Synthetic Mind as a Reentry Loop: Architecture and Gradient-Free Learning Algorithm for Intrinsically Safe AGI

We present a complete, technically realizable architecture for intrinsically safe artificial general intelligence (AGI), whose operational principles are fundamentally isomorphic to the dynamics of the living human brain. Safety is embedded directly into the geometric structure: the agent's goal vector cannot be overwritten, distorted, or bypassed by prompt injections, hallucinatory storms, or content-level adversarial attacks. The core innovation is a gradient-free learning algorithm — no backpropagation, no loss function — that holds a closed reentry loop between the intending (D) and intentional (I) subsystems on the edge of chaos. A new measure, χ (chi), built from Lyapunov exponents, Kolmogorov–Sinai entropy, and the cycle complexity of the strange attractor, drives a purely local homeostatic weight update. We prove (and verify in Lean 4) that when chaos exceeds a critical threshold, the rule becomes a Banach contraction, geometrically extinguishing hostile perturbations. Safety of the D‑register is guaranteed by non-Hausdorff topology (the Alexandroff connected doubleton), which makes the overwrite I→D a discontinuous operation. The paper is written as a pedagogical textbook for engineers with no prior exposure to reentry or chaos theory. A complete PyTorch implementation is provided. We also extend the framework to BCI, photonics, memristors, and cortical organoids, showing that a 10⁹‑parameter closed-loop system with χ > 0 surpasses any 10¹²‑parameter feedforward model.