The CRI Framework (Extended Version): A Universal Metric for Empirical Validation of Structural Hardness

The CRI Framework (Compression Resistance Index) introduces a universal empirical metric for assessing structural hardness across 36 formal languages spanning 15 computational complexity classes.Unlike traditional uncomputable metrics, CRI estimates structural resistance through inference failure after structure-preserving compression, measured via a standardized pipeline of classical, quantum, and hybrid compressors. Key contributions: A unified β̃(L) protocol using a 7-layer hierarchical Graph Neural Network (GNN) Machine learning-based estimation of circuit size to derive realistic θ(n) thresholds Evaluation on over 680,000 labeled instances, from P to QMA Clear structural separation between classes (e.g., 3SAT ∉ BPP/poly, LWE cryptographically hard) This framework enables quantifiable differentiation of borderline languages (e.g., Graph Isomorphism) and supports measurable assumptions in post-quantum cryptography. The full implementation of the CRI protocol—compressor modules, inference models, and dataset handlers—will be released in the final paper of this series.All source code and benchmark data will be published on the official GitHub repository: