Zero-Leap Mathematical System A Self-Consistent Axiomatic System Allowing Singularity Division-by-Zero and Modeling of the Big Bang Singularity

This submission includes two closely related preprints that systematically establish the Zero-Leap Mathematical System and apply it to unify core problems in cosmology, quantum gravity, black hole physics, string theory, and cosmic origin. The first paper constructs a self-consistent axiomatic system that legalizes singularity division-by-zero, redefines singular zero \(0_{\Omega}\) and leap infinity \(\infty^{*}\), builds a three-layer nested number domain and Zero-Leap Calculus, and realizes rigorous mathematical modeling for the Big Bang singularity and black hole singularities, resolving the divergence breakdown of classical mathematics and general relativity at spacetime singularities. The second paper extends the Zero-Leap framework to a unified physical theory, explaining dark energy as transdimensional flux from the high‑dimensional leap field, modeling black holes as mini-bubble universes, providing a foundational mathematical basis for quantum gravity, reconstructing branes and higher dimensions in string/M‑theory, mathematically describing cosmic creation “from nothing to everything” via singular zero normalization, and deriving the origin of time, dimensions, and causality from Zero-Leap operations. Together, these two works form a complete, closed, and computable system that unifies mathematics, general relativity, quantum mechanics, string theory, and cosmology, offering a new path toward the ultimate unified theory of physics.