ONE AXIOM : ABC: Alignment-Based Coherence

ABC: Alignment-Based Coherence – From Primordial Quartet to Dimensional Emergence (Version 2. 0) This document establishes the mathematical foundations for the ₅ layer, an 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, , F, 1 gives rise to the symmetry group G = S₄ Z₂³. Version 2. 0 proves 136 results (91 theorems, 36 corollaries, 9 propositions) and adds a complete bibliography of upstream ONE AXIOM documents with Zenodo DOIs. 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 S₄ symmetry framework necessitates four ontological dimensions. State Partition and Density Factorization: The identification of the 51: 13 ratio of Allowed to Forbidden states, together with the density factorization identity |F|ₑ₄₋ = 13/64 relating the partition to the resolution quantum