The 0S-A-U Triad: An Algebraic Exploration of Duality and Synthesis
Creators: name: GenAI, LLMs
name: Ivliev, Ihor
Publisher: name: Zenodo
Year: 2025
Subjects: subject: 0S-A-U Triad
subject: Axiomatic Systems
subject: Universal Algebra
subject: Abstract Algebra
subject: Non-Standard Algebra
subject: ¬-Duality
subject: Duality Involution
subject: Synthesis Axiom
subject: Modified Associativity
subject: Polar Distributivity
subject: Foundations of Mathematics
subject: Mathematical Logic
subject: Axiomatics
subject: Paradox Resolution
subject: Proof Theory
subject: Philosophy of Mathematics
subject: Mathematical logic, set theory, lattices and universal algebra
subject: Mathematical logic
subject: Pure mathematics
subject: Applied mathematics
subject: Mathematics
subject: FOS: Mathematics
subject: Mathematics
subject: Algebra
subject: Commutative algebra
subject: Operator algebra
subject: Logic

description: At its heart, this work develops an algebra where "zero" (0S) and its dual "plenitude" (A) multiply to synthesize a new "universal" element (U), aiming to address a symmetry gap in classical structures.

 

Standard mathematical frameworks, while extraordinarily powerful, exhibit a potential foundational asymmetry. These frameworks are often powerfully centered on the concept of Zero (denoted herein as 0S), the symbol of absence, origin, or additive nullity. Yet, they arguably lack an equally fundamental, axiomatically integrated counterpart representing the complementary pole of plenitude, totality, or the boundless context—a concept this exploration terms Apeiron (denoted as A). This document presents an initial, rigorous exploration into addressing this perceived asymmetry. It proposes a novel algebraic structure systematically grounded in the core principle of ¬-Duality, conceptualized as a symmetric relationship 0S↔A mediated by a formal involution operator, ¬.

The journey to this proposed structure involved confronting significant theoretical challenges. Early attempts to integrate such a duality, particularly through defining a multiplicative synthesis 0S⋅A that would yield an outcome mathematically distinct from either 0S or A (thereby reflecting their co-equal status), encountered insurmountable logical contradictions. These contradictions emerged consistently when such a synthesis was constrained by the simultaneous imposition of universally applied standard algebraic axioms—most notably, universal associativity for all defined multiplications and universal distributivity across all relevant pairings of multiplicative and additive operations.

A critical re-evaluation led to the interpretation of these contradictions not as failures of the underlying vision of polar symmetry, but as profound indicators—indeed, fundamental discoveries—regarding the potential limitations inherent in standard algebraic frameworks when tasked with coherently accommodating such a deep symmetry alongside a potent, synthesizing polar interaction. This interpretive shift precipitated a pivotal philosophical reorientation: 0S (representing Void) and A (representing Plenitude) are herein conceptualized as co-equal, primordial peers. Their fundamental interaction, specifically via primary multiplication, 0S⋅A, is postulated to uniquely synthesize a transcendent, qualitatively distinct entity—the Universal (denoted as U). This Universal is characterized as self-dual (i.e., ¬U=U) and as a universally absorbing element for all operations within the algebra. This conceptualization forms the core ontological and structural framework of the 0S-A-U Triad.

The proposed algebraic system, formally defined by a set of eight axiom groups (A1-A8), is derived from the primacy of this Triad, the Synthesis Axiom (0S⋅A=U), and the overarching ¬-Duality principle. Internal analyses detailed herein argue that this system appears coherent with respect to previously identified paradoxes. Specifically, a critical inconsistency leading to the collapse U=0S, which plagued precursor formulations, is demonstrably resolved through the adoption of a meticulously refined dual additive structure (Axiom A3').

Achieving this level of coherence, however, demonstrably necessitates specific, principled departures from standard universal algebraic laws. These include:



The adoption of Modified Associativity (Axiom A4.4) for the dual multiplication operation (denoted as ⊗). This specific non-standard associative law, x⊗(y⊗z)=(x⊗z)⊗y, is argued to be a necessary structural consequence of maintaining standard associativity for primary multiplication (denoted as ⋅) when linked via ¬-Duality (Axiom A6.3). The complete justification for this adoption also transparently relies on its established consistency with primal associativity under the ¬-Duality mapping, combined with a crucial (but currently unproven and explicitly stated as such) Conjecture 1 concerning the system-level incompatibility of standard dual associativity with the full axiomatic system due to constraints imposed by the synthesis and distributivity axioms.

The implementation of Polar Distributivity (Axiom A8). This principle restricts the universal application of distributive laws, positing that they hold universally only within the same polarity (i.e., primary multiplication ⋅ over primary addition +; dual multiplication ⊗ over dual addition ⊕). This restriction, excluding universal cross-polar distributivity, is argued as being structurally mandated for consistency with the 0S⋅A=U synthesis.


While the internal arguments presented support the proposed structure's coherence relative to these specific identified paradoxes and justify its axiomatic form, its formal mathematical validation remains an outstanding requirement. This validation critically depends on: (a) the formal proof of consistency via the construction of at least one concrete mathematical model satisfying all axioms A1-A8, and (b) the formal resolution (proof or disproof) of the aforementioned Conjecture 1. These tasks constitute essential future work.

This exploration, therefore, aims to rigorously define the proposed algebraic structure, detail its justification narrative (including paradox resolution and the arguments for its necessitated non-standard features), analyze its preliminary structural properties and operational calculus, explore its potential philosophical and logical interpretations, and clearly delineate the critical path forward for its formal validation by the broader scientific and mathematical community.