Triadic Neurosymbolic Engine: Prime Factorization as a Neurosymbolic Bridge: Projecting Continuous Embeddings into Discrete Algebraic Space for Deterministic Verification

We propose a method for bridging continuous neural embeddings and discrete symbolic reasoning bymapping dense vectors to composite prime integers via Locality Sensitive Hashing (LSH), effectively enabling exact set operations on LSH signatures via unique factorization. Each LSH hyperplane is assigneda unique prime; a concept’s integer is the product of primes for its active hyperplanes. This encodingyields three algebraic operations impossible under cosine similarity or Hamming distance: (1) logical subsumption via divisibility, (2) concept composition via LCM, and (3) abductive gap analysis via GCD factorization. We benchmark on a 107-word vocabulary across a series of 9 experiments, achieving 28.4×faster pairwise verification than cosine similarity, and provide honest analysis of limitations. Code: https://github.com/arturoornelasb/Triadic-Neurosymbolic-Engine.