Abstract Formal verification is paramount for neural networks in safety-critical domains yet remains constrained by the trade-off between precision and scalability, especially with modern high frequency activation functions. However, the inherent NP-hardness of the problem forces a fundamental trade-off between scalability and precision in existing methods, which fail to adequately capture the high-frequency nonlinearities of modern activations. To address this, Quadratic Approximation based Verification via Fourier series and Taylor truncation (QAV-FT) is proposed as a framework unifying Fourier-enhanced quadratic approximation and Taylor remainder-aware error control. Specifically: (i) A Fourier-based quadratic abstraction is formulated for arbitrary activations, with a rigorous error bound established via Jackson’s theorem and Lebesgue constant analysis; (ii) A second-order propagation scheme utilizing Lagrange remainder theory is devised to analytically derive sound, tight bounds with reduced symbolic complexity; and (iii) These mechanisms are integrated into a scalable full-network verification algorithm. Empirical results on MNIST-FC and other benchmarks demonstrate that QAV-FT achieves an average certification accuracy of 96.8% across complex activations like Swish, GELU, and Mish, with favorable accuracy and efficiency over comparable methods on partial models.