QAM-W: Joint 2D Codebook Quantization for LLM Weights via Hadamard Rotation and Activation-Aware Scaling

Scalar post-training quantizers discard pairwise coordinate structure within weight rows. We introduce QAM-W (Quadrature Amplitude Modulation for Weights), a codec that recovers this structure: each row is L2-normalized, block-Hadamard rotated, paired into 2D coordinates, and quantized against a single Lloyd-Max codebook trained on the unit circular Gaussian, with activation-aware per-channel scaling. In a cross-model study spanning five LLMs from four families (1.1B--13B parameters) and eight quantized configurations, the activation-aware variant at ≈ 5.5 bpw stays within ± 0.4% of BF16 WikiText-2 perplexity on every model, matching the SmoothQuant W8A8 quality envelope at 32% fewer weight bits. Joint 2D coding outperforms polar (amplitude × phase) coding by 2--15~pp DeltaPPL at equal bitrate, and paired KL against BF16 tracks DeltaPPL% at Spearman ρ = 0.99 across 37 (method, model) rows, consistent with a monotone composite bound from codec distortion to KL divergence. A 3.5~bpw variant is competitive on quantization-tolerant architectures. At strict 4~bpw, the rotated-codebook frontier method QTIP outperforms QAM-W; the contribution is the quality-preserving 5--6~bpw band.