Practical Limits of Lossless Compression for bf16 Transformer LLM Weights With Companion Measurements on fp16 and on Q4_K-typed Tensors in GGUF Q4_K_M Files, Under Strict Profile-Byte Accounting

Abstract. We measure the lossless compressibility of bf16 transformer LLM weights and of Q4_K-typed tensors in GGUF Q4_K_M files under a strict fairness contract that counts profile bytes against trained methods on every file, verifies byte-exact roundtrips on every run, and uses a model-level train/test split. Across 12 source models and 11,942 verified roundtrip method-evaluations over 7,960 unique benchmark rows (zero roundtrip failures), we establish three results.

First, the measured practical ceiling for bf16 weights under iid marginal-byte coding is ≈1.495× byte-weighted (model-level 95% CI [1.487, 1.502]), derivable from a one-line Shannon proposition at measured byte entropies (median H_high = 2.74 bits, H_low = 7.97 bits) and consistent with α-stable theory of SGD-trained networks. Our bf16_split predictor reaches 1.488×; the official OpenZL le-u16 result marginally exceeds the bound (1.499×) by using context coding within byte planes, a regime our proposition does not cover. Result validated through Qwen2.5-7B-Instruct (byte-weighted +0.0001 vs the small-model corpus).

Second, the measured practical ceiling for Q4_K-typed tensors in GGUF Q4_K_M files is ≈1.076× at tensor-stream level (best method 1.052×). At the full GGUF artifact level this becomes ≈1.041–1.045× because Q4_K_M files include Q6_K tensors that compress at ~1.01×. Mechanism: optimized per-block scaling produces near-uniform nibble distributions (median 3.86 of 4.0 bits/symbol).

Third, we find no evidence of usable linear redundancy between adjacent same-role transformer-layer weight matrices at bf16 precision (median Pearson +0.0004 across 250 pairs in two source models from the Qwen2.5 family), ruling out simple scalar-affine cross-layer compression but not nonlinear schemes or cross-checkpoint deltas.

This paper does not propose a new compressor. The two methods we built (bf16_split and a Q4_K block mixture-CDF coder) are used only to test whether the measured entropy ceilings are practically reachable; both cluster at the ceiling rather than beating it. The methodology contribution is the fairness contract under which prior methods can be compared directly.