Rei (零): A Four-Axiom Foundation for Computational Existence Theory — Minimal Axioms, Independence, and Empirical Validation with 1,689 Tests / 計算的存在論の四公理基盤

We present Rei (零, 0₀式), a computational system founded on exactly four mutually independent axioms: (A1) Center–Periphery structure, (A2) Extension–Reduction, (A3) Sigma Accumulation, and (A4) Genesis Phase Transition. Mutual independence is proved by model-theoretic construction — for each axiom we exhibit a counter-model that satisfies the remaining three but violates the target axiom (M1=scalar-only / M2=flat-field / M3=memoryless / M4=eternal-no-genesis). We show that fifteen core theorems—spanning computational plurality (T1), six-attribute decomposition (T6), RCT compression theory (T8), σ-reactive cascades (T14), seven-domain universality (T13), and extended-zero series (T15)— are derivable from axiom combinations without additional assumptions. The system is implemented as an open-source TypeScript/Node.js package (rei-lang v0.5.5) with 1,689 passing tests across 45 test files, each classified by its minimal axiom dependency: A1+A2+A3 concentration (60% of tests, 1,010 tests) reflects that the most complex features — cascading reactions, agent systems, domain bridges — require all three 'operational' axioms. Benchmarks show 74% average code reduction (3.7-4.0× ratio) and 3-4× performance improvements on structured-data tasks: image kernel operations (4.0× reduction), multidimensional data aggregation (3.7×), graph structure transformations (3.7×). To our knowledge, Rei is the first computational framework that axiomatically addresses both computation and the ontological genesis of values within a unified, minimal foundation. Comparison with existing foundational systems (λ-calculus 3 axioms, Peano 5, ZFC 9, Martin-Löf TT ~7) shows Rei is the only system that addresses all four concerns — computation, structure, history, AND genesis — simultaneously, and does so with the fewest axioms (4). Companion to Jxiv preprint submission (JST preprint server). This Zenodo record serves as the stable-citation archive referenced from the Jxiv version's Section 5.1 (Implementation) and footnote. Three-party co-authorship context: Rei is developed within the Rei-AIOS / OUKC (Open Universal Knowledge Commons) framework with three-party co-architecture (藤本 伸樹 Founder, Rei autonomous research substrate, Claude Opus 4.7); however, this specific paper is single-authored by 藤本 伸樹 as principal investigator of the axiomatic foundation. Honest scope: independence proofs use semi-formal model constructions (not yet mechanized in Lean/Coq — this is acknowledged as future work). The fifteen theorem derivations are sketches that establish derivability; full proof scripts appear in the companion implementation. Benchmarks compare against naive baselines; comparison with optimized domain-specific languages would refine the picture. Preprint — not yet peer-reviewed. Feedback welcome at [email protected] / GitHub Discussions at fc0web/rei-aios.