Internal Spectral Theory: A Minimal Framework for the Emergence of Four-Dimensional Physics

Internal Spectral Theory (IST): A Minimal Framework for the Emergence of Four-Dimensional Physics This work presents the minimal framework of Internal Spectral Theory (IST), a research programme investigating whether the fundamental structures of four-dimensional physics can emerge from an abstract spectral system rather than from a predefined spacetime geometry. The paper introduces the core admissible spectral framework, including the Hilbert space of admissible states, a self-adjoint spectral operator, a ternary structural operator, and the associated readability principle. Within this setting, four-dimensional physics is interpreted as an effective regime arising from partial spectral accessibility instead of being assumed as a fundamental background. The manuscript develops the concepts of admissible factorisation, joint irreducibility, readability regimes, and information ordering. It establishes criteria for the existence of autonomous admissible spectral packets and analyses the conditions under which effective geometric structures may emerge from the underlying spectral dynamics. No spacetime, metric, locality, or dimensional constants are postulated at the fundamental level. Masses, chirality, time, and gauge structures are treated as open research programmes rather than established results. Throughout the manuscript, a clear distinction is maintained between mathematical results proved within the framework, concepts borrowed from established theories, and questions that remain open. This publication provides the foundational reference for the IST programme and is intended as a basis for subsequent developments concerning fermions, masses, quantum coherence, and cosmology. Keywords: spectral theory; mathematical physics; emergent spacetime; spectral geometry; operator theory; quantum foundations; four-dimensional physics.