Fractal Information Dynamics – Part 1

This paper develops a mathematical framework for studying how derivability changes when a formal system is recursively extended. The starting intuition is geometric: statements are represented as information nodes, inference relations as directed branches, and rule-bearing junctions as local transformation structures. The geometric picture is not taken as a theorem about formal systems; rather, it motivates a precise sequence of recursively related theories F₀, F₁, … together with derivability-state maps, independence sets, transition sets, complexity coordinates, and extension operators. The central proposal is to treat changes in unresolved or independent information as a dynamical object. Two distinct information measures are introduced: a distributional entropy over derivability classes and a configurational unresolved complexity = ln|| for finite truncations. Their inter-level differences define meta-information flux. External information injection is formalized not as the removal of Gödel incompleteness, but as an extension → =(,) that may decide statements independent of the weaker theory while generating a new unresolved frontier in the stronger theory. A three-coordinate logical phase space—statement complexity, proof complexity, and meta-system level—is proposed as a visualization and analysis device. A finite computational experiment illustrates the measurable quantities without claiming to establish fractality or asymptotic laws for arithmetic. The framework therefore separates established incompleteness results from new definitions and testable hypotheses concerning recursive scaling, information flux, and possible fractal structure across formal-system extensions.