The Geometric Complexity Index

This paper introduces the Geometric Complexity Index (GCI), a novel mathematical framework that maps algorithmic complexity into a continuous 3D manifold known as Functional Phase Space. While traditional computational complexity theory (Big O notation) relies on discrete, disjointed classes (e.g., O(n), O(n2)), the GCI provides a heuristic coordinate system Rank (Z), Magnitude (X), and Rate (Y) to vectorize algorithmic cost. By utilizing Level-Index Arithmetic, the GCI allows for the numerical comparison of hyper-exponential structures without the constraints of computational overflow, effectively acting as a "Geiger Counter" for technical debt and algorithmic security. Key Contributions Vectorization of Asymptotic Notation: Transforms qualitative Big O labels into quantitative coordinates (Z,X,Y), preserving constant factors often lost in traditional analysis. The GCI Metric (GCI1M): Defines a standardized scalar score for complexity evaluated at a reference load, allowing for direct comparison between disparate algorithms. Continuous Rank Interpolation: Proposes an engineering-focused heuristic for interpolating between complexity classes (e.g., moving continuously from Polynomial to Exponential regimes). Practical Applications: Outlines use cases in CI/CD pipelines for detecting ReDoS vulnerabilities, Static Analysis for measuring refactoring distance, and AI Refinement by using GCI as a regularization term in code-generation models.