Core and Observable Cosmological Signatures

This work presents a mathematical and cosmological formulation of the Timeless Energy Principle (TEP), introducing the constitutive invariant Ξ = E/(tS), where energy, entropy, and emergent time are coupled within a phenomenological framework. The paper develops scalar-field realizations, modified cosmological evolution equations, and potential observational signatures related to dark energy dynamics and gravitational-wave propagation.