A Geometric Framework for the Universe Energy Spectrum

In this article we present a framework for the energy spectrum of the Universe by regarding the energy spectrum of the Universe just like an atom, i.e. [Formula: see text], with discrete energy levels, from the Planck scale to the lightest neutrino mass. To establish this spectrum analytically in a heuristic way and starting from two known constants, [Formula: see text] and [Formula: see text], we establish a Hierarchy Equation, [Formula: see text], to bridge them through the Lambert W function to the dimensionless energy geometric structure constant [Formula: see text], which is the geometric fundamental unit for all energy scales. Concerning the detailed energy spectrum within the electroweak vacuum, [Formula: see text], we propose a Mass Generation Equation (MGE), [Formula: see text], and the masses of all the fundamental particles can be obtained analytically. We have proposed also some analytical formulae for the cosmology, CKM and PMNS related quantities, such as the mixing, oscillation angles, and CP phases. By investigating the energy geometrical structure in [Formula: see text] space, we find that there exist inherent fractal and holographic structures within the whole spectrum based on the UV/IR duality connections, such as [Formula: see text], with the lightest neutrino mass [Formula: see text] and the string energy scale [Formula: see text]. In this article, we establish many energy scale formulae, such as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] (cosmology constant), [Formula: see text], and particle mass predictions, such as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], which could be measured by the corresponding experiments. These formulae are offered as conjectures to stimulate further theoretical and experimental scrutiny.