Encapsulation Domains: Deterministic Structures for Consistent Multi-Scale Systems

This work introduces the Encapsulation Domain (ED), a deterministic mathematical framework for modeling systems whose internal structure may vary while their externally observable identity remains invariant. The framework formalizes structural identity, closure, composability, and scale-dependent appearance as axiomatic requirements. These principles guarantee consistency without exceptions and provide a unified formal basis for discrete and continuous systems. The ED framework defines encapsulated objects as entities possessing internal state, observable projection, and invariant identity. The resulting structure ensures deterministic evolution, preservation of identity under transformation, and observational equivalence across internal configurations. This document establishes the foundational definitions, axioms, and structural theorems of the Encapsulation Domain and provides a minimal formal system suitable for multi-scale modeling, computational geometry, and deterministic system analysis. First public disclosure: February 27, 2026.