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Providing protection without jeopardising a system's functionality and flexibility is a profoundly difficult challenge for researchers and practitioners. Approximate circuits (AxCs) trade off increased numerical precision for reduced hardware area, latency, and energy consumption. IP key vendors who want to develop such circuits must persuade customers of the quality of the resulting approximation. Propose PUF-based proof-carrying AxCs as a workaround. The provider produces an estimated IP core along with a certificate that verifies the accuracy of the approximation. The evidence certificate is submitted to the customer along with an estimated IP heart.The chip's feature is activated using the PUF answer. The user will formally validate the IP core's approximation accuracy for a fraction of the expense of conventional formal verification.