Omega Interlevel Numbers: Hierarchical Numerical Identity, Path Algebra, and Canonical Geometry

Omega Interlevel Numbers is an exploratory mathematical extension of Omega-NR — The Universe That Never Repeats. The framework investigates numerical objects whose identity is not determined by visible value alone, but may also include hierarchical structure, ordered path/history, realization identity, and local emergence or geometric data. The current research candidate develops the Omega Numerical Uniqueness Axiom, hierarchical depth projections, reconstructive structural signatures, a first-difference metric, the Local Emergence Law, canonical structural mass, a canonical nested geometric representation, a noncommutative Omega Path Algebra, computational stress tests, and explicit limitations. A central principle of the framework is: “Structure may be reconstructed; identity cannot be replayed.” This release is a preliminary research framework. It does not claim to establish a new branch of mathematics, complete novelty relative to all prior mathematical literature, or a physical proof of Omega-NR. Independent criticism, counterexamples, formal proofs, reproducible experiments, and alternative formulations are welcome. Parent Omega-NR:Version DOI: 10.5281/zenodo.23624320Concept DOI: 10.5281/zenodo.23624319 GitHub:https://github.com/saydofs/omega-research-pack/tree/main/omega-interlevel-numbers GitHub release:https://github.com/saydofs/omega-research-pack/releases/tag/interlevel-v0.3