A Cyber-Physical Sovereign Architecture for Autonomous Grid-Inspection Humanoids: Formal Verification, Lyapunov Dissipation Bounds, and Immutable Zero-Trust Sandboxing

Autonomous humanoid robotics deployed in high-voltage grid inspection and critical infrastructure require deterministic stability guarantees and cryptographic resilience against physical, cyber, and adversarial AI intrusion. This paper presents the mathematical formalization, proof architecture, and runtime assurance framework of the ZARQA Grid Inspection Humanoid Core (zarqa_gih_math_core.py). I formalize a cyber-physical sovereignty pipeline that unifies five distinct mathematical guarantees: 1. Phase-locked synchronization of distributed control oscillators via the Kuramoto model (). 2. Lyapunov asymptotic stability of the Linear Inverted Pendulum Model (LIPM) under Continuous Algebraic Riccati Equation (CARE) optimal feedback (). 3. Stochastic Zero-Moment Point (ZMP) safety verification bounded under Ornstein-Uhlenbeck proprioceptive drift (). 4. Cryptographic attestation of Dynamic Movement Primitives (DMPs) via SHA3-256 Merkle root verification. 5. High-dimensional 3D occupancy grid compression bounded by Tensor-Train (TT-SLAM) low-rank manifolds (). I prove that combining these physical invariants with a Trusted Platform Module (TPM 2.0) seeded hash-chain audit log, blue-green immutable execution sandboxes, and automated Lean 4 theorem verification satisfies the safety and security mandates of IEC 63439 and IEC 62443.