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 $\pi_5$ 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 ${0, \heartsuit, \mathfrak{F}, 1}$ gives rise to the symmetry group $G = S_4 \times \mathbb{Z}_2^3$. 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 $S_4$ 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 $|\mathcal{F}|\cdot\delta_{\mathrm{rel}} = 13/64$ relating the partition to the resolution quantum $