A Formal Framework for the Definition of 'State': Hierarchical Representation and Meta-Universe Interpretation

TL;DR

This paper develops a rigorous, hierarchical formal framework for the concept of 'state' applicable across disciplines, introducing the Intermediate Meta-Universe for explicit meta-level descriptions and expanding inter-universal theory.

Contribution

It introduces a unified hierarchical state grid and the Intermediate Meta-Universe, enabling explicit meta-level descriptions and broadening inter-universal theory beyond mathematics.

Findings

Proposes a hierarchical state grid for unified notation.

Introduces the Intermediate Meta-Universe for meta-level operations.

Expands inter-universal theory to include linguistic and agent interactions.

Abstract

This study aims to reinforce the theoretical foundation for diverse systems--including the axiomatic definition of intelligence--by introducing a mathematically rigorous and unified formal structure for the concept of 'state,' which has long been used without consensus or formal clarity. First, a 'hierarchical state grid' composed of two axes--state depth and mapping hierarchy--is proposed to provide a unified notational system applicable across mathematical, physical, and linguistic domains. Next, the 'Intermediate Meta-Universe (IMU)' is introduced to enable explicit descriptions of definers (ourselves) and the languages we use, thereby allowing conscious meta-level operations while avoiding self-reference and logical inconsistency. Building on this meta-theoretical foundation, this study expands inter-universal theory beyond mathematics to include linguistic translation and agent integration, introducing the conceptual division between macrocosm-inter-universal and microcosm-inter-universal operations for broader expressivity. Through these contributions, this paper presents a meta-formal logical framework--grounded in the principle of definition = state--that spans time, language, agents, and operations, providing a mathematically robust foundation applicable to the definition of intelligence, formal logic, and scientific theory at large.