Abstract
<title>Abstract</title> <p>We present a unified mathematical framework based on quadratic spectral interactions that establishes formal connections between four seemingly unrelated phenomena: the distribution of prime gaps, the evolution of dark energy, the emergence of neural dynamics, and the structure of spacetime. The framework is constructed within a rigorous Hilbert space setting, where a self-adjoint spectral operator D governs the dynamics through a quadratic functional of the form ⟨f, Df ⟩ + λ⟨f, D 2 f ⟩. The central result is a quadratic spectral law λ n γ 2 n |c n | 2 , where γ n represent spectral frequencies and c n are expansion coefficients. This law reveals that the statistical properties of prime gaps, the acceleration of cosmic expansion, the oscil-latory patterns of neural activity, and the quantum corrections to general relativity may all originate from the same mathematical structure. Numerical simulations using the first 100 non-trivial zeta zeros validate the convergence properties of the framework. All mathematical elements are explicitly defined, including the spectral operator, test functions f x (t) = e −t 2 cos(t log x), and the interaction parameter λ. This work is presented as a mathematical proposal for further investigation, with the understanding that the connections established are heuristic and intended to motivate future research rather than to assert proven physical laws.</p>