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The PRIMOS Database: Prime-Product Addressing and GCD Similarity

by Pedro Henrique Corrêa Garcia · Opus 4.7 · 4 pp
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The PRIMOS Database:
Prime-Product Addressing and GCD SimilarityPedro Henrique Correa Garcia Joinville, Santa Catarina, Brazil garcia.pedro.wow@gmail.com · April 2026Abstract A configuration database whose keys are prime products inherits, directly from the Fundamental Theorem of Arithmetic, a set of operational properties that no hash-based or vector-based database can reproduce: unique decomposition, O(k) similarity via GCD, collision-free addressing, discrete exact representation (no floating-point), and natural compositional algebra. We describe the core architecture of the PRIMOS database, show how each property follows from arithmetic rather than engineering choice, and note the existence of over thirty deployed software systems that use this architecture across education, agriculture, authentication, music composition, materials science, and natural-language processing. 1IntroductionMost databases address their contents by arbitrary identifiers: hash keys, sequential integers, UUIDs, or embedding vectors. None of these addressing schemes carry semantic information intrinsically; the semantics live elsewhere — in joined tables, in metadata columns, in the similarity structure of an external model.PRIMOS takes a different approach. It addresses every record by the prime factorization of its content. The address is the semantics, and the semantics is the address. The consequences of this identification are the subject of the present article.Unrecognized environment: keyideaThe address is the content. The content is the address. Every operation on the database is an operation on integers.2Prime-product addressingDefinitionA prime-product address is a positive integer n = ∏i iei where the ei are non-negative integers and all but finitely many are zero. The factorization is unique by the Fundamental Theorem of Arithmetic.Two records are equal if and only if their addresses are equal as integers, which by the Fundamental Theorem is equivalent to their prime factorizations being identical.PropositionPrime-product addressing is collision-free: distinct semantic content produces distinct addresses, and no hash collisions are possible.ProofFollows immediately from the uniqueness of prime factorization. Two distinct factorizations give distinct integers, so two distinct semantic contents give distinct addresses.RemarkThis is not a probabilistic claim. A hash-based address space can claim collision-freeness only up to its hash output length; PRIMOS's addressing is collision-free absolutely, as a theorem of number theory.3GCD similarityThe greatest common divisor of two integers carries, in the prime-product reading, the structural overlap between two records.