Algebraic Realizations of the 0S-A-U Triad: The A1-A8 Proposal and the N=5 Modeled Adjusted Triadic System

This repository contains the full manuscript detailing two axiomatic frameworks exploring the 0S-A-U Triad. The primary framework, the <b>0S-A-U Triadic System (Axioms A1-A8)</b>, is structured around the interaction of three core elements: the additive identity 0S (Void), its ¬-dual counterpart A (Apeiron/Plenitude), and their unique synthesis product, the Universal U (0S⋅A=U). This system is defined by eight axiom groups governing ¬-duality, specific interaction rules including the synthesis of U, Modified Associativity for its dual multiplication (⊗), and a principle of Polar Distributivity. This system is presented as an open proposal awaiting full mathematical validation, including model construction and resolution of key conjectures regarding its non-standard axioms.The <b>Bonus Part</b> of the manuscript introduces the <b>Adjusted Triadic System (ATS v3)</b>, a distinct axiomatic framework (Axioms ATS.1-ATS.5) also designed to coherently realize the 0S-A-U Triad but employing standard associativity for both primary and dual multiplication operations. ATS achieves coherence by positing that the 0S⋅A=U synthesis and U-absorption take precedence, thereby conditioning universal distributivity. This results in a characteristic Non-Distributive Locus (NDL), where distributivity fails precisely in those instances necessary to preserve the distinctness of the core elements. This V3 appendix presents the exhaustive computational verification of <b>two distinct 5-element ATS models (L.11-ATS-v1, with </b><b>E</b><b>⋅</b><b>E</b><b>=</b><b>1</b><b>S</b><b>; and L.11-ATS-v2, with </b><b>E</b><b>⋅</b><b>E</b><b>=</b><b>E</b><b>) which satisfy all core ATS axioms (ATS.1-ATS.5)</b>. A crucial finding from these models is that the 'normal element' subdomain (Snorm​=S∖{U}), despite forming an Abelian group under addition, <b>does not form a commutative ring</b> due to specific, structurally determined failures of distributivity even within Snorm​ (Conditional Distributivity Hypothesis failure).This research combines rigorous algebraic design with the pursuit of conceptual symmetry, offering two distinct axiomatic foundations (the A1-A8 proposal and the N=5-modeled ATS v3) for studying synthesis-based systems featuring ¬-duality, universal absorption, and precisely conditioned distributivity. It aims to contribute to the exploration of non-standard algebraic structures capable of addressing foundational mathematical and logical concepts concerning void, plenitude, and their unification.This revised version aims to be more accurate, comprehensive, and highlight the latest significant findings regarding ATS v3.