Closure, Stabilization, and the Emergence of Physical Description: A Unified Framework for Finite Resolution Physics

This treatise develops a closure-first framework in which admissible composition, closure, stabilization, refinement, and stability plateaux are treated as primitive, while equations of motion, symmetry, geometry, and Einstein-like field equations are derived as representational consequences rather than assumed foundations. The work introduces a carrier-dependent continuation program, FRP($G$), showing that stabilized continuation may be encoded through different continuation carriers, including linear, discrete, compact periodic, hierarchical, and mixed classes. Within this broader landscape, Finite Resolution Physics proper is identified as the distinguished real additive branch, where global stitching together with arbitrarily fine subdivision forces finite resolution. On this branch, the treatise develops additive representation theorems, regime activation, generator-based dynamics, symmetry from stabilized compatibility via the SILM construction, geometry as a projection codec, and Einstein-like equations as minimal geometric stabilizers on smooth plateaux. The work also analyzes failure modes, plateau boundaries, higher-order and nonlocal codec extensions, non-commuting projections, and multi-plateaux structure. Overall, the treatise proposes a unified structural order of explanation: closure before projection, projection before representation, and representation before equations.