Abstract: This document specifies the Machine-Checkable Blockchain Execution Specification(BNES) for BearNetworkChain a formally verifiable, deterministic blockchain execution machine. It defines a complete formal correctness predicate layer that harmonizes Post-Quantum Cryptography (PQC) trust-root authentication, Zero-Knowledge (ZK) verifiable computation, Clique deterministic ordering, EVM execution semantics, and Γ (Gamma) execution invariant observation. Key Features & Architecture: BNES Formal Correctness Layer: Provides the sole deterministic predicate system for correctness evaluation and automated auditing.PQC Trust Root Layer: Establishes post-quantum secure identity, transaction authorization, and signature verification boundaries resistant to quantum threats. ZK Verifiable Computation Layer (ZKEngine): Defines deterministic circuit representations and execution witness binding for absolute execution integrity. Γ Execution Invariant Observer Layer: Implements a global invariant scalar based on core dynamical equations to monitor overall system coherence during block finalization. State vs. Ownership Separation: Strictly isolates system-level state representation from external legal ownership and claims. This specification serves as the canonical, machine-checkable, and replayable execution standard tailored for AI agents, compilers, auditors, and node runtimes.