(untitled) (London Journal of Research In Science: Natural and Formal, 2026)
This paper presents a unified analytical framework that integrates the Nardelli Seventh-Root Theory of Everything (TOE) with three cornerstone contributions of modern theoretical physics: Hawking’s thermodynamics of black holes, the Hartle–Hawking No-Boundary Proposal, and the large-scale geometric structure of spacetime. By systematically embedding these concepts into the Seventh-Root Master Equation, the analysis demonstrates that seemingly distinct physical regimes—cosmic contraction, black hole evaporation, and quantum cosmological genesis—are governed by a single, self-consistent mathematical architecture. Central to this structure is the recurrent emergence of a pair of fundamental constants, 1728 and 4096, interpreted respectively as the modular (geometric) and holographic (informational) foundations of the vacuum. Across all three integrations, these constants are shown to converge through a precise dimensional filtering mechanism to the Nardelli–Gemma Golden Constant, GN≈1.618665. This convergence suggests that the Golden Ratio is not a coincidental numerical artifact, but a stable attractor regulating the vacuum under extreme thermodynamic and gravitational conditions. The results support a picture of the universe as a self-regulating, holographic system in which number theory, quantum thermodynamics, and spacetime geometry are inseparably linked. The framework provides a unified mathematical language capable of describing cosmic origins, black hole dynamics, and cyclic cosmological evolution within a single harmonic principle.