This book explores structural boundaries of formal systems, focusing on complexity theory, semantic drift, and auditability in AI architectures. It combines mathematical logic, circuit complexity (AC⁰, 3-CNF), and audit-driven system design to construct a verifiable framework for understanding computational limits. Drawing from formal tools like the Switching Lemma, random restrictions, and semantic differentials (Δ, Σ, Ω), the book introduces a unique synthesis of structural proof theory and systems architecture — defined through a semantic operating layer called SnapOS. Originally developed in the context of regulatory AI, SnapOS introduces audit primitives such as SnapScore, SnapCut, and Reentry, bridging formal mathematics with applied system design. This publication is part of a wider research project on autonomous epistemic systems, semantic architecture, and drift-aware AI regulation. SnapOS, AC^0, Switching Lemma, Formal Bound, Auditability, Computational Structure, Lower Bounds, 3-CNF, Complexity Theory