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We propose a novel method to increase shift invariance and prediction accuracy in convolutional neural networks. Specifically, we replace the first-layer combination "real-valued convolutions + max pooling" (RMax) by "complex-valued convolutions + modulus" (CMod), which is stable to translations, or shifts. To justify our approach, we claim that CMod and RMax produce comparable outputs when the convolution kernel is band-pass and oriented (Gabor-like filter). In this context, CMod can therefore be considered as a stable alternative to RMax. To enforce this property, we constrain the convolution kernels to adopt such a Gabor-like structure. The corresponding architecture is called mathematical twin, because it employs a well-defined mathematical operator to mimic the behavior of the original, freely-trained model. Our approach achieves superior accuracy on ImageNet and CIFAR-10 classification tasks, compared to prior methods based on low-pass filtering. Arguably, our approach's emphasis on retaining high-frequency details contributes to a better balance between shift invariance and information preservation, resulting in improved performance. Furthermore, it has a lower computational cost and memory footprint than concurrent work, making it a promising solution for practical implementation.