Pre-Registering the Detectable Effect: A Paired-MDE Budget for 4-bit Quantization Benchmarks, with a Pilot Audit

We develop a second-order theory of quantization noise in matrix multiplication in which the quantization format is characterized by the variance it assigns to each element. The constant variance profile of integer quantization recovers existing integer-noise theory, while the multiplicative profile of floating-point rounding reduces the data dependence to a scalar, the participation factor $κ$, yielding a closed-form signal-to-noise-ratio law. The resulting functional also admits a closed-form upper bound $κ^{*}$ that no function-preserving linear transform can exceed and that is attained by a recent state-of-the-art method. Building on this analysis, we introduce KBBQ (\textbf{K}appa-\textbf{B}raked \textbf{B}lockwise \textbf{Q}uantization), which parameterizes the extent to which a transform approaches this ceiling. At W4A4, across four base models and two FP4 formats, KBBQ outperforms the prior state of the art without additional deployment-time computation.

Post-training quantization to 4-bit weights is widely reported to be nearly lossless. We test this claim for multi-turn, tool-calling agents, where it now matters most. On $τ^2$-bench, across two open-weight model families in dense and MoE variants and two domains (eight cells, 456 episodes each, at 16-, 8-, and 4-bit weights), quantization indeed looks free on the standard metric. No cell shows a score change that survives multiple-comparison correction, and in the cell that carries the largest process damage, equivalence testing bounds the change within $\pm$7.5 points. The process tells a different story. Quantization amplifies the failure the model already exhibits at full precision (tool-name hallucination in telecom, with the same directional trend in retail entity errors) by up to 2.5$\times$ in volume (+17.6 points per task), while creating essentially no new failures. The failure set is the same at every precision (rank correlation $\geq$ 0.94, 0.18% novel events). The score stays flat because the benchmark's ten-error budget absorbs the extra failures. Shrinking the budget to two errors re-exposes a score gap of 17 points, and it does so only in the one cell where quantization added error volume, exactly as the masking account predicts. A targeted error-repair prompt, run for five telecom models at every precision, removes the damage exactly and only where it lives. Both diagnostics, the per-channel error rate and success under a shrinking budget, come from logs benchmarks already collect; we suggest reporting them alongside task reward.

Quantization is known to hurt below four bits, but nobody can say which of a model's decisions will change at a given bit-width. This matters most where a model acts rather than answers: a compressed agent stops calling its tools and, one bit lower, loses roughly half its safety refusals, while benchmark scores barely move. Prior work assumes the added noise has a roughly fixed size, which would make confident decisions safe. We measure the decision instead: the margin, the picked option's score minus its best alternative's, tracked before and after quantization across 16 models from 8 families under round-to-nearest, seven under AWQ, two under GPTQ and one under GGUF, at 8 down to 2 bits. The damage is proportional, not fixed in size: the margin is multiplied by a factor that collapses with bit-width (median 0.86 at 4 bits, 0.33 at 3, 0.00 at 2), which we call margin shrinkage. Contraction removes the protection a large margin affords; the model's own biases pick the direction: at 3 bits the decision to call a tool collapses toward inaction while the choice of which tool is untouched. No additive account, including one whose noise grows with the margin, wins a single damaged whether-to-call or safety cell (378 of 378). Given a condition's own constants the relation predicts held-out flip rates to a median 1.7 points, calibrated per decision (error 0.004 over 161,744 predictions), no flip used in the fit. Borrowed constants are wrong by 18-33 points at 3 bits, so the paired margin set has to be measured per model and bit-width: it locates breaking decisions without replacing measurement. At 4 bits the measurement is anchored to behaviour (the most likely token over the whole vocabulary is one of the item's two options in 85% of tool items); we treat the 2-bit floor as where the instrument stops measuring. No label-free repair we tested recovers more than one more bit does.