Deep Homology of Mathematical Objects (DHMO): A Unified Framework for Cross-Domain Universal Roots

Modern mathematics has traditionally been characterized by structural specialization, where disparate branches such as algebra, geometry, topology, and number theory develop independent languages and methodologies. Although categorical frameworks have successfully bridged various subfields, a comprehensive foundational paradigm that systematically extracts the deepest common origins of seemingly unrelated mathematical structures remains incomplete. This paper introduces the Deep Homology of Mathematical Objects (DHMO) theory. The core thesis of DHMO is that complex, specialized mathematical objects descend from a shared, abstract universal precursor space, denoted as the Z-space. By formalizing the relational dynamic through the universal mapping schema A <- Z -> B, DHMO provides a rigorous methodology for tracing distinct mathematical entities back to a common ancestral source. This framework unifies diverse mathematical domains, reveals hidden structural symmetries, and establishes a new epistemological foundation for the unity of mathematics without relying on empirical validation.