A Multidimensional Arithmetical Ontology for High-Parameter Cytometry: Gaussian Primes and Cryptographic Hashing for Cell Population Standardization
Background. The rapid expansion of high-dimensional single-cell technologies has outpaced the development of standardized metadata infrastructure. Current text-based nomenclature for cell populations and gating strategies lacks computational rigor, hindering cross-study reproducibility and automated machine learning. Our previous mathematical frameworks, the Prime Population System (PPS) and the Unique Gating Identifier (UGI), proposed arithmetical solutions but were limited by biological fragility (no inherent link between a protein and its structural isoforms) and by computational bottlenecks caused by exponential integer growth. Objective. To develop a scalable, multidimensional arithmetical ontology that unambiguously identifies cell populations and hierarchical gating strategies without biological ambiguity or computational overflow. Methods. We transition the base marker dictionary from the one-dimensional real axis to the complex plane of Gaussian integers (a+bi). The reviewed human proteome of the UniProt Knowledgebase is paired with Gaussian primes satisfying a > b > 0, each list being totally ordered by a key intrinsic to it, so that the pairing is determined by position alone. Each protein receives a contiguous reserved block of 64 primes: slot 0 addresses the protein with isoform unresolved, and slot k addresses UniProt isoform k, the slot index being read directly off the accession. A Bipartite Complex Ontology encodes positive (a+bi) and negative (b+ai) expression states through Conjugate Pair Symmetry. To bypass the 64-bit integer limits of Goedel exponentiation, hierarchical gating sequences are held as vectorized arrays of complex numbers and compressed by SHA-256 deterministic hashing. Results. Pairing against the reference release assigns 53,296 identifiers, 20,431 protein-level and 32,865 isoform-level, with no collisions and no manual curation at any step. Every isoform record in the release resolves under a single rule, including the 329 records whose accession prefix differs from the entry that cites them. The geometric symmetry conserves biological state, distinguishing positive, negative and isoform-resolved expression upon factorization over [Z][i]. Localized vectorization with SHA-256 hashing yields immutable 64-character hexadecimal fingerprints for entire gating strategies, resolving the integer explosion problem. Conclusion. By combining complex plane topology with cryptographic hashing, this framework provides a universally scalable, computationally lightweight and biologically accurate registry system. The ontology eliminates nomenclature ambiguity and opens the way to machine learning interoperability and centralized global cytometry databases.