Stabilized Closure Operators and Their FRP Representations. The Mathematical Core of Gravitational Regimes

This work develops a mathematical framework linking two complementary approaches to the foundations of physics: Finite Resolution Physics (FRP) and closure-based analyses of operational descriptions. Finite Resolution Physics classifies physical regimes through representations of the additive structure (R,+)(\mathbb{R},+)(R,+) under finite-resolution constraints, yielding classical, quantum-like, and geometric regimes as enforced representation-theoretic outcomes. Independently, closure-based approaches emphasize that physically meaningful descriptions arise only as stabilized fixed points of probe-induced completion under finite resolution. The present paper provides a structural bridge between these perspectives. It introduces probe-induced closure operators on finite-resolution operational description spaces, formulates precise notions of weak and strong stabilization, and defines closure-fixed structures and inheritance under resolution refinement. It then characterizes the conditions under which stabilized closure regimes admit FRP representations and shows that boundary coherence of such representations enforces an internal stabilizing structure, formalized as Simultaneously Interlocked Lagrangian Multipliers (SILM). Within this framework, invariant decomposition of stabilized FRP representations yields channel structures classified by closure order. Second-order closure invariants—identified in FRP terminology as spin–2 channels—emerge inevitably in regimes admitting geometric interpretation. Gravity is therefore treated not as a primitive dynamical interaction, but as a regime property of stabilized closure, with metric gravity (including General Relativity) corresponding to a particularly rigid and well-inherited subclass of such regimes. The paper further introduces a classification of gravitational regimes, analyzes resolution-induced regime transitions and failure modes of geometric encoding, and presents an illustrative toy model demonstrating how spin–2 structure and metricity can be lost under refinement without invoking quantization or new dynamical degrees of freedom. This work should be understood as a foundational step toward a systematic theory of stabilized closure and its representations. It does not propose a complete theory of gravity, but rather a structural framework clarifying when geometric descriptions are valid, how they emerge, and how they fail. The results are relevant to foundational physics, mathematical physics, and philosophy of spacetime.