Axiometry: Prolegomena to a Metric Science of Value (Transcendental Foundations and Formal Architecture – Part I)

This article, the first part of a foundational series, establishes the mathematical and formal pillars of Axiometry, conceived as a metric science of value capable of transcending the divide between qualitative philosophy and computational engineering. Unlike classical axiologies (Scheler, Hartmann, Hartman), which limited themselves to describing values as static entities, Axiometry proposes an operational formalism that explains how values originate, organize, and transform within cognition and social systems. The core of this volume is the derivation of the Supra‑Boolean Algebra, a four‑valued system {0, 1, 0̸, 1̸} that models the ontological tension between Existence (E) and Manifestation (M). Built upon this algebraic foundation, the article demonstrates the transcendental necessity of the Axon (α) as the minimal and irreducible unit of valuative donation, and formalizes the principles of Axiometric Correlation and Phenomenological Coupling, which integrate the subject's noesis with the concrete state of the phenomenon within a shared Paradigmatic Space. The architecture culminates in a far‑reaching mathematical result: the Completeness Theorem of the Phenomenological System (S₅), proving that the supra‑boolean algebra is sufficient to categorize not only classical states (material and immaterial) but also the overflowing phenomena of quantum mechanics and liminal experience. Additionally, the Axiometric Hyperspace (Ξ) is introduced as a unifying framework to map the topology of value within the space‑time continuum. This work does not seek immediate applications, but rather the rigorous mathematical scaffolding upon which future AI agent architectures, hallucination detection systems, and social dynamics models will be built. It is the formal toolbox that transforms value theory into a computable science.