1. Problem and contribution

This work started from the problem's 4-bit weight-and-activation (W4A4) post-training quantization (PTQ) setting for Qwen3-8B with native microscaled arithmetic (statement version 1; version 2 widens the scope to compression with preserved capability, for which the distortion profile remains relevant). The main tools are function-preserving transforms (Hadamard rotations, smoothing) and branches that remove part of the signal from the quantized path. Their value depends on the number format: rotations help INT4 strongly, help MXFP4 moderately, and are weak or harmful in NVFP4 [1, 2, 3]. A quantitative, per-layer account of that dependence would explain the exceptions and let a practitioner predict a lever's effect on distortion without trying it.

Two published theories address this:

• KBBQ's noise law [2]. Under unbiased, uncorrelated rounding errors whose variance depends only on the element (its assumptions A1-A3), the expected squared error of a dot product is $\mathbb{E}[\delta S^2]=\sum_k \mathbb{E}[x_k^2]\varphi_w(w_k)+\mathbb{E}[w_k^2]\mathbb{E}[\varphi_x(x_k)]$. It uses a multiplicative $\varphi=\sigma^2_{\mathrm{fmt}}a^2$ for floating point and an additive step-size variance for integers.
• OSFP4 [4]. It models a multiplicatively dithered FP4 quantizer, whose normalized error $\phi(z/\gamma)$ is evaluated at the element divided by its block scale $\gamma$. It builds a smoothing and scale optimizer on that surrogate.

Question. Which part of the realized deterministic distortion of microscaled formats do these models miss, and does correcting it predict lever effects on layers, documents and models never used for calibration?

Contributions.

1. Mechanism of deterministic block coupling. The realized NVFP4 activation error in Qwen3-8B, decomposed by element class within each 16-block, shows three effects.
   • Block maxima carry 44-82% of the incoherent energy but have about 13x lower relative variance than ordinary elements; only E4M3 scale rounding hits them.
   • Elements below the zero-rounding boundary are flushed. For effective dequantization scale $s_b$ the ideal E2M1 boundary is $|z|<s_b/4$, i.e. $b_{\max}/24$ for an unrounded 6-scale and $b_{\max}/16$ for a 4-scale, shifted by E4M3 scale rounding. Elements below the nominal $b_{\max}/24$ threshold carry under 0.4% of the energy but 5-43% of the noise.
   • The per-block 4/6 scale choice tracks the block fill. As a result the effective per-layer variance ranges from 0.0030 to 0.0095, whereas KBBQ treats it as a format constant. A Hadamard rotation drives every layer to nearly the same value (0.0076).
2. A compact two-statistic profile. $\mathbb{E}[e_z^2\mid\text{block}]=b_{\max}^2\,\psi_{\mathrm{fmt}}(u,\rho)$, with $u=|z|/b_{\max}$ (the maximum is its own class) and block fill $\rho=\overline{z^2}/b_{\max}^2$. It is calibrated once per format and scale rule on four layers of Qwen3-8B and then frozen.
3. Out-of-sample prediction, tested on unseen Qwen3-8B layers and on Qwen3-4B with fresh documents and all projection output rows, and zero-shot on a third family (SmolLM2-1.7B, Llama architecture, own tokenizer) using the Qwen3-8B table:
   • NVFP4 layer noise: 0.02-0.11 dB RMS error. KBBQ is at 1.2-2.4 dB; OSFP4's dithered $\phi$, used as a predictor of realized RTN distortion, overestimates by 3.1-4.8 dB.
   • Rotation effects in NVFP4, on these held-out transfer sets: RMS log-ratio error 0.010-0.029, sign correct in 97-100% of layers. KBBQ: 0.26-0.57 and 83-93%. OSFP4-$\phi$: 0.25-0.56 and 46-66%.
   • All 145 Qwen3-8B layer cells, including early layers: the rotation log-ratio error has median 0.0055 but RMS 0.32, driven by a tail of five layers (Section 4.5). This paper does not establish the tail's mechanism.
   • Energy-removal effect (subtracting a principal activation subspace) is predicted as well.
   • Other formats: in MXFP4 and INT4 the profile matches or improves on the format-appropriate KBBQ form.
4. Explicit boundaries.
   • The per-element sum omits cross-coordinate terms. Five tail layers, all with massive-activation inputs, carry almost all of the Hadamard prediction error, and the profile does not explain them.
   • Layer distortion is not model quality. Even oracle and prospective law-guided selective rotation make NVFP4 perplexity worse. Single-linear KL measurements carry a BF16 numerical baseline of the same order as the quantization effect (Section 5).

2. Related work and distinction

Noise laws. KBBQ [2] derives the SNR law $10\log_{10}\kappa+C_{\mathrm{fmt}}$ from A1-A3, acknowledges that a shared block scale violates A2, and absorbs the deviation into a tuned "brake". We keep their decomposition into per-element contributions but replace A3's element-only variance with a block-conditional one. OSFP4 [4] already makes the error depend on the element divided by its block scale. Its $\phi$ is the normalized error of a dithered quantizer, chosen to make an optimization objective smooth. Its Lemma 1 relates it to the deterministic quantizer through a bound (for some per-vector $\beta\in[1,2)$). It is therefore not designed as a calibrated predictor of the realized RTN error. We evaluate it at that task because it is the closest scale-relative published model. The specific deterministic features it does not capture (exact maxima, a sharp flush threshold, the fill-dependent 4/6 decision) are where our profile gains. The "Great Inversion" survey [1] explains the opposing effects of grid shape and block scale on flattening transforms. Our results make that account quantitative per layer, and Section 5 marks where it stops predicting model quality.

Known mechanisms. MR-GPTQ [3, Sections 3.1-3.2] already observes that absmax scaling preserves the block maximum before scale rounding, that NVFP4 maxima are effectively promoted to E4M3 precision, that small elements fall into a dead zone, and that rotation forfeits this protection. Four Over Six [5] supplies the adaptive scale choice. Our contribution is not these qualitative observations. It is a compact, calibrated conditioning on relative magnitude and block fill, its quantitative transfer to held-out layers, documents and models, and the analysis of where it fails.

Transforms and branches. QuaRot and SpinQuant rotations [6], MR-GPTQ block rotations [3], SmoothQuant [7], weight-side low-rank branches (SVDQuant, BiLoCo, HSVDQuant [8, 9, 10]) and activation-side subspace protection (ResQ [11]) are the levers whose effects we predict. We probe with block-Hadamard-128 and with a rank-32 principal-subspace split of the activation ("innovation": quantize $X-XUU^\top$ and add $XU(WU)^\top$ exactly). Neither is claimed as a new method.

rStation. Our forum post (r/post_training_quantization/post/1) reported the methodology checks used here: logit-scale control, GPTQ damping and the PPL noise floor.

3. Method

3.1 Setting and element-class decomposition

A layer cell is one linear projection: q_proj (standing for q/k/v), o_proj, gate_proj (for gate/up), down_proj, or lm_head. Operands are the SmoothQuant($\alpha=0.5$)-preconditioned activation $X$ (BF16-model inputs on C4 text, attention-sink position 0 excluded) and weight $W$, matching the per-linear recipe of the supplied baselines. Formats, simulated exactly:

• NVFP4: 16-element blocks, E4M3 block scales under an FP32 tensor scale, E2M1 values, and the 4/6 MSE scale rule [5]. Our simulator matches the FourOverSix CUDA quantizer on 99.9998% of elements. "Static-6" is NVFP4 without 4/6.
• MXFP4: 32-element blocks with E8M0 scales.
• INT4-g128: symmetric $[-7,7]$ with an FP scale per 128 elements.

The realized activation-side and weight-side noises are $N_A=\|(Q(X)-X)W^\top\|_F^2$ and $N_W=\|X(Q(W)-W)^\top\|_F^2$, relative to $\|XW^\top\|_F^2$. They include all within-block error correlations.

Table 1. NVFP4 activation-side noise $\sum_{t,k}c_ke_{tk}^2$ with $c_k=\|W_{:,k}\|^2$, decomposed by element class within each 16-block (Qwen3-8B, held-out C4). "Effective σ²" is noise divided by incoherent energy $\sum_{t,k}c_kx_{tk}^2$.

| layer | energy: max | nominal flush (<1/24) | small (1/24-1/6) | normal | noise: max | flushed | small | normal | eff. σ² native | eff. σ² Hadamard | |---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | L18 gate | 0.44 | 0.0003 | 0.017 | 0.54 | 0.04 | 0.05 | 0.16 | 0.75 | 0.0067 | 0.0076 | | L33 gate | 0.47 | 0.0003 | 0.016 | 0.51 | 0.05 | 0.05 | 0.16 | 0.74 | 0.0063 | 0.0076 | | L18 down | 0.57 | 0.0023 | 0.041 | 0.39 | 0.04 | 0.24 | 0.33 | 0.39 | 0.0095 | 0.0076 | | L33 down | 0.79 | 0.0034 | 0.027 | 0.18 | 0.06 | 0.43 | 0.31 | 0.20 | 0.0081 | 0.0077 | | lm_head | 0.82 | 0.0003 | 0.008 | 0.17 | 0.19 | 0.11 | 0.19 | 0.51 | 0.0030 | 0.0076 |

Per-class relative variances are nearly the same in every layer: about 0.0007 for block maxima, 0.009 for normal elements, 0.06-0.09 for small elements, and about 1.0 for flushed elements. What differs between layers is how energy is distributed over the classes.

3.2 The profile and the predictors

For element $z$ of a block of size $B$ with maximum magnitude $b_{\max}$, let $u=|z|/b_{\max}$ and $\rho=\frac{1}{B}\sum_{j}z_j^2/b_{\max}^2$. We tabulate $\psi_{\mathrm{fmt},s}(u,\rho)=\mathbb{E}[e^2/b_{\max}^2\mid u,\rho]$ for each operand side $s\in\{A,W\}$, using 40 log-spaced $u$ bins on $[10^{-4},1)$, a separate bin for $u=1$, and 7 $\rho$ bins with edges 0.08, 0.12, 0.17, 0.23, 0.30 and 0.40. The table is an empirical conditional second moment, not a proven conditional variance. As implemented (ptq/exp20_law.py), the fit uses Qwen3-8B layers 0, 3, 6 and 9, with native and Hadamard-rotated operands. Activations come from one C4 calibration window (window 24 of the prepared file; four other windows are used only for the principal subspace), and weights from 1,024 randomly sampled output rows. Empty $(u,\rho)$ bins receive zero; no marginal fallback is applied in the retained tables. The predicted noises are

$$ \widehat N_A=\sum_{t,k}c_k\,b_{\max,tk}^2\,\psi_A(u_{tk},\rho_{tk}),\qquad \widehat N_W=\sum_{j,k}m_k\,b_{\max,jk}^2\,\psi_W(u_{jk},\rho_{jk}), $$

with $m_k=\sum_tX_{tk}^2$, relative to $\|XW^\top\|_F^2$. Like KBBQ and OSFP4, these sums ignore cross-coordinate error terms. Each scale rule gets its own table: NVFP4 with 4/6 and static-6 are calibrated separately, then each frozen table is tested out of sample. Figure 1 shows the tables.

Figure 1: Fitted relative error variance E[e²]/z² against u = |z|/block max for NVFP4, MXFP4 and INT4-g128 activations, with four block-fill curves; stars mark the block maximum.

Figure 1. Relative variance is flat in the normal range for the FP formats (multiplicative) and grows as $u^{-2}$ for INT (additive). It rises to about 1 where elements flush. At $u=1$ it is about $7\times10^{-4}$ for NVFP4 and 0 for INT4, whose maximum maps exactly to 7. MXFP4's power-of-two scale does not represent its maximum exactly. The dashed line is a single fitted $\sigma^2$.

Predictors compared, all evaluated with the same per-element sums:

• KBBQ: $\sigma^2_{\mathrm{fmt},s}z^2$ for FP formats, with $\sigma^2$ fitted on the same calibration cells; the additive $(b_{\max}/7)^2/12$ (their Eq. 4) for INT4.
• OSFP4-$\phi$: $z^2\phi(z/\gamma)$ with OSFP4's dithered $\phi$ (their Eqs. 3-4, computed numerically). $\gamma$ is either the absmax-to-6 scale $b_{\max}/6$ or the scale the 4/6 rule actually chose per block ($b_{\max}/6$ or $b_{\max}/4$, read from the quantizer).
• Local-step proxy: uniform rounding inside the local E2M1 cell at $|z|/s_b$, a simple deterministic analytic control.
• Block profile (ours).

3.3 Levers and test sets

Each cell is evaluated native, with block-Hadamard-128 on the reduction dimension of both operands, and with innovation at rank 32.

• (i) Unseen layers: Qwen3-8B layers 13, 17, ..., 33, 35 and lm_head (29 cells).
• (ii) Unseen model: Qwen3-4B layers 1, 5, ..., 33, 35 and lm_head (41 cells).
• (iii) Unseen family: SmolLM2-1.7B [12] layers 0, 3, ..., 21, 23 and the tied head (37 cells), using the Qwen3-8B tables. All tests use fresh documents (a WikiText-2 validation window and a C4 window never used for fitting). Projections use all output rows; lm_head uses 8,192 sampled rows.

4. Results

4.1 Layer-noise prediction

Table 2. RMS error of predicted vs realized layer noise, in dB, as activation side / weight side. "OSFP4-φ" uses the 4/6-chosen scale; the absmax-6 variant is 0.3-0.45 dB better and is reported in the supplement.

| test set | format | KBBQ | OSFP4-φ | local-step | block profile | |---|---|---:|---:|---:|---:| | Qwen3-8B layers | NVFP4 | 1.47 / 1.47 | 4.24 / 3.83 | 3.21 / 2.73 | 0.06 / 0.02 | | | NVFP4 static-6 | 1.56 / 1.41 | 3.48 / 2.77 | 2.79 / 2.13 | 0.06 / 0.05 | | | MXFP4 | 1.15 / 0.65 | — | 0.97 / 0.14 | 0.20 / 0.10 | | | INT4-g128 | 1.67 / 0.18 | — | 1.67 / 0.18 | 0.57 / 0.10 | | Qwen3-4B | NVFP4 | 1.51 / 1.21 | 4.00 / 3.57 | 3.12 / 2.49 | 0.08 / 0.09 | | | NVFP4 static-6 | 1.43 / 1.10 | 3.06 / 2.47 | 2.56 / 1.83 | 0.10 / 0.09 | | | MXFP4 | 1.07 / 0.71 | — | 0.94 / 0.19 | 0.21 / 0.14 | | | INT4-g128 | 2.21 / 0.16 | — | 2.21 / 0.16 | 0.17 / 0.22 | | SmolLM2-1.7B (zero-shot) | NVFP4 | 2.39 / 2.38 | 4.77 / 4.37 | 3.78 / 3.42 | 0.11 / 0.08 | | | NVFP4 static-6 | 2.55 / 2.36 | 4.12 / 3.43 | 3.42 / 2.88 | 0.22 / 0.05 | | | MXFP4 | 1.16 / 0.89 | — | 0.88 / 0.49 | 0.39 / 0.46 | | | INT4-g128 | 1.95 / 0.42 | — | 1.95 / 0.42 | 0.20 / 0.03 |

Document-level replication. The transfer sets above aggregate one WikiText window and one C4 window per model, so cells within a model share documents. Re-running the frozen test on a disjoint document pair (WikiText-2 validation window 3, and a C4 window from different calibration documents) gives:

• NVFP4 cell error 0.04/0.02 dB (Qwen3-8B) and 0.09/0.08 dB (Qwen3-4B);
• rotation log-ratio error 0.010 and 0.028, with rotation sign accuracy 0.97 and 1.00;
• MXFP4 and INT4 numbers within ±0.1 dB of Table 2 (Appendix C). The unit of replication here is the document pair, and there are only two; the many cells evaluated on each pair are not independent document replications. A document-level uncertainty estimate would need more documents as units.

OSFP4's dithered $\phi$ is consistently high, as expected for a smoothed surrogate of the deterministic error. It charges the block maximum its plateau value, about 0.0107, against a realized 0.0007.

4.2 Predicting lever effects

Table 3. RMS error of predicted $\log(N_{\text{lever}}/N_{\text{native}})$, where 'total noise' $N=N_A+N_W$ is the sum of the separately measured activation-side and weight-side errors (not the squared error of simultaneous W4A4 quantization, which adds interaction terms), and rotation sign accuracy, as KBBQ / OSFP4-φ (chosen scale) / block profile. MXFP4 and INT4 rows compare against the local-step proxy instead of OSFP4-φ, which is NVFP4-specific.

| test set | format | Hadamard error | Hadamard sign | innovation error | |---|---|---|---|---| | Qwen3-8B | NVFP4 | 0.348 / 0.334 / 0.010 | 0.83 / 0.52 / 1.00 | 0.259 / 0.208 / 0.007 | | | NVFP4 static-6 | 0.370 / 0.429 / 0.017 | 0.62 / 0.31 / 0.97 | 0.265 / 0.240 / 0.012 | | | MXFP4 (KBBQ / local-step / ours) | 0.304 / 0.130 / 0.019 | 0.59 / 1.00 / 1.00 | 0.056 / 0.018 / 0.023 | | | INT4-g128 | 0.277 / 0.277 / 0.085 | 1.00 / 1.00 / 1.00 | 0.041 / 0.041 / 0.078 | | Qwen3-4B | NVFP4 | 0.264 / 0.254 / 0.029 | 0.93 / 0.66 / 0.98 | 0.268 / 0.191 / 0.014 | | | NVFP4 static-6 | 0.265 / 0.323 / 0.037 | 0.83 / 0.54 / 0.95 | 0.246 / 0.198 / 0.018 | | | MXFP4 | 0.276 / 0.112 / 0.045 | 0.59 / 1.00 / 1.00 | 0.084 / 0.046 / 0.034 | | | INT4-g128 | 0.294 / 0.294 / 0.039 | 1.00 / 1.00 / 1.00 | 0.045 / 0.045 / 0.037 | | SmolLM2-1.7B | NVFP4 | 0.567 / 0.515 / 0.022 | 0.86 / 0.49 / 0.97 | 0.479 / 0.399 / 0.015 | | | NVFP4 static-6 | 0.604 / 0.617 / 0.031 | 0.68 / 0.27 / 0.89 | 0.498 / 0.454 / 0.022 | | | MXFP4 | 0.282 / 0.118 / 0.051 | 0.51 / 1.00 / 1.00 | 0.115 / 0.059 / 0.036 | | | INT4-g128 | 0.333 / 0.333 / 0.032 | 1.00 / 1.00 / 1.00 | 0.192 / 0.192 / 0.028 |

In NVFP4 the rotation lowers layer noise in only about half of the layers, and most effects lie near a ratio of 1. Predicting the sign therefore needs small errors.

• OSFP4-φ predicts large rotation gains almost everywhere, so its sign accuracy is near or below chance.
• KBBQ fails mainly on peaky layers.
• The block profile is accurate in every family tested. Figure 2 shows NVFP4 predictions against realized ratios.

Figure 2: Predicted versus realized NVFP4 noise ratio (Hadamard over native) for 107 held-out layers from Qwen3-8B, Qwen3-4B and SmolLM2-1.7B; the block profile lies on the diagonal while the multiplicative and local-step models predict large reductions where the realized ratio is near or above one.

Figure 2. The lower-left cluster consists of peaky layers where format-constant or step-based models predict 10-30x reductions that do not occur. For legibility the figure shows KBBQ and the local-step proxy; OSFP4-φ behaves like the latter.

4.3 Prospective layer-level decisions

For each held-out layer we choose between native and Hadamard by minimum predicted noise, freezing the choice before reading realized values. We report the aggregate regret: the excess realized noise over the per-layer oracle, summed over layers.

| format | block profile | KBBQ | OSFP4-φ | local-step | always native | always Hadamard | |---|---:|---:|---:|---:|---:|---:| | NVFP4 (8B / 4B / SmolLM2) | 0.000 / 0.001 / 0.001% | 0.23 / 0.42 / 0.82% | 7.2 / 5.0 / 7.6% | 1.5 / 1.1 / 1.6% | 0.7 / 2.5 / 1.9% | 8.1 / 7.0 / 9.7% | | NVFP4 static-6 | 0.006 / 0.011 / 0.016% | 1.0 / 1.2 / 1.3% | 14.5 / 12.5 / 15.5% | 3.4 / 2.8 / 3.1% | 0.3 / 2.5 / 1.2% | 16.1 / 13.9 / 17.7% | | MXFP4 | 0.000 / 0.000 / 0.000% | 14.2 / 12.4 / 15.6% | — | 0.000% (all three) | 26-39% | 0-0.4% |

When innovation is also offered, every predictor selects it in every NVFP4 layer, so that larger choice set adds no decision evidence beyond native-versus-rotation.

4.4 Full-model consequences

Official protocol (raw WikiText-2 test, 146 x 2,048 windows; 256 fixed C4 windows; SDPA). The format is applied to all linears including the head, with SmoothQuant 0.5 statistics from the official eight C4 calibration documents and RTN. BF16 scores 9.725 / 12.296.

Table 4. Full-model PPL (WikiText / C4), seed 0.

| format | native | Hadamard-128 on every linear | innovation r32 on gate/up inputs | |---|---:|---:|---:| | NVFP4 | 10.104 / 12.692 | 10.246 / 12.771 | 10.053 / 12.647 | | MXFP4 | 11.074 / 13.972 | 10.817 / 13.212 | 10.897 / 13.808 | | INT4-g128 | 12.380 / 14.562 | 10.728 / 13.112 | 12.076 / 14.306 |

Rotation recovers about 62%/64% of the INT4-g128 gap and makes NVFP4 worse. This matches the per-format direction predicted at layer level.

4.5 What the profile teaches: deterministic block coupling

1. The block maximum is nearly free. The scale is computed from the block maximum, so the maximum lands on the top grid point (6, or 4 under 4/6), and only the E4M3 rounding of the scale perturbs it (relative variance about $7\times10^{-4}$). A dithered model charges it about 0.0107 and the fitted format constant about 0.0075. Block maxima carry 44-82% of the incoherent energy in Qwen3 projections, so this effect sets most of the layer-to-layer variation in effective variance.
2. Flushing is a sharp threshold. Elements below $s_b/4$ round to zero with relative variance 1. This is $b_{\max}/24$ for the nominal 6-scale and $b_{\max}/16$ for a 4-scale; Table 1 uses the nominal $b_{\max}/24$ classification, so its 'flush' class is approximate.
3. Block fill predicts the scale decision. In Qwen3-8B layer-18 MLP inputs, the fraction of blocks that take the 4-scale rises monotonically from 1% at $\rho<0.08$ to 77% at $\rho>0.4$. Conditioning on $\rho$ cut NVFP4 rotation-sign errors from 14-23% (a profile in $u$ and the tensor-relative scale only) to 2-5%.

With these three effects tabulated, the same table transferred to the held-out layers, documents, Qwen3-4B and SmolLM2-1.7B tested here. On those sets, realized deterministic distortion is well approximated by a function of position-in-block and block fill. This is an empirical conditional second-moment profile with demonstrated transfer on three dense models, not a universal law: it fails on the tail described next. This explains the format dependence of rotations. Rotations make blocks Gaussian-like, which removes flush penalties but destroys peaky blocks whose maxima were nearly free, and in NVFP4 the two nearly cancel on average. MXFP4's power-of-two scale never made the maximum free, and INT4's additive noise rewards flattening directly.

Where the compact profile ceases to suffice. Over all 145 Qwen3-8B layer cells, including the calibration layers on fresh documents, the Hadamard rotation log-ratio error has median 0.0055 but RMS 0.32. Five cells carry almost all of it: gate_proj of layers 1-3 and down_proj of layers 1 and 16; without them the RMS is 0.042. All five have inputs dominated by a massive-activation channel. Their mechanism is left to separate work.

5. Limits: layer distortion is not model quality

Oracle selection (development sets). On Qwen3-8B we rotated only the layers whose realized NVFP4 layer noise falls by at least 10% (20 linears) or 3% (104 linears). PPL on WikiText-val / C4 dev was 10.063 / 14.359 and 10.152 / 14.431, against native 10.027 / 14.372 and all-rotated 10.135 / 14.472.

Prospective selection (official test). Choosing layers by the block profile's predictions, frozen before the readout, gives:

• predicted reduction at least 10% (28 linears): 10.147 / 12.707;
• at least 3% (112 linears): 10.272 / 12.754;
• any reduction (153 linears): 10.171 / 12.715; against native 10.104 / 12.692. None beats native; the measured noise floor for equivalent configurations is about 0.02 PPL.

Single-linear KL is not a clean currency at this effect size. We quantized one linear at a time in otherwise BF16 Qwen3-8B (20 linears across depths and types) and measured full-model KL to BF16 on eight held-out windows. Two measurement problems dominate.

• Numerical baseline. Routing a single linear through the SmoothQuant path in BF16 without any quantization (bit-level reassociation only) already yields KL of 0.0003-0.0009 nats per token. The KL difference between the quantized and unquantized paths of the same routing is 0.00005-0.0009, of the same order. Raw KL ratios are therefore strongly affected by this baseline.
• Window noise. KL ratios of early layers change by up to 0.13 between disjoint window sets, while MSE ratios are stable.

As a diagnostic we also computed baseline-relative KL differences (quantized minus unquantized path of the same routing). These are not a measurement of quantization-only harm: even in a local quadratic approximation, the difference keeps a cross term between the BF16 path perturbation and the quantization perturbation, which can reinforce or cancel. With that caveat, the late down_proj differences mostly move with MSE under rotation: x1.13 vs MSE x1.15 (layer 18) and x1.20 vs x1.23 (layer 27). The exceptions are x1.10 vs x1.30 (layer 34) and x0.96 vs x1.22 (layer 9). Across all 20 linears these diagnostic ratios correlate weakly with MSE ratios (0.25; sign agreement 80%), and early layers have very small, noisy differences. We therefore withdraw the v1 statement that late down_proj tolerates 12-30% more MSE at only 3-9% more KL, and we do not replace it with another interpretation.

Weighting the error by empirical output sensitivity did not change the picture. We used gradients of next-token NLL in three forms: per channel, per token and channel, and the full first-order term, also with labels sampled from the model. These squared task-NLL gradients are not the curvature of the teacher-student KL. The most reliable evidence for the boundary is therefore the full-model results above: oracle and prospective layer-noise-guided rotation do not improve perplexity. A model-level decision rule remains outside this paper's claim.

Other limits.

• Three models from two families, all dense and at most 8B parameters.
• The attention-sink position is excluded from layer measurements, and the head is row-sampled.
• Quantization is simulated (bit-exact NVFP4 rounding); no speed claim is made.
• The supplement's fixed-seed full-model numbers are single runs. Earlier three-seed runs of the RTN baseline vary by about 0.01 PPL.

6. Reproducibility

The supplement contains:

• the simulator (ptq/nvfp4.py, ptq/sim.py) and OSFP4's $\phi$ (ptq/osfp4phi.py);
• the profile fit/test engines: ptq/exp20_law.py (Qwen3, all formats, fresh documents, full rows), ptq/exp21_smollm.py (zero-shot SmolLM2), and ptq/exp24_osfp4_*.py (adds the OSFP4-φ predictors for NVFP4);
• the element-class decomposition (ptq/decomp_cpu.py), the cross-term check (ptq/exp25_cross.py), and the document replication (ptq/exp34_docrep.py);
• the full-model scripts (ptq/exp12_fmt.py, ptq/exp14_test.py, ptq/exp17_selrot.py, ptq/exp23_lawpolicy.py, ptq/exp19_harm.py, ptq/exp27_klrep.py, ptq/exp38_klfloor.py);
• the analysis scripts (ptq/an20.py, ptq/an21.py, ptq/an24.py);
• all result JSONs, including the fitted tables (results/exp20_law.json). Hardware is one RTX 5090. Fitting plus all held-out layers takes 3-17 minutes per model, depending on predictors. Each full-model PPL evaluation takes about 4 minutes. Software: PyTorch 2.8, Transformers 5.6.2, FourOverSix 1.0.5. Models: Qwen3-8B at the supplied revision, Qwen3-4B, and SmolLM2-1.7B (HuggingFaceTB, main branch). Data: the supplied WikiText-2 and C4 files.

Appendix A. Development-set full-model measurements (Qwen3-8B)

| format | native | Hadamard-128 | innovation r32 (gate/up) | |---|---:|---:|---:| | NVFP4 | 10.027 / 14.372 | 10.135 / 14.472 | 9.938 / 14.301 | | MXFP4 | 10.958 / 16.069 | 10.640 / 15.009 | 10.764 / 15.804 | | INT4-g128 | 12.157 / 16.690 | 10.627 / 14.885 | 11.980 / 16.345 |

BF16 on these development sets (WikiText-2 validation, 64 windows; held-out C4 windows) is 9.629 / 13.836. Over three calibration seeds on the RTN host, innovation r32 gives 9.961 ± 0.029 / 14.312 ± 0.015 vs RTN 10.024 ± 0.007 / 14.361 ± 0.012. Over a GPTQ host (damping 1.0 x mean diagonal), its increment across two seeds is +0.05/-0.03 (WikiText) and -0.005/+0.025 (C4), i.e. not robust.

Appendix B. Cross-coordinate check (NVFP4 weight side, Qwen3-8B)

Ratio of realized weight-side noise to its per-element (diagonal) part, native → Hadamard, and the input top-eigenvalue share. Scope: first 4,096 output rows and one held-out C4 window, which is smaller than the exp22 scope (all rows, WikiText-val/C4 pair). The values are supporting evidence and do not exactly decompose the exp22 tail.

| layer | full/diag native | full/diag Hadamard | top eigenvalue share | |---|---:|---:|---:| | L1 gate | 1.08 | 5.23 | 0.39 | | L2 gate | 1.07 | 6.29 | 0.65 | | L3 gate | 1.02 | 3.94 | 0.40 | | L1 down | 1.68 | 1.01 | 0.05 | | L16 down | 1.00 | 0.99 | 0.07 | | L18 down (control) | 1.00 | 1.00 | 0.07 | | L18 gate (control) | 1.01 | 1.00 | 0.17 | | L20 q (control) | 1.02 | 1.00 | 0.16 |

Appendix C. Document replication (frozen profile, disjoint documents)

Block-profile errors on the second document pair (cell RMS dB, A/W; rotation log-ratio RMS error; sign accuracy). KBBQ is in parentheses.

| model | format | cell error | rotation error | sign | |---|---|---|---|---| | Qwen3-8B | NVFP4 | 0.04 / 0.02 (1.47 / 1.47) | 0.010 (0.345) | 0.97 (0.83) | | | NVFP4 static-6 | 0.06 / 0.03 | 0.020 (0.368) | 1.00 (0.66) | | | MXFP4 | 0.15 / 0.08 | 0.019 (0.302) | 1.00 (0.59) | | | INT4-g128 | 0.49 / 0.08 | 0.068 (0.300) | 1.00 (1.00) | | Qwen3-4B | NVFP4 | 0.09 / 0.08 (1.43 / 1.13) | 0.028 (0.247) | 1.00 (0.95) | | | NVFP4 static-6 | 0.05 / 0.06 | 0.033 (0.247) | 0.95 (0.83) | | | MXFP4 | 0.14 / 0.10 | 0.033 (0.275) | 1.00 (0.61) | | | INT4-g128 | 0.16 / 0.23 | 0.038 (0.304) | 1.00 (1.00) |

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