On the Non-Perturbative Nature of the 0-Sphere Model and the Fine-Structure Constant: A Supplement on Methodological Scope and Realist Foundations

Description 36 Supplement This document is a supplementary theoretical and methodological exposition of the 0-Sphere model, focusing on its non-perturbative character, its geometric derivation of internal dynamical quantities, and its explicit delimitation from perturbative quantum electrodynamics, particularly with respect to the fine-structure constant. The present supplement clarifies why certain quantities—most notably the Zitterbewegung velocity scale and the geometric structure underlying the electron anomalous magnetic moment—can be derived within the 0-Sphere framework from first principles, while others, such as the fine-structure constant and electric charge, currently lie outside its formal scope. Rather than proposing an alternative perturbative calculation, this document articulates the methodological boundary conditions of the 0-Sphere model and situates it as a realist, background-independent, non-perturbative research program grounded in closed geometric relations and line-integral ontology. Purpose and Conceptual Scope This supplement is intended to be read alongside the main 0-Sphere manuscripts, including earlier supplements on emergent gravitational-like behavior and internal phase dynamics. Its primary aims are: to clarify the non-perturbative nature of the 0-Sphere model and its consequences for what can be meaningfully derived, to provide a concrete example of first-principles derivation (the Zitterbewegung velocity) without invoking perturbative series or coupling constants, to explain why the fine-structure constant α is not addressed within the current framework, and why this omission is methodological rather than accidental, and to articulate the philosophical stance—realist rather than instrumentalist—that guides the model’s development. Conceptual Motivation In the 0-Sphere framework, physical observables arise from deterministic geometric structures, energy conservation identities, and accumulated phase along actual trajectories. The model does not employ weak-coupling expansions, virtual particle loops, or perturbative corrections. As a result, quantities that are operationally defined in conventional quantum electrodynamics through perturbative expansions—such as the fine-structure constant—do not naturally appear within the present formalism. This supplement explains why attempting to “derive α” directly from the 0-Sphere model would be conceptually inconsistent at its current stage. By contrast, quantities that depend only on relativistic kinematics, internal oscillatory geometry, and experimentally measured inputs can be treated non-perturbatively. The derivation of the Zitterbewegung RMS velocity from the measured anomalous magnetic moment serves as a paradigmatic example. Key Themes Addressed This supplement elaborates on the following points: Non-perturbative derivation of internal dynamicsDemonstrating how the Zitterbewegung velocity scale follows directly from Lorentz kinematics and geometric interpretation, without reference to α or loop corrections. Distinction between peak and RMS internal velocitiesClarifying how time-averaged relativistic effects relate to oscillatory internal motion and observable quantities. Geometric origin of the anomalous magnetic momentInterpreting g − 2 as a manifestation of Lorentz contraction of a circulating photon structure rather than a radiative correction. Reinterpretation of spin and Thomas precessionTreating spin as an emergent, phase-dependent geometric quantity rather than a pre-assigned intrinsic label. Explicit scope delimitationIdentifying which quantities are currently derivable within the model and which remain open problems. Methodological realismContrasting the realist commitments of the 0-Sphere model with the instrumentalist character of perturbative QED. Relation to the Fine-Structure Constant A central clarification of this supplement concerns the status of the fine-structure constant. In conventional QED, α functions as a perturbative expansion parameter tied to electric charge and loop corrections. The 0-Sphere model, being non-perturbative and not yet equipped with a quantitative theory of electric charge, does not incorporate such a parameter. This document emphasizes that the absence of α is not a deficiency but a transparent statement of scope. Any future connection between the 0-Sphere framework and perturbative constants would require an explicit effective-theory bridge, which remains a subject for future work. Relation to the Main Manuscript Series This supplementary document should be cited together with the main 0-Sphere manuscripts and prior supplements. Taken together, these works aim to establish a coherent conceptual and methodological foundation for evaluating emergent-force and internal-dynamics models based on phase, geometry, and energy flow, without introducing unobserved mediators or perturbative assumptions. Keywords 0-Sphere modelnon-perturbative frameworkZitterbewegunganomalous magnetic momentphase-dependent spingeometric realismfine-structure constant (scope delimitation)no perturbative expansion