Weight-only post-training quantization (PTQ) for large language models (LLMs) typically ends with the projection of floating-point weights onto a uniform integer lattice, introducing structured output error due to non-zero activation means. Existing bias correction methods compensate for this shift by adding bias terms, which is incompatible with modern bias-less architectures such as LLaMA and Mistral and only addresses the error indirectly. We propose **Complementary Lattice Correction (CLC)**, a lightweight post-quantization method that corrects projection error directly on the integer lattice without modifying the model architecture. CLC performs a budget-constrained greedy descent that selectively flips quantized weights to adjacent lattice levels. We further show that CLC provably reduces the first-moment error and admits a worst-case bound on the layer-wise mean-squared error change, which certifies non-increase under a verifiable budget condition. Experiments across multiple models, quantizers, and bit widths show that CLC consistently improves perplexity and downstream accuracy with negligible overhead.