Resolution-Dependent Structure and the Emergence of Physical Observables

We propose a conceptual framework in which physical structure emerges from observational
projection acting on an underlying non-closed configuration.
Observation is formulated as a projection with finite resolution λ, generating distinguishability and defining effective physical structure. Within this framework, existence is defined
operationally through energy, and information arises from distinguishable differences.
We introduce a dual-layer structure consisting of a non-closed general domain and a resolutiondefined observable domain. This structure provides a unified perspective on quantum measurement, information, and emergent physical behavior.
We show that the framework constrains admissible probability measures and naturally motivates the quadratic form associated with the Born rule.
This work establishes the foundational structure of a resolution-based approach to physical
observables.