The paper introduces the first efficiently verifiable Fully Homomorphic Encryption (FHE) scheme that supports arbitrary-depth homomorphic circuits. The authors propose a novel approach that leverages the double-CRT representation, where FHE schemes are typically computed, and lattice-based SNARKs to prove components of the computation separately, including maintenance operations such as modulus switching and key switching. This construction can handle bootstrapping operations and is the first to implement verifiable computation on encrypted data for a neural network with multiple ciphertext-ciphertext multiplications, achieving verification times of less than 1 second while maintaining reasonable prover costs. The paper also discusses the efficiency of the scheme, including optimizations for reducing proof size and verification time, and provides a detailed security analysis, demonstrating that the scheme meets the required correctness, completeness, soundness, and IND-CPA security properties.The paper introduces the first efficiently verifiable Fully Homomorphic Encryption (FHE) scheme that supports arbitrary-depth homomorphic circuits. The authors propose a novel approach that leverages the double-CRT representation, where FHE schemes are typically computed, and lattice-based SNARKs to prove components of the computation separately, including maintenance operations such as modulus switching and key switching. This construction can handle bootstrapping operations and is the first to implement verifiable computation on encrypted data for a neural network with multiple ciphertext-ciphertext multiplications, achieving verification times of less than 1 second while maintaining reasonable prover costs. The paper also discusses the efficiency of the scheme, including optimizations for reducing proof size and verification time, and provides a detailed security analysis, demonstrating that the scheme meets the required correctness, completeness, soundness, and IND-CPA security properties.