ONE AXIOM : ABC: Alignment-Based Coherence
Robert Spychalski
Abstract
ABC: Alignment-Based Coherence – From Primordial Quartet to Dimensional Emergence (Version 1.0) Abstract: This document establishes the mathematical foundations for the $\pi_5$ layer, a ontological pre-physical level of structure that precedes physical manifestation. Using the "Alignment-Based Coherence" (ABC) framework, the work demonstrates how a fundamental quartet of elements $\{0, \heartsuit, \mathfrak{F}, 1\}$ gives rise to the symmetry group $G = S_4 \times \mathbb{Z}_2^3$. The treatise employs a rigorous Dual-Track (Cyclic-Binary Convergence) methodology to identify the structural constraints that govern this layer. Key achievements include: Mathematical Necessity of 4-Fold Structure: A formal demonstration of why a coherent alignment within the $S_4$ symmetry framework necessitates four ontological dimensions. State Partition Analysis: The identification of the $51:13$ ratio of Allowed to Forbidden states, derived from the interaction of cyclic and binary perspectives. Isotropic Gradient Equilibrium (The Float Principle): The formalization of Theorem 27, which defines $\pi_5$ as a dynamic, flow-through reservoir. This principle explains how internal directional balance is maintained via informational pressure, allowing for stable coherence transfer without assuming a closed system. Combinatorial Foundations: The identification of the structural precursors to fundamental coupling, specifically the emergence of the value $\alpha \approx 137$ from the cardinality of group orbits and state density. ABC serves as a "Handoff Contract," providing the necessary mathematical blueprint for the subsequent emergence of spacetime and physical properties in lower layers ($\pi_6$).