The resource requirements of neural networks can be significantly reduced through pruning - the removal of seemingly less important parameters. However, for LLMs, full retraining to recover pruning-induced performance degradation is often prohibitive and classical approaches such as magnitude pruning are suboptimal on Transformers. State-of-the-art methods hence solve a layer-wise mask selection problem: finding a pruning mask that minimizes per-layer pruning error on a small set of calibration data. Exactly solving this problem is computationally infeasible due to its combinatorial nature and the size of the search space, and existing approaches rely on approximations or heuristics. We demonstrate that the mask selection problem can be made drastically more tractable at LLM scale. To that end, we decouple the rows by enforcing equal sparsity levels per row. This allows us to derive optimal 1-swaps (exchanging one kept and one pruned weight) computable efficiently via the Gram matrix. We propose a simple 1-swap algorithm that warmstarts from any pruning mask, runs efficiently on GPUs at LLM scale, and is essentially hyperparameter-free. Our approach reduces per-layer pruning error by up to 60% over Wanda (Sun et al., 2024) and consistently improves perplexity and zero-shot accuracy across state-of-the-art GPT architectures.