KBBQ: A Predictive Noise Law and the Limits of Spectrum Flattening in FP4 Quantization

We study low-precision computation of C=AB with both factors quantized. We derive an exact finite-dimensional identity for the expected squared product error under independent, zero-mean entrywise errors with known variance fields; it holds exactly for non-overloading subtractive dither and for independent stochastic rounding, and we empirically assess deterministic round-to-nearest (RTN). Using the product-preserving equivalence AB=(AT)(T^{-1}B), we formulate contraction-gauge preconditioning: jointly choosing a factor representation and its sharing pattern before quantization. Preconditioning can reduce product error but may require extra transformed, quantized copies of the opposite operand: a shared transform needs one copy, a block-specific transform up to one per block. Within the bounded family of positive diagonal gauges (folds), a geometric program computes a globally optimal shared fold and a linear program decides whether the identity fold is already optimal. For other families we derive computable selection statistics -- tail index for scaling, profile spread for partitioning, coherence and weighted-Gram energy for rotations, slice-energy covariance for hierarchy depth -- with upper bounds for ranking heuristic candidates. Across twelve linear products from a trained three-block image classifier, median within-product rank correlations between dither-model predictions and deterministic-RTN errors are 0.937 at 8 bits and 0.918 at 4 bits. The GP fold cuts held-out product error over the identity fold by 18.0% (8-bit) and 20.5% (4-bit) in geometric mean, beats a SmoothQuant-style grid baseline at both precisions and on ten of twelve products, and lowers composed logit MSE by 15.4% and 26.4%. We thus provide exact stochastic product-error accounting, certified selection within the diagonal family, and a common objective for evaluating reusable transform candidates under RTN.

This is a planning-method note with an unpaired pilot audit. We adapt the classical paired-binary sample-size calculation (Miettinen, 1968) to quantization benchmarks, giving a conservative minimum detectable effect (MDE) bound $δ^{*} \le (z_{1-α/2}+z_{1-β})\sqrt{ρ_d/m}$ in the paired item count $m$ and the FP16-NF4 disagreement rate $ρ_d$. The bound turns "how reliable is my quantization claim?" into a one-line budget a benchmark designer can commit to before running. We illustrate the bound on four models and four benchmarks ($k=5$ splits of $n=100$), and add a parallel MMLU prompt-template study to put the bound's quantization-noise scale alongside the prompt-noise scale. Assuming $ρ_d=0.10$ (an unmeasured planning value), all observed NF4-FP16 deltas fall below the implied MDE, and most cross-split SDs lie within $\pm 1.5$ pp of the binomial reference $\sqrt{p(1-p)/n}$, so much of the variance reported as "benchmark unreliability" on $n=100$ subsamples is binomial sampling noise. The single borderline cell (OPT-WinoGrande, $|Δ|=3.2$ pp) is below the implied MDE at $ρ_d=0.10$ but above it at $ρ_d=0.05$, illustrating the planning trade-off the bound makes explicit. On MMLU, prompt-template ranges of 2-10 pp meet or exceed the largest observed quantization delta (3.2 pp), so a quantization audit that does not first fix the prompt template absorbs template variance into its noise floor. We complement the bound with a five-line pre-registration template.

Mixed-precision quantization requires discrete allocation of weight and activation bit-widths, followed by recovery of the selected network. We develop a task-aware quadratic unconstrained binary optimization (QUBO) surrogate with separate weight and activation profiles, a bit-operation (BOP) cost, and selected structural priors. QUBO provides a network-wide allocation that can be refined through direct validation-based PROTES search. On a compact NAFBlock-based denoiser, the refined route achieves 37.192 dB after LSQ+ at 4.035% routed-layer BOPs, versus 37.092 dB at 4.101% for a HAWQ-style baseline. The repeated-search primary experiment shows that LSQ+ largely closes the quality gap between QUBO allocation and expensive direct refinement. An additional restoration architecture retains a larger recovered gain, indicating that refinement's value depends on architecture and recovery. We evaluate quality, achieved cost, routing stability and optimization expense together. Deployment measurements characterize a fake-quantized floating-point implementation; BOP reductions describe analytical allocation savings.