Adaptive Convergence for Chaotic Randomness (ACCR) v1.1

Adaptive Convergence for Chaotic Randomness (ACCR) is a unified operator framework designed to stabilize chaotic or divergent iterative processes through a structured hierarchy of invariants. The system defines a fixed family of endomorphisms that act as renormalization steps, progressively reducing instability and taming chaos without relying on differential equations, probabilistic priors, or domain‑specific physical assumptions. Invariant Architecture ACCR operates across a three‑layer structural blueprint—topology, shape, and variance—preserving deep structural information during state evolution. Empirical Stability A nine‑test empirical stability suite identifies a universal divergence boundary at dc = 0.60 ± 0.03, with highly reproducible behavior across Euclidean, circular, and graph‑based substrates. Formal Verification A full Lean 4 verification layer provides machine‑checked proofs for the foundational components of the architecture, including operator idempotence, left‑absorption, and invariant preservation. Cross‑Domain Reach ACCR bridges continuous fields, discrete systems, and network structures through spectral graph theory and Laplacian‑driven dynamics, enabling consistent behavior across heterogeneous substrates. This monograph and repository serve as the canonical archival specification for ACCR v1.1. Included Assets Reference Python implementations Stability test suites Formal verification components Reproducible datasets All materials are provided for open‑source use and citation.