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Robustness with respect to weight perturbations underpins guarantees for generalization, pruning and quantization. Existingguarantees rely on *Lipschitz bounds in parameter space*, cover only plain feed-forward MLPs, and break under the ubiquitous neuron-wise rescaling symmetry of ReLU networks. We prove a new Lipschitz inequality expressed through the $\ell^{1}$-*path-metric* of the weights. The bound is (i) *rescaling-invariant* by construction and (ii) applies to any ReLU-DAG architecture with any combination of convolutions, skip connections, pooling, and frozen (inference-time) batch-normalization —thus encompassing ResNets, U-Nets, VGG-style CNNs, and more. By respecting the network’s natural symmetries, the new bound strictly sharpens prior parameter-space bounds and can be computed in two forward passes. To illustrate its utility, we derive from it a symmetry-aware pruning criterion andshow—through a proof-of-concept experiment on a ResNet-18 trained on ImageNet—that its pruning performance matches that of classical magnitude pruning, while becoming totally immune to arbitrary neuron-wise rescalings.