Saturation Makes Quantization Error Additive: A Coverage Model with a Certificate

# Saturation Makes Quantization Error Additive: A Coverage Model with a Certificate
Source: [https://arxiv.org/html/2607.12266](https://arxiv.org/html/2607.12266)
Inc\. David R\. Cheriton School of Computer ScienceUniversity of Waterloo

###### Abstract

Mixed\-precision quantization must decide which parts of a model to keep at higher precision\. A common premise, shared by sensitivity\-based methods such as HAWQ and CoopQ, is that the loss from quantizing a set of layers can be reconstructed from per\-layer or pairwise sensitivities measured in isolation\. We test this premise at the 4\-bit weight\-and\-activation precisions now being deployed, treating the change in lossf​\(S\)f\(S\)from quantizing a layer setSSas a set function on the Boolean cube and analyzing it through two classical changes of basis\. This analysis yields two findings\. First, across configurations drawn from the deployment distribution, 85–93% of the variance offfis explained by per\-layer effects alone\. Second, a monotone transform of a sum of per\-layer terms reproducesff’s ranking of configurations, misordering at most 2% of pairs\.

Both findings are consequences of the*saturation*of the loss\. Because the loss is bounded, each quantized layer consumes a fractionaia\_\{i\}\(its break\-rate\) of the remaining headroom toward a ceilingcc; the resulting coverage modelf​\(S\)=c​\(1−∏i∈S\(1−ai\)\)f\(S\)=c\\bigl\(1\-\\prod\_\{i\\in S\}\(1\-a\_\{i\}\)\\bigr\)reproduces the measured variance profile offfto within a few percent from itsLLfitted break\-rates\. This structure supports two predictors of a configuration’s loss, each withL\+1L\+1parameters\. The additive model is the optimal first\-order predictor\. By Parseval’s identity its mean\-squared error equals the variance offfleft unexplained by per\-layer effects, which we measure on full lattices, estimate out of sample at full\-network scale, and report with every result as a certificate of how well any additive model can do\. The coverage model itself is the second predictor, and the two predict comparably\.

Saturation makes a layer’s marginal damage depend on how many other layers are quantized, so sensitivities measured in isolation, including a Hessian\-trace HAWQ\-v2, mis\-price a configuration’s loss by 71–376%\. An Aumann–Shapley path integral recovers the optimal additive coefficient up to a layer\-specific effective deployment density, preserving its ranking of layers exactly, a recovery we prove under the coverage model\. Both predictors price a configuration before deployment, match black\-box surrogates at small measurement budgets, degrade at most2×2\\timesbeyond the configurations fit on, where the surrogates degrade22–5×5\\times\. As allocators at matched memory, they attain the lowest KL divergence among the compared allocators at the contended budgets of mixture\-of\-experts models from 30B to 355B parameters \(17–21% lower KL divergence than the production ModelOpt baseline on Qwen3\-30B\)\. Below four bits, the resulting allocations continue to solve code and reasoning tasks at budgets where allocations from gradient sensitivities no longer produce terminating generations\.

## 1Introduction

Quantizing a transformer to low bit\-widths is the dominant route to cheap inference, and the formats now being deployed quantize both weights and activations to 4 bits\. What remains a design choice ismixed precision: which layers \(or tensors\) to keep at higher precision under a memory budget\. The choice is consequential, since allocations at the same memory budget can differ in output fidelity by tens of percent \(Section[7](https://arxiv.org/html/2607.12266#S7)\), and it is currently made by per\-layer sensitivity heuristics\.

These heuristics share a structural premise\. Each scores a layer by a sensitivity measured largely in isolation \(i\.e\. with other layers unquantized\), such as a Hessian eigenvalue or trace\(HAWQ; Donget al\.,[2019](https://arxiv.org/html/2607.12266#bib.bib43),[2020](https://arxiv.org/html/2607.12266#bib.bib10)\), a squared\-gradient Fisher score\(ModelOpt; NVIDIA,[2024](https://arxiv.org/html/2607.12266#bib.bib48)\), or at most a pairwise interaction term\(Denget al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib58)\), and then aggregates these scores to rank layers or to estimate the loss of a joint configuration\. To our knowledge, to what extent the loss of quantizing a*set*of layers is in fact recoverable from such low\-order measurements has never been tested directly\. Throughout, the*order*of a term is the number of layers it involves jointly, so per\-layer terms are first\-order and pairwise terms second\-order\. The loss from quantization is a*set function*defined on all2L2^\{L\}subsets ofLLlayers \(equivalently, the boolean cube\{0,1\}L\\\{0,1\\\}^\{L\}\), which cannot be enumerated for models of any realistic depth\.

To measure the order of this function directly, we writef​\(S\)=LCE​\(S​quantized\)−LCE​\(full precision\)f\(S\)=L\_\{\\mathrm\{CE\}\}\(S\\ \\text\{quantized\}\)\-L\_\{\\mathrm\{CE\}\}\(\\text\{full precision\}\)for the damage of quantizing the layer setSS, obtained by evaluating the partially quantized model, and analyze the measuredffthrough two classical changes of basis\. The Walsh transform \(the functional\-ANOVA decomposition\) splits the variance offfby order under the*deployment distribution*, the distribution in which each layer is quantized independently with probabilitypp\. The Möbius expansion reconstructsfffrom signed terms of increasing order\. This analysis yields two findings\.

The first finding is that the variance of the damage is almost entirely first\-order\. Across configurations drawn from the deployment distribution, 85–93% of the variance offfis explained by per\-layer effects, rising to 98–99% on exact lattices \(blocks ofLLlayers for which every one of the2L2^\{L\}configurations is measured\) once measurement noise is corrected for \(Fig\.[1](https://arxiv.org/html/2607.12266#S3.F1), Section[3](https://arxiv.org/html/2607.12266#S3)\)\. The unexplained remainder therefore serves as a*certificate*: a single number, measured on full lattices \(an identity, by Theorem 1\) or forecast in closed form from fitted break\-rates at full\-network scale \(Section[4\.3](https://arxiv.org/html/2607.12266#S4.SS3)\), that states in advance how well any additive model can do, and which we report alongside every result\. The additive model attaining this error floor also has a closed form \(Section[5](https://arxiv.org/html/2607.12266#S5)\)\.

The second finding is that the variance offfexplained by higher\-order terms does not reflect a failure of per\-layer structure\. The order\-≥2\\geq 2share, the fraction of the variance that per\-layer effects leave unexplained, equals by Theorem 1 the mean\-squared error of the best additive fit divided by the variance \(Section[3](https://arxiv.org/html/2607.12266#S3)\)\. This share can be nonzero even when the damage is perfectly described by per\-layer terms\. Ifffis additive after a strictly increasing transformation, that is, it has the formg​\(∑i∈Sbi\)g\\bigl\(\\sum\_\{i\\in S\}b\_\{i\}\\bigr\)for per\-layer contributionsbib\_\{i\}and a strictly increasinggg, then composing withg−1g^\{\-1\}yields a function whose order\-≥2\\geq 2energy is exactly zero\. Whetherffbelongs to this class is testable on rankings alone \(Theorem 2\), and we measure the smallest fraction of pairwise configuration comparisons that any additive index must order incorrectly, certified against measurement noise \(Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)\)\. This fraction is at most 2%, on some models indistinguishable from zero, and a direct search over transformations agrees\. The measured damage is therefore of the formf​\(S\)≈g​\(∑i∈Sbi\)f\(S\)\\approx g\\bigl\(\\sum\_\{i\\in S\}b\_\{i\}\\bigr\)\. This model class recurs throughout the paper\. It justifies the form fitted in Section[4](https://arxiv.org/html/2607.12266#S4), and it caps how much any interaction\-aware allocator can improve on additive configuration scoring \(Section[7](https://arxiv.org/html/2607.12266#S7)\)\.

We show that a single property,*saturation*, explains both findings exactly\. Because the loss is bounded, one can define the*headroom*h​\(S\)=c−f​\(S\)h\(S\)=c\-f\(S\), the damage still available below a ceilingcc\. The measurements are well described by a model in which each quantized layer consumes a fixed fractionaia\_\{i\}of the remaining headroom,h​\(S∪\{i\}\)=\(1−ai\)​h​\(S\)h\(S\\cup\\\{i\\\}\)=\(1\-a\_\{i\}\)\\,h\(S\), independently of which layers preceded it\. Equivalently,f​\(S\)=c​\(1−∏i∈S\(1−ai\)\)f\(S\)=c\\bigl\(1\-\\prod\_\{i\\in S\}\(1\-a\_\{i\}\)\\bigr\), which we call the*coverage model*\. The model is fitted to the measured losses of sampled configurations by least squares\.

The model class of the second finding is not arbitrary\. By a representation theorem of Aczél and Ling\(Aczél,[1966](https://arxiv.org/html/2607.12266#bib.bib73); Ling,[1965](https://arxiv.org/html/2607.12266#bib.bib74)\), any rule that accumulates damage one layer at a time through a fixed operation that is associative, continuous, and strictly increasing can be rewritten in the formg​\(∑ibi\)g\\bigl\(\\sum\_\{i\}b\_\{i\}\\bigr\)\(App\.[A](https://arxiv.org/html/2607.12266#A1)\)\.

Aggregating isolated sensitivities is equivalent to fitting an additive model constrained to predict zero damage at the unquantized network, and this constraint is what fails in the incumbent heuristics: freeing the single intercept parameter raisesR2R^\{2\}from0\.130\.13to0\.940\.94on the exact lattice where FP4 damage is large \(Section[3](https://arxiv.org/html/2607.12266#S3)\)\. Because saturation shrinks a layer’s marginal damage as more layers are quantized, a sensitivity measured with every other layer at full precision is inflated by a closed\-form factor, the quantitative form of a context dependence known qualitatively\(Anconaet al\.,[2020](https://arxiv.org/html/2607.12266#bib.bib56); McGrathet al\.,[2023](https://arxiv.org/html/2607.12266#bib.bib57); Zhaoet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib23)\), and summingLLinflated coefficients compounds the error to 71–376%, which we measure for isolate\-and\-sum \(i\.e\. the quantization loss is the sum of the damages from quantizing each layer separately\) and for a Hessian\-trace HAWQ\-v2 \(Section[5](https://arxiv.org/html/2607.12266#S5)\)\. A single*Aumann–Shapley path integral*recovers the additive model’s optimal coefficient evaluated at a layer\-specific effective deployment density, and with it the exact ranking of layers, with no sampled configurations, a recovery we prove under the coverage model \(Section[6](https://arxiv.org/html/2607.12266#S6), App\.[B](https://arxiv.org/html/2607.12266#A2)\)\.

Because both models predict rather than rank, they can*price*a configuration, that is, quote its loss before committing to it, which a ranking cannot do\. On unseen configurations that protect the most fragile layers and quantize the rest, the quoted loss is accurate to within 4% \(App\.[E](https://arxiv.org/html/2607.12266#A5)\)\. Against the black\-box surrogates used for the same task, random forests\(Jiang and Lyu,[2025](https://arxiv.org/html/2607.12266#bib.bib37)\), Gaussian processes\(Zhouet al\.,[2026](https://arxiv.org/html/2607.12266#bib.bib39)\), and RBF kernels\(Leeet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib38)\), the\(L\+1\)\(L\{\+\}1\)\-parameter models are best or tied at small measurement budgets, the regime that matters when every measurement is a forward pass\. The strongest of these, the Gaussian process, succeeds for a reason our analysis identifies and fails where the theory predicts, degrading22–5×5\\timesunder extrapolation to differently quantized configurations than those it was fit on, where the additive and coverage models stay within2×2\\times\. That direction of failure matters because an allocator must price configurations quantized at various levels \(Section[7](https://arxiv.org/html/2607.12266#S7)\)\.

Used as allocators at matched memory, the same fitted models attain the lowest KL divergence among the compared allocators \(sensitivity scores, structural rules, and the black\-box surrogates converted into allocators\) on mixture\-of\-experts language models across models from 0\.6B to 355B parameters \(Section[7](https://arxiv.org/html/2607.12266#S7)\)\. Below four bits configurations allocated from the fitted models continue to solve code and reasoning tasks at effective bit\-widths where configurations from other methods of the same memory budget no longer produce terminating generations, at both 30B and 235B \(Section[7\.4](https://arxiv.org/html/2607.12266#S7.SS4)\)\.

#### Contributions\.

1. 1\.Aclosed\-form theory: the coverage model’s exact Möbius and Walsh spectra \(Theorems 3 and 4\), corroborated by the confirmed divergence of the signed expansion \(Fig\.[2](https://arxiv.org/html/2607.12266#S4.F2)\); the certificate identity \(Theorem 1\) with an exactO​\(L\)O\(L\)heterogeneous form \(Proposition 5\); and the ordinal characterization of additivity up to an increasing transformation \(Theorem 2\) \(Sections[3](https://arxiv.org/html/2607.12266#S3)–[4](https://arxiv.org/html/2607.12266#S4), App\.[F](https://arxiv.org/html/2607.12266#A6)\)\.
2. 2\.Anegative result with a closed\-form account: sensitivities measured in isolation, including HAWQ\-v2, mis\-price a configuration’s loss by 71–376%, and the inflation factor responsible is derived in closed form \(Section[5](https://arxiv.org/html/2607.12266#S5)\)\.
3. 3\.Afit\-free estimator: one Aumann–Shapley path integral recovers the additive model’s optimal coefficient at a layer\-specific effective density, and its layer ranking exactly, proven under the coverage model \(Section[6](https://arxiv.org/html/2607.12266#S6), App\.[B](https://arxiv.org/html/2607.12266#A2)\)\.
4. 4\.Aninterpretable predictorthat is best or tied against RF/GP/RBF/pairwise baselines at small budgets, degrades at most2×2\\timesunder configuration extrapolation where the GP degrades22–5×5\\times, and is validated on LLMs 0\.6B–355B and a diffusion DiT \(Section[6](https://arxiv.org/html/2607.12266#S6), Fig\.[4](https://arxiv.org/html/2607.12266#S6.F4)\)\.
5. 5\.Anallocator at matched memorythat attains the lowest KL among sensitivity\-, rule\-, and surrogate\-based allocators \(17–21% below the production ModelOpt baseline on Qwen3\-30B, 13–27% on Qwen3\-235B, 20–37% on GLM\-4\.6\) and decides served task survival below four bits\.

## 2Related work

#### Mixed\-precision allocation via per\-layer sensitivity\.

AWQ\(Linet al\.,[2024](https://arxiv.org/html/2607.12266#bib.bib31)\)uses activation\-aware scaling, HAWQ and HAWQ\-v2\(Donget al\.,[2019](https://arxiv.org/html/2607.12266#bib.bib43),[2020](https://arxiv.org/html/2607.12266#bib.bib10)\)a Hessian eigenvalue or trace, BRECQ\(Liet al\.,[2021](https://arxiv.org/html/2607.12266#bib.bib11)\)and GPTQ\(Frantaret al\.,[2022](https://arxiv.org/html/2607.12266#bib.bib5)\)second\-order weight reconstruction, and NVIDIA ModelOpt’sauto\_quantize\(NVIDIA,[2024](https://arxiv.org/html/2607.12266#bib.bib48)\), the production method we benchmark, a Taylor/Fisher score combined with a linear program over a sum of isolated per\-layer scores\. All of these aggregate first\- or second\-order single\-layer signals\. In our deployment regime the closest neighbor is FGMP\(Hooperet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib34)\), which selects FP8\-versus\-NVFP4 blocks for weights and activations by a Fisher\-weighted perturbation score\. Its score belongs to the same Taylor/Fisher family whose production embodiment, ModelOpt, we benchmark throughout, while its intra\-tensor block granularity requires co\-designed kernels that no serving stack we target provides, so it composes with layer\-level allocation rather than competing with it\. ScaleBITS\(Liet al\.,[2026](https://arxiv.org/html/2607.12266#bib.bib35)\)performs hardware\-aligned bitwidth allocation for weights only\.

#### Interaction\-aware allocation\.

CoopQ\(Zhaoet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib23)\), first circulated under the name IMPQ, is the closest neighbor among interaction\-aware methods\. Its SPQE estimator samples layer permutations and progressively quantizes along each, thereby measuring marginal contributions in context \(permutation\-sampled Shapley, at a cost of roughly100​L100Lquantized evaluations\), then fits a linear\-plus\-pairwise surrogate \(a⊤​q\+q⊤​K​qa^\{\\top\}q\+q^\{\\top\}Kq\) and allocates by solving a binary quadratic program via MILP linearization\. CLADO\(Denget al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib58)\)measures pairwise cross\-layer effects by joint\-quantization finite differences on CNNs and ViTs and rejects the independence assumption\. Both fix the expansion order at two and do not test whether that order suffices\. GuidedQuant\(Kimet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib21)\)keeps within\-output\-channel dependencies but ignores cross\-layer terms, and QEP\(Arai and Ichikawa,[2025](https://arxiv.org/html/2607.12266#bib.bib22)\)models cross\-layer accumulation as a compensation term\. Sections[3](https://arxiv.org/html/2607.12266#S3)–[4](https://arxiv.org/html/2607.12266#S4)supply what none of these test, namely the order of the expansion, and Section[5](https://arxiv.org/html/2607.12266#S5)gives the closed\-form account of the isolated\-anchor bias they implicitly work around\.

#### Configuration\-quality surrogates\.

For the prediction task itself, MiCo\(Jiang and Lyu,[2025](https://arxiv.org/html/2607.12266#bib.bib37)\)fits a random\-forest accuracy predictor for edge\-scale networks, AMQ\(Leeet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib38)\)an RBF kernel model for LLM weight\-only quantization, and AutoQRA\(Zhouet al\.,[2026](https://arxiv.org/html/2607.12266#bib.bib39)\)a Gaussian\-process surrogate inside a joint search over bit\-widths and adapter ranks\. These predictor classes are accurate in\-distribution but opaque, and we show they degrade22–5×5\\timeswhen extrapolating to more heavily quantized configurations \(Section[6](https://arxiv.org/html/2607.12266#S6)\)\.

Additional related works are surveyed in App\.[H](https://arxiv.org/html/2607.12266#A8)\.

## 3The measured structure of quantization damage

This section reports the two measurements on which the rest of the paper builds: the decomposition of the damage variance by order, and the ordinal defect, which measures how much of that structure survives every increasing transformation of the loss\.

#### Set\-up\.

For a contiguous block ofLLlayers \(finer units are quantized in Section[7](https://arxiv.org/html/2607.12266#S7)\), centered at the network’s mid\-depth \(App\.[D](https://arxiv.org/html/2607.12266#A4)\), we measuref​\(S\)=Δ​LCEf\(S\)=\\Delta L\_\{\\mathrm\{CE\}\}for the full subset lattice of2L2^\{L\}subsetsSS, running the model with exactly the layers inSSquantized\. The quantizer rounds stochastically, so eachf​\(S\)f\(S\)is the average of several independent rounding draws \(16 for the lattices in this section; App\.[D](https://arxiv.org/html/2607.12266#A4)\)\. A measure over configurations must be fixed before variances or prediction errors can be defined, and it should weigh the configurations that matter in practice, which are mostly quantized\. We therefore analyzeffunder the*deployment distribution*μp\\mu\_\{p\}, the product measure in which each layer is quantized independently with probabilitypp, and takep=0\.6p=0\.6throughout unless stated otherwise\. We expand the measuredffin the*Walsh basis*\{χT\}\\\{\\chi\_\{T\}\\\}, with coefficientsf^​\(T\)\\hat\{f\}\(T\)\(O’Donnell,[2014](https://arxiv.org/html/2607.12266#bib.bib40)\)\. We denote the*order\-kkenergy*Wk=∑\|T\|=kf^​\(T\)2W\_\{k\}=\\sum\_\{\|T\|=k\}\\hat\{f\}\(T\)^\{2\}, and by Parseval’s identity,Var⁡\(f\)=∑k≥1Wk\\operatorname\{Var\}\(f\)=\\sum\_\{k\\geq 1\}W\_\{k\}, so the ratioWk/Var⁡\(f\)W\_\{k\}/\\operatorname\{Var\}\(f\)is the fraction of the loss variance attributable to order\-kkstructure\. The following identity gives these energies their operational meaning\.

###### Theorem 1\(additive error equals order\-≥2\\geq 2energy\)\.

For anyffon\{0,1\}L\\\{0,1\\\}^\{L\}and anyp∈\(0,1\)p\\in\(0,1\), the best degree\-≤1\\leq 1predictorf1f\_\{1\}underμp\\mu\_\{p\}has squared error exactly equal to the high\-order Walsh energy:𝔼μp​\[\(f−f1\)2\]=∑k≥2Wk\\mathbb\{E\}\_\{\\mu\_\{p\}\}\\bigl\[\(f\-f\_\{1\}\)^\{2\}\\bigr\]=\\sum\_\{k\\geq 2\}W\_\{k\}\.

###### Proof\.

The degree\-≤1\\leq 1functions are exactlyspan​\{χ∅,χ\{1\},…,χ\{L\}\}\\mathrm\{span\}\\\{\\chi\_\{\\emptyset\},\\chi\_\{\\\{1\\\}\},\\ldots,\\chi\_\{\\\{L\\\}\}\\\}, least squares inL2​\(μp\)L^\{2\}\(\\mu\_\{p\}\)is the orthogonal projection onto that span, and by Parseval the squared residual is∑\|T\|≥2f^​\(T\)2\\sum\_\{\|T\|\\geq 2\}\\hat\{f\}\(T\)^\{2\}\. ∎

This is thed=1d=1case of low\-degree learning\(O’Donnell,[2014](https://arxiv.org/html/2607.12266#bib.bib40); Linialet al\.,[1993](https://arxiv.org/html/2607.12266#bib.bib41); Sobol,[2001](https://arxiv.org/html/2607.12266#bib.bib44)\)\.

#### Result \(Fig\.[1\(a\)](https://arxiv.org/html/2607.12266#S3.F1.sf1)\)\.

Roughly 90% of the variance is order\-1, with order\-≥2\\geq 2contributing 7–15% across models and datasets\. On a 10\-layer Qwen3\-8B block, order\-1 accounts for87\.3%87\.3\\%of the variance as measured and98\.1%98\.1\\%once the noise floor contributed by the stochastic\-rounding draws is subtracted \(the split\-half estimate of App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\), and on an 8\-layer Qwen3\-0\.6B block for93\.8%93\.8\\%and99\.2%99\.2\\%\. Across\{\\\{Qwen3\-0\.6B, Qwen2\.5\-3B, Qwen3\-4B, Qwen3\-8B, Llama\-3\.2\-3B, Qwen3\-30B\-A3B\}×\{\\\}\\times\\\{wiki, code\}\\\}\(WikiText\-2 and a code corpus, App\.[D](https://arxiv.org/html/2607.12266#A4)\) the sampled order\-1 share is 85–93% when whole layers are quantized at FP4\. The concentration is not an artifact of whole\-layer units or of small scale \(Fig\.[1\(a\)](https://arxiv.org/html/2607.12266#S3.F1.sf1)\): on exact282^\{8\}lattices over eight consecutive linear units at W4A4, the order\-1 share is 97–99% on models from 0\.6B to Qwen2\.5\-72B, and 90\.7% on gpt\-oss\-120B\.

#### The intercept diagnostic \(Fig\.[1\(b\)](https://arxiv.org/html/2607.12266#S3.F1.sf2)\)\.

An additive model with no intercept, which is the constraint that aggregating isolated sensitivities imposes \(f​\(∅\)=0f\(\\emptyset\)=0\), reachesR2=0\.13R^\{2\}=0\.13on the 0\.6B block, where FP4 damage is large, against0\.780\.78on the 8B block, where it is small\. Relaxing this constraint raises both to0\.940\.94and0\.870\.87\. Both fits are least squares underμp\\mu\_\{p\}on the full lattice\.

The remaining questions are why the small order\-≥2\\geq 2content is structured the way it is and what generates it\. The answer we propose is saturation \(Section[4](https://arxiv.org/html/2607.12266#S4)\)\. Before turning to it, we ask how much of the order\-≥2\\geq 2energy reflects structure among layers at all\.

### 3\.1The ordinal defect: additivity up to an increasing transformation

The order\-≥2\\geq 2energy has two caveats as an indicator of structure among layers\. First, it is measure\-relative\. Second, and more fundamentally, it is not invariant under a change of coordinates on the loss values, since replacingffbyψ∘f\\psi\\circ ffor a strictly increasingψ:ℝ→ℝ\\psi:\\mathbb\{R\}\\to\\mathbb\{R\}can change the energy ifψ\\psiis curved, so an exactly additiveffcould acquire nonzero higher\-order energy\. A principled notion of irreducible structure should therefore quantify over all strictly increasingψ\\psi\. For binary factors this is possible, because additivity up to an increasing transformation is a purely ordinal property\.

###### Theorem 2\(additivity up to an increasing transformation is ordinal\)\.

f​\(x\)=g​\(b⋅x\)f\(x\)=g\(b\\cdot x\)for someb∈ℝLb\\in\\mathbb\{R\}^\{L\}and strictly increasingggif and only if some additive index reproducesff’s ranking of subsets, that is,f​\(x\)<f​\(y\)⇔b⋅x<b⋅yf\(x\)<f\(y\)\\iff b\\cdot x<b\\cdot yfor allx,y∈\{0,1\}Lx,y\\in\\\{0,1\\\}^\{L\}\.

The theorem is the finite binary case of additive conjoint measurement\(Luce and Tukey,[1964](https://arxiv.org/html/2607.12266#bib.bib77); Debreu,[1960](https://arxiv.org/html/2607.12266#bib.bib78)\)\. App\.[A](https://arxiv.org/html/2607.12266#A1)gives the short proof\. Throughout, an*additive index*is a linear scorex↦b⋅xx\\mapsto b\\cdot xused only through the ordering it induces on configurations\. Whether such an index exists is a linear program, with one inequalityb⋅\(x−y\)\>0b\\cdot\(x\-y\)\>0per ordered pair\(Scott,[1964](https://arxiv.org/html/2607.12266#bib.bib79)\)\. The smallest witnesses of infeasibility are*rank flips*, pairs of comparisons that order the same two layers oppositely in two contexts:f​\(S∪i\)\>f​\(S∪j\)f\(S\\cup i\)\>f\(S\\cup j\)butf​\(T∪i\)<f​\(T∪j\)f\(T\\cup i\)<f\(T\\cup j\)forces bothbi\>bjb\_\{i\}\>b\_\{j\}andbi<bjb\_\{i\}<b\_\{j\}, so a single true flip refutes every strictly increasingψ\\psi\. We call the minimal fraction of ranking constraints that any additive index must violate the*ordinal defect*offf\.

#### Measurement\.

On full282^\{8\}configuration lattices, measured with 16 stochastic\-rounding draws per configuration \(App\.[D](https://arxiv.org/html/2607.12266#A4)\), we bound the defect from both sides, certifying each statistic against measurement noise\. For the lower bound we count rank flips\. Splitting the 16 draws into two halves gives two independent estimates of every configuration’s loss, and a flip is counted only if it is detected above the noise floor on the first half and recurs with the same sign on the second\. The surviving count is then read against the count that the identical pipeline produces on a matched\-noise null, a synthetic lattice constructed to contain no true flips \(the coverage model of Section[4](https://arxiv.org/html/2607.12266#S4)fitted to the measured losses, plus noise at each configuration’s measured level\)\. For the upper bound we exhibit an index, since the fraction of ranking constraints violated by any single index bounds the defect from above\. The exhibited index minimizes the total slack of the constraints of one half of the draws, again a linear program, and its violated fraction is evaluated both on that half and on the held\-out half’s constraints, with the larger of the two quoted \(App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1), per\-model results in Table[7](https://arxiv.org/html/2607.12266#A1.T7)\)\. In the deployed per\-block regime the certified ordinal defect is at most1\.4%1\.4\\%\(the largest cell, Llama\-3\.2\-3B\)\. The energy counterpart agrees: minimizing the order\-≥2\\geq 2share over all strictly increasing transformations of the loss \(the estimatorE∗E^\{\*\}of App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\) yields an upper bound of0–5%5\\%across the deployed cells \(last column of Table[7](https://arxiv.org/html/2607.12266#A1.T7)\)\. The zeros are not blindness of the procedure\. Applied to the non\-deployed per\-tensor regime of Section[8](https://arxiv.org/html/2607.12266#S8), the same procedure finds flips well in excess of its matched\-noise null \(74 against 18, Table[7](https://arxiv.org/html/2607.12266#A1.T7)\), so genuine reversals are detected where a known mechanism produces them, and even there an additive index orders at least98%98\\%of the comparisons above the noise floor correctly\. The measuredffis therefore, to that accuracy, of the formg​\(∑i∈Sbi\)g\\bigl\(\\sum\_\{i\\in S\}b\_\{i\}\\bigr\)for a strictly increasinggg, and the apparent high\-order energy reflects the curvature ofggrather than structure among layers\.

![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig1a_order1_share.png)\(a\)Order\-1 share, every exact lattice\.
![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig1b_intercept.png)\(b\)The intercept diagnostic \(in sample\)\.

Figure 1:\([1\(a\)](https://arxiv.org/html/2607.12266#S3.F1.sf1)\) Order\-1 share of the damage variance on every exact lattice measured, under the deployment measureμp\\mu\_\{p\},p=0\.6p=0\.6\. Red: the two whole\-layer FP4 lattices of this section \(Qwen3\-0\.6B, 8 layers,282^\{8\}configurations; Qwen3\-8B, 10 layers,2102^\{10\}\), noise\-corrected as in App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\. Teal: exact282^\{8\}lattices at per\-linear W4A4 granularity \(NVFP4, block 16\), each over eight consecutive mid\-depth linear units, with damage measured as the KL divergence to the full\-precision model on a WikiText\-2 calibration set \(light bars: code corpus\)\. The share is 96\.9–99\.7% on dense models from 0\.6B to 72B and 90\.7% on the sparse\-MoE gpt\-oss\-120B, where fused expert blocks add coupling\. \([1\(b\)](https://arxiv.org/html/2607.12266#S3.F1.sf2)\) The additive fit without an intercept, the constraint that isolated\-sensitivity aggregation imposes \(f​\(∅\)=0f\(\\emptyset\)=0\), against the same model class with a free intercept, which by Theorem 1 is the degree\-≤1\\leq 1projection; least squares underμp\\mu\_\{p\}on the two whole\-layer lattices\.

## 4The reconciling theory: saturation generates both views

### 4\.1The coverage model

Section[3](https://arxiv.org/html/2607.12266#S3)left two measurements in need of a common explanation: per\-layer effects carry most of the variance of the measured damageff, and some additive index reproduces at least98%98\\%of its pairwise ranking of configurations\. This section proposes saturation as the origin of both\. We describe the damage with a saturating*coverage model*,

f​\(S\)=c​\(1−∏i∈S\(1−ai\)\),f\(S\)=c\\Bigl\(1\-\\prod\_\{i\\in S\}\(1\-a\_\{i\}\)\\Bigr\),\(1\)with a per\-layer*break\-rate*ai∈\[0,1\]a\_\{i\}\\in\[0,1\]and a*ceiling*c\>0c\>0\. Define the*headroom*h​\(S\):=c−f​\(S\)h\(S\):=c\-f\(S\), the remaining distance to the ceiling\. Imposing the coverage model is exactly the statement that each quantized layer consumes a fixed fractionaia\_\{i\}of whatever headroom remains,h​\(S∪\{i\}\)=\(1−ai\)​h​\(S\)h\(S\\cup\\\{i\\\}\)=\(1\-a\_\{i\}\)\\,h\(S\)regardless of which layers came before\.

This model is grounded in a measured property of the damage\. Saturation, as we use the term, is the conjunction of two facts visible on the configuration lattices of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1): the loss of a heavily quantized configuration is finite, and the marginal damage of one more quantized layer shrinks as more layers are already quantized\. Table[8](https://arxiv.org/html/2607.12266#A4.T8)reports the measurement for each of the five certification cells, where a*cell*is one model and quantizer setting measured as a full282^\{8\}lattice\. Averagingf​\(S∪\{i\}\)−f​\(S\)f\(S\\cup\\\{i\\\}\)\-f\(S\)over layersiiand contextsS∌iS\\not\\ni iof a fixed size, we see that every cell’s average marginal quantization damage is largest when no other layer is quantized and falls once other layers are quantized concurrently\. The evidence for coverage is assembled with each claim’s standing in App\.[G](https://arxiv.org/html/2607.12266#A7)\.

Equivalently, the coverage model is the statement thatffis additive in log\-headroom,−log⁡\(1−f​\(S\)/c\)=∑i∈Sbi\-\\log\(1\-f\(S\)/c\)=\\sum\_\{i\\in S\}b\_\{i\}withbi:=−log⁡\(1−ai\)b\_\{i\}:=\-\\log\(1\-a\_\{i\}\), hence coverage lies in the classg​\(∑i∈Sbi\)g\\bigl\(\\sum\_\{i\\in S\}b\_\{i\}\\bigr\)of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)with generatorg​\(u\)=c​\(1−e−u\)g\(u\)=c\(1\-e^\{\-u\}\)\. The measurements of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)placeffwithin a∼2%\{\\sim\}2\\%ordinal defect of this class across measured models\.

### 4\.2The Möbius and Walsh\-Hadamard expansions

The coverage model was introduced to account for the variance and ranking measurements of Section[3](https://arxiv.org/html/2607.12266#S3), and it is fitted to measured loss values \(Section[4\.4](https://arxiv.org/html/2607.12266#S4.SS4)\)\. It makes a second prediction about a quantity that plays no role in that fit\. The*Möbius/Harsanyi transform*gives signed coordinatesφ​\(T\)=∑R⊆T\(−1\)\|T\|−\|R\|​f​\(R\)\\varphi\(T\)=\\sum\_\{R\\subseteq T\}\(\-1\)^\{\|T\|\-\|R\|\}f\(R\)that reconstruct the damage asf​\(T\)=∑R⊆Tφ​\(R\)f\(T\)=\\sum\_\{R\\subseteq T\}\\varphi\(R\)\. Restricting the reconstruction sums to subsets of size≤1\\leq 1is precisely the isolate\-and\-sum aggregation of Section[5](https://arxiv.org/html/2607.12266#S5), sinceφ​\(∅\)=f​\(∅\)=0\\varphi\(\\emptyset\)=f\(\\emptyset\)=0andφ​\(\{i\}\)=f​\(\{i\}\)\\varphi\(\\\{i\\\}\)=f\(\\\{i\\\}\)\. Restricting them to size≤2\\leq 2adds exactly the pairwise finite differencesφ​\(\{i,j\}\)=f​\(\{i,j\}\)−f​\(\{i\}\)−f​\(\{j\}\)\\varphi\(\\\{i,j\\\}\)=f\(\\\{i,j\\\}\)\-f\(\\\{i\\\}\)\-f\(\\\{j\\\}\)that CLADO\(Denget al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib58)\)measures as its cross\-layer sensitivities\.

###### Theorem 3\(Möbius spectrum of coverage\)\.

The Möbius transform of the coverage model is, exactly,φ​\(T\)=c​\(−1\)\|T\|\+1​∏i∈Tai\\varphi\(T\)=c\\,\(\-1\)^\{\|T\|\+1\}\\prod\_\{i\\in T\}a\_\{i\}for\|T\|≥1\|T\|\\geq 1andφ​\(∅\)=0\\varphi\(\\emptyset\)=0\.

Coverage predicts a distinct form for this expansion\. Its coefficients are nonzero at every order and alternate in sign, so low\-order truncations should overshoot and oscillate\. The measured lattices show exactly this behavior \(Fig\.[2](https://arxiv.org/html/2607.12266#S4.F2)\)\. On Qwen3\-0\.6B at FP4 \(8\-layer exact lattice\) the partial sums alternate, converging only atk=Lk=L\. On a 10\-layer Qwen3\-8B block with the full2102^\{10\}\-configuration lattice the same divergence\-then\-cancellation appears at larger amplitude, swinging to±4\.8\\pm 4\.8against a true loss of0\.1480\.148, an overshoot of up to32×32\\times\. Because an order\-kkMöbius coefficient is an alternating sum of up to2k2^\{k\}measured values, the partial sums amplify measurement noise, so we certify the swing by the split\-half device of App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1): partial sums computed from two disjoint halves of the rounding draws independently reproduce the alternating overshoot \(both swing beyond±4\\pm 4\), while the same sums computed from the half\-difference table, which contains noise only, stay within±1\.2\\pm 1\.2\.

###### Theorem 4\(Walsh spectrum of coverage\)\.

Under thepp\-biased measureμp\\mu\_\{p\}, the Walsh coefficients are, exactly,

f^​\(T\)=c​\(−1\)\|T\|\+1​\(p​\(1−p\)\)\|T\|/2​\(∏i∈Tai\)​∏i∉T\(1−ai​p\)for​T≠∅,\\hat\{f\}\(T\)=c\\,\(\-1\)^\{\|T\|\+1\}\\,\(p\(1\-p\)\)^\{\|T\|/2\}\\Bigl\(\\prod\_\{i\\in T\}a\_\{i\}\\Bigr\)\\prod\_\{i\\notin T\}\(1\-a\_\{i\}p\)\\quad\\text\{for \}T\\neq\\emptyset,andf^​\(∅\)=𝔼μp​\[f\]=c​\(1−∏i\(1−ai​p\)\)\\hat\{f\}\(\\emptyset\)=\\mathbb\{E\}\_\{\\mu\_\{p\}\}\[f\]=c\\bigl\(1\-\\prod\_\{i\}\(1\-a\_\{i\}p\)\\bigr\)\.

Proofs of both transforms are in App\.[F](https://arxiv.org/html/2607.12266#A6)\.

![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig1_mobius_divergence.png)Figure 2:The signed Möbius expansion behaves as coverage predicts: order\-kktruncations alternate and overshoot before full\-order cancellation\. Qwen3\-8B 10\-layer exact 1024\-configuration lattice, swings to±4\.8\\pm 4\.8\(up to32×32\\timesthe true0\.1480\.148\), converging only atk=10k=10\(right\)\. The oscillation is certified against measurement noise: noise\-only partial sums from the split\-half difference table stay within±1\.2\\pm 1\.2\(Section[4\.2](https://arxiv.org/html/2607.12266#S4.SS2)\)\.
### 4\.3The saturation\-depth law

By Theorem 1, the mean\-squared error of the best additive predictor equals the order\-≥2\\geq 2energy∑k≥2Wk\\sum\_\{k\\geq 2\}W\_\{k\}\. Dividing by the variance turns it into the order\-≥2\\geq 2share

share≥2:=∑k≥2WkVar⁡\(f\),\\mathrm\{share\}\_\{\\geq 2\}:=\\frac\{\\sum\_\{k\\geq 2\}W\_\{k\}\}\{\\operatorname\{Var\}\(f\)\},the fraction of the damage variance that per\-layer effects leave unexplained, which is the normalized form in which we report the certificate\. The certificate columns of the benchmark and allocation tables quote the share, which is comparable across cells \(Sections[6](https://arxiv.org/html/2607.12266#S6)and[7](https://arxiv.org/html/2607.12266#S7)\), while statements in loss units use the absolute energies, namely the additive error floor of Theorem 1 and the sampling variance of the additive\-coefficient estimator, which is the order\-≥2\\geq 2energy plus noise \(App\.[D](https://arxiv.org/html/2607.12266#A4)\)\. On a measured lattice the share is computed exactly, but at full\-network scale, where no2L2^\{L\}lattice can be measured, it must be forecast from theL\+1L\+1fitted parameters of the coverage model\. This subsection derives that forecast, and that of the higher\-order Walsh energies, from Proposition 5 in closed form\. For a break\-rate profile\(a1,…,aL\)\(a\_\{1\},\\ldots,a\_\{L\}\)define the per\-layer energy ratios and their aggregates

βi:=p​\(1−p\)​ai2\(1−ai​p\)2,τ2:=∑iβi,Leff:=\(∑iβi\)2∑iβi2∈\[1,L\],\\beta\_\{i\}:=\\frac\{p\(1\-p\)\\,a\_\{i\}^\{2\}\}\{\(1\-a\_\{i\}p\)^\{2\}\},\\qquad\\tau^\{2\}:=\\sum\_\{i\}\\beta\_\{i\},\\qquad L\_\{\\mathrm\{eff\}\}:=\\frac\{\\bigl\(\\sum\_\{i\}\\beta\_\{i\}\\bigr\)^\{2\}\}\{\\sum\_\{i\}\\beta\_\{i\}^\{2\}\}\\in\[1,L\],withτ\\tauthe*saturation depth*andLeffL\_\{\\mathrm\{eff\}\}the participation ratio of the profile, which equalsLLexactly when allβi\\beta\_\{i\}are equal and approaches11as one layer dominates\. Letek:=∑\|T\|=k∏i∈Tβie\_\{k\}:=\\sum\_\{\|T\|=k\}\\prod\_\{i\\in T\}\\beta\_\{i\}be the elementary symmetric polynomials in theβi\\beta\_\{i\}\.

###### Proposition 5\(heterogeneous certificate\)\.

Assume not allaia\_\{i\}are zero\. Under the coverage model,

1. \(i\)\(product form\)Wk=c2​\[∏i\(1−ai​p\)2\]​ekW\_\{k\}=c^\{2\}\\bigl\[\\prod\_\{i\}\(1\-a\_\{i\}p\)^\{2\}\\bigr\]\\,e\_\{k\}, hence computable in closed form from theL\+1L\+1fitted parameters, which are themselves fit from𝒪​\(L\)\\mathcal\{O\}\(L\)measured configurations, and, share≥2=1−τ2∏i\(1\+βi\)−1;\\mathrm\{share\}\_\{\\geq 2\}=1\-\\frac\{\\tau^\{2\}\}\{\\prod\_\{i\}\(1\+\\beta\_\{i\}\)\-1\};\(2\)
2. \(ii\)\(leading order\)share≥2=Leff−12​Leff​τ2\+R\\mathrm\{share\}\_\{\\geq 2\}=\\frac\{L\_\{\\mathrm\{eff\}\}\-1\}\{2L\_\{\\mathrm\{eff\}\}\}\\,\\tau^\{2\}\+Rwith\|R\|≤14​τ4​eτ2\|R\|\\leq\\tfrac\{1\}\{4\}\\tau^\{4\}e^\{\\tau^\{2\}\}, so to second order in the per\-layer amplitudes heterogeneity enters the certificate only through the effective depthLeffL\_\{\\mathrm\{eff\}\};
3. \(iii\)at fixedτ2\\tau^\{2\},share≥2\\mathrm\{share\}\_\{\\geq 2\}is Schur\-concave in\(β1,…,βL\)\(\\beta\_\{1\},\\ldots,\\beta\_\{L\}\), hence share≥2≤1−τ2\(1\+τ2/L\)L−1≤1−τ2eτ2−1,\\mathrm\{share\}\_\{\\geq 2\}\\;\\leq\\;1\-\\frac\{\\tau^\{2\}\}\{\(1\+\\tau^\{2\}/L\)^\{L\}\-1\}\\;\\leq\\;1\-\\frac\{\\tau^\{2\}\}\{e^\{\\tau^\{2\}\}\-1\},with equality on the left exactly when allβi\\beta\_\{i\}are equal\.

###### Proof\.

For \(i\), Theorem 4 givesf^​\(T\)2=c2​\(p​\(1−p\)\)\|T\|​∏i∈Tai2​∏i∉T\(1−ai​p\)2\\hat\{f\}\(T\)^\{2\}=c^\{2\}\(p\(1\-p\)\)^\{\|T\|\}\\prod\_\{i\\in T\}a\_\{i\}^\{2\}\\prod\_\{i\\notin T\}\(1\-a\_\{i\}p\)^\{2\}\. Multiplying and dividing by∏i∈T\(1−ai​p\)2\\prod\_\{i\\in T\}\(1\-a\_\{i\}p\)^\{2\}factors every squared coefficient into theTT\-independent constantc2​∏i=1L\(1−ai​p\)2c^\{2\}\\prod\_\{i=1\}^\{L\}\(1\-a\_\{i\}p\)^\{2\}times∏i∈Tβi\\prod\_\{i\\in T\}\\beta\_\{i\}, soWkW\_\{k\}is that constant timeseke\_\{k\}, the variance∑k≥1Wk\\sum\_\{k\\geq 1\}W\_\{k\}is the constant times∏i\(1\+βi\)−1\\prod\_\{i\}\(1\+\\beta\_\{i\}\)\-1by the generating identity∏i\(1\+βi\)=∑k≥0ek\\prod\_\{i\}\(1\+\\beta\_\{i\}\)=\\sum\_\{k\\geq 0\}e\_\{k\}\. The claim follows\. Parts \(ii\) and \(iii\) are proven in App\.[F](https://arxiv.org/html/2607.12266#A6)\. ∎

Combining Theorem 1 with part \(ii\), for smallτ\\tauthe best additive predictor’s mean\-squared error is at most≈12​τ2​Var⁡\(f\)\{\\approx\}\\tfrac\{1\}\{2\}\\tau^\{2\}\\,\\operatorname\{Var\}\(f\), so even an exact coverage fit, whose error under the model is zero, can improve on additive prediction by no more than that\. The symmetric case is stated as a corollary, with its binomial order spectrum, in App\.[F](https://arxiv.org/html/2607.12266#A6)\.

### 4\.4Validation of the certificate

#### Theτ\\tau\-collapse \(Fig\.[3](https://arxiv.org/html/2607.12266#S4.F3)\)\.

We empirically test the saturation\-depth law\. Each cell contributes one point, its measured order\-≥2\\geq 2share against its fittedτ\\tau, and Proposition 5\(iii\) requires any coverage\-generated profile to fall on or below the parameter\-free curve, with balanced break\-rate profiles on it\. We measured 36 full282^\{8\}lattices, three models \(Qwen3\-0\.6B, Qwen2\.5\-3B, Llama\-3\.2\-3B\) crossed with twelve quantizer settings, integer round\-to\-nearest at 8/6/5/4/3 bits with per\-channel or per\-tensor scales and 4\-bit floating point \(FP4\) with per\-block or per\-tensor scales \(App\.[D](https://arxiv.org/html/2607.12266#A4)\)\. For each cell we noise\-correct the order energies \(App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\), fit the coverage model to the same lattice by nonlinear least squares \(App\.[D](https://arxiv.org/html/2607.12266#A4)\), and compare the measured order\-≥2\\geq 2share with the fit\. The measured share tracks the fittedτ=\(∑iβi\)1/2\\tau=\(\\sum\_\{i\}\\beta\_\{i\}\)^\{1/2\}at Spearmanρ=0\.958\\rho=0\.958, and the exact share of Proposition 5\(i\), computed from the fitted\(c,ai\)\(c,a\_\{i\}\), predicts the measured share at Pearsonρ=0\.984\\rho=0\.984\. Ten of the 36 cells exceed the curve, by at most 8 percentage points\. The largest exceedances are 8\-bit cells whose absolute damage sits at the measurement noise floor, where the noise\-corrected share is unstable, and Llama\-3\.2\-3B cells, the one model with certified rank reversals \(Table[7](https://arxiv.org/html/2607.12266#A1.T7)\), so the exceedances are consistent with share noise plus the residual structure that coverage does not capture\. Repeating the comparison on Qwen3\-30B\-A3B blocks \(32 further cells, 8 block positions by 4 quantizer formats\) gives Pearsonr=0\.997r=0\.997\.

#### Out\-of\-sample validation\.

A certificate measured on a full lattice uses only Theorem 1\. When enumeration is intractable, we instead assume the coverage model and forecast the certificate from break\-rates fitted on sampled configurations\. We test the forecasting pipeline directly against held\-out lattices, with per\-cell results in Table[11](https://arxiv.org/html/2607.12266#A4.T11)of App\.[D](https://arxiv.org/html/2607.12266#A4)\. The forecast classifies the near\-additive and heavily coupled regimes correctly \(App\.[D](https://arxiv.org/html/2607.12266#A4)\)\.

![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig_tau_collapse.png)Figure 3:One parameter organizes the order\-≥2\\geq 2share\. Left: measured \(noise\-corrected\) order\-≥2\\geq 2energy share vs\. fitted saturation depthτ=\(∑βi\)1/2\\tau=\(\\sum\\beta\_\{i\}\)^\{1/2\}for 36 cells, against the parameter\-free symmetric lawshare≥2​\(τ\)=1−τ2/\[\(1\+τ2/L\)L−1\]\\mathrm\{share\}\_\{\\geq 2\}\(\\tau\)=1\-\\tau^\{2\}/\[\(1\+\\tau^\{2\}/L\)^\{L\}\-1\]\. The curve is parameter\-free: by Proposition 5\(iii\) it is the exact worst case at matchedτ\\taufor a coverage\-generated profile\. Most per\-channel and per\-block cells sit on or below it \(ten cells exceed it by at most 8 percentage points, noise\-floor and Llama\-3\.2\-3B cells, Section[4\.4](https://arxiv.org/html/2607.12266#S4.SS4)\), while per\-tensor cells deviate below it, in the regime the theory flags as beyond coverage\. Right: measured share vs\. the exact heterogeneous Proposition 5 prediction from the fitted\(c,ai\)\(c,a\_\{i\}\)\(Pearsonr=0\.984r=0\.984; 30B\-block extensionr=0\.997r=0\.997\)\.

## 5Why isolated sensitivity mis\-prices a configuration: the wrong anchor

Throughout, a predictor’s*anchor*is the configuration, or distribution over configurations, at which its coefficients are measured: isolated sensitivities are anchored at the unquantized network, while the coefficients of a fitted additive model are anchored at the deployment distribution\.

#### The measurement\.

An estimate of a configuration’s loss built from per\-layer sensitivities alone is the sum∑i∈Sf​\(\{i\}\)\\sum\_\{i\\in S\}f\(\\\{i\\\}\), which we call*isolate\-and\-sum*throughout\. On the fully quantized configuration of the exact whole\-layer FP4 lattices \(the two lattices of Section[3](https://arxiv.org/html/2607.12266#S3)and the four certification cells of Table[7](https://arxiv.org/html/2607.12266#A1.T7)\) this aggregation overstates the measured loss by 71–346%\. The mis\-pricing persists on full networks\. Predictingffon configurations that protect a sampled subset of layers and quantize the rest \(all 36 Qwen3\-8B and 48 Qwen3\-30B\-A3B layers, protection sizes 1 toL/2L/2\), isolate\-and\-sum’s median relative error is 51% on Qwen3\-8B and 203% on Qwen3\-30B\-A3B, while additive and coverage models fitted on 80 sampled configurations attain 2–3% on the same held\-out protection sets\. HAWQ\-v2, scoring each layer by its Hessian trace \(Hutchinson estimator\) times its mean squared quantization perturbation\(Donget al\.,[2020](https://arxiv.org/html/2607.12266#bib.bib10)\), inherits the same anchor: summed over each of the two Section[3](https://arxiv.org/html/2607.12266#S3)lattices it overstates the fully quantized corner by 89% \(Qwen3\-8B\) and 376% \(Qwen3\-0\.6B\), against isolate\-and\-sum’s 155% and 250% on the same corners\.

#### Marginal scale\.

The failure is caused by two distinct order\-1 coefficients\. The*in\-context*slope, the coefficient of the optimal additive fit, iswi=𝔼p​\[f∣xi=1\]−𝔼p​\[f∣xi=0\]w\_\{i\}=\\mathbb\{E\}\_\{p\}\[f\\mid x\_\{i\}=1\]\-\\mathbb\{E\}\_\{p\}\[f\\mid x\_\{i\}=0\], the marginal effect of layeriiaveraged over deployment configurations\. The*isolated*slope, the quantity that HAWQ\-style scores approximate, issi=f​\(\{i\}\)s\_\{i\}=f\(\\\{i\\\}\), the marginal at the unquantized network\. Under coverage,si=c​ais\_\{i\}=c\\,a\_\{i\}whilewi=c​ai​∏j≠i\(1−p​aj\)w\_\{i\}=c\\,a\_\{i\}\\prod\_\{j\\neq i\}\(1\-pa\_\{j\}\), so

siwi=1∏j≠i\(1−p​aj\)≈ep​∑jaj,\\frac\{s\_\{i\}\}\{w\_\{i\}\}=\\frac\{1\}\{\\prod\_\{j\\neq i\}\(1\-pa\_\{j\}\)\}\\approx e^\{\\,p\\sum\_\{j\}a\_\{j\}\},a ratio that grows exponentially withp​∑jajp\\sum\_\{j\}a\_\{j\}\. We validate the formula directly on the full282^\{8\}lattices, comparing the measured ratiosi/wis\_\{i\}/w\_\{i\}against the value the fitted coverage model predicts\. The measured medians are 1\.9 / 3\.0 / 3\.9 / 9\.2 \(Llama\-3\.2\-3B / Qwen3\-30B\-A3B / Qwen3\-8B / Qwen3\-0\.6B\) against predicted 1\.8 / 2\.3 / 2\.5 / 7\.7, with more detail in App\.[C](https://arxiv.org/html/2607.12266#A3)\. The two\-slope identity is a one\-line corollary of Theorems 1 and 4 \(App\.[F](https://arxiv.org/html/2607.12266#A6)\)\.

## 6An interpretable predictor, in and out of distribution

Two predictors are fit throughout: the coverage model, and the additive model\. Each hasL\+1L\+1parameters, fit by least squares in loss space onN≈LN\\approx Lmeasured configurations\. The choice of estimator \(joint regression rather than per\-coefficient sampling\), its consistency under misspecification, and the cost comparison with the∼100​L\{\\sim\}100Lprogressive\-quantization evaluations of CoopQ’s SPQE estimator and with the fit\-free Aumann–Shapley variant are discussed in App\.[D](https://arxiv.org/html/2607.12266#A4)\.

### 6\.1Benchmark

The benchmark task is to predictf​\(S\)f\(S\)for held\-out configurations fromNNmeasured ones\. Per \(model, corpus\) cell, every method is fit onNNconfigurations sampled fromμp\\mu\_\{p\}atp=0\.6p=0\.6and scored by median relative error on 120 held\-out configurations, against the surrogate predictors of Section[2](https://arxiv.org/html/2607.12266#S2)\(random forest, Gaussian process, RBF kernel model, and CoopQ’s linear\-plus\-pairwise model\), at W4A4 on the six models and two corpora of Table[9](https://arxiv.org/html/2607.12266#A4.T9)\(App\.[D](https://arxiv.org/html/2607.12266#A4), with the error\-versus\-NNsweep in Fig\.[7](https://arxiv.org/html/2607.12266#A4.F7)\)\. At theN=L/2N=L/2budget, coverage or additive is best in 5 of 11 cells and within 0\.03 of best in 8, with small margins separating the leading methods, while isolate\-and\-sum, at the same measurement cost, fails at 0\.38–0\.63 error on the FP4\-damaged models\.

The deciding comparisons are extrapolation to differently quantized configurations, where coverage degrades at most2×2\\timesagainst the Gaussian process’s22–5×5\\times\(Section[6\.2](https://arxiv.org/html/2607.12266#S6.SS2)\), allocation, where the surrogates fail outright \(Section[7\.3](https://arxiv.org/html/2607.12266#S7.SS3)\), and pricing, which only the parametric models provide \(App\.[E](https://arxiv.org/html/2607.12266#A5)\)\.

### 6\.2Out of distribution: stationary kernels do not encode saturation

On binary configuration vectors the RBF kernel\(Rasmussen and Williams,[2006](https://arxiv.org/html/2607.12266#bib.bib49)\)reduces toK​\(S,S′\)=exp⁡\(−γ​\|S​△​S′\|\)K\(S,S^\{\\prime\}\)=\\exp\(\-\\gamma\\,\|S\\triangle S^\{\\prime\}\|\), a stationary kernel in the Hamming metric\. Under the uniform measure, kernels of this form are diagonal in the Walsh basis with eigenvalues that depend only on the order\|T\|\|T\|and decay as\|T\|\|T\|grows \(O’Donnell,[2014](https://arxiv.org/html/2607.12266#bib.bib40); derivation in App\.[F](https://arxiv.org/html/2607.12266#A6)\), and approximately so atp=0\.6p=0\.6, so Gaussian\-process regression with this kernel is shrinkage of high\-order Walsh coefficients\(Rasmussen and Williams,[2006](https://arxiv.org/html/2607.12266#bib.bib49), Section 2\.6\), the same low\-order preference our measurements justify\. This is why the Gaussian process matches our predictors at largeNNin distribution, where order\-1 terms carry most of the damage variance \(Section[3](https://arxiv.org/html/2607.12266#S3)\)\. The difference emerges out of distribution\. Stationarity applies the same similarity rule everywhere on the cube, while saturation does not: under coverage the marginal effectf​\(S∪\{i\}\)−f​\(S\)=c​ai​∏j∈S\(1−aj\)f\(S\\cup\\\{i\\\}\)\-f\(S\)=c\\,a\_\{i\}\\prod\_\{j\\in S\}\(1\-a\_\{j\}\)shrinks as the contextSSgrows\. The stationary prior does not encode this structure\. We test this on the two cells of Table[16](https://arxiv.org/html/2607.12266#A5.T16)\(Qwen3\-0\.6B and Qwen2\.5\-3B, wiki\), fitting every method on configurations drawn atp=0\.6p=0\.6as before and scoring each on 120 test configurations drawn atp=0\.85p=0\.85or atp=0\.35p=0\.35\. At theN=L/2N=L/2budget, coverage’s error grows by1\.31\.3–2\.0×2\.0\\timesand the additive model’s by1\.11\.1–1\.9×1\.9\\timesrelative to their in\-distribution errors across the four \(cell, shift\) pairs, while the Gaussian process’s grows by2\.12\.1–4\.9×4\.9\\timesand the random forest’s by2\.02\.0–3\.9×3\.9\\times\(Fig\.[4](https://arxiv.org/html/2607.12266#S6.F4)\)\.

The heavily quantized direction is the one allocation exercises, hence an explanation for its performance in allocation\.

![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig4_ood.png)Figure 4:Fit atp=0\.6p=0\.6, tested atp=0\.85p=0\.85andp=0\.35p=0\.35\. Coverage degrades1\.31\.3–2\.0×2\.0\\timesand the additive fit1\.11\.1–1\.9×1\.9\\timesrelative to their in\-distribution errors, while the Gaussian process degrades2\.12\.1–4\.9×4\.9\\timesand the random forest2\.02\.0–3\.9×3\.9\\times; the stationary priors of the surrogates do not encode saturation\.
### 6\.3A diffusion transformer

The benchmark repeats on the Qwen\-Image\-2512 diffusion transformer \(L=60L=60blocks, per\-block NVFP4\), withf​\(S\)f\(S\)the mean squared error between the quantized and full\-precision model outputs at the first denoising step \(per\-budget results in App\.[D](https://arxiv.org/html/2607.12266#A4), Table[10](https://arxiv.org/html/2607.12266#A4.T10)\)\. The measured order\-≥2\\geq 2share is 0\.36, above every LLM cell \(0\.016–0\.30, Tables[9](https://arxiv.org/html/2607.12266#A4.T9)and[16](https://arxiv.org/html/2607.12266#A5.T16)\)\. At small budgets, coverage and random forest edge the other methods, with performance converging across methods as budget size increases\.

## 7Allocation at matched memory

### 7\.1Setup

Given a memory budget in effective bits,eff=∑numel⋅bits/∑numel\\mathrm\{eff\}=\\sum\\mathrm\{numel\}\\cdot\\mathrm\{bits\}/\\sum\\mathrm\{numel\}, the allocator assigns each unit \(a weight tensor, or a fused group that must share one serving format, App\.[D](https://arxiv.org/html/2607.12266#A4)\) a bit\-width in\{16,8,4\}\\\{16,8,4\\\}to minimize predictedΔ​LCE\\Delta L\_\{\\mathrm\{CE\}\}\(Section[7\.4](https://arxiv.org/html/2607.12266#S7.SS4)extends the alphabet below four bits\)\. Under the coverage model, predicted damage increases in the summed log\-headroom costsbib\_\{i\}of the demotions taken, so minimizing it under a bit budget is a knapsack over those per\-demotion costs \(App\.[C](https://arxiv.org/html/2607.12266#A3)\)\.

#### Converting the baselines into allocators\.

The Section[6](https://arxiv.org/html/2607.12266#S6)baselines are predictors, so each is fit on the sameN=300N=300sampled configurations and run through the same knapsack over its predicted marginals \(fit\-addfor the fitted additive model,gp\-allocandrf\-allocfor the surrogates\)\. We fit the coverage and additive scores either with least squares or an Aumann\-Shapley path integral\. The Aumann–Shapley score is the loss gradient averaged over increasingly quantized contexts, and under the coverage model it equals the optimal in\-context coefficient evaluated at an effective deployment density \(App\.[B](https://arxiv.org/html/2607.12266#A2)\)\.

### 7\.2Serving against vendor NVFP4 checkpoints

On the dense Qwen3\.6\-27B we compare the additive allocation at 5\.41 effective bits, derived from measured per\-unit step damages on a task\-aligned calibration set \(App\.[D](https://arxiv.org/html/2607.12266#A4)\), against the two public NVFP4 checkpoints, NVIDIA’s at 6\.31 effective bits and Unsloth’s at 6\.73, all served by vLLM on one B200 at concurrency 128\. Ours and Unsloth’s checkpoints serve W4A4 NVFP4, quantizing activations at runtime against per\-block scales; NVIDIA’s published checkpoint keeps its NVFP4 linears weight\-only \(W4A16\)\. The additive configuration matches the vendor checkpoints downstream \(Table[1](https://arxiv.org/html/2607.12266#S7.T1)\), and the smaller footprint is realized as speed: it is the fastest system at every prompt/generation shape,1\.51\.5–2\.0×2\.0\\timesthe bf16 throughput and 8–24% above the Unsloth checkpoint \(Fig\.[5](https://arxiv.org/html/2607.12266#S7.F5)\)\.

Table 1:Qwen3\.6\-27B served accuracy \(pass@1; AIME25 averaged over 8 seeds; thinking sampling\)\.![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig_race27b.png)Figure 5:Qwen3\.6\-27B output throughput at four prompt/generation shapes, with the speedup over bf16 marked on the additive bars\. The additive configuration is fastest at every shape at the lowest effective bits\.The comparison repeats at 753B\. On GLM\-5\.2 we compare the additive allocation \(4\.84 stored weight bits per parameter, 474 GB\) against NVIDIA’s published NVFP4 checkpoint \(4\.76 bits, 465 GB\) and the FP8 base model, served by SGLang on4×4\\timesB200 at concurrency 8\. Both NVFP4 configurations are W4A4\. Accuracy matches at matched stored size \(Fig\.[6](https://arxiv.org/html/2607.12266#S7.F6)\), and the additive configuration reads 28\.3 GB of weights per decoded token against the checkpoint’s 45\.9, so it decodes1\.19×1\.19\\timesfaster: the allocation places its low\-precision bits on the units decode reads most often\.

![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig_glm52.png)Figure 6:GLM\-5\.2 \(753B\) decode throughput, SGLang on4×4\\timesB200 at concurrency 8, with total stored weight and weight bytes read per decoded token \(batch size 1\) below each bar\. Accuracy is matched: MMLU 0\.842 \(additive\) against 0\.838 \(NVIDIA NVFP4\) and 0\.864 \(the FP8 base\), GPQA diamond 0\.470 against 0\.455 and 0\.465 \(zero\-shot letter\-logit scoring; 228\-question stratified MMLU subset, 198 GPQA questions\)\.
### 7\.3MoE language models

We benchmark our methods against the incumbent methods\. The certificate accompanying these results depends on the units it is measured over\. On the eight whole\-layer FP4 block lattices of Qwen3\-30B \(Section[4\.4](https://arxiv.org/html/2607.12266#S4.SS4)\) the measured order\-≥2\\geq 2share has median 0\.013\. At the allocation granularity itself \(per\-linear units over the\{16,8,4\}\\\{16,8,4\\\}alphabet\) no lattice is enumerable, and the split\-half cross\-estimate \(coefficient products of two disjoint halves, as in App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\) from the 300 sampled configurations of the fitted baselines, unbiased but not constrained to be nonnegative, gives−0\.09±0\.10\-0\.09\\pm 0\.10on the 30B pool and0\.24±0\.140\.24\\pm 0\.14on the 235B pool\. AtN=300N=300the sampling error is of the order of the share itself, so the allocation\-granularity certificate is an order\-of\-magnitude statement, not a precise value\. Per\-method values are in Tables[2](https://arxiv.org/html/2607.12266#S7.T2)and[3](https://arxiv.org/html/2607.12266#S7.T3), each with a paired test over the 16 shared evaluation sequences, Holm\-corrected within each budget row at the 5% level\. On Qwen3\-30B the additive\- and coverage\-scored allocators lead at every measured budget\.

Table 2:Qwen3\-30B allocation comparison at matched effective bits, calibration KL on the task\-aligned corpus of App\.[D](https://arxiv.org/html/2607.12266#A4)\(BFCL and RULER prompts\)\.∗\*marks methods significantly different from AS\-additive under a pairedtt\-test over the 16 shared sequences, Holm\-corrected within each budget row at the 5% level\.†marks MC\-MoE’s per\-expert form, which assigns each expert its own format and cannot be loaded by the fused serving kernels, so the fusion column is its servable form\. int8\-outlier ranks units by an llm\.int8\-style activation\-outlier severity score\. Marginal 95% intervals on the entries span±\.001\\pm\.001–\.007\.007\.The predictor skill of a black\-box surrogate does not transfer:gp\-alloctrails the AS and fitted allocators by 3–18% at the contended budgets \(the extrapolation failure of Section[6\.2](https://arxiv.org/html/2607.12266#S6.SS2)in allocation form\), andrf\-allocfalls below naive\-uniform\. CoopQ trails AS by\+19\+19to\+54%\+54\\%despite 1,205 sampled KL evaluations against AS’s zero, and it scores whole layers rather than the 240 per\-linear units of the other methods, the only granularity its measurement design affords, since its permutation\-walk cost scales with the unit count\.

Table[12](https://arxiv.org/html/2607.12266#A4.T12)shows AIME\-2025 scores for the evaluated methods on Qwen3\-30B, with every method between 66 and 76 against the bf16 anchor 73\.3 and no significant ordering at any budget\. This finding survives both W4A4 and W4A16 quantization regimes\.

#### Scale replication \(Qwen3\-235B, Table[3](https://arxiv.org/html/2607.12266#S7.T3)\)\.

The comparison was re\-run at8×8\\timesscale on Qwen3\-235B\. The in\-context family again carries the contended budgets: AS\-additive is best at eff 4\.2, the fitted additive and coverage variants at 4\.5–6\.0, and every method outside the family is significantly worse at eff 4\.2, 4\.5, and 6\.0 \(at 5\.0, naive\-uniform, ModelOpt, andgp\-allocare not separated after the Holm correction\)\. The comparison extends once more to GLM\-4\.6 \(355B\): a three\-level\{16,8,4\}\\\{16,8,4\\\}allocation over its 43,364 units gives AS\-coverage calibration KLs of 0\.059, 0\.045, and 0\.036 at effective bits 5, 6, and 7, against ModelOpt’s 0\.088, 0\.071, and 0\.045, margins of 34, 37, and 20%, with AS\-additive within 0\.006 of AS\-coverage at every budget\.

Table 3:Qwen3\-235B, all methods measured jointly on the task\-aligned corpus of App\.[D](https://arxiv.org/html/2607.12266#A4), with paired significance vs\. AS\-additive over the 16 shared sequences \(∗\*= significant after Holm correction within each budget row at the 5% level\)\.

### 7\.4Allocation below four bits

Below four bits the base quantizer decides whether allocation has anything to work with: under round\-to\-nearest quantization, every allocator we measured collapses at every sub\-4 budget regardless of model scale, so we use an error\-compensating base\. We construct a per\-unit GPTQ\(Frantaret al\.,[2022](https://arxiv.org/html/2607.12266#bib.bib5)\)ladder on Qwen3\-30B, quantizing every unit at 4, 3, and 2 bits \(weight\-only; details in App\.[D](https://arxiv.org/html/2607.12266#A4)\)\.

In Table[5](https://arxiv.org/html/2607.12266#S7.T5), the additive allocator’s KL sits3\.03\.0–6\.6×6\.6\\timesbelow ModelOpt’s at the servable budgets \(≥2\.75\\geq 2\.75, established by the serving results below\)\. CoopQ and the isolated ranking, which reorders the additive allocator’s own per\-unit damages, occupy the same tier, while MC\-MoE and int8\-outlier sit above ModelOpt\.

The construction replicates at scale on Qwen3\-235B\-A22B \(Table[4](https://arxiv.org/html/2607.12266#S7.T4)\): of the three constructed allocations the additive one holds the lowest calibration KL at every sub\-4 budget, preserves reasoning when served at effective bits 3\.5, and degrades through 3\.0\. At effective bits 2\.9 the additive scoring rule appears on both sides of the survival boundary: derived on the evaluation\-task corpus itself, the additive and coverage allocations remain functional \(42\.9 and 41\.2 AIME\-2025\), while the additive allocation of identical memory derived on a separately measured corpus collapses \(2\.1, as does the random mix at 5\.0\)\. The gap opens before the boundary \(58\.3 against 39\.6 at effective bits 3\.0\), so at this scale the derivation corpus, not the scoring rule, sets the usable budget; every arm collapses at 2\.75\.

Table 4:Qwen3\-235B below four bits on the evaluation\-task corpus\. Top: calibration KL of the three directly constructed allocations \(same ladder and corpus protocol as Table[5](https://arxiv.org/html/2607.12266#S7.T5)\)\. Bottom: served accuracy, AIME\-2025 \(8 seeds\) / GPQA diamond / MMLU\-Pro, for the five listed configurations \(GPTQ weight\-only checkpoints\); bf16 anchors 80\.0/71\.7/82\.9\. The additive allocation preserves reasoning at 3\.5 effective bits and degrades through 3\.0; every earlier\-derived method collapses by 2\.9\. The rows marked eval\.\-task deriv\. are allocations derived on the evaluation\-task corpus itself\. Both remain functional at 2\.9 \(42\.9 and 41\.2 AIME with no generation reaching the token cap\) The surviving configurations are those whose calibration KL \(0\.44\) sits below the termination threshold of Section[7\.4](https://arxiv.org/html/2607.12266#S7.SS4)\.Table 5:Sub\-4 allocation on the GPTQ ladder \(Qwen3\-30B\), calibration KL on the evaluation\-task corpus \(96 MBPP, 64 gsm8k\-train, 64 MATH, 64 MMLU\-Pro validation prompts; evaluation half, 16 shared sequences\)\. Every method is derived and measured on this corpus\.∗\*marks methods significantly different from the additive allocator under a pairedtt\-test over the 16 shared sequences, Holm\-corrected within each budget row at the 5% level\. Parentheticals give each method’s measurement cost:*meas\.*counts isolated per\-unit damage measurements on the ladder \(additive prices each unit’s4→34\\to 3and3→23\\to 2steps; isolated\-rank ranks by the same damages\),*cfg\.*counts sampled mixed configurations the fitted, GP, and RF predictors are fit on, and CoopQ’s count is its permutation\-walk evaluations\.Serving these configurations tests whether the calibration\-KL margins are realized as served accuracy\. Table[6](https://arxiv.org/html/2607.12266#S7.T6)reports HumanEval at both scales, and the full reasoning grid is Table[14](https://arxiv.org/html/2607.12266#A4.T14)\(App\.[D](https://arxiv.org/html/2607.12266#A4)\)\. The failure mode has a common boundary, a*termination threshold*: above a model\-specific calibration KL a configuration stops emitting its end\-of\-sequence token, every generation runs to the token cap, and no answer is produced\. On the 30B grid the boundary sits near 0\.45 nats \(rf\-allocat KL 0\.449 collapses on AIME\-2025 while the additive configuration at 0\.415 continues to score, Table[14](https://arxiv.org/html/2607.12266#A4.T14)\), and the threshold governs termination rather than accuracy\. At effective bits 2\.9 the four HumanEval methods order exactly as their calibration KLs \(0\.215, 0\.228, 0\.236, and 1\.042 giving 73\.2, 44\.5, 38\.4, and 0\.0\)\. The accuracy gaps do not scale with the KL gaps: CoopQ sits within 0\.013 nats of the additive configuration yet 29 points below it, so which units receive the 2\-bit assignments changes served accuracy at fixed memory and near\-fixed KL\. At 2\.75 the ordering itself breaks: the uniform mix has the lowest KL \(0\.319 against additive’s 0\.415\) yet scores 19\.5 against 54\.9\.

Table 6:Served HumanEval accuracy \(pass rate on base tests, all 164 problems, greedy decoding\)\. Cells at 0\.0 are collapsed configurations, with all or nearly all generations reaching the 8,192\-token cap without terminating\. The two columns per model share one anchor measurement\. The 235B CoopQ configurations were not constructed due to measurement cost\.

## 8Scope

#### The activation channel carries the structure\.

Measuring the damage set function on the same layers with weights only \(W4A16\), activations only \(W16A4\), and both \(W4A4\) quantized, the weight channel is non\-saturating \(isolate\-and\-sum error≈0\.10\\approx 0\.10,∑if​\(\{i\}\)/f​\(all\)≈1\.0\\sum\_\{i\}f\(\\\{i\\\}\)/f\(\\text\{all\}\)\\approx 1\.0\) while the activation channel saturates strongly \(isolate\-and\-sum 2\.80 against coverage’s 0\.098 on the 3B W16A4 cell, Table[16](https://arxiv.org/html/2607.12266#A5.T16)\)\. The saturation therefore lives in the activation channel, which is why the method targets activation and W4A4 quantization\.

#### Outlier\-borne coupling under coarse activation scaling\.

At the deployed NVFP4 per\-block format the order\-≥2\\geq 2share is 0\.07 and the additive and coverage models attain the lowest errors, while under per\-tensor activation scales \(a coarse setting that is not deployed\) the share reaches 0\.67 on the 0\.6B certification lattice of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)\(the per\-tensor cell of Table[7](https://arxiv.org/html/2607.12266#A1.T7); the separately measured per\-tensor lattice of Table[11](https://arxiv.org/html/2607.12266#A4.T11)puts its own share at 0\.81\), and the nonparametric surrogates attain lower error, a breakdown the certificate forecasts\. The coupling is outlier\-borne: localizing the scale \(per\-block\-256\) drops the share to 0\.01, clipping the top 0\.1% of activations drops it to 0\.16, and SmoothQuant\-style scaling before quantizing drops it to 0\.46\. The fitted break\-rates themselves are depth\-structured rather than outlier\-driven \(on Qwen3\-0\.6B they correlate with depth at Spearmanρ=\+0\.68\\rho=\+0\.68and with output RMS atρ=\+0\.50\\rho=\+0\.50, but not with activation kurtosis\)\.

#### Measurement protocol\.

Headline cells use 3 seeds \(std≤0\.03\\leq 0\.03\), and the per\-tensor W4A4 isolate\-and\-sum failure is also 3\-seed stable \(0\.6B20\.6±0\.320\.6\\pm 0\.3, 3B14\.1±0\.314\.1\\pm 0\.3\), so the rankings are robust\.

## 9Future work

#### Data randomness in the theory\.

We treatf​\(S\)=Δ​Lf\(S\)=\\Delta Las deterministic, but it is an expectation over the calibration distribution, with data\-dependent break\-rates\. A two\-source variance decomposition, layer\-selection variance \(thepp\-biased Walsh account\) plus data\-sampling variance, would give a principled law connecting calibration size to prediction variance and a data\-aware certificate\.

#### Hardware\- and framework\-aware allocation\.

Effective bits equal memory is the wrong cost for realized speedup, which is a property of the serving stack: fusion granularity sets the alphabet \(one format per fused qkv or expert block\), kernel availability determines if certain bit\-widths \(e\.g\. sub\-4\) are servable\. Co\-designing the cost function of the allocator with the serving stack would lead to better realized speed gains\.

#### Composing allocation with outlier and rotation methods\.

Allocation chooses how many bits each unit gets, while rotations \(QuaRot, SpinQuant, QuIP\#\), SmoothQuant, and llm\.int8\(\) change how much a unit is damaged at a given bit\-width, Our framework operates on whatever base quantizer producesf​\(S\)f\(S\), so the two compose: allocate over a rotated or outlier\-handled base\.

## 10Conclusion

Viewed through its signed Möbius expansion, low\-bit quantization damage appears to carry irreducible structure at every order, yet in the Walsh basis roughly 90% of its variance is order\-1, and a saturating coverage model with one parameter per layer generates both behaviors exactly, the divergent series and the concentrated spectrum alike\. Because the order\-≥2\\geq 2share equals the best additive predictor’s mean\-squared error as a fraction of the variance \(Theorem 1\), every result is paired with a certificate of when additive machinery suffices for prediction, measured on exact lattices or forecast in closed form from fitted break\-rates, and the ordinal defect places the damage within a few percent of a monotone function of a per\-layer sum, so the residual interaction is largely the curvature of the loss scale rather than structure among layers\.

The practical consequences follow\. Sensitivities measured in isolation overstate configuration losses by 71–376%, by a closed\-form inflation factor\. The fitted coverage and additive models price a configuration’s loss before deployment, are best or tied against black\-box surrogates at small measurement budgets, and degrade at most2×2\\timesunder extrapolation to differently quantized configurations where Gaussian\-process surrogates degrade22–5×5\\times\. One Aumann–Shapley path integral recovers the optimal additive coefficient at a layer\-specific effective density, and the ordering of layers exactly, with no sampled configurations\. Used as allocators at matched memory, the same models attain the lowest KL among the compared allocators,1717–21%21\\%below the production ModelOpt baseline on Qwen3\-30B, with consistent margins at 235B and 355B\. The margins appear where per\-unit sensitivities are heterogeneous and budgets are contended, and they vanish in the served≥4\\geq 4\-bit regime, which is near\-lossless by construction\. Below four bits, where an RTN base collapses every allocator, an error\-compensating base restores the choice, and allocation decides whether the served model produces terminating generations at all\.

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## Appendix AThe model classg​\(∑ibi\)g\\bigl\(\\sum\_\{i\}b\_\{i\}\\bigr\): canonicity, and details for the ordinal analysis

The Walsh–Hadamard expansion offfis exactly the functional\-ANOVA decomposition on thepp\-biased Boolean cube: order 0 is the mean, order 1 the per\-layer main effects, orderkkthekk\-way components\. The tempting reading, that order\-≥2\\geq 2energy measures interaction among layers, does not follow, because ANOVA additivity is additivity in the loss values and the loss saturates, so a model that is additive in the loss must spend order\-≥2\\geq 2energy merely to represent the curvature of saturation, even with no coupling among layers\. Order\-≥2\\geq 2energy therefore conflates real coupling with the shape of a scalar transformation\. The classL​\(S\)=g​\(∑i∈Sbi\)L\(S\)=g\\bigl\(\\sum\_\{i\\in S\}b\_\{i\}\\bigr\), additive after a strictly increasinggg, partially disentangles the two, since if a single suchgglinearizesff\(that is,g−1∘fg^\{\-1\}\\circ fis order\-1\), the order\-≥2\\geq 2energy was produced byggrather than by interaction among layers\. The residual test that survives this critique is the order\-≥2\\geq 2energy after the best transformation, measured in Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1), with protocol details below\.

#### Why searching over transformations is canonical\.

The classg​\(∑ibi\)g\\bigl\(\\sum\_\{i\}b\_\{i\}\\bigr\)is not a parametric convenience\. It is exactly the class of damage functions that accumulate one layer at a time by a fixed rule,f​\(S∪\{i\}\)=f​\(S\)⊕dif\(S\\cup\\\{i\\\}\)=f\(S\)\\oplus d\_\{i\}, with a per\-layer incrementdid\_\{i\}that does not depend onSSand an operation⊕\\oplusthat is associative, continuous, and strictly increasing with identity0\. By the Aczél–Ling representation theorem, every such⊕\\oplusis conjugate to ordinary addition: there is a strictly increasing generatorggwithx⊕y=g​\(g−1​\(x\)\+g−1​\(y\)\)x\\oplus y=g\\bigl\(g^\{\-1\}\(x\)\+g^\{\-1\}\(y\)\\bigr\), unique up to scale\[Aczél,[1966](https://arxiv.org/html/2607.12266#bib.bib73), Ling,[1965](https://arxiv.org/html/2607.12266#bib.bib74)\]\. Accumulating damage by any rule with these properties is therefore equivalent toffbeing additive after some strictly increasing transformation, and the search of App\.[C](https://arxiv.org/html/2607.12266#A3)ranges over such rules\. Coverage is the instancex⊕y=x\+y−x​y/cx\\oplus y=x\+y\-xy/c, the probabilistic sum \(noisy\-OR\)\[Pearl,[1988](https://arxiv.org/html/2607.12266#bib.bib45), Klementet al\.,[2000](https://arxiv.org/html/2607.12266#bib.bib75)\], whose generator is the log\-headroomg−1​\(f\)=−log⁡\(1−f/c\)g^\{\-1\}\(f\)=\-\\log\(1\-f/c\)\. The essential interaction of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)is then precisely what no rule with these properties can absorb, namely the ordinal defect\.

### A\.1Details for Section[3](https://arxiv.org/html/2607.12266#S3): the ordinal certification and the noise\-corrected energies

###### Proof of Theorem 2\.

\(⇒\\Rightarrow\) A strictly increasingggpreserves and reflects strict order\. \(⇐\\Leftarrow\) On the finite setV=\{b⋅x\}V=\\\{b\\cdot x\\\}defineg0​\(b⋅x\):=f​\(x\)g\_\{0\}\(b\\cdot x\):=f\(x\)\. This is well\-defined \(ifb⋅x=b⋅yb\\cdot x=b\\cdot y, the order condition forcesf​\(x\)=f​\(y\)f\(x\)=f\(y\)\) and strictly increasing \(ifb⋅x<b⋅yb\\cdot x<b\\cdot ythenf​\(x\)<f​\(y\)f\(x\)<f\(y\)\), and it extends piecewise\-linearly to a strictly increasingggonℝ\\mathbb\{R\}\. ∎

The claim is the finite binary case of additive conjoint measurement\[Luce and Tukey,[1964](https://arxiv.org/html/2607.12266#bib.bib77), Debreu,[1960](https://arxiv.org/html/2607.12266#bib.bib78), Scott,[1964](https://arxiv.org/html/2607.12266#bib.bib79)\], and is stated for completeness\.

#### Certification protocol: the estimand and the noise floor\.

This and the next three paragraphs detail how the ordinal defect of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)is measured and certified\. The estimand is the draw\-expected damagef~\\tilde\{f\}: where the measuredf​\(S\)f\(S\)of Section[3](https://arxiv.org/html/2607.12266#S3)averages 16 stochastic\-rounding draws,f~​\(S\)\\tilde\{f\}\(S\)is its expectation over draws, and all randomness below is over those draws\. The ranking constraints off~\\tilde\{f\}are the inequalitiesb⋅\(x−y\)\>0b\\cdot\(x\-y\)\>0, one per ordered pair of configurations withf~​\(x\)\>f~​\(y\)\\tilde\{f\}\(x\)\>\\tilde\{f\}\(y\)\. By Theorem 2, an index satisfying every constraint exists exactly whenf~\\tilde\{f\}is additive up to a strictly increasing transformation, and the ordinal defect is the smallest fraction of the constraints that any index must violate\. The complication is thatf~\\tilde\{f\}is known only up to the draw noise\. Both bounds below therefore share two devices\. First, the draws are split into two halves, whose meansf¯A\\bar\{f\}\_\{A\}andf¯B\\bar\{f\}\_\{B\}are independent estimates off~\\tilde\{f\}at every configuration, the split\-half device of reliability theory\[Spearman,[1910](https://arxiv.org/html/2607.12266#bib.bib81), Brown,[1910](https://arxiv.org/html/2607.12266#bib.bib82)\]\. Second, a pair of configurations contributes a constraint on a given half \(is*eligible*there\) only if its gap on that half exceeds twice its standard error\. A half\-mean at configurationSShas standard errorσ^​\(S\)/8\\hat\{\\sigma\}\(S\)/\\sqrt\{8\}, withσ^​\(S\)\\hat\{\\sigma\}\(S\)the sample standard deviation of the draws, so the gap between configurationsxxandyyhas standard errorσ^​\(x\)2\+σ^​\(y\)2/8\\sqrt\{\\hat\{\\sigma\}\(x\)^\{2\}\+\\hat\{\\sigma\}\(y\)^\{2\}\}/\\sqrt\{8\}\.

#### The upper bound: an exhibited index\.

Because the defect is a minimum over indices, the fraction of constraints violated by any single index bounds it from above\. The tightest such bound would come from the index minimizing the violated fraction, which is NP\-hard to find\[Amaldi and Kann,[1995](https://arxiv.org/html/2607.12266#bib.bib85)\], so we exhibit instead the solution\(b,ξ\)\(b,\\xi\)of the hinge relaxation

minb,ξ​∑pξpsubject tob⋅\(xp−yp\)≥1−ξp,ξp≥0,\\min\_\{b,\\,\\xi\}\\ \\sum\_\{p\}\\xi\_\{p\}\\qquad\\text\{subject to\}\\qquad b\\cdot\(x\_\{p\}\-y\_\{p\}\)\\ \\geq\\ 1\-\\xi\_\{p\},\\quad\\xi\_\{p\}\\geq 0,the pairwise\-preference construction of ranking with slack variables\[Joachims,[2002](https://arxiv.org/html/2607.12266#bib.bib86)\], here without the norm term, so the problem is a linear program, fit on the eligible constraints of half A\. The fitted index’s violated fraction is then evaluated twice: in\-sample, on the half\-A constraints it was fit on, and cross\-half, on half B’s own eligible constraints, which the fit never saw\. The two evaluations play different roles\. With noisy draws the reported number is instead an estimate of the index’s violation rate onf~\\tilde\{f\}’s constraints, and the in\-sample rate is biased low for that quantity\. The cross\-half rate removes this adaptation, evaluating a fixed index on independently selected constraints\. We report both and quote the larger\. On most cells the two agree to within about a tenth of a percentage point\.

#### The lower bound: certified witnesses\.

The witnesses are rank flips, two contexts that order the same two layers oppositely \(the infeasibility witnesses of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)\)\. For a layer pair\(i,j\)\(i,j\)and a contextSScontaining neither, the constraint betweenf~​\(S∪\{i\}\)\\tilde\{f\}\(S\\cup\\\{i\\\}\)andf~​\(S∪\{j\}\)\\tilde\{f\}\(S\\cup\\\{j\\\}\)involvesbbonly through the sign ofbi−bjb\_\{i\}\-b\_\{j\}, which is the same for every context\. If, amongf~\\tilde\{f\}’s constraints,mmcontexts orderiiabovejjandnnorderjjaboveii, then every additive index violates at leastmin⁡\(m,n\)\\min\(m,n\)of them\. Summingmin⁡\(m,n\)\\min\(m,n\)over layer pairs, whose constraint families are disjoint, counts constraints that every index must violate, so this count divided by the number of eligible constraints is a lower bound on the defect, and by Theorem 2 a single opposed pair of contexts already refutes every increasing transformation\. To certify a context against noise, it must be eligible on half A and recur with the same sign on half B\. The witness count of Table[7](https://arxiv.org/html/2607.12266#A1.T7)is the sum over layer pairs of the smaller number of certified contexts of each preference direction\.

#### The matched\-noise null\.

The noise floor and the eligible constraint check still pass a small number of false positives, so every statistic above, for both bounds, is read against a matched\-noise null lattice pushed through the identical pipeline\. The null is the coverage model fitted to the measured means, which is additive after an increasing transformation and so contains no true flips, with the 16 draws regenerated as independent Gaussian samples centered on the fit, scaled by each configuration’s measured per\-draw sample standard deviationσ^​\(S\)\\hat\{\\sigma\}\(S\)\. This is a parametric bootstrap under the hypothesis of no essential interaction\[Efron and Tibshirani,[1993](https://arxiv.org/html/2607.12266#bib.bib87), Section 6\.5\]\. The Gaussian form is an assumption, and a mild one here, because every statistic consumes the draws only through the two half\-means, each an average of 8 draws that are themselves means of per\-token losses\. The null measures the false\-positive rate of the whole procedure, and a cell is certified only where the measured statistic exceeds its null\.

Table 7:Certification of the ordinal defect on full282^\{8\}lattices\. Only comparisons whose loss gap exceeds twice its standard error enter any count\. Witnesses count is the sum over layer\-pairs of the smaller number of contexts of each preference direction that sign\-replicate across the two halves of the draws\. Sign replication is the replicated fraction of half\-A comparisons\. LP upper bound is the fraction of eligible ranking constraints violated by the additive index fit on half A, evaluated in\-sample on half A and cross\-half on half B\. The range spans the two evaluations, and the text quotes the larger\. Every statistic is compared with the identical statistic on the matched\-noise null, which contains no true flips\.E∗E^\{\*\}is the order\-≥2\\geq 2energy share remaining after the best strictly increasing transformation of the loss \(the free\-monotone estimator of App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\)\.
#### Per\-model detail\.

Table[7](https://arxiv.org/html/2607.12266#A1.T7)reports every cell against its null\. In the deployed per\-block regime, Qwen3\-0\.6B and Qwen3\-8B sit at or below their nulls and are statistically indistinguishable from zero, Qwen3\-30B\-A3B has a defect of at most0\.8%0\.8\\%, at its null, and Llama\-3\.2\-3B carries a small genuine residue \(56 witnesses against a null of 6\)\. In the per\-tensor regime the witnesses exceed their null \(74 against 18, at0\.950\.95sign replication\), which is the outlier coupling of Section[8](https://arxiv.org/html/2607.12266#S8)appearing as certified rank reversals\. An additive index nevertheless still orders at least98%98\\%of the ranking constraints above the noise floor correctly\. Nearly all of the apparent high\-order energy in this worst cell is therefore curvature of the loss coordinates, as seen byE⋆E^\{\\star\}\.

#### Removing the stochastic\-rounding noise floor from measured order energies\.

The split of the draws serves a second, separate role for the order energies of Section[3](https://arxiv.org/html/2607.12266#S3), where the halves are differenced rather than held out\. Each measuredf​\(S\)f\(S\)averagesKKstochastic\-rounding draws and so equals the draw\-expected damagef~​\(S\)\\tilde\{f\}\(S\)plus a mean\-zero Monte\-Carlo error with varianceσ2​\(S\)/K\\sigma^\{2\}\(S\)/K, whereσ2​\(S\)\\sigma^\{2\}\(S\)is the per\-draw variance at configurationSS\. The Walsh transform is linear, so the measured coefficients are likewise unbiased\. Their squares are not, since𝔼​\[\(a\+Z\)2\]=a2\+Var⁡\(Z\)\\mathbb\{E\}\[\(a\+Z\)^\{2\}\]=a^\{2\}\+\\operatorname\{Var\}\(Z\)for constantaaand mean\-zeroZZ, so each measured order energyWkW\_\{k\}exceeds the energy of the draw\-expectation by the summed error variances of its order\-kkcoefficients, a floorνk\>0\\nu\_\{k\}\>0\. The per\-coefficient error variance is nearly the same for every basis function \(independent per\-configuration noise is white in an orthonormal basis: exactly even atp=12p=\\tfrac\{1\}\{2\}, within a small per\-order factor atp=0\.6p=0\.6\), soνk\\nu\_\{k\}is close to proportional to the number\(Lk\)\\binom\{L\}\{k\}of order\-kkbasis functions\. All butLLof the2L−12^\{L\}\-1non\-constant functions have order≥2\\geq 2, so the floor’s mass lands almost entirely in the order\-≥2\\geq 2energies and deflates the measured order\-1 share\. To estimate it, we average two disjoint halves of the draws separately,f¯A\\bar\{f\}\_\{A\}andf¯B\\bar\{f\}\_\{B\}, and form the difference tabled​\(S\)=12​\(f¯A​\(S\)−f¯B​\(S\)\)d\(S\)=\\tfrac\{1\}\{2\}\\bigl\(\\bar\{f\}\_\{A\}\(S\)\-\\bar\{f\}\_\{B\}\(S\)\\bigr\), in which the draw\-expectation cancels exactly and a pure mean\-zero error of the same varianceσ2​\(S\)/K\\sigma^\{2\}\(S\)/Kremains, so the order energies ofddsatisfy𝔼​\[Wk\(d\)\]=νk\\mathbb\{E\}\[W^\{\(d\)\}\_\{k\}\]=\\nu\_\{k\}\. This requires only that the two halves are independent \(fresh rounding draws per configuration and draw\)\. The corrected energies reported in Sections[3](https://arxiv.org/html/2607.12266#S3)and[4\.4](https://arxiv.org/html/2607.12266#S4.SS4)aremax⁡\(Wk−Wk\(d\),0\)\\max\(W\_\{k\}\-W^\{\(d\)\}\_\{k\},\\,0\)\. Subtracting replicate\-estimated error variance from an observed second moment is the classical variance\-components device of the analysis of variance\[Scheffé,[1959](https://arxiv.org/html/2607.12266#bib.bib80), Ch\. 7\], applied here to split\-half replicates\[Spearman,[1910](https://arxiv.org/html/2607.12266#bib.bib81), Brown,[1910](https://arxiv.org/html/2607.12266#bib.bib82)\]\.

#### Estimating the transformation\-independent energy floor\.

For strictly increasingggletE​\(g\)E\(g\)be the relative order\-≥2\\geq 2Walsh energy of the transformed damageg−1∘f~g^\{\-1\}\\circ\\tilde\{f\}\. By Theorem 1 applied tog−1∘f~g^\{\-1\}\\circ\\tilde\{f\},E​\(g\)E\(g\)is the fraction of the transformed damage’s variance that the best additive model leaves unexplained, andE∗:=infgE​\(g\)E^\{\*\}:=\\inf\_\{g\}E\(g\)is the floor of that fraction over all strictly increasing transformations\. A smallE∗E^\{\*\}therefore certifies thatg−1∘f~g^\{\-1\}\\circ\\tilde\{f\}is nearly additive for some strictly increasinggg, so the measured high\-order energy is curvature of the loss coordinates rather than coupling among layers\. Estimation must respect two hazards: Monte\-Carlo noise spreads nearly evenly across the Walsh basis, as above \(247 of 255 basis functions have order≥2\\geq 2, so raw noise masquerades as high\-order structure\), andg−1g^\{\-1\}is nonlinear, amplifying noise where the inverse is steep\. The protocol therefore splits the 16 rounding draws into three parts and fits every transform on part 1 only\. The parametric families are fit by nonlinear least squares, and the free monotone transform by alternating a weighted degree\-≤1\\leq 1fit with isotonic regression against the ordering offf’s values, which searches all increasing transformations at once and is the fit that estimatesE∗E^\{\*\}: isotonic regression returns the least\-squares best monotone nondecreasing fit to a scatter, computed exactly by the pool\-adjacent\-violators algorithm\[Ayeret al\.,[1955](https://arxiv.org/html/2607.12266#bib.bib83), Barlowet al\.,[1972](https://arxiv.org/html/2607.12266#bib.bib84)\]\. Each squared Walsh coefficient of the transformed damage is then estimated by multiplying the coefficient computed from part 2 by the coefficient computed from part 3, an unbiased estimate of the squared coefficient because the two parts’ noises are independent and mean zero\. The product is unbiased but not constrained to be nonnegative, so a cell whose true share is near zero can estimate slightly negative, as the−0\.4%\-0\.4\\%entry of Table[7](https://arxiv.org/html/2607.12266#A1.T7)does\. The headline numbers appear in Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1), and the per\-cell values in the last column of Table[7](https://arxiv.org/html/2607.12266#A1.T7)\.

#### What depends on the measure\.

Being exactly degree\-≤1\\leq 1, raw or after a transformation, is an algebraic property offf\(of its unique multilinear extension\), independent of the product measure, since the span of the degree\-≤k\\leq kμp\\mu\_\{p\}\-Walsh functions is the same subspace for everypp\. What depends onppis the accounting: the best degree\-≤1\\leq 1projection, its coefficientswi​\(p\)w\_\{i\}\(p\), and the order\-≥2\\geq 2energy are all priced under the chosen measure\. Two consequences follow\. Theorem 1’s certificate is stated, and must be read, at the deploymentμp\\mu\_\{p\}\. The wrong anchor of Section[5](https://arxiv.org/html/2607.12266#S5)is ultimately a wrong measure, the isolated slopesis\_\{i\}being thep→0p\\to 0limit of the deployment\-point slopewi​\(p\)w\_\{i\}\(p\)\.

## Appendix BAumann–Shapley recovers the in\-context coefficient at an effective density \(proof under coverage\)

There are three per\-layer slopes, all gradients of the saturating loss at different anchors: the empty\-anchor slope\[HAWQ, isolated; Donget al\.,[2019](https://arxiv.org/html/2607.12266#bib.bib43),[2020](https://arxiv.org/html/2607.12266#bib.bib10)\],si=f​\(\{i\}\)=c​ais\_\{i\}=f\(\\\{i\\\}\)=c\\,a\_\{i\}, inflated by the factor of Section[5](https://arxiv.org/html/2607.12266#S5); the deployment\-point slope, which is the optimal additive coefficient,wi=c​ai​∏j≠i\(1−p​aj\)w\_\{i\}=c\\,a\_\{i\}\\prod\_\{j\\neq i\}\(1\-p\\,a\_\{j\}\), optimal but requiring a fit at thepp\-biased operating point; and the path\-average slope\[Aumann–Shapley; Aumann and Shapley,[1974](https://arxiv.org/html/2607.12266#bib.bib42), Sundararajanet al\.,[2017](https://arxiv.org/html/2607.12266#bib.bib17)\], which is fit\-free,

A​Si=c​ai​∫01∏j≠i\(1−t​aj\)​d​t\.AS\_\{i\}=c\\,a\_\{i\}\\int\_\{0\}^\{1\}\\prod\_\{j\\neq i\}\(1\-t\\,a\_\{j\}\)\\,dt\.Completeness, the property that the coefficients sum exactly to the total damage, is the classical efficiency property of the Aumann–Shapley value and does not require the coverage form\. Under coverage it is immediate, sincedd​t​\[c−c​∏j\(1−t​aj\)\]=∑ic​ai​∏j≠i\(1−t​aj\)\\frac\{d\}\{dt\}\\bigl\[c\-c\\prod\_\{j\}\(1\-t\\,a\_\{j\}\)\\bigr\]=\\sum\_\{i\}c\\,a\_\{i\}\\prod\_\{j\\neq i\}\(1\-t\\,a\_\{j\}\), and integrating overt∈\[0,1\]t\\in\[0,1\]gives∑iA​Si=f​\(all\)\\sum\_\{i\}AS\_\{i\}=f\(\\text\{all\}\)with no remainder \(measured residual on real models 0\.06%\)\. To state what the path integral recovers, writewi​\(q\)=c​ai​∏j≠i\(1−q​aj\)w\_\{i\}\(q\)=c\\,a\_\{i\}\\prod\_\{j\\neq i\}\(1\-q\\,a\_\{j\}\)for the optimal additive coefficient when the deployment density isqq\.

###### Proposition B\.1\.

Letffbe the coverage model with break\-ratesa1,…,aL∈\[0,1\)a\_\{1\},\\ldots,a\_\{L\}\\in\[0,1\)and letp∈\(0,1\)p\\in\(0,1\)\. \(i\) For every layeriithere is an effective densityti∗∈\(0,12\]t\_\{i\}^\{\*\}\\in\(0,\\tfrac\{1\}\{2\}\]such thatA​Si=wi​\(ti∗\)AS\_\{i\}=w\_\{i\}\(t\_\{i\}^\{\*\}\), so the fit\-free path integral equals the optimal additive coefficient evaluated atti∗t\_\{i\}^\{\*\}in place ofpp\. \(ii\) For all layersiiandkk,A​Si\>A​SkAS\_\{i\}\>AS\_\{k\}if and only ifwi\>wkw\_\{i\}\>w\_\{k\}, if and only ifai\>aka\_\{i\}\>a\_\{k\}, so the path\-average and deployment\-point coefficients order the layers identically at every deployment density\.

###### Proof\.

\(i\) Writegi​\(t\)=∏j≠i\(1−t​aj\)g\_\{i\}\(t\)=\\prod\_\{j\\neq i\}\(1\-t\\,a\_\{j\}\), so thatA​Si=c​ai​∫01giAS\_\{i\}=c\\,a\_\{i\}\\int\_\{0\}^\{1\}g\_\{i\}andwi​\(q\)=c​ai​gi​\(q\)w\_\{i\}\(q\)=c\\,a\_\{i\}\\,g\_\{i\}\(q\)\. Ifaj=0a\_\{j\}=0for everyj≠ij\\neq ithengi≡1g\_\{i\}\\equiv 1and anyti∗t\_\{i\}^\{\*\}serves\. Otherwisegig\_\{i\}is positive, continuous, strictly decreasing, and convex on\[0,1\]\[0,1\]: withhj​\(t\)=−aj/\(1−t​aj\)h\_\{j\}\(t\)=\-a\_\{j\}/\(1\-t\\,a\_\{j\}\), differentiating the product givesgi′=gi​∑jhjg\_\{i\}^\{\\prime\}=g\_\{i\}\\sum\_\{j\}h\_\{j\}andgi′′=gi​∑j≠lhj​hl≥0g\_\{i\}^\{\\prime\\prime\}=g\_\{i\}\\sum\_\{j\\neq l\}h\_\{j\}h\_\{l\}\\geq 0, where the sums run over indices other thaniiand thehjh\_\{j\}agree in sign\. Jensen’s inequality for the uniform measure on\[0,1\]\[0,1\]gives∫01gi≥gi​\(12\)\\int\_\{0\}^\{1\}g\_\{i\}\\geq g\_\{i\}\(\\tfrac\{1\}\{2\}\), strict monotonicity gives∫01gi<gi​\(0\)\\int\_\{0\}^\{1\}g\_\{i\}<g\_\{i\}\(0\), and the intermediate value theorem then yields a uniqueti∗∈\(0,12\]t\_\{i\}^\{\*\}\\in\(0,\\tfrac\{1\}\{2\}\]withgi​\(ti∗\)=∫01gig\_\{i\}\(t\_\{i\}^\{\*\}\)=\\int\_\{0\}^\{1\}g\_\{i\}\. Multiplying byc​aic\\,a\_\{i\}givesA​Si=wi​\(ti∗\)AS\_\{i\}=w\_\{i\}\(t\_\{i\}^\{\*\}\)\. \(ii\) Factor out the full survival productG​\(t\)=∏j\(1−t​aj\)G\(t\)=\\prod\_\{j\}\(1\-t\\,a\_\{j\}\), positive on\[0,1\]\[0,1\]because everyaj<1a\_\{j\}<1\. Then

wi=c​G​\(p\)​ai1−p​ai,A​Si=c​∫01G​\(t\)​ai1−t​ai​𝑑t,w\_\{i\}=c\\,G\(p\)\\,\\frac\{a\_\{i\}\}\{1\-p\\,a\_\{i\}\},\\qquad AS\_\{i\}=c\\int\_\{0\}^\{1\}G\(t\)\\,\\frac\{a\_\{i\}\}\{1\-t\\,a\_\{i\}\}\\,dt,and for each fixedt∈\[0,1\]t\\in\[0,1\]the mapx↦x/\(1−t​x\)x\\mapsto x/\(1\-t\\,x\)is strictly increasing on\[0,1\)\[0,1\)\. Henceai\>aka\_\{i\}\>a\_\{k\}implieswi\>wkw\_\{i\}\>w\_\{k\}andA​Si\>A​SkAS\_\{i\}\>AS\_\{k\}, whileai=aka\_\{i\}=a\_\{k\}forces equality in both, and the two orderings coincide with the ordering by break\-rate\. ∎

Part \(i\) places the fit\-free coefficient in the same family as the fitted one, evaluated at the effective densityti∗t\_\{i\}^\{\*\}rather than at the deployment densitypp\(sinceti∗≤12<pt\_\{i\}^\{\*\}\\leq\\tfrac\{1\}\{2\}<pat ourp=0\.6p=0\.6, the path\-average coefficient sits above the deployment\-point one whenever other layers have nonzero break\-rates\)\. Part \(ii\) shows that this difference in evaluation point never reorders layers\. What remains numerical is the magnitude agreement betweenA​SiAS\_\{i\}andwi​\(p\)w\_\{i\}\(p\): the correlation is 0\.9997 on coverage\-generated data and 0\.785 on real models, while the empty\-anchorsis\_\{i\}is inflated by the factor≈ep​∑jaj\{\\approx\}\\,e^\{p\\sum\_\{j\}a\_\{j\}\}\. Sincesi=c​ais\_\{i\}=c\\,a\_\{i\}is itself increasing in the break\-rate, part \(ii\) extends to the empty anchor under the coverage model, so the failure of isolated sensitivity documented in Section[5](https://arxiv.org/html/2607.12266#S5)is a failure of magnitude rather than of order\.

## Appendix CThe log\-headroom knapsack, and the generator the data selects

#### Knapsack certificate\.

Because coverage is linear inbb, a service\-level constraint of the formΔ​LCE≤ε\\Delta L\_\{\\mathrm\{CE\}\}\\leq\\varepsilonmaps to a scalar budget∑i∈Sbi≤B\\sum\_\{i\\in S\}b\_\{i\}\\leq BwithB=−log⁡\(1−ε/c\)B=\-\\log\(1\-\\varepsilon/c\)\. The cheapest configuration meeting the constraint protects the setPPminimizing protection cost subject to∑i∉Pbi≤B\\sum\_\{i\\notin P\}b\_\{i\}\\leq Bis a 0/1 knapsack, solvable exactly by dynamic programming\. The deliverable is a predictedΔ​LCE\\Delta L\_\{\\mathrm\{CE\}\}for the chosen configuration, quotable before committing \(≤4%\\leq 4\\%error, App\.[E](https://arxiv.org/html/2607.12266#A5)\), which a ranking structurally cannot produce and which a Gaussian process prices with1\.61\.6–2\.9×2\.9\\timescoverage’s error near the heavily quantized corner where optimal configurations live \(Section[6\.2](https://arxiv.org/html/2607.12266#S6.SS2)\)\.

#### Which generator the data selects\.

The ceilingccand membership in the classg​\(∑ibi\)g\\bigl\(\\sum\_\{i\}b\_\{i\}\\bigr\)are established empirically, by the marginal\-damage measurements of Section[4\.1](https://arxiv.org/html/2607.12266#S4.SS1)and the ordinal certification of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1), while the exponential generator \(coverage/noisy\-OR\), which is what makes the headroom consumption multiplicative, is justified only by fit quality\. Refitting the lattices with candidate strictly increasing generators \(all withL\+1L\+1parameters,p=0\.6p=0\.6\) shows that prediction is insensitive to the choice \(held\-out relative error≈0\.064\{\\approx\}\\,0\.064–0\.0710\.071for all\), but the slope calibration, the ratiowmodel/wdataw\_\{\\text\{model\}\}/w\_\{\\text\{data\}\}of model\-implied to measured in\-context slopes, selects one: across the five non\-degenerate of six \(model, corpus\) cells, the probit\-⋅\\sqrt\{\\cdot\}ratio is 0\.93–1\.03, while the exponential ratio spans 0\.86 down to≈0\{\\approx\}0\(exp, probit, tanh, and rational sit at 0\.58–0\.67 on Qwen3\-0\.6B\)\. The saturation rule best supported by the data is therefore the probit\-⋅\\sqrt\{\\cdot\}generatorg​\(u\)=c​erfc⁡\(1/2​u\)g\(u\)=c\\,\\operatorname\{erfc\}\\bigl\(1/\\sqrt\{2u\}\\bigr\), the probability that zero\-mean Gaussian corruption of varianceuuexceeds a fixed threshold, scaled by the ceiling; the exponential coverage model is its tractable approximation\.

## Appendix DExperimental setup

We collect the protocol shared across Sections[3](https://arxiv.org/html/2607.12266#S3)–[8](https://arxiv.org/html/2607.12266#S8)here\. Section\-specific deviations are noted in place\.

#### Models\.

*LLMs*: Qwen3\-0\.6B, Qwen2\.5\-3B, Qwen3\-4B, Qwen3\-8B, Llama\-3\.2\-3B, with the per\-linear order\-share cells of Section[3](https://arxiv.org/html/2607.12266#S3)extending to Qwen3\-14B, Qwen3\-32B, Llama\-3\.3\-70B, Qwen2\.5\-72B, and gpt\-oss\-120B; Qwen3\-30B\-A3B, Qwen3\-235B\-A22B, and GLM\-4\.6 \(355B MoE\) for allocation at scale; Qwen3\.6\-27B for the vendor\-checkpoint serving comparison, with damage measured and ModelOpt calibrated on a chat\-templated task corpus \(96 MATH\-train, 64 gsm8k\-train, 96 MMLU\-Pro validation prompts\); Qwen3\-235B and GLM\-5\.2 \(753B\) for the scale\-robustness checks\.*Diffusion*: Qwen\-Image\-2512 DiT \(L=60L=60transformer blocks\) for the diffusion prediction results\. All weights are the public checkpoints\.

#### Quantization formats\.

The measurement regime throughout is W4A4 with NVFP4 per\-block scales\. We also analyze W16A4 \(activation\-only, isolating the channel that carries the non\-additivity\) and per\-tensor activation scaling \(a coarse setting that induces interactions of every order among layers\)\. The allocation alphabet is\{16,8,4\}\\\{16,8,4\\\}bits \(bf16 / FP8 / NVFP4\) for the servable studies and is extended to\{16,8,4,2,1\}\\\{16,8,4,2,1\\\}for the below\-4\-bit collapse study\. Serving uses three pipelines\. NVIDIA ModelOpt PTQ with TensorRT\-LLM serves the≥4\\geq 4\-bit MoE deployment recipes, and vLLM the vendor\-checkpoint comparison of Section[7\.2](https://arxiv.org/html/2607.12266#S7.SS2); these paths serve W4A4 ModelOpt exports with runtime activation quantization on the NVFP4 units \(NVIDIA’s published 27B checkpoint, which keeps its NVFP4 linears weight\-only, is the exception\), as does the SGLang\-served GLM\-5\.2 comparison\.

#### The damage set function and the lattice\.

For a contiguous block ofLLlayers we formf​\(S\)=LCE​\(S​quantized\)−LCE​\(bf16\)f\(S\)=L\_\{\\mathrm\{CE\}\}\(S\\ \\text\{quantized\}\)\-L\_\{\\mathrm\{CE\}\}\(\\text\{bf16\}\)by running the model with exactly the units inSSquantized\. Every exact lattice in the paper measures such a block centered at the network’s mid\-depth \(block offset⌊\(L′−L\)/2⌋\\lfloor\(L^\{\\prime\}\-L\)/2\\rfloorin anL′L^\{\\prime\}\-layer network, so the middle eight of Qwen3\-0\.6B’s 28 layers\), except the Qwen3\-30B\-A3B extension of Fig\.[3](https://arxiv.org/html/2607.12266#S4.F3), which repeats the measurement at eight block positions\. Restricting to a small block is what makes the2L2^\{L\}enumeration feasible, and the enumerated lattice is a reference against which sampled estimates, fitted models, and the noise corrections of App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)are certified\. The full\-network cells \(Section[3](https://arxiv.org/html/2607.12266#S3), the prediction benchmark, and the allocation studies\) instead quantize units anywhere in the network, where enumeration is impossible, and sample configurations from thepp\-biased measureμp\\mu\_\{p\}\. Eachf​\(S\)f\(S\)is averaged over stochastic\-rounding draws \(3 seeds for the headline sampled cells, std≤0\.03\\leq 0\.03; 8–16 draws for the exact lattices of Section[3](https://arxiv.org/html/2607.12266#S3), whose noise treatment is in App\.[A\.1](https://arxiv.org/html/2607.12266#A1.SS1)\)\. The Walsh/Möbius transforms are computed from measuredf​\(S\)f\(S\)\. The order\-≥2\\geq 2energy certificate is the measured∑k≥2Wk/Var⁡\(f\)\\sum\_\{k\\geq 2\}W\_\{k\}/\\operatorname\{Var\}\(f\)where a lattice is affordable and, at scale, the closed\-form forecast from the fitted break\-rates, namely the exact heterogeneous product form1−∑βi/\[∏\(1\+βi\)−1\]1\-\\sum\\beta\_\{i\}/\[\\prod\(1\+\\beta\_\{i\}\)\-1\]of Proposition 5 \(Section[4\.3](https://arxiv.org/html/2607.12266#S4.SS3)\), whose symmetric summary isτ\\tau\(share≈12​τ2\\approx\\tfrac\{1\}\{2\}\\tau^\{2\}\)\. The “τ\\tau\-forecast” label refers to this pipeline\. Where a measured share exists for comparison, the forecast understates it \(Qwen3\-30B: forecast3×10−43\{\\times\}10^\{\-4\}against measured 0\.013\)\. Even at that ratio the 235B and GLM\-4\.6 forecasts \(∼10−4\{\\sim\}10^\{\-4\}\) stay below 0\.01, so they certify additive dominance, not a precise share\.

#### Calibration and evaluation data\.

Prediction and interaction\-order studies use WikiText\-2 \(“wiki”\) and a Python code corpus, CodeSearchNet functions \(“code”\)\. The MoE allocation studies use a task\-aligned calibration set: BFCL\-v3 \(live\_simple, live\_multiple, simple\_python, multiple\) tool\-use prompts plus RULER validation prompts, tokenized to 16 sequences of 2,048 tokens\. The fake\-quant metric is the KL divergenceKL​\(bf16∥quantized\)\\mathrm\{KL\}\(\\text\{bf16\}\\,\\\|\\,\\text\{quantized\}\)of next\-token distributions on the disjoint eval half of this set\. Served downstream evaluation uses AIME\-2025, GPQA diamond, MMLU\-Pro, HumanEval, and gsm8k \(Section[7](https://arxiv.org/html/2607.12266#S7)\)\.

Table 8:Saturation measured directly: mean marginal damage by context size on the certification cells of Section[3\.1](https://arxiv.org/html/2607.12266#S3.SS1)\. Each entry averagesf​\(S∪\{i\}\)−f​\(S\)f\(S\\cup\\\{i\\\}\)\-f\(S\)over the layer\-context pairs with\|S\|\|S\|layers already quantized\. In every cell the marginal damage is largest at\|S\|=0\|S\|=0and falls once other layers are quantized concurrently\.
#### Predictor fitting\.

The coverage model\(c,\{ai\}\)\(c,\\\{a\_\{i\}\\\}\)and the additive model are fit by joint least squares rather than per\-coefficient sampling, for the variance and consistency reasons given in the next two paragraphs\. Fits useNNsampled configurations\. Fits are initialized by linearizing through the generator \(ordinary least squares on the log\-headroom values at a provisional ceiling\) and refined by bounded trust\-region least squares in loss space\. Fitting in loss space avoids the alternative of transforming the data and then regressing, which is biased because noise pushed through the nonlinear transform no longer averages to zero\. The surrogate baselines \(RF/MiCo, GP\-RBF/AutoQRA, RBF kernel regression/AMQ, CoopQ pairwise\) are fit on the same configurations, and GP/RF are additionally converted into allocators \(knapsack over their predicted marginals\) for the allocator\-to\-allocator comparison in Section[7](https://arxiv.org/html/2607.12266#S7)\. Aumann–Shapley scores are computed fit\-free by the path\-integral estimator: the loss gradient is integrated by a 4\-node midpoint rule along the path that scales weights and unit\-input activations linearly from full precision \(t=0t=0\) to fully quantized \(t=1t=1\), one integral per demotion level \(16→816\\to 8and16→416\\to 4\), with MoE routing frozen along the path since the router’s argmax is not differentiable\.

#### Why joint least squares rather than per\-coefficient sampling\.

The plug\-in estimate of each Walsh coefficient from samples,\(1/N\)​∑nf​\(Sn\)​χT​\(Sn\)\(1/N\)\\sum\_\{n\}f\(S\_\{n\}\)\\chi\_\{T\}\(S\_\{n\}\)\[Linialet al\.,[1993](https://arxiv.org/html/2607.12266#bib.bib41)\], is unbiased, but its variance carries the total second moment𝔼μp​\[f2\]\\mathbb\{E\}\_\{\\mu\_\{p\}\}\[f^\{2\}\], because every other coefficient’s energy leaks in through the sampled, non\-orthogonal design\. Jointly regressing on allL\+1L\+1degree\-≤1\\leq 1terms absorbs that leakage into the fitted terms\. Ordinary least squares is the classical control\-variate regression estimator, whose asymptotic variance is only the energy offforthogonal to the fitted span, that is, the order\-≥2\\geq 2energy plus noise, rather thanVar⁡\(f\)\\operatorname\{Var\}\(f\)\[Lavenberg and Welch,[1981](https://arxiv.org/html/2607.12266#bib.bib67), Glynn and Szechtman,[2002](https://arxiv.org/html/2607.12266#bib.bib68), Owen,[2013](https://arxiv.org/html/2607.12266#bib.bib69)\]\. Random\-design least squares is stable and near\-optimal already atN∝L​log⁡LN\\propto L\\log Lsamples\[Cohen and Migliorati,[2017](https://arxiv.org/html/2607.12266#bib.bib70)\]\. With∼90%\{\\sim\}90\\%of the variance at order 1 \(Section[3](https://arxiv.org/html/2607.12266#S3)\), regression gives roughly a10×10\\timesper\-coefficient variance reduction at equalNN, the reciprocal of the order\-≥2\\geq 2share, so the certificate also quantifies the gain from regression\. The same regression\-over\-sampling argument recurs in Shapley estimation \(KernelSHAP against permutation sampling,Covert and Lee,[2021](https://arxiv.org/html/2607.12266#bib.bib76)\)\.

#### Statistical footing of the fits\.

The coverage fit is ordinary nonlinear least squares: consistent for the best coverage\-family approximation offfeven when the model is misspecified\[Jennrich,[1969](https://arxiv.org/html/2607.12266#bib.bib71), White,[1981](https://arxiv.org/html/2607.12266#bib.bib72)\], and the maximum\-likelihood estimate under Gaussian noise when it is correct\. Two caveats: the objective is non\-convex, so we initialize by linearizing through the generator as described above, and on near\-linear data the ceiling is weakly identified \(only the productsc​aic\\,a\_\{i\}are pinned down\), which perturbs\(c,ai\)\(c,a\_\{i\}\)individually but not predictions\.

#### Measurement cost\.

Cost is counted in measured configurations, since each is a forward pass over the calibration set\. The headline budget isN≈LN\\approx Lconfigurations, against roughly100​L100Lprogressive\-quantization evaluations for SPQE’s permutation\-sampled Shapley\[Zhaoet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib23)\]\.

#### Prediction benchmark in full\.

Table[9](https://arxiv.org/html/2607.12266#A4.T9)reports the per\-cell results of the Section[6\.1](https://arxiv.org/html/2607.12266#S6.SS1)benchmark at the small budget, and Fig\.[7](https://arxiv.org/html/2607.12266#A4.F7)the error\-versus\-NNsweep on the 3B\-wiki cell\. The certificate column is the held\-out residual of an additive fit, the direct Theorem 1 estimate: 300 configurations are sampled per cell fromμ0\.6\\mu\_\{0\.6\}, an additive model is fit on the first 150, and the column reports the mean squared residual on the remaining 150 divided by the damage variance\.

Table 9:Prediction benchmark at the small\-NNbudget \(N=L/2N=L/2\): Median held\-out relative error at half theLLsingle\-layer measurements that isolate\-and\-sum requires\. Entries are single\-seed measurements over 120 held\-out configurations\. Table[16](https://arxiv.org/html/2607.12266#A5.T16)reports 3\-seed spreads on two W4A4 cells\. Isolate\-and\-sum reaches 0\.38–0\.63 on the FP4\-damaged models \(0\.6B, 4B, 30B\), against 0\.08–0\.19 for the better of coverage and additive on 0\.6B and 4B\. Two cells require comment\. On 8B the damage of the fully quantized model is small \(f​\(all\)≈0\.46f\(\\text\{all\}\)\\approx 0\.46\), all methods are approximately tied, and isolate\-and\-sum looks adequate because there is little to mis\-price\. The 30B MoE row is hard for every method \(best 0\.252\)\. The last column is the cell’s order\-≥2\\geq 2share underμ0\.6\\mu\_\{0\.6\}\. It is largest on the rows where every method’s error is worst \(4B\-wiki at 0\.21 and 30B at 0\.30\)\.![Refer to caption](https://arxiv.org/html/2607.12266v1/figures/fig3_predict_vs_N.png)Figure 7:Median held\-out relative error vs\. number of measured configurations on the 3B\-wiki cell of Table[9](https://arxiv.org/html/2607.12266#A4.T9)\. Coverage \(L\+1L\+1parameters\) sits at its error floor fromN=L/2N=L/2\(0\.093, tied with the random forest at 0\.094\), the additive fit sits at 0\.29–0\.32 forN≤LN\\leq Land ties with coverage fromN≈4​LN\\approx 4L, and the Gaussian process needs roughly4​L4Lconfigurations to match\. FromN≈LN\\approx Lthe random forest attains the lowest error on this cell, reaching 0\.03 atN≈8​LN\\approx 8Lagainst coverage’s plateau at 0\.08\. The large\-NNwinner varies by cell \(on 0\.6B\-wiki the Gaussian process leads and the forest trails every other method\)\. Isolate\-and\-sum is constant at 0\.11, since it has no fitted parameters, and its failures occur on the damaged cells of Table[9](https://arxiv.org/html/2607.12266#A4.T9)\. CoopQ overfits at largeNN\.Table 10:Qwen\-Image DiT \(Section[6\.3](https://arxiv.org/html/2607.12266#S6.SS3)\): median held\-out relative error vs\.NN,L=60L=60blocks at per\-block NVFP4\. The final column is the measured order\-≥2\\geq 2share\.
#### Certificate estimators\.

Table[11](https://arxiv.org/html/2607.12266#A4.T11)reports the per\-cell detail behind the estimator comparison of Section[4\.4](https://arxiv.org/html/2607.12266#S4.SS4): the root\-mean\-squared error, over 8 sampling seeds, of estimating a lattice’s true order\-≥2\\geq 2share fromNNconfigurations sampled atp=0\.6p=0\.6, for the Proposition 5\(i\) forecast from the coverage fit and the held\-out residual of an additive fit\. The residual estimator’s agreement with the forecast, available at no extra measurement cost, is a per\-instance check of the coverage assumption itself\.

Table 11:Estimating a cell’s order\-≥2\\geq 2share fromNNsampled configurations\. Each estimator receives onlyNNconfigurations drawn fromμp\\mu\_\{p\}atp=0\.6p=0\.6, and the draw is repeated with 8 independent seeds\. An entry is the root\-mean\-squared error of the 8 resulting estimates against the true share\. The two estimators are the Proposition 5\(i\) forecast computed from a coverage fit \(fcst\) and the held\-out residual of an additive fit \(resid\), both defined in Section[4\.4](https://arxiv.org/html/2607.12266#S4.SS4)\. The forecast is best at 14 of 15 cell\-budget pairs\.
#### Allocation\.

Given a memory budgeteff=∑numel⋅bits/∑numel\\mathrm\{eff\}=\\sum\\mathrm\{numel\}\\cdot\\mathrm\{bits\}/\\sum\\mathrm\{numel\}\(kernel\-independent, with numel a tensor’s element count\), a 3\-level\{16,8,4\}\\\{16,8,4\\\}knapsack minimizes predictedΔ​LCE\\Delta L\_\{\\mathrm\{CE\}\}per effective bit\. A unit is a weight tensor or a fused group that must share one format, such as the q/k/v projections or one MoE layer’s experts\. The served recipes additionally tie fusion groups \(q/k/v into oneqkv\_proj, one expert format per MoE layer block\) so the per\-layer\-mixed checkpoint loads in TensorRT\-LLM\. Baselines are matched to the sameeff\\mathrm\{eff\}\. The two fit\-free variants price demotions differently: AS\-add uses the path\-integral coefficients directly, so predicted damage adds across units, while AS\-cov converts the same coefficients to break\-rates, with the ceiling calibrated from one measured corner \(the all\-4\-bit configuration\), so damage composes through the coverage form\. The two structural rules use no damage measurements\. Naive\-uniform lowers every unit together through the\{16,8,4\}\\\{16,8,4\\\}ladder and, if the resulting uniform level falls below the target effective bits, restores the smallest tensors to 16 bits until the next restoration would exceed the budget\. Naive\-routed places routed experts at the lowest level and protects attention\.

Table 12:Served AIME\-2025 accuracy \(average of 8 sampled seeds\) of every Qwen3\-30B allocation at effective bits 4\.2–8\.0 over the\{16,8,4\}\\\{16,8,4\\\}levels, served by SGLang; bf16 anchor 73\.3\. GPQA diamond spans 59\.1–63\.1 and MMLU\-Pro 76\.2–78\.3 across the same grid, no method ordering is significant at any budget \(a 16\-seed replication of the AS\-additive–uniform gap at eff 4\.5 gives 73\.3 against 69\.8, about 1\.4 standard errors\)\.Table 13:Served accuracy of the Qwen3\-235B allocations at effective bits 4\.2–8\.0 over the\{16,8,4\}\\\{16,8,4\\\}levels \(allocations derived on the evaluation\-task corpus of Table[5](https://arxiv.org/html/2607.12266#S7.T5)\)\. Each cell gives AIME\-2025 \(average of 8 sampled seeds\) / GPQA diamond / MMLU\-Pro; bf16 baselines 80\.0/71\.7/82\.9\. Every measured cell is within seed noise of the anchor on all three benchmarks\.
#### GPTQ ladder details \(Section[7\.4](https://arxiv.org/html/2607.12266#S7.SS4)\)\.

Grids are symmetric with one scale per 128 input columns, and each linear layer’s Hessian is accumulated from the scoring half of the task calibration set\. The all\-units GPTQ\-4 calibration KL of 0\.054 is comparable to NVFP4 RTN at 0\.047, and at 2 bits RTN gives 7\.0 against GPTQ’s 13\.6, so 2 bits is unusable as a uniform level under either quantizer\. For the fused expert blocks, all experts in a layer share the Hessian of the block’s full token stream: the first projection’s Hessian is accumulated from the block input, and the second projection’s from intermediate activations recomputed on that input, a standard simplification that ignores the router’s token\-to\-expert assignment\. At the 235B scale the all\-units GPTQ\-3 calibration KL is 0\.115\.

#### Sub\-4 served detail \(Section[7\.4](https://arxiv.org/html/2607.12266#S7.SS4)\)\.

Table[14](https://arxiv.org/html/2607.12266#A4.T14)gives the full served grid on the reasoning benchmarks for every method and budget\. The reasoning benchmarks are AIME\-2025 \(30 problems\), GPQA diamond \(198 questions\), and MMLU\-Pro \(a fixed 1,000\-question subset\), decoded at temperature 0\.6, top\-pp0\.95, top\-kk20, with a 32,768\-token budget on AIME\-2025 and 16,384 elsewhere\. The ModelOpt configuration scores at the chance level of each multiple\-choice format \(23\.2 on four\-option GPQA, 9\.0 on ten\-option MMLU\-Pro\)\.

Table 14:Served accuracy of every sub\-4 Qwen3\-30B configuration on the reasoning benchmarks\. Each cell gives AIME\-2025 \(average of 8 sampled seeds\) / GPQA diamond / MMLU\-Pro pass rates for the method’s configuration at that effective\-bit budget All allocations are derived from damages measured on the evaluation\-task calibration corpus of Table[5](https://arxiv.org/html/2607.12266#S7.T5)\. bf16 baselines: 73\.3/61\.6/76\.2\.
#### Joint measurement of the allocation tables\.

CoopQ’s quadratic program is solved by greedy plus local search \(SCIP unavailable\)\. Its published∼100​L\{\\sim\}100Lbudget \(∼24\{\\sim\}24k evaluations\) was not run \(Section[7\.3](https://arxiv.org/html/2607.12266#S7.SS3)reports the 1,205\- and 942\-evaluation tiers\)\. An ablation isolates the cause of its deficit: the additive and coverage models fit on the 942 configurations CoopQ’s permutation walks visited, allocating on its alphabet, land at CoopQ’s level \(\+\.004\+\.004–\.008\.008, paired\-significant\), because configurations visited along a permutation walk are efficient for averaging marginal contributions but nearly collinear as regression data, while the same regression on 300 independently sampled configurations lands within\.002\.002of AS at every contended budget \(within\.001\.001at eff≥4\.5\\geq 4\.5\)\.

#### Hardware and software\.

Scoring and fake\-quant run on a single B200 \(device\-map pipeline\-parallel across8×8\\timesB200 for the 235B/355B/753B models\), and serving runs on1×1\\timesB200 \(30B, TP=1\) up to8×8\\timesB200 \(GLM, TP=8\)\. PyTorch and Hugging Face Transformers handle fake\-quant, NVIDIA ModelOpt and TensorRT\-LLM handle the≥4\\geq 4\-bit deployment recipes\. SGLang serves the sub\-4 benchmarks from bf16\-exported checkpoints and the GLM 5\.2 benchmarks \(Section[7\.4](https://arxiv.org/html/2607.12266#S7.SS4)\), vLLM serves the Qwen3\.6\-27B comparison \(Section[7\.2](https://arxiv.org/html/2607.12266#S7.SS2)\), and scikit\-learn provides the surrogate baselines\.

## Appendix EInterpretability of the fitted parameters

#### Budget pricing\.

Fitting coverage on∼200\{\\sim\}200configurations of Qwen3\-0\.6B \(all 28 layers, activations quantized at FP4 per\-block\-32, configurations drawn atp=0\.6p=0\.6\), we price unseen configurations of the form “protect themmmost fragile layers and quantize the rest”, comparing the predictedΔ​L\\Delta L, which costs no forward passes once the model is fit, against the measured value \(Table[15](https://arxiv.org/html/2607.12266#A5.T15)\)\. Across the deployment range, protecting a few to about half the layers, coverage prices the configuration to within 4%, whereas isolate\-and\-sum’s prediction is22–3×3\\timesthe truth\. The lone breakdown is quantizing a single layer, where the model, fit on configurations drawn atp=0\.6p=0\.6, extrapolates far outside its fit distribution and misses by∼95%\{\\sim\}95\\%\.

Table 15:Budget pricing on unseen protect\-mmconfigurations \(Qwen3\-0\.6B\)\.
#### The fragility scores are corpus\-portable\.

Fitting the break\-ratesaia\_\{i\}independently on wiki and on code yields strongly agreeing vectors \(Spearmanρ=0\.74\\rho=0\.74, Pearsonr=0\.86r=0\.86on Qwen3\-0\.6B;ρ=0\.78\\rho=0\.78,r=0\.93r=0\.93on Qwen2\.5\-3B\), with the same deep layers fragile in both corpora\. We propose to fitaia\_\{i\}once on a source corpus, and to deploy on a new corpus refit only the single scalar ceilingcc\. At small budgets this one\-parameter transfer attains lower error than the\(L\+1\)\(L\{\+\}1\)\-parameter in\-domain fit, 0\.058 against 0\.098 at 9 configurations for code\-to\-wiki on both 0\.6B and 3B, and 0\.076 against 0\.110 at 36 configurations for wiki\-to\-code on 3B\.

#### Multi\-seed robustness\.

Table[16](https://arxiv.org/html/2607.12266#A5.T16)reports median held\-out relative error over 3 seeds at theN≈LN\\approx Lbudget\. Coverage attains the lowest error on 0\.6B and ties the Gaussian process on 3B, isolate\-and\-sum fails uniformly across seeds, and 3B’s higher order\-≥2\\geq 2share \(0\.26 against 0\.6B’s 0\.05\) is exactly where coverage’s margin over the additive fit is largest, so the certificate forecasts where the saturation structure pays off before any predictor is fit\.

Table 16:Multi\-seed robustness at theN≈LN\\approx Lbudget, with the certificate per row\.

## Appendix FProofs \(Theorems 3 and 4, Proposition 5, the two anchors, kernel eigenvalues\)

#### Notation\.

Under the product measureμp\\mu\_\{p\}\(eachxi∼Bernoulli​\(p\)x\_\{i\}\\sim\\mathrm\{Bernoulli\}\(p\), independent\), the space of functions on\{0,1\}L\\\{0,1\\\}^\{L\}carries the inner product⟨g,h⟩=𝔼μp​\[g​h\]\\langle g,h\\rangle=\\mathbb\{E\}\_\{\\mu\_\{p\}\}\[gh\]\. With the standardized coordinatesϕi​\(x\):=\(xi−p\)/p​\(1−p\)\\phi\_\{i\}\(x\):=\(x\_\{i\}\-p\)/\\sqrt\{p\(1\-p\)\}\(𝔼​\[ϕi\]=0\\mathbb\{E\}\[\\phi\_\{i\}\]=0,𝔼​\[ϕi2\]=1\\mathbb\{E\}\[\\phi\_\{i\}^\{2\}\]=1\), the Walsh functionsχT:=∏i∈Tϕi\\chi\_\{T\}:=\\prod\_\{i\\in T\}\\phi\_\{i\}\(χ∅:=1\\chi\_\{\\emptyset\}:=1\) form an orthonormal basis: forT≠T′T\\neq T^\{\\prime\}, independence factorizes𝔼​\[χT​χT′\]\\mathbb\{E\}\[\\chi\_\{T\}\\chi\_\{T^\{\\prime\}\}\]per coordinate, and each coordinate inT​△​T′T\\triangle T^\{\\prime\}contributes𝔼​\[ϕi\]=0\\mathbb\{E\}\[\\phi\_\{i\}\]=0\. Hencef=∑Tf^​\(T\)​χTf=\\sum\_\{T\}\\hat\{f\}\(T\)\\chi\_\{T\}withf^​\(T\)=⟨f,χT⟩\\hat\{f\}\(T\)=\\langle f,\\chi\_\{T\}\\rangle,𝔼​\[f\]=f^​\(∅\)\\mathbb\{E\}\[f\]=\\hat\{f\}\(\\emptyset\), and \(Parseval\)Varp⁡\(f\)=∑T≠∅f^​\(T\)2\\operatorname\{Var\}\_\{p\}\(f\)=\\sum\_\{T\\neq\\emptyset\}\\hat\{f\}\(T\)^\{2\}\[O’Donnell,[2014](https://arxiv.org/html/2607.12266#bib.bib40)\]\.

#### Proof of Theorem 3 \(Möbius spectrum\)\.

ForT≠∅T\\neq\\emptyset,φ​\(T\)=∑R⊆T\(−1\)\|T\|−\|R\|​f​\(R\)\\varphi\(T\)=\\sum\_\{R\\subseteq T\}\(\-1\)^\{\|T\|\-\|R\|\}f\(R\)withf​\(R\)=c−c​∏i∈R\(1−ai\)f\(R\)=c\-c\\prod\_\{i\\in R\}\(1\-a\_\{i\}\)\. The constantccalternates to zero over the subsets of a nonemptyTT\. For the product part, the alternating subset\-sum of a product factorizes coordinatewise,

∑R⊆T\(−1\)\|T\|−\|R\|​∏i∈Rgi=∏i∈T\(gi−1\)\.\\sum\_\{R\\subseteq T\}\(\-1\)^\{\|T\|\-\|R\|\}\\prod\_\{i\\in R\}g\_\{i\}=\\prod\_\{i\\in T\}\(g\_\{i\}\-1\)\.Withgi=1−aig\_\{i\}=1\-a\_\{i\}this gives−c​∏i∈T\(−ai\)=c​\(−1\)\|T\|\+1​∏i∈Tai\-c\\prod\_\{i\\in T\}\(\-a\_\{i\}\)=c\\,\(\-1\)^\{\|T\|\+1\}\\prod\_\{i\\in T\}a\_\{i\}\. ∎

#### Proof of Theorem 4 \(Walsh spectrum\)\.

Writef=c−Hf=c\-Hwith the headroomH​\(x\)=c​∏igi​\(xi\)H\(x\)=c\\prod\_\{i\}g\_\{i\}\(x\_\{i\}\),gi​\(xi\)=1−ai​xig\_\{i\}\(x\_\{i\}\)=1\-a\_\{i\}x\_\{i\}\. Constants are orthogonal to everyχT\\chi\_\{T\}withT≠∅T\\neq\\emptyset, sof^​\(T\)=−H^​\(T\)\\hat\{f\}\(T\)=\-\\hat\{H\}\(T\)forT≠∅T\\neq\\emptyset, whilef^​\(∅\)=𝔼​\[f\]=c−H^​\(∅\)=c​\(1−∏i\(1−ai​p\)\)\\hat\{f\}\(\\emptyset\)=\\mathbb\{E\}\[f\]=c\-\\hat\{H\}\(\\emptyset\)=c\\bigl\(1\-\\prod\_\{i\}\(1\-a\_\{i\}p\)\\bigr\)\. BecauseHHandχT\\chi\_\{T\}are both products over*independent*coordinates, the expectation factorizes:H^​\(T\)=c​∏i∈T𝔼​\[gi​ϕi\]⋅∏i∉T𝔼​\[gi\]\\hat\{H\}\(T\)=c\\prod\_\{i\\in T\}\\mathbb\{E\}\[g\_\{i\}\\phi\_\{i\}\]\\cdot\\prod\_\{i\\notin T\}\\mathbb\{E\}\[g\_\{i\}\]\. The one\-coordinate integrals \(usingxi2=xix\_\{i\}^\{2\}=x\_\{i\}\):𝔼​\[gi\]=1−ai​p\\mathbb\{E\}\[g\_\{i\}\]=1\-a\_\{i\}pand𝔼​\[gi​ϕi\]=−ai​𝔼​\[xi​\(xi−p\)\]/p​\(1−p\)=−ai​p​\(1−p\)\\mathbb\{E\}\[g\_\{i\}\\phi\_\{i\}\]=\-a\_\{i\}\\,\\mathbb\{E\}\[x\_\{i\}\(x\_\{i\}\-p\)\]/\\sqrt\{p\(1\-p\)\}=\-a\_\{i\}\\sqrt\{p\(1\-p\)\}\. Substituting and negating givesf^​\(T\)=c​\(−1\)\|T\|\+1​\(p​\(1−p\)\)\|T\|/2​∏i∈Tai⋅∏i∉T\(1−ai​p\)\\hat\{f\}\(T\)=c\\,\(\-1\)^\{\|T\|\+1\}\(p\(1\-p\)\)^\{\|T\|/2\}\\prod\_\{i\\in T\}a\_\{i\}\\cdot\\prod\_\{i\\notin T\}\(1\-a\_\{i\}p\)\. ∎

###### Proof of Proposition 5\(ii\) and \(iii\)\.

Notation as in Section[4\.3](https://arxiv.org/html/2607.12266#S4.SS3), withB:=τ2=∑iβiB:=\\tau^\{2\}=\\sum\_\{i\}\\beta\_\{i\}\. By Proposition 5\(i\),share≥2=S2/S1\\mathrm\{share\}\_\{\\geq 2\}=S\_\{2\}/S\_\{1\}withS1:=∑k≥1ek=∏i\(1\+βi\)−1S\_\{1\}:=\\sum\_\{k\\geq 1\}e\_\{k\}=\\prod\_\{i\}\(1\+\\beta\_\{i\}\)\-1andS2:=∑k≥2ekS\_\{2\}:=\\sum\_\{k\\geq 2\}e\_\{k\}\.

*\(ii\)*The leading term is exact algebra:\(∑βi\)2=∑βi2\+2​e2\(\\sum\\beta\_\{i\}\)^\{2\}=\\sum\\beta\_\{i\}^\{2\}\+2e\_\{2\}, soe2=12​\(B2−∑βi2\)=12​B2​\(1−1/Leff\)e\_\{2\}=\\tfrac\{1\}\{2\}\(B^\{2\}\-\\sum\\beta\_\{i\}^\{2\}\)=\\tfrac\{1\}\{2\}B^\{2\}\(1\-1/L\_\{\\mathrm\{eff\}\}\), hencee2/e1=12​B​\(1−1/Leff\)e\_\{2\}/e\_\{1\}=\\tfrac\{1\}\{2\}B\(1\-1/L\_\{\\mathrm\{eff\}\}\)\. For the error we usek\!​ek≤Bkk\!\\,e\_\{k\}\\leq B^\{k\}\(expandBk=\(∑β\)kB^\{k\}=\(\\sum\\beta\)^\{k\}: thek\!​ekk\!\\,e\_\{k\}injective monomials are a subset of its nonnegative terms\), whenceS2−e2=∑k≥3ek≤∑k≥3Bk/k\!≤\(B3/6\)​eBS\_\{2\}\-e\_\{2\}=\\sum\_\{k\\geq 3\}e\_\{k\}\\leq\\sum\_\{k\\geq 3\}B^\{k\}/k\!\\leq\(B^\{3\}/6\)e^\{B\}\(termwise,3\!​\(k−3\)\!≤k\!3\!\(k\-3\)\!\\leq k\!\) andS1≤eB−1S\_\{1\}\\leq e^\{B\}\-1\.*Upper:*share≥2≤S2/e1≤e2/B\+\(B2/6\)​eB\\mathrm\{share\}\_\{\\geq 2\}\\leq S\_\{2\}/e\_\{1\}\\leq e\_\{2\}/B\+\(B^\{2\}/6\)e^\{B\}\.*Lower:*share≥2≥e2/S1≥e2/\(eB−1\)=\(e2/B\)⋅B/\(eB−1\)≥\(e2/B\)​\(1−B/2\)≥e2/B−B2/4\\mathrm\{share\}\_\{\\geq 2\}\\geq e\_\{2\}/S\_\{1\}\\geq e\_\{2\}/\(e^\{B\}\-1\)=\(e\_\{2\}/B\)\\cdot B/\(e^\{B\}\-1\)\\geq\(e\_\{2\}/B\)\(1\-B/2\)\\geq e\_\{2\}/B\-B^\{2\}/4, using the elementary inequalityx/\(ex−1\)≥1−x/2x/\(e^\{x\}\-1\)\\geq 1\-x/2forx≥0x\\geq 0\(the power series ofx−\(ex−1\)​\(1−x/2\)x\-\(e^\{x\}\-1\)\(1\-x/2\)is∑k≥3k−22​k\!​xk\\sum\_\{k\\geq 3\}\\frac\{k\-2\}\{2\\,k\!\}x^\{k\}, with nonnegative coefficients\) ande2/B≤B/2e\_\{2\}/B\\leq B/2\. Both one\-sided deviations are≤\(B2/4\)​eB\\leq\(B^\{2\}/4\)e^\{B\}, which is the stated remainder bound\. The profile’s third moment \(viae3e\_\{3\}\) first appears atO​\(τ4\)O\(\\tau^\{4\}\)\.

*\(iii\)*At fixedBB,share≥2=1−B/\(P−1\)\\mathrm\{share\}\_\{\\geq 2\}=1\-B/\(P\-1\)is strictly increasing inP:=∏i\(1\+βi\)P:=\\prod\_\{i\}\(1\+\\beta\_\{i\}\), andlog⁡P=∑ilog⁡\(1\+βi\)\\log P=\\sum\_\{i\}\\log\(1\+\\beta\_\{i\}\)is a symmetric sum of one concave scalar function over the coordinates, the canonical Schur\-concave form\[Marshallet al\.,[2011](https://arxiv.org/html/2607.12266#bib.bib47)\], soshare≥2\\mathrm\{share\}\_\{\\geq 2\}is Schur\-concave at fixedBB\. The uniform profileβi≡B/L\\beta\_\{i\}\\equiv B/Lis majorized by every profile of totalBB, hence is the maximizer \(equivalently, Jensen on the strictly concavelog\(1\+⋅\)\\log\(1\+\\cdot\)giveslog⁡P≤L​log⁡\(1\+B/L\)\\log P\\leq L\\log\(1\+B/L\), with equality iff allβi\\beta\_\{i\}are equal\)\. The outer bound follows sinceL↦\(1\+B/L\)LL\\mapsto\(1\+B/L\)^\{L\}increases toeBe^\{B\}\(AM–GM onLLcopies of1\+B/L1\+B/Land one copy of11\)\. AsLeff→1L\_\{\\mathrm\{eff\}\}\\to 1the share vanishes, since a one\-fragile\-layerf≈c​ai​xif\\approx c\\,a\_\{i\}\\,x\_\{i\}is a one\-coordinate, hence exactly degree\-1, function\. ∎

#### Corollary \(the two anchors of Section[5](https://arxiv.org/html/2607.12266#S5)\)\.

By Theorem 1 the optimal degree\-≤1\\leq 1predictor is the projectionf^​\(∅\)\+∑if^​\(\{i\}\)​χ\{i\}\\hat\{f\}\(\\emptyset\)\+\\sum\_\{i\}\\hat\{f\}\(\\\{i\\\}\)\\chi\_\{\\\{i\\\}\}\. Convertingχ\{i\}\\chi\_\{\\\{i\\\}\}back to the\{0,1\}\\\{0,1\\\}coordinate gives the optimal additive coefficientwi=f^​\(\{i\}\)/p​\(1−p\)=𝔼p​\[f∣xi=1\]−𝔼p​\[f∣xi=0\]w\_\{i\}=\\hat\{f\}\(\\\{i\\\}\)/\\sqrt\{p\(1\-p\)\}=\\mathbb\{E\}\_\{p\}\[f\\mid x\_\{i\}=1\]\-\\mathbb\{E\}\_\{p\}\[f\\mid x\_\{i\}=0\], the in\-context marginal, which under coverage \(Theorem 4 withT=\{i\}T=\\\{i\\\}\) iswi=c​ai​∏j≠i\(1−aj​p\)w\_\{i\}=c\\,a\_\{i\}\\prod\_\{j\\neq i\}\(1\-a\_\{j\}p\)\. The empty\-anchor slope issi=f​\(\{i\}\)=c​ais\_\{i\}=f\(\\\{i\\\}\)=c\\,a\_\{i\}directly from the coverage model\. Hence the inflation factorsi/wi=1/∏j≠i\(1−aj​p\)≈ep​∑jajs\_\{i\}/w\_\{i\}=1/\\prod\_\{j\\neq i\}\(1\-a\_\{j\}p\)\\approx e^\{p\\sum\_\{j\}a\_\{j\}\}quoted in Section[5](https://arxiv.org/html/2607.12266#S5)\. ∎

#### The symmetric case\.

For completeness, the equality case of Proposition 5\(iii\) with its order spectrum\.

###### Corollary\(symmetric saturation law\)\.

Ifai≡aa\_\{i\}\\equiv a, thenβi≡β=p​\(1−p\)​a2/\(1−a​p\)2\\beta\_\{i\}\\equiv\\beta=p\(1\-p\)\\,a^\{2\}/\(1\-ap\)^\{2\}andτ2=L​β\\tau^\{2\}=L\\beta, the order spectrum is binomial,

Wk=c2​\(Lk\)​\(p​\(1−p\)\)k​a2​k​\(1−a​p\)2​\(L−k\),W\_\{k\}=c^\{2\}\\binom\{L\}\{k\}\\,\(p\(1\-p\)\)^\{k\}a^\{2k\}\(1\-ap\)^\{2\(L\-k\)\},and the certificate collapses to the one\-parameter law

share≥2=1−τ2\(1\+τ2/L\)L−1→τ→0L−12​L​τ2≈12​τ2\.\\mathrm\{share\}\_\{\\geq 2\}=1\-\\frac\{\\tau^\{2\}\}\{\(1\+\\tau^\{2\}/L\)^\{L\}\-1\}\\;\\xrightarrow\{\\;\\tau\\to 0\\;\}\\;\\frac\{L\-1\}\{2L\}\\,\\tau^\{2\}\\;\\approx\\;\\tfrac\{1\}\{2\}\\tau^\{2\}\.

###### Proof\.

With equal break\-rates every size\-kksubset contributes the same product, soek=\(Lk\)​βke\_\{k\}=\\binom\{L\}\{k\}\\beta^\{k\}and∏i\(1\+βi\)=\(1\+β\)L\\prod\_\{i\}\(1\+\\beta\_\{i\}\)=\(1\+\\beta\)^\{L\}, and Proposition 5\(i\) reduces to the stated law, which is also the equality case of Proposition 5\(iii\)\. TheWkW\_\{k\}formula is Theorem 4 with equalaia\_\{i\}times the\(Lk\)\\binom\{L\}\{k\}count of size\-kksets, and the small\-τ\\taulimit is Proposition 5\(ii\) withLeff=LL\_\{\\mathrm\{eff\}\}=L\. ∎

#### Walsh eigenvalues of the RBF kernel \(Section[6\.2](https://arxiv.org/html/2607.12266#S6.SS2)\)\.

On binary vectors the squared Euclidean distance collapses to the size of the symmetric difference,∥x−x′∥2=∑i\(xi−xi′\)2=\|x⊕x′\|\\lVert x\-x^\{\\prime\}\\rVert^\{2\}=\\sum\_\{i\}\(x\_\{i\}\-x\_\{i\}^\{\\prime\}\)^\{2\}=\|x\\oplus x^\{\\prime\}\|, so the RBF kernel is a function of the XOR alone,K​\(x,x′\)=κ​\(x⊕x′\)K\(x,x^\{\\prime\}\)=\\kappa\(x\\oplus x^\{\\prime\}\)withκ​\(u\)=e−γ​∑iui\\kappa\(u\)=e^\{\-\\gamma\\sum\_\{i\}u\_\{i\}\}\. For any such kernel, the operator\(TK​f\)​\(x\)=𝔼x′​\[K​\(x,x′\)​f​\(x′\)\]\(T\_\{K\}f\)\(x\)=\\mathbb\{E\}\_\{x^\{\\prime\}\}\[K\(x,x^\{\\prime\}\)f\(x^\{\\prime\}\)\]under the uniform measure \(p=12p=\\tfrac\{1\}\{2\}\) is diagonal in the Walsh basis\. Fixxxand reparametrize the average overx′x^\{\\prime\}byu:=x⊕x′u:=x\\oplus x^\{\\prime\}\. Hence

\(TK​χT\)​\(x\)=𝔼x′​\[κ​\(x⊕x′\)​χT​\(x′\)\]=𝔼u​\[κ​\(u\)​χT​\(x⊕u\)\]\.\(T\_\{K\}\\chi\_\{T\}\)\(x\)=\\mathbb\{E\}\_\{x^\{\\prime\}\}\\bigl\[\\kappa\(x\\oplus x^\{\\prime\}\)\\,\\chi\_\{T\}\(x^\{\\prime\}\)\\bigr\]=\\mathbb\{E\}\_\{u\}\\bigl\[\\kappa\(u\)\\,\\chi\_\{T\}\(x\\oplus u\)\\bigr\]\.The character identityχT​\(x⊕u\)=χT​\(x\)​\(−1\)∑i∈Tui\\chi\_\{T\}\(x\\oplus u\)=\\chi\_\{T\}\(x\)\\,\(\-1\)^\{\\sum\_\{i\\in T\}u\_\{i\}\}then givesTK​χT=λT​χTT\_\{K\}\\chi\_\{T\}=\\lambda\_\{T\}\\,\\chi\_\{T\}withλT=𝔼u​\[κ​\(u\)​\(−1\)∑i∈Tui\]\\lambda\_\{T\}=\\mathbb\{E\}\_\{u\}\\bigl\[\\kappa\(u\)\\,\(\-1\)^\{\\sum\_\{i\\in T\}u\_\{i\}\}\\bigr\]\. This expectation factorizes over the independent uniform bits,

λT=∏i∈T𝔼\[e−γ​ui\(−1\)ui\]∏i∉T𝔼\[e−γ​ui\]=\(1−e−γ2\)\|T\|\(1\+e−γ2\)L−\|T\|=\(1\+e−γ2\)Ltanh\(γ/2\)\|T\|\.\\lambda\_\{T\}\\;=\\;\\prod\_\{i\\in T\}\\mathbb\{E\}\\bigl\[e^\{\-\\gamma u\_\{i\}\}\(\-1\)^\{u\_\{i\}\}\\bigr\]\\prod\_\{i\\notin T\}\\mathbb\{E\}\\bigl\[e^\{\-\\gamma u\_\{i\}\}\\bigr\]\\;=\\;\\Bigl\(\\tfrac\{1\-e^\{\-\\gamma\}\}\{2\}\\Bigr\)^\{\|T\|\}\\Bigl\(\\tfrac\{1\+e^\{\-\\gamma\}\}\{2\}\\Bigr\)^\{L\-\|T\|\}\\;=\\;\\Bigl\(\\tfrac\{1\+e^\{\-\\gamma\}\}\{2\}\\Bigr\)^\{L\}\\,\\tanh\(\\gamma/2\)^\{\|T\|\}\.The eigenvalues are positive, depend onTTonly through\|T\|\|T\|, and decay geometrically in\|T\|\|T\|with ratiotanh⁡\(γ/2\)<1\\tanh\(\\gamma/2\)<1, which is the high\-order shrinkage quoted in Section[6\.2](https://arxiv.org/html/2607.12266#S6.SS2)\. Atp≠12p\\neq\\tfrac\{1\}\{2\}the kernel is still a function ofx⊕x′x\\oplus x^\{\\prime\}, but theμp\\mu\_\{p\}\-Walsh basis is no longer exactly its eigenbasis, which is why Section[6\.2](https://arxiv.org/html/2607.12266#S6.SS2)states the diagonalization as approximate atp=0\.6p=0\.6\.

## Appendix GThe assembled evidence for the coverage model

Table[17](https://arxiv.org/html/2607.12266#A7.T17)collects the evidence for the coverage model in one place, with each claim’s standing stated explicitly, since the support is deliberately heterogeneous: a representation theorem, certified measurements, a confirmed prediction, and a set of benchmark results\. Detailed treatments are at the listed locations\.

Table 17:The evidence for the coverage model, with each claim’s standing\. Proven means a theorem \(numerically verified where marked\), measured means a lattice or benchmark result, and a confirmed prediction is a property of the data that the fit did not use\.
## Appendix HExtended related work

#### MoE\-specific mixed precision\.

For mixture\-of\-experts models, MC\-MoE\[Huanget al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib61)\]solves a linear program over expert routing and frequency importance\. It allocates by an additive importance score and does not analyze the order of the damage or report a validity certificate\. The head\-to\-head comparisons against CoopQ and MC\-MoE appear in Section[7](https://arxiv.org/html/2607.12266#S7)\.

#### Path\-integral sensitivity\.

Integrated Gradients\[Sundararajanet al\.,[2017](https://arxiv.org/html/2607.12266#bib.bib17)\]and the Aumann–Shapley value\[Aumann and Shapley,[1974](https://arxiv.org/html/2607.12266#bib.bib42)\]are the classical methods\. PQI/ReQuant\[Huet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib18)\]applies the path integral to weight\-level sensitivity but assumes per\-weight effects sum, and QIG\[Xianget al\.,[2026](https://arxiv.org/html/2607.12266#bib.bib19)\]is token\-level for vision\-language models\. We apply the path integral to the layer\-selection set function \(App\.[B](https://arxiv.org/html/2607.12266#A2)\)\.

#### Rotation methods\.

QuaRot, SpinQuant, and QuIP\#\[Ashkbooset al\.,[2024](https://arxiv.org/html/2607.12266#bib.bib15), Liuet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib16), Tsenget al\.,[2024](https://arxiv.org/html/2607.12266#bib.bib14)\]flatten the representation before quantizing rather than allocating precision\. They alter the base quantizer on which any allocator operates, so they compose with allocation \(Section[9](https://arxiv.org/html/2607.12266#S9)\)\. A rotation that flattens per\-layer heterogeneity also removes the per\-layer differences that allocation exploits\.

#### Low\-order surrogates of subset functions outside quantization\.

The analytical approach has precedents in several adjacent literatures: functional ANOVA over configuration spaces\[Hutteret al\.,[2014](https://arxiv.org/html/2607.12266#bib.bib50)\]; Datamodels\[Ilyaset al\.,[2022](https://arxiv.org/html/2607.12266#bib.bib51)\], whose finding that linear surrogates of training\-data\-subset effects work well is the data\-subset analog of our first\-order finding; SPEX\[Kanget al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib52)\], sparse Boolean\-Fourier surrogates of LLM outputs over input\-masking subsets; the Harsanyi\-interaction line\[Renet al\.,[2023](https://arxiv.org/html/2607.12266#bib.bib53)\], which applies Möbius machinery to input coalitions; and Walsh surrogates for pseudo\-Boolean black\-box optimization\[Leprêtreet al\.,[2019](https://arxiv.org/html/2607.12266#bib.bib54), Baptista and Poloczek,[2018](https://arxiv.org/html/2607.12266#bib.bib55)\]\.

#### In\-context and isolated measurements\.

That an effect measured in isolation differs from the effect in context is known qualitatively from coalition\-value pruning\[Anconaet al\.,[2020](https://arxiv.org/html/2607.12266#bib.bib56)\], the joint\-versus\-sum analysis of Combinatorial Brain Surgeon for weight pruning\[Yuet al\.,[2022](https://arxiv.org/html/2607.12266#bib.bib59)\], the Hydra\-effect self\-repair mechanism\[McGrathet al\.,[2023](https://arxiv.org/html/2607.12266#bib.bib57)\], the permutation\-sampled marginals of CoopQ’s SPQE estimator\[Zhaoet al\.,[2025](https://arxiv.org/html/2607.12266#bib.bib23)\], and the layer\-subset surrogates now used for LLM pruning\[Dinget al\.,[2026](https://arxiv.org/html/2607.12266#bib.bib60)\], all of which work around the same anchor problem\. What we contribute is the application to mixed\-precision quantization, specifically, the closed\-form inflation factor under saturation, its measured magnitude on modern LLMs, and a fit\-free estimator of the correct coefficient \(Section[6](https://arxiv.org/html/2607.12266#S6)\), one Aumann–Shapley path integral in place of the SPQE estimator’s∼100​L\{\\sim\}100Lquantized evaluations\.