Finite-Resolution Physics from Additivity Symmetry

This paper presents a comprehensive construction of the additivity–symmetry framework, a mathematical and conceptual model unifying quantum, statistical, and deterministic (topological) regimes under a single algebraic principle of finite resolution. The framework shows that the fundamental structures of modern physics—Hilbert spaces, probability measures, and deterministic flows—emerge naturally from the four finite-resolution features (FR1–FR4) of the additive group, without invoking separate postulates of quantization or probability. Through harmonic analysis, GNS representations, and invariant–mean methods, the paper demonstrates how operational probes activate different physical regimes from the same additive algebra. Extensions to multidimensional groups introduce spinorial representations and internal symmetries as consequences of automorphism-naturality. At higher structural levels, the Activation Theorem formalizes the hierarchy of admissible regimes, while renormalization and anomaly cancellation are reinterpreted as intrinsic consequences of closure under admissible automorphisms. The final sections analyze the boundary and geometric implications of finite resolution, showing how gravitational coherence emerges as a boundary phenomenon. The result is a unified ontology of fields and interactions in which all physical regimes are finite and admissible, and no external quantization or probabilistic assumptions are required. Versioning and peer-review status Versions 2 and higher of this Zenodo record were uploaded prior to the peer-review acceptance of the manuscript and should not be interpreted as post-acceptance revisions or corrections. The manuscript corresponding to Version 1 was published by the International Journal of Geometric Methods in Modern Physics with DOI https://doi.org/10.1142/S0219887826501173 .The acceptance applies specifically to that submitted manuscript version; the existence of later Zenodo versions does not indicate deficiencies in the accepted work, but reflects parallel author-initiated developments and exploratory extensions prepared independently of the review process. The final published journal version will be linked here upon publication. Note This work supersedes and extends the previous series Additivity Symmetry I–III, integrating and generalizing their results into a single coherent formulation. It provides the full mathematical development and interpretative synthesis of the finite-resolution framework first outlined in those earlier papers.