Large language models with billions of parameters are often over-provisioned: many layers contribute little unique information yet dominate the memory and energy footprint during inference. We present LieQ (Layer-wise information effectiveness Quantization), a hardware-native, metric-driven post-training quantization framework that addresses the critical challenge of maintaining accuracy in sub-8B models, model parameters less than 8B, under extreme low-bit compression. LieQ keeps uniform bit-width within each layer while mixing precision across layers, preserving standard multiplication kernels and avoiding irregular memory access, codebooks, or irregular formats at inference time. Our method uncovers a strong correlation between layer-wise functional saliency and representational compactness, revealing that layers with higher training-induced energy concentration are functionally irreplaceable. Leveraging this insight, we propose a purely geometry-driven sensitivity proxy that enables automatic bit-width allocation under a target average-bit budget without expensive gradient updates or inference-based perplexity probing. Under an average weight bit-width approaching two bits per parameter, LieQ consistently reduces the large accuracy gap typically observed for naive uniform 2-bit baselines on Qwen3 and LLaMA3.x families, while retaining standard-kernel efficiency. These properties make LieQ a practical path toward deploying small language models on resource-constrained edge devices. Code will be available at: https://github.com/HeXiao-55/LieQ-official.git.