Error Diffusion Replaces Latent Weights: Training Binary-Weight Language Models on a Few Bits of Dithered State

Extreme low-bit inference offers a route toward smaller models and constrained deployment. Ternary language models restrict weights to $\{-1,0,+1\}$, approaching the limit of $\log_2 3 \approx 1.585$ bits/weight. The practical question for a pretrained model is not simply whether weights can be quantised but which capabilities survive and whether it remains useful for adaptation. We explore this by converting Qwen3.5-0.8B (752M parameters) to ternary weights using 72.4M tokens of quantisation-aware training (QAT). The resulting model, Cloe, is evaluated across 29 benchmarks, representation diagnostics, and downstream fine-tuning. The evidence shows non-uniform degradation. A linear probe recovers 43.76% of MMLU answers from the full-precision teacher's representations but only 26.19% from Cloe (near chance), indicating specialist factual information is lost. However, Cloe retains measurable performance on ten tasks, averaging 77.1% of teacher performance. Crucially, fine-tuning raises Cloe to 89.8% on SST-2 (95.6% of the matched teacher) and reaches 79.4% teacher retention on XSum. We attribute degradation to a combination of quantisation-induced information loss and incomplete recovery due to the limited QAT budget. We also highlight an evaluation pitfall: standard answer-letter scoring failed (Cloe emitted "A" on 98.6% of MMLU questions), necessitating continuation scoring. Ultimately, ternary conversion is unsuitable as a drop-in general replacement yet remains valuable as a compact substrate for task-specific models.

Low-bit quantization reduces the memory footprint of large language models, but preserving accuracy with efficient training remains challenging. Scalar quantization encodes weights independently, while explicit-codebook vector quantization introduces costly search during training. We propose E8-QAT, a 2-bit quantization-aware training framework that uses the $E_8$ lattice as an implicit eight-dimensional codebook. It jointly quantizes weight groups through arithmetic projection without storing or searching an explicit codebook during the training forward pass. A 2-bit core encoding and sparse extension preserve the selected lattice points exactly, adding less than 0.1% to core index storage. On Qwen3.6-35B-A3B, E8-QAT matches the performance of the full-precision model in terms of the average score across the four tasks. E8-QAT improves over the state-of-the-art baseline by 2.03% on IFEval and by 1.51% on LiveCodeBench, under the same data, schedule, and bit width. The 2-bit representation reduces peak memory by 80% and supports a $4\times$ larger batch. At each representation's largest feasible batch, E8-QAT reaches $1.97\times$ the decoding throughput of BF16.

Low-bit quantization shrinks language models but treats precision as a single global hyper-parameter: every weight uses the same bit-width. We introduce Variable Bit-width Quantization (VBQ), a training-time method in which each contiguous group of 64 weights learns its own resolution from {1,2,4,8} bits via a Gumbel-Softmax relaxation, trained jointly by an alternating optimization that gives the precision logits a clean, task-aligned signal. VBQ discovers a consistent, strongly heterogeneous allocation within individual projection types, not merely across layers, impossible to express with per-layer methods: 69% of groups collapse to 1 bit, the LM head averages 1.09 bits, while the first MLP block keeps ~2.5 bits. This pattern is stable enough to freeze into a fixed recipe and reuse without further search. The recipe yields a "bigger-but-smaller" regime: a 131M model at 1.82 mean bits reaches perplexity 4.2 on TinyStories, beating a 55M FP16 model (PPL 4.4) at 3.8x less storage, and lets a 1.46B model on FineWeb-Edu match a 593M FP16 control at ~3.7x less storage with 2.5x more parameters. As quality-per-byte, VBQ is 3.9-8.4x more efficient than FP16. The recipe maps directly to packed low-bit storage, so it also accelerates inference: with custom fused dequantize-and-multiply kernels, memory-bandwidth-bound autoregressive decode is faster at equal output, and the speedup grows with scale (parity at 131M, 1.9x at 1.0B, 4.7x at 9B on Apple silicon). A distributional analysis (KL divergence and argmax-flip rate) reveals a striking mechanism: deeper layers progressively self-heal the quantization error injected by early layers. The win is a from-scratch, train-time phenomenon; scaling the search economically beyond 1.5B parameters remains open. VBQ reframes precision as a learnable, non-uniform resource and shows that spending a fixed bit budget unevenly beats spending it uniformly.