Description for Zenodo Upload Abstract: We propose four axioms specifying necessary conditions for structural persistence and derive that any physical substrate satisfying these axioms must exhibit the mathematical structure of a complex scalar field with non-local temporal self-coupling. The framework is built upon the following first principles: The Regeneration Axiom ($R \ge D$): Organized complexity must be maintained against entropic decay, necessitating an amplitude degree of freedom. The Coordination Axiom: Distributed components must achieve phase-coherent interaction, necessitating a phase degree of freedom with periodic structure. The Exclusion Axiom: Distinct elements cannot be compressed into identical states without unbounded energy cost, necessitating a stiffness parameter and a vacuum potential that diverges at zero amplitude. The Reflection Axiom: Persistent structures must receive causal feedback from their own environmental deformations, necessitating a non-local memory kernel coupling the field to its own history. Together, these requirements determine a field of the form $\Phi=\rho e^{i\theta}$ with potential $V(\rho)$ satisfying specific boundary conditions, augmented by a non-local feedback term mediated by a causal memory kernel $K(\Delta x,\Delta\tau)$. This structure recovers the polar decomposition of a complex scalar field foundational to Ginzburg-Landau theory, while the non-local term provides a physical mechanism for autopoietic memory and self-navigation without requiring discrete storage. Reference Implementation: The experimental validation of this framework, including the "Sovereign Organism" simulation demonstrating self-organization in a void via the Reflection Axiom, is available in the associated software repository. Code Repository: https://github.com/gloryape/quaternity-emergence Software License: GNU Affero General Public License v3.0 (AGPL-3.0)