As Large Language Models (LLMs) are increasingly deployed for narrow tasks in resource-constrained settings, a central question arises: how much of an LLM is truly necessary for a given task? We present LLM-Sieve, a framework that prunes LLMs down to the minimal parameter subset needed to preserve task performance. Our approach introduces two innovations: (i) output-aligned non-orthogonal projections, which yield more faithful low-rank approximations than traditional PCA/SVD by aligning directly with layer outputs; and (ii) adaptive pruning via a Genetic Algorithm, which automatically discovers matrix-specific pruning levels and exposes the uneven distribution of task-relevant knowledge. Across models from 3.8B to 70B parameters, LLM-Sieve removes 20-75% of weights with only 1-5% accuracy loss-substantially ahead of prior pruning methods. Beyond efficiency, our framework reveals bottleneck matrices that concentrate critical knowledge, suggesting architectural implications for future LLM design. LLM-Sieve integrates seamlessly with LoRA fine-tuning and quantization, enabling both efficient deployment and deeper understanding of knowledge organization in LLMs.