Self–Fibre Bundle Geometry: A Mathematical–Physical Framework for Local Spacetime Superposition and Emergent Quantum Dynamics

We develop a mathematical–physical framework in which the degrees of freedom of spacetime arise from the infinite–dimensional manifold of Lorentzian geometries Lor(3, 1) on a fixed smooth four–manifold M. The central hypothesis of the Self–Fibre picture is that spacetime provides its own fibre: each point of a base space B (representing “local spacetimes”) is assigned a Lorentzian metric gb ∈ Lor(3, 1), and the dynamics of geometry across different local spacetimes are encoded in a connection A : TB → TLor(3, 1). The curvature F = dA + 1/2 [A,A] measures the failure of local spacetimes to glue consistently across B. A natural action functional containing a Yang–Mills term on B, a vertical Einstein–Hilbert term on each fibre, and a vacuum functional V [g] is constructed. Its Euler–Lagrange equations yield a coupled system of generalized Yang–Mills equations and Einstein–type equations. We argue that quantum–like phenomena—including geometric phases, interference, vacuum fluctuations, and Born–rule–type probability weights—can be interpreted as emerging from the infinite–dimensional geometry of the fibre and the DeWitt supermetric. In this reading, holonomies of the Self–Fibre connection generate phase structures analogous to quantum interference, and the Gaussian measures on Lor(3, 1) induced by the supermetric suggest an emergent probability interpretation without postulating a Hilbert space. We further construct a solvable toy model with one–dimensional base space B = R, illustrating explicitly how holonomies, effective Planck constants, and Born–rule–type 1 behaviour arise from the Self–Fibre geometry at a heuristic level. Qualitative physical predictions are outlined at cosmological, quantum, and gravity–quantum–interplay scales (black holes, singularity resolution, variation of effective Planck constant, etc.). The present work is exploratory in nature: we do not claim a complete quantum theory of gravity, but rather propose that quantum theory and classical general relativity may arise as effective limits of a single geometric principle, based on the curvature and measure structure of a Self–Fibre bundle with infinite–dimensional fibre Lor(3, 1).