Signed Symmetric Quantization for Few-Bit Integers

# Signed Symmetric Quantization for Few-Bit Integers
Source: [https://arxiv.org/html/2607.08779](https://arxiv.org/html/2607.08779)
Ian Colbert, Eashan Dash11footnotemark:1, Pablo Monteagudo\-Lago, Juan Amboage, Srinidhi N, Giuseppe Franco, Nicholas J\. Fraser, Arun Ramachandran AMD

###### Abstract

The signed integer alphabet contains one more negative representable value than positive\. Yet, by convention, the standard symmetric integer quantizer fixes its scale to be strictly positive, which assigns this extra representable value to the negative tail and can force clipping of positive outliers\. In this work, we show that, at few\-bit precision, such clipping is a non\-trivial source of quantization error\. Asymmetric quantization addresses this problem with a zero point, shifting the grid toward the observed data range; however, this flexibility is well\-known to carry a runtime penalty\. For example, inllama\.cppon an AMD EPYC™ “Turin” CPU, a 4\-bit symmetric format uses up to9%9\\%less memory with up to2\.45×2\.45\\timeshigher throughput than its asymmetric counterpart\. We highlight signed symmetric quantization as a third option that retains the runtime profile of symmetric quantization without the penalty of the asymmetric format: our signed absmax grid places the extra representable value on the dominant\-outlier tail through a principled and lightweight sign selection rule while keeping the zero point at zero\. Our theoretical analysis offers two main results\. First, we establish the signed absmax grid as conditionally bound\-optimal onℓ2\\ell\_\{2\}quantization error, and show that the condition holds for8888–99%99\\%of weight groups across pre\-trained large language models \(LLMs\) at low bit widths\. Second, we show that negating the scale of a standard symmetric quantizer is analytically equivalent to a unit zero point shift on the same signed integer alphabet\. We empirically validate our proposal on models from the Qwen3, Qwen3\.5, and Llama3 families, and observe improvement in perplexity and downstream few\-shot accuracy over the standard unsigned symmetric quantizer at no extra inference cost\.

## 1Introduction

Few\-bit quantization relies heavily on uniform integer grids uniquely defined by an alphabet, a scale factor, and \(optionally\) a zero point\. A small but often overlooked detail is that the standard signedqq\-bit alphabet𝒜q=\{−2q−1,…,2q−1−1\}\\mathcal\{A\}\_\{q\}=\\\{\-2^\{q\-1\},\\ldots,2^\{q\-1\}\-1\\\}is not symmetric: it contains one more negative value than positive\. Yet, by convention, the standard symmetric quantizer takes the scalessto be positiveGerganov \([2023](https://arxiv.org/html/2607.08779#bib.bib47)\); Gholamiet al\.\([2022](https://arxiv.org/html/2607.08779#bib.bib2)\); Xilinx \([2018](https://arxiv.org/html/2607.08779#bib.bib45)\), assigning the extra representable value to the negative tail of the data and thereby possibly clipping large positive values\. We show in this work that this can become a material source of quantization error at few\-bit precision\.

Asymmetric quantization introduces a zero point to align the grid with the data range, but this flexibility is well\-known to carry an inference costGholamiet al\.\([2022](https://arxiv.org/html/2607.08779#bib.bib2)\); Jacobet al\.\([2018](https://arxiv.org/html/2607.08779#bib.bib8)\); Krishnamoorthi \([2018](https://arxiv.org/html/2607.08779#bib.bib49)\)\. Kernels must account for the offset, and the corresponding metadata must be stored and loaded\. For example, on Llama3 8B, we find that a symmetric 4\-bit format uses 9% less memory and provides2\.21×2\.21\\timeshigher prefill and2\.03×2\.03\\timeshigher decode throughput than its asymmetric counterpart as measured on an AMD EPYC™ “Turin” CPU \(Table[1](https://arxiv.org/html/2607.08779#S3.T1)\)\. This motivates a narrower question: can we recover the useful endpoint alignment of an asymmetric grid while keeping the inference path operationally symmetric?

We answer this question with signed symmetric quantization, where we choose the sign of the scale to align the dominant outlier in the data with the extra negative representable value in the signed integer alphabet\. The resulting sign rule is closed\-form, data\-free, and metadata\-free when scales are already stored as signed real values, as inllama\.cpp\. Our analyses show that signed symmetric quantization recovers a restricted asymmetric alignment while preserving the symmetric deployment path\. In particular, on Llama3 8B at 2 bits, it reduces WikiText2 perplexity from103\.1103\.1to17\.817\.8and improves few\-shot accuracy recovery by7\.97\.9percentage points \(Table[3](https://arxiv.org/html/2607.08779#S5.T3)\), while retaining1\.24×1\.24\\timesprefill,1\.08×1\.08\\timesdecode, and14%14\\%memory advantages over asymmetric quantization \(Table[1](https://arxiv.org/html/2607.08779#S3.T1)\)\.

#### Contributions\.

Our contributions are four\-fold\.First,we define signed symmetric quantization and introduce a principled, lightweight sign selection rule that preserves the dominant outlier \(Section[3](https://arxiv.org/html/2607.08779#S3)\)\.Second,we prove a worst\-caseℓ2\\ell\_\{2\}error bound that yields conditional bound\-optimality of our proposed sign selection rule \(Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)and Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)\)\.Third,we prove that a scale sign flip is analytically equivalent to a unit zero\-point shift on the same signed alphabet \(Theorem[4\.6](https://arxiv.org/html/2607.08779#S4.Thmtheorem6)\)\.Fourth,we evaluate models in the Qwen3, Qwen3\.5, and Llama3 families, showing that signed symmetric quantization improves over the conventional strictly positive symmetric grid, especially at low bit widths \(Tables[2](https://arxiv.org/html/2607.08779#S5.T2),[3](https://arxiv.org/html/2607.08779#S5.T3), and[4](https://arxiv.org/html/2607.08779#S5.T4)\), while preserving the symmetric deployment profile \(Table[1](https://arxiv.org/html/2607.08779#S3.T1)\)\.

#### Notation\.

Letw∈ℝdw\\in\\mathbb\{R\}^\{d\}denote add\-dimensional weight vector, with elementswj∈ℝw\_\{j\}\\in\\mathbb\{R\}forj∈\[d\]j\\in\[d\], where\[d\]=\{1,…,d\}\[d\]=\\\{1,\\ldots,d\\\}\. A quantizer𝒬\\mathcal\{Q\}maps real values to a discreteqq\-bit alphabet𝒜q⊂s​ℤ\\mathcal\{A\}\_\{q\}\\subset s\\mathbb\{Z\}with a step size \(or resolution\) ofs∈ℝs\\in\\mathbb\{R\}, where\|𝒜q\|=2q\|\\mathcal\{A\}\_\{q\}\|=2^\{q\}\. In particular, the standard integer quantizer is

𝒬\(w\)=s⋅\(clip\(⌊ws\+z⌉;𝒜q\)−z\),\\mathcal\{Q\}\(w\)=s\\cdot\\left\(\\text\{clip\}\\Big\(\\left\\lfloor\\frac\{w\}\{s\}\+z\\right\\rceil;\\mathcal\{A\}\_\{q\}\\Big\)\-z\\right\),\(1\)wheressis the non\-zero scaling factor,z∈ℝz\\in\\mathbb\{R\}is the zero point,⌊⋅⌉\\lfloor\\cdot\\rceilis the round\-to\-nearest \(RTN\) operator, andclip​\(x;𝒜q\):=max⁡\(min⁡\(x,max⁡𝒜q\),min⁡𝒜q\)\\text\{clip\}\(x;\\mathcal\{A\}\_\{q\}\):=\\max\(\\min\(x,\\max\\mathcal\{A\}\_\{q\}\),\\min\\mathcal\{A\}\_\{q\}\)\. Different integer quantization schemes \(e\.g\., asymmetric or symmetric\) are distinguished by hows,z,s,z,and𝒜q\\mathcal\{A\}\_\{q\}are chosen, and we refer to unique selections ofs,z,s,z,and𝒜q\\mathcal\{A\}\_\{q\}as a grid\.

## 2Background and Related Work

Quantization seeks to map a full\-precision tensor onto a discrete, low\-precision grid while minimizing reconstruction error, often in the form‖w−𝒬​\(w~\)‖22\\\|w\-\\mathcal\{Q\}\(\\tilde\{w\}\)\\\|\_\{2\}^\{2\}\. Prior work attacks this objective along two complementary axes: \(i\) for a fixed grid, search for the assignmentw~\\tilde\{w\}that minimizes the error; \(ii\) for a fixedw~\\tilde\{w\}, search for the grid that minimizes the error\. The two are not mutually exclusive, as we demonstrate empirically in Section[5](https://arxiv.org/html/2607.08779#S5)\.

#### Optimizingw~\\tilde\{w\}for a fixed grid\.

Frantar et al\.Frantaret al\.\([2023](https://arxiv.org/html/2607.08779#bib.bib11)\)introduce GPTQ, a greedy sequential rounding scheme that uses approximate second\-order information from calibration data to update weights on a fixed grid, correcting for local quantization error\. More recently, Zhang et al\.Zhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\)introduce Qronos, which generalizes this idea to also account for quantization error propagated from previously quantized layers\. A complementary line of work transforms the weight tensor before rounding: Xiao et al\.Xiaoet al\.\([2023](https://arxiv.org/html/2607.08779#bib.bib13)\)rescale channels to migrate quantization difficulty from activations to weights, Ashkboos et al\.Ashkbooset al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib16)\)apply Hadamard rotations to suppress outliers, and Liu et al\.Liuet al\.\([2025](https://arxiv.org/html/2607.08779#bib.bib17)\)extend this with learned rotations\. This family of algorithms is not the focus of our work, but we show that signed symmetric quantization composes constructively with it \(Section[5](https://arxiv.org/html/2607.08779#S5)\)\.

#### Optimizing the grid for a fixedw~\\tilde\{w\}\.

Esser et al\.Esseret al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib48)\)learn scaling factors via gradient descent, extended by Bhalgat et al\.Bhalgatet al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib46)\)to learnable zero points\. In LLM quantization, however, the scale is typically derived from data range using heuristicsGholamiet al\.\([2022](https://arxiv.org/html/2607.08779#bib.bib2)\); Krishnamoorthi \([2018](https://arxiv.org/html/2607.08779#bib.bib49)\); Zhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\), or by greedily minimizing local quantization errorFrantaret al\.\([2023](https://arxiv.org/html/2607.08779#bib.bib11)\)\. Recent works instead optimize the scale to minimize local output error without explicitly constraining its sign, yet in practice restrict evaluation to positive solutionsAmboageet al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib57)\); Zhang and Saab \([2026](https://arxiv.org/html/2607.08779#bib.bib53)\)\. In any case, the scale is assumed \(implicitly or explicitly\) to be strictly positive; this convention is widespread but not a theoretical requirement\. While some implementations differ on whether to use the full representation range of the formatXilinx \([2018](https://arxiv.org/html/2607.08779#bib.bib45)\)or restrict to a symmetric subset \(i\.e\., narrow range\)Gerganov \([2023](https://arxiv.org/html/2607.08779#bib.bib47)\); Xilinx \([2018](https://arxiv.org/html/2607.08779#bib.bib45)\), this choice determines*whether*the extra negative level is used, not*which tail*receives it\. To our knowledge, no prior work treats the sign of the scale factor as an explicit degree of freedom\. Our contribution therefore sits in this second strategy: we provide the first theoretical analysis and systematic evaluation of signed scale factors for few\-bit integer quantization\.

## 3Signed Symmetric Quantization

Signed symmetric quantization exploits the native asymmetry of the signedqq\-bit alphabet: it has one more negative value than positive\. Standard symmetric quantization fixes the scale to be positive by convention, which always assigns this extra representable value to the negative tail, possibly clipping large positive values\. This is often negligible at higher precision, but this clipping can dominate quantization error as the bit widthqqis reduced, as later made explicit in Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)\.

Asymmetric quantization shifts the grid with a zero point, improving range alignment but changing the deployed format: kernels must handle the offset and store its metadata\. We instead ask whether the signed alphabet’s native asymmetry can be matched to the data while keepingz=0z=0\. Signed symmetric quantization does exactly this, as we detail below\.

Table 1:Inference cost of the symmetric grid relative to the asymmetric grid inllama\.cppon an AMD EPYC™ “Turin” 128\-core CPU\. Each cell reports the symmetric\-to\-asymmetric ratio of model memory footprint \(GB\), prefill throughput \(tokens/s\), and decode throughput \(tokens/s\) at the same precision\. Values<1<1for memory and\>1\>1for throughput favor the symmetric format\.#### Asymmetric quantizer\.

Asymmetric quantization allows both the scale and zero point to be free parameters \(i\.e\.,s,z∈ℝs,z\\in\\mathbb\{R\}\)\. A common choice is the min\-max grid,

s=max⁡\(w\)−min⁡\(w\)2q−1,z=−min⁡\(w\)s,s=\\frac\{\\max\(w\)\-\\min\(w\)\}\{2^\{q\}\-1\},\\qquad z=\-\\frac\{\\min\(w\)\}\{s\},\(2\)on the unsigned alphabet𝒜q:=\{0,…,2q−1\}\\mathcal\{A\}\_\{q\}:=\\\{0,\\ldots,2^\{q\}\-1\\\}\.

#### Symmetric quantizer\.

Symmetric quantization fixes the zero point to zero \(i\.e\.,z=0z=0\) and commonly defines the scale as the maximum absolute value \(or absmax\),

s=‖w‖∞2q−1,s=\\frac\{\\\|w\\\|\_\{\\infty\}\}\{2^\{q\-1\}\},\(3\)on the signed alphabet𝒜q:=\{−2q−1,…,2q−1−1\}\\mathcal\{A\}\_\{q\}:=\\\{\-2^\{q\-1\},\\ldots,2^\{q\-1\}\-1\\\}\. We refer to this as the absmax grid\.

These grid choices imply different inference costsGholamiet al\.\([2022](https://arxiv.org/html/2607.08779#bib.bib2)\); Jacobet al\.\([2018](https://arxiv.org/html/2607.08779#bib.bib8)\); Jainet al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib50)\)\. We can observe this distinction directly in GGUF models executed withllama\.cppGerganov \([2023](https://arxiv.org/html/2607.08779#bib.bib47)\)on an AMD EPYC™ “Turin” CPU\. At 4\-bit precision,Q4\_1defines an asymmetric quantizer per group of 32 elements, storing a 16\-bit scale, a 16\-bit zero point, and 32 4\-bit weights per group;Q4\_0is the corresponding symmetric format, storing only a scale per group of 32 4\-bit weights\. We implement analogous 2\-bit formats,Q2\_1andQ2\_0, to further characterize the low\-bit regime; these are experimental formats introduced for this study, with implementation details and further discussion deferred to Appendix[B](https://arxiv.org/html/2607.08779#A2)\. Table[1](https://arxiv.org/html/2607.08779#S3.T1)shows that the symmetric formats use significantly less memory on larger models \(up to9%9\\%at Q4 and up to15%15\\%at Q2\) and consistently improve prefill throughput \(1\.101\.10–2\.45×2\.45\\timesat Q4 and1\.021\.02–1\.39×1\.39\\timesat Q2\), with decode throughput mostly following the same trend\. This is the price of the flexibility of the asymmetric format, and it motivates a quantizer that retains the runtime profile of a symmetric format while \(partially\) recovering the endpoint alignment of the asymmetric format\.

#### Signed symmetric quantizer\.

We define signed symmetric quantization as the family of uniform integer quantizers on the signed alphabet𝒜q=\{−2q−1,…,2q−1−1\}\\mathcal\{A\}\_\{q\}=\\\{\-2^\{q\-1\},\\ldots,2^\{q\-1\}\-1\\\}with zero pointz=0z=0and non\-zero scale \(i\.e\.,s≠0s\\neq 0\)\. Given scale magnitudeα\>0\\alpha\>0and signγ∈\{−1,\+1\}\\gamma\\in\\\{\-1,\+1\\\}, lets:=γ​αs:=\\gamma\\alphaand

𝒬γ​\(w\):=𝒬​\(w;γ​α,0\)=𝒬​\(w;s,0\)\.\\mathcal\{Q\}\_\{\\gamma\}\(w\):=\\mathcal\{Q\}\(w;\\,\\gamma\\alpha,\\,0\)=\\mathcal\{Q\}\(w;\\,s,\\,0\)\.\(4\)Note that the standard symmetric quantizer is a degenerate case of the signed symmetric quantizer withγ=\+1\\gamma=\+1\. Furthermore, allowingγ=−1\\gamma=\-1does not change the grid resolution; it only determines which tail receives the extra representable endpoint in the signed alphabet\.

#### The signed absmax grid\.

Within this family, we introduce the signed absmax grid\. LetM:=‖w‖∞\>0M:=\\\|w\\\|\_\{\\infty\}\>0andα:=M/2q−1\\alpha:=M/2^\{q\-1\}\. Choose

γ⋆=−sign⁡\(wi⋆\),i⋆∈arg​maxi∈\[d\]⁡\|wi\|,\\gamma^\{\\star\}=\-\\operatorname\{sign\}\(w\_\{i^\{\\star\}\}\),\\qquad i^\{\\star\}\\in\\operatorname\*\{arg\\,max\}\_\{i\\in\[d\]\}\|w\_\{i\}\|,\(5\)wherei⋆i^\{\\star\}indexes the largest\-magnitude coordinate, with deterministic tie\-breaking\. Intuitively, this sign choice maps the largest representable value in𝒜q\\mathcal\{A\}\_\{q\}to the dominant outlier inww\. In particular,

𝒬γ⋆​\(wi⋆\)=γ⋆​α⋅\(−2q−1\)=−γ⋆​M=sign⁡\(wi⋆\)​M=wi⋆\.\\mathcal\{Q\}\_\{\\gamma^\{\\star\}\}\(w\_\{i^\{\\star\}\}\)\\;=\\;\\gamma^\{\\star\}\\alpha\\cdot\(\-2^\{q\-1\}\)\\;=\\;\-\\gamma^\{\\star\}M\\;=\\;\\operatorname\{sign\}\(w\_\{i^\{\\star\}\}\)\\,M\\;=\\;w\_\{i^\{\\star\}\}\.Hence, the dominant outlier is represented exactly\. The opposite sign choice would instead place the extra endpoint on the other tail, potentially clipping the largest\-magnitude coordinate\. Indeed, in Section[4](https://arxiv.org/html/2607.08779#S4), we prove that our sign choice conditionally minimizes worst\-case quantization error \(Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)\); a condition that holds for 88–99% of evaluated LLM weight groups \(Remark[4\.5](https://arxiv.org/html/2607.08779#S4.Thmtheorem5)\)\.

This is the key distinction from min\-max asymmetric quantization\. Signed absmax does not attempt to cover the entire interval\[min⁡w,max⁡w\]\[\\min w,\\max w\]\. Instead, it uses the asymmetry already present in the signed alphabet to place the exact endpoint on the tail containing the dominant outlier, while retaining the runtime profile measured above for symmetric formats\. Section[4](https://arxiv.org/html/2607.08779#S4)analyzes this choice, and Section[5](https://arxiv.org/html/2607.08779#S5)evaluates how the proposed endpoint alignment translates into model\-level accuracy\.

## 4Theoretical Analysis

In Section[3](https://arxiv.org/html/2607.08779#S3), we defined the signed symmetric quantization family and argued informally that the signed absmax grid reduces clipping error by aligning the representable endpoint with the dominant outlier\. We now formalize this intuition\.

We first define the clipped set \(Definition[4\.1](https://arxiv.org/html/2607.08779#S4.Thmtheorem1)\), then decompose theℓ2\\ell\_\{2\}error to isolate the two roles of the scale: \(1\) its magnitude governs grid resolution and rounding error, while \(2\) its sign governs grid polarity and clipping error \(Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)\)\. We bound theℓ2\\ell\_\{2\}error in terms of the clipped set \(Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)\), then prove that the signed absmax grid minimizes this bound under a mild cardinality condition \(Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)\)\. We empirically verify this condition on real LLM weights and show that reducing the clipped set cardinality is an effective surrogate for reducingℓ2\\ell\_\{2\}quantization error \(Remark[4\.5](https://arxiv.org/html/2607.08779#S4.Thmtheorem5)\)\. We close by showing that the signed symmetric quantization family is a non\-trivial special case of the asymmetric quantization family \(Theorem[4\.6](https://arxiv.org/html/2607.08779#S4.Thmtheorem6)\)\.

Throughout our analysis, we assume weight vectorw∈ℝdw\\in\\mathbb\{R\}^\{d\}is quantized to theqq\-bit signed integer alphabet𝒜q:=\{−2q−1,…,2q−1−1\}\\mathcal\{A\}\_\{q\}:=\\\{\-2^\{q\-1\},\\ldots,2^\{q\-1\}\-1\\\}under round\-to\-nearest with step sizeα:=M/2q−1\\alpha:=M/2^\{q\-1\}, whereM:=‖w‖∞\>0M:=\\\|w\\\|\_\{\\infty\}\>0\. We also assumewi/α∉ℤ\+1/2w\_\{i\}/\\alpha\\notin\\mathbb\{Z\}\+1/2for alli∈\[d\]i\\in\[d\]\(a measure\-zero condition under any continuous weight distribution\) so that⌊−v⌉=−⌊v⌉\\lfloor\-v\\rceil=\-\\lfloor v\\rceilforv∈ℝv\\in\\mathbb\{R\}\. For brevity, all proofs are deferred to Appendix[A](https://arxiv.org/html/2607.08779#A1)\.

###### Definition 4\.1\(Clipped Set\)\.

For eachγ∈\{−1,\+1\}\\gamma\\in\\\{\-1,\+1\\\}, we define the*clipped set*CγC\_\{\\gamma\}as the set of indices whose rounded elements are clipped under𝒬γ\\mathcal\{Q\}\_\{\\gamma\}\(Equation[4](https://arxiv.org/html/2607.08779#S3.E4)\) such that⌊wi/\(γα\)⌉∉𝒜q\\lfloor w\_\{i\}/\(\\gamma\\alpha\)\\rceil\\notin\\mathcal\{A\}\_\{q\}\. In particular, for the signed integer alphabet, weights are clipped whenγ​wi\>M−α/2\\gamma w\_\{i\}\>M\-\\alpha/2, and therefore

Cγ=\{i∈\[d\]:γ​wi\>M​\(1−2−q\)\},C\_\{\\gamma\}=\\bigl\\\{i\\in\[d\]:\\gamma w\_\{i\}\>M\(1\-2^\{\-q\}\)\\bigr\\\},\(6\)and its complement\[d\]∖Cγ\[d\]\\setminus C\_\{\\gamma\}is the set of indices whose weights are rounded without clipping\.

###### Lemma 4\.2\(Error Decomposition\)\.

LetR\(wi\):=α⌊wi/α⌉R\(w\_\{i\}\):=\\alpha\\lfloor w\_\{i\}/\\alpha\\rceildenote rounding to the unbounded gridα​ℤ\\alpha\\mathbb\{Z\}\. Define rounding errorE​\(w\)E\(w\)and clipping penaltyΔγ​\(w\)\\Delta\_\{\\gamma\}\(w\)as

E​\(w\):=∑i=1d\(wi−R​\(wi\)\)2,Δγ​\(w\):=∑i∈Cγ\[\(wi−𝒬γ​\(wi\)\)2−\(wi−R​\(wi\)\)2\]\.E\(w\):=\\sum\_\{i=1\}^\{d\}\\bigl\(w\_\{i\}\-R\(w\_\{i\}\)\\bigr\)^\{2\},\\qquad\\Delta\_\{\\gamma\}\(w\):=\\sum\_\{i\\in C\_\{\\gamma\}\}\\Bigl\[\\bigl\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\\bigr\)^\{2\}\-\\bigl\(w\_\{i\}\-R\(w\_\{i\}\)\\bigr\)^\{2\}\\Bigr\]\.LetT:=M​\(1−2−q\)T:=M\(1\-2^\{\-q\}\)\. Then,E​\(w\)E\(w\)is independent ofγ\\gamma, and theℓ2\\ell\_\{2\}error decomposes as

‖w−𝒬γ​\(w\)‖22=E​\(w\)\+Δγ​\(w\)=E​\(w\)\+2​α​∑i∈Cγ\(γ​wi−T\)\.\\\|w\-\\mathcal\{Q\}\_\{\\gamma\}\(w\)\\\|\_\{2\}^\{2\}=E\(w\)\+\\Delta\_\{\\gamma\}\(w\)=E\(w\)\+2\\alpha\\sum\_\{i\\in C\_\{\\gamma\}\}\(\\gamma w\_\{i\}\-T\)\.\(7\)

Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)separates the two roles of the scale: \(1\) its magnitudeα\\alphadetermines the grid resolution and rounding errorE​\(w\)E\(w\), and \(2\) its signγ\\gammadetermines the grid polarity and clipping penaltyΔγ​\(w\)\\Delta\_\{\\gamma\}\(w\)\. The following theorem converts this exact expression to a deterministic worst\-case bound\.

![Refer to caption](https://arxiv.org/html/2607.08779v1/x1.png)Figure 1:Empirical validation of signed symmetric quantization on Llama3 8B and Qwen3 8B, quantizing all linear projection weights with group size 32 \(over 200M groups per model; no rotations or error correction\)\.\(a\)Per\-model fraction of weight groups satisfying the cardinality condition in Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4),\|Cγ⋆\|≤\|C−γ⋆\|\|C\_\{\\gamma^\{\\star\}\}\|\\leq\|C\_\{\-\\gamma^\{\\star\}\}\|, at bit widthsq∈\{2,3,4\}q\\in\\\{2,3,4\\\}\.\(b\)–\(d\)For eachq∈\{2,3,4\}q\\in\\\{2,3,4\\\}\(one panel per bit width, indicated in the panel title\), pooled scatter of cardinality marginm​\(w\)=\|C−γ⋆\|−\|Cγ⋆\|m\(w\)=\|C\_\{\-\\gamma^\{\\star\}\}\|\-\|C\_\{\\gamma^\{\\star\}\}\|against realized error gaing​\(w\)=Δ−γ⋆​\(w\)−Δγ⋆​\(w\)g\(w\)=\\Delta\_\{\-\\gamma^\{\\star\}\}\(w\)\-\\Delta\_\{\\gamma^\{\\star\}\}\(w\)\(Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)\)\. The cardinality condition holds for points right ofm=0m=0; the sign rule strictly reducesℓ2\\ell\_\{2\}error for points aboveg=0g=0\. Colors encode the quadrant assignment of each point\. Quadrant percentages are computed from exact full counts; the scatter displays a 1% subsample for visual clarity\.###### Theorem 4\.3\(Worst\-Case Error Bound\)\.

Givenw∈ℝdw\\in\\mathbb\{R\}^\{d\}, each per\-coordinate error term satisfies

\(wi−𝒬γ​\(wi\)\)2≤α24​for​i∈\[d\]∖Cγ,α24<\(wi−𝒬γ​\(wi\)\)2≤α2​for​i∈Cγ\.\\bigl\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\\bigr\)^\{2\}\\leq\\tfrac\{\\alpha^\{2\}\}\{4\}\\;\\;\\text\{for \}\\,i\\in\[d\]\\setminus C\_\{\\gamma\},\\qquad\\tfrac\{\\alpha^\{2\}\}\{4\}<\\bigl\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\\bigr\)^\{2\}\\leq\\alpha^\{2\}\\;\\;\\text\{for \}\\,i\\in C\_\{\\gamma\}\.In particular, substitutingα=M/2q−1\\alpha=M/2^\{q\-1\}and writingργ=\|Cγ\|/d\\rho\_\{\\gamma\}=\|C\_\{\\gamma\}\|/d, the totalℓ2\\ell\_\{2\}error satisfies

‖w−𝒬γ​\(w\)‖22≤2−2​q​M2​d​\(1\+3​ργ\)\.\\\|w\-\\mathcal\{Q\}\_\{\\gamma\}\(w\)\\\|\_\{2\}^\{2\}\\;\\leq\\;2^\{\-2q\}\\,M^\{2\}\\,d\\,\(1\+3\\rho\_\{\\gamma\}\)\.\(8\)

Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)turns the decomposition identified in Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)into a worst\-case objective for sign selection\. Givenww, Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)shows that the sign of the scale affects theℓ2\\ell\_\{2\}error only through clipping\. Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)then upper bounds the total squared error by a quantity whose onlyγ\\gamma\-dependent term is the clipped fractionργ=\|Cγ\|/d\\rho\_\{\\gamma\}=\|C\_\{\\gamma\}\|/d\. Therefore, under the deterministic worst\-case bound, sign selection reduces to minimizing the size of the clipped set\.

Importantly, as outliers are well\-known to dominate few\-bit quantization errorAshkbooset al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib16)\); Liuet al\.\([2025](https://arxiv.org/html/2607.08779#bib.bib17)\), worst\-case error bounds are effective optimization targets for post\-training quantizationSanjeetet al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib5)\)\. The following corollary and remark build on this intuition to justify the signed absmax grid \(Equation[5](https://arxiv.org/html/2607.08779#S3.E5)\), both analytically and empirically: Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)characterizes when the proposed sign choice minimizes the worst\-case bound, and Remark[4\.5](https://arxiv.org/html/2607.08779#S4.Thmtheorem5)shows that this condition holds for 88–99% of evaluated LLM weight groups, with a cardinality margin that is highly predictive of realizedℓ2\\ell\_\{2\}gains\.

###### Corollary 4\.4\(Conditional Bound\-Optimal Sign Choice\)\.

Leti⋆∈arg​maxi∈\[d\]⁡\|wi\|i^\{\\star\}\\in\\operatorname\*\{arg\\,max\}\_\{i\\in\[d\]\}\|w\_\{i\}\|index the dominant outlier inww\. Defineγ⋆:=−sign⁡\(wi⋆\)\\gamma^\{\\star\}:=\-\\operatorname\{sign\}\(w\_\{i^\{\\star\}\}\)\. Since the worst\-case bound from Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)depends onγ\\gammaonly through\|Cγ\|\|C\_\{\\gamma\}\|,

arg​minγ∈\{−1,\+1\}⁡2−2​q​M2​d​\(1\+3​ργ\)=arg​minγ∈\{−1,\+1\}⁡\|Cγ\|\.\\operatorname\*\{arg\\,min\}\_\{\\gamma\\in\\\{\-1,\+1\\\}\}2^\{\-2q\}\\,M^\{2\}\\,d\\,\(1\+3\\rho\_\{\\gamma\}\)\\;=\\;\\operatorname\*\{arg\\,min\}\_\{\\gamma\\in\\\{\-1,\+1\\\}\}\|C\_\{\\gamma\}\|\.Then, whenever\|Cγ⋆\|≤\|C−γ⋆\|\|C\_\{\\gamma^\{\\star\}\}\|\\leq\|C\_\{\-\\gamma^\{\\star\}\}\|,γ⋆\\gamma^\{\\star\}is a minimizer of the worst\-case bound\.

Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)links the practical sign rule from Section[3](https://arxiv.org/html/2607.08779#S3)to the worst\-case analysis in Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3): whenever\|Cγ⋆\|≤\|C−γ⋆\|\|C\_\{\\gamma^\{\\star\}\}\|\\leq\|C\_\{\-\\gamma^\{\\star\}\}\|, the choiceγ⋆\\gamma^\{\\star\}minimizes Equation[8](https://arxiv.org/html/2607.08779#S4.E8)\. This yields a directly interpretable optimization criterion: aligning the extra representable endpoint with the dominant outlier should not increase the number of clipped coordinates\. It also establishes the size of the clipped set\|Cγ\|\|C\_\{\\gamma\}\|as a surrogate for the exact sign\-dependent clipping penaltyΔγ​\(w\)\\Delta\_\{\\gamma\}\(w\)in Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)\. Both relationships are directly measurable on LLM weight groups, allowing us to evaluate not only how often the condition holds, but also how well the cardinality margin predicts realizedℓ2\\ell\_\{2\}error reduction\.

We note that our choiceγ⋆=−sign⁡\(wi⋆\)\\gamma^\{\\star\}=\-\\operatorname\{sign\}\(w\_\{i^\{\\star\}\}\)also carries two structural properties that hold unconditionally, independent of whether the cardinality condition is met\. First, as detailed in Section[3](https://arxiv.org/html/2607.08779#S3), it represents the dominant outlier exactly:𝒬γ⋆​\(wi⋆\)=wi⋆\\mathcal\{Q\}\_\{\\gamma^\{\\star\}\}\(w\_\{i^\{\\star\}\}\)=w\_\{i^\{\\star\}\}, since\|wi⋆\|=M\|w\_\{i^\{\\star\}\}\|=Mcoincides with the representable endpoint placed on its side\. Second, it confines clipping to the opposite tail:Cγ⋆⊆\{i∈\[d\]:γ⋆​wi\>0\}C\_\{\\gamma^\{\\star\}\}\\subseteq\\\{i\\in\[d\]:\\gamma^\{\\star\}w\_\{i\}\>0\\\}\. These properties help explain why the sign rule remains effective in practice even on the small fraction of groups where the condition fails \(Figure[1](https://arxiv.org/html/2607.08779#S4.F1)\): preserving the dominant outlier exactly and confining clipping to the opposite tail makeγ⋆\\gamma^\{\\star\}a natural choice when a clear dominant tail is present\. Tables[3](https://arxiv.org/html/2607.08779#S5.T3)and[4](https://arxiv.org/html/2607.08779#S5.T4)corroborate this at the model level: the largest gains appear at 2 bits, precisely where the cardinality condition holds least often \(Figure[1](https://arxiv.org/html/2607.08779#S4.F1)\)\.

Interestingly, asymmetric quantization pursues the same goal of aligning the grid endpoint with the dominant outlier, but does so by introducing a zero point that carries a non\-trivial inference costJainet al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib50)\); Jacobet al\.\([2018](https://arxiv.org/html/2607.08779#bib.bib8)\); Gholamiet al\.\([2022](https://arxiv.org/html/2607.08779#bib.bib2)\)\(Table[1](https://arxiv.org/html/2607.08779#S3.T1)\)\. The next theorem shows that the signed symmetric quantizer is identical to a unit zero point shift on the same signed integer alphabet, showing that signed symmetric quantization is analytically a non\-trivial instance of the family of asymmetric quantizers\.

###### Theorem 4\.6\(Sign Flip as Unit Zero\-Point Shift\)\.

Letα\>0\\alpha\>0\. For anyw∈ℝw\\in\\mathbb\{R\}withw/α∉ℤ\+12w/\\alpha\\notin\\mathbb\{Z\}\+\\tfrac\{1\}\{2\}, the quantizers𝒬​\(⋅;−α,0\)\\mathcal\{Q\}\(\\cdot;\-\\alpha,0\)and𝒬​\(⋅;α,−1\)\\mathcal\{Q\}\(\\cdot;\\alpha,\-1\)induce the same representable grid\. In particular,

𝒬​\(w;−α,0\)=𝒬​\(w;α,−1\),\\mathcal\{Q\}\(w;\-\\alpha,0\)=\\mathcal\{Q\}\(w;\\alpha,\-1\),where both quantizers share the same signed integer alphabet with a grid resolution ofα\\alpha\.

Theorem[4\.6](https://arxiv.org/html/2607.08779#S4.Thmtheorem6)shows that, at a shared step sizeα\\alpha, negating the scale underz=0z=0is identical to settingz=−1z=\-1with a strictly positive scale on the same signed integer alphabet\. Signed symmetric quantization therefore admits two equivalent views: \(1\) operationally a symmetric quantizer \(z=0z=0\) that incurs no zero point overhead at inference, and \(2\) analytically an asymmetric quantizer with a unit zero point shift \(z=−1z=\-1\) that absorbs the shift into the sign of the scale\.

It is worth noting the relationship to the standard min\-max asymmetric grid \(Equation[2](https://arxiv.org/html/2607.08779#S3.E2)\), which is not addressed by Theorem[4\.6](https://arxiv.org/html/2607.08779#S4.Thmtheorem6): the min\-max grid uses an unsigned alphabet and a scale derived from the full data range,i\.e\.,max⁡w−min⁡w\\max w\-\\min w, rather than from‖w‖∞\\\|w\\\|\_\{\\infty\}\. Both schemes preserve the dominant outlier exactly, but the min\-max grid additionally preserves the opposite tail by covering\[min⁡w,max⁡w\]\[\\min w,\\max w\]exactly, eliminating clipping error and leaving rounding as the only source of error \(Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)\)\. Furthermore, its step size depends on the full data range, so its relationship withα\\alphavaries with the distribution ofww\. An analytical comparison between the signed symmetric and min\-max asymmetric grids is therefore non\-trivial\. However, we compare them empirically in Section[5](https://arxiv.org/html/2607.08779#S5)\.

## 5Experimental Results

Sections[3](https://arxiv.org/html/2607.08779#S3)and[4](https://arxiv.org/html/2607.08779#S4)show that the sign of the scale is a useful degree of freedom: by choosing the sign appropriately, the dominant outlier in the data can be mapped to the extra negative value in the signed integer alphabet rather than being clipped\. We now evaluate whether this local choice of grid polarity yields benefits for end\-to\-end model accuracy\.

We organize our evaluation in two parts\. First, we isolate the sign rule by removing post\-training quantization machinery \(no rotations, error correction, or activation quantization\) and show that the signed absmax grid improves over the conventional strictly positive absmax grid \(Section[4](https://arxiv.org/html/2607.08779#S4)\)\. Second, we use Llama3 8B as a representative case study to show that this improvement survives composition with state\-of\-the\-art transformations and rounding methods\.

Models and datasets\. Our experiments span three model families: Qwen3Team \([2025](https://arxiv.org/html/2607.08779#bib.bib43)\), Qwen3\.5Qwen Team \([2026](https://arxiv.org/html/2607.08779#bib.bib42)\), and Llama3Grattafioriet al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib7)\), all sourced as instruction fine\-tuned checkpoints from HuggingfaceWolfet al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib24)\)and used without modification\. For calibration, we sample 128 random sequences of 2048 tokens from the WikiText2Merityet al\.\([2016](https://arxiv.org/html/2607.08779#bib.bib22)\)training split and report perplexity on its test split\. To evaluate downstream reasoning, we report zero\-shot accuracy via LightEvalHabibet al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib41)\)on ARC \(Challenge and Easy\)Clarket al\.\([2018](https://arxiv.org/html/2607.08779#bib.bib33)\), HellaSwagZellerset al\.\([2019](https://arxiv.org/html/2607.08779#bib.bib34)\), PIQABisket al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib38)\), and WinoGrandeSakaguchiet al\.\([2021](https://arxiv.org/html/2607.08779#bib.bib35)\)\. We further report few\-shot accuracy on MMLU\-Redux \(MMLU\-R\)Gemaet al\.\([2025](https://arxiv.org/html/2607.08779#bib.bib52)\)and GSM8KCobbeet al\.\([2021](https://arxiv.org/html/2607.08779#bib.bib39)\)\. We quantize all models using the Brevitas quantization libraryXilinx \([2018](https://arxiv.org/html/2607.08779#bib.bib45)\)\.

#### Sign rule in isolation\.

We first isolate the effect of the sign rule by direct casting all linear projection weights with group size 32, matching thellama\.cppblock convention\. This experiment uses no rotations, no error correction methods \(e\.g\., Qronos\), no activation quantization, and no calibration or optimization\. Thus, the only difference between the absmax and signed rows in Table[2](https://arxiv.org/html/2607.08779#S5.T2)is whether the scale is constrained to be strictly positive or allowed to choose its sign per group\.

Table 2:Sign rule in isolation\. WikiText2 perplexity \(↓\\downarrow\) and average zero\-shot accuracy \(↑\\uparrow\) across five reasoning tasks, ARC \(Challenge and Easy\), HellaSwag, WinoGrande, and PIQA, under direct casting\. We compare the signed absmax grid \(denoted*signed*\) against the strictly positive absmax grid \(denoted*absmax*\)\. BF16 baselines provided as reference \(q=16q=16\)\.The results show that our sign rule \(Equation[5](https://arxiv.org/html/2607.08779#S3.E5)\) gives a consistent improvement over the standard absmax grid: signed absmax improves WikiText2 perplexity in 23 of the 24 configurations in Table[2](https://arxiv.org/html/2607.08779#S5.T2), with gains concentrated at lower precision\. This is consistent with the2−2​q2^\{\-2q\}dependence in the error bound of Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3), Equation[8](https://arxiv.org/html/2607.08779#S4.E8)\. In particular, the largest reductions occur on Qwen3\-1\.7B \(81\.681\.6to25\.925\.9\) and Llama3\-1B \(32\.332\.3to21\.921\.9\) when quantizing weights to 3 bits \(q=3q=3\)\.

The downstream zero\-shot accuracy results, averaged over ARC \(Challenge and Easy\), HellaSwag, Winogrande, and PIQA, follow the same trend: signed absmax improves average zero\-shot accuracy by0\.240\.24points at INT4 and1\.111\.11points at INT3\. Across all 24 configurations, signed absmax improves 18, ties 2, and trails in 4; every regression is at most0\.20\.2points\. These results support the central claim from Section[4](https://arxiv.org/html/2607.08779#S4): the sign of the scale is a small local choice but, at few\-bit precision, it has measurable model\-level consequences\.

#### Composition with PTQ methods\.

Direct casting isolates the sign rule, but practical quantization pipelines typically include transformations, error correction, and activation quantization\. For example, inllama\.cpp, few\-bit weight formats are commonly paired with dynamicQ8\_0activation quantization at inference\. We therefore evaluate Llama3 8B as a case study under a practical PTQ pipeline\. We first insert normalized Hadamard rotations into rotation\-invariant regions within the modelAshkbooset al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib16)\)\(Figure[2](https://arxiv.org/html/2607.08779#A3.F2)\)\. Weight grids are then calculated on the rotated weights, while activation quantization grids are computed dynamically from the rotated activations underQ8\_0quantization\. Qronos is then applied as the adaptive rounding step to reduce the mismatch between the quantized model and its full\-precision counterpartZhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\)\. We defer further algorithmic details to Appendix[C](https://arxiv.org/html/2607.08779#A3)\.

Table[3](https://arxiv.org/html/2607.08779#S5.T3)presents results under three few\-bit formats in extended GGUF notation:Q4\_0denotes the standard symmetric format, using the absmax grid from Equation[3](https://arxiv.org/html/2607.08779#S3.E3),Q4\_0sdenotes our signed symmetric variant, using the signed absmax grid from Equation[5](https://arxiv.org/html/2607.08779#S3.E5), andQ4\_1denotes the asymmetric format, using the min\-max grid from Equation[2](https://arxiv.org/html/2607.08779#S3.E2)\. We use analogous notation for Q3 and Q2\.111Q4\_0andQ4\_1are standard GGUF formats; the Q3 and Q2 variants are introduced for this study \(Appendix[B](https://arxiv.org/html/2607.08779#A2)\)\.

Table 3:Composition with PTQ methods\. WikiText2 perplexity \(↓\\downarrow\), few\-shot accuracy \(%,↑\\uparrow\), and average recovery \(↑\\uparrow\) on Llama3 8B for few\-bit GGUF formats\. All quantized rows use mergeable normalized Hadamard rotations, QronosZhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\), and dynamicQ8\_0activation quantization\. Recovery is the average downstream accuracy divided by the BF16 average over all tasks\.Table 4:Composition with jointly learned rotations and grid parameters\. WikiText2 perplexity \(↓\\downarrow\), few\-shot accuracy \(%,↑\\uparrow\), and recovery \(↑\\uparrow\) on Llama3 8B for few\-bit GGUF formats\. We jointly learn rotations and grid parameters on FineWebPenedoet al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib54)\)using Table[3](https://arxiv.org/html/2607.08779#S5.T3)as an initialization, then freeze them before applying QronosZhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\)with dynamicQ8\_0activation quantization\. Recovery is the average downstream accuracy divided by the BF16 average over all tasks\.At W4, all three formats are close in accuracy\.Q4\_1andQ4\_0sboth achieve6\.66\.6WikiText2 perplexity, whileQ4\_0gives6\.76\.7\. Downstream recovery is similarly saturated:98\.9%98\.9\\%forQ4\_1,98\.6%98\.6\\%forQ4\_0s, and98\.7%98\.7\\%forQ4\_0\. In this regime, the choice between symmetric and asymmetric formats is dominated less by accuracy than by deployment cost\. SinceQ4\_0suses the same runtime path asQ4\_0, it inherits the memory and throughput advantages of the symmetric format measured in Table[1](https://arxiv.org/html/2607.08779#S3.T1)\. On Llama3 8B at W4, this corresponds to a2\.21×2\.21\\timesprefill and2\.03×2\.03\\timesdecode throughput advantage overQ4\_1, whileQ4\_0strailsQ4\_1by only0\.30\.3recovery points\.

At lower bit widths, the sign choice remains valuable but does not fully eliminate the advantage of an asymmetric zero point\. At W3, signed symmetric improves the conventional symmetric baseline from7\.77\.7to7\.27\.2perplexity and from89\.0%89\.0\\%to94\.6%94\.6\\%recovery, closing the recovery gap toQ3\_1\(which achieves7\.07\.0perplexity and95\.9%95\.9\\%recovery\) from6\.96\.9to1\.31\.3percentage points\. At W2, the effect is even more pronounced in perplexity:Q2\_0collapses to103\.1103\.1, whileQ2\_0sreduces this to17\.817\.8\. The asymmetricQ2\_1format remains stronger at12\.012\.0perplexity, but signed symmetric closes the few\-shot accuracy recovery gap from15\.615\.6to7\.77\.7percentage points\.

#### Composition with learned rotations and grid parameters\.

We next ask whether the same degree of freedom remains useful when the quantization pipeline is allowed to adapt from data\. To this end, we jointly learn rotations and grid parameters on FineWebPenedoet al\.\([2024](https://arxiv.org/html/2607.08779#bib.bib54)\)at the same insertion points as the fixed pipeline \(Figure[2](https://arxiv.org/html/2607.08779#A3.F2)\), using Table[3](https://arxiv.org/html/2607.08779#S5.T3)as an initialization, then freeze them before applying QronosZhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\)with dynamicQ8\_0activation quantization\. Rotations are optimized on the Stiefel manifold using Cayley SGD as in SpinQuantLiuet al\.\([2025](https://arxiv.org/html/2607.08779#bib.bib17)\)\. For the grid parameters,Q4\_1learns the scale and zero\-point offset jointly as in LSQ\+Bhalgatet al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib46)\), whileQ4\_0andQ4\_0sfollow LSQEsseret al\.\([2020](https://arxiv.org/html/2607.08779#bib.bib48)\)with the scale constrained to be strictly positive \(s\>0s\>0\) and non\-zero \(s≠0s\\neq 0\), respectively\. We use analogous parameterizations for Q3 and Q2, with full optimization details deferred to Appendix[C](https://arxiv.org/html/2607.08779#A3)\.

Table[4](https://arxiv.org/html/2607.08779#S5.T4)shows that signed symmetric formats continue to improve over their conventional symmetric counterparts after joint rotation and grid learning\. At W4,Q4\_0sslightly improves overQ4\_0in both WikiText2 perplexity,6\.76\.7to6\.66\.6, and recovery,98\.7%98\.7\\%to98\.8%98\.8\\%\. The effect is larger at lower precision:Q3\_0simproves overQ3\_0from7\.37\.3to7\.17\.1perplexity and from90\.9%90\.9\\%to95\.3%95\.3\\%recovery;Q2\_0sreducesQ2\_0perplexity from103\.8103\.8to12\.912\.9and improves recovery from42\.7%42\.7\\%to56\.1%56\.1\\%\. The comparison to the asymmetric format is mixed but informative:Q4\_1andQ3\_1maintain superior recovery at W4 and W3, whileQ2\_0sslightly exceedsQ2\_1in aggregate recovery at W2 \(56\.1%56\.1\\%versus54\.3%54\.3\\%\) despite worse perplexity \(12\.912\.9versus12\.212\.2\)\. This perplexity\-versus\-recovery inversion is consistent with prior observations that the two metrics can decouple at low precisionZhanget al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib18)\); Sanjeetet al\.\([2026](https://arxiv.org/html/2607.08779#bib.bib5)\), and is analogous to overfitting: Zhang et al\.Zhanget al\.\([2025](https://arxiv.org/html/2607.08779#bib.bib4)\)\(Remark 3\.8\) decompose generalization error into a calibration reconstruction term and a parameter\-proximity term\|w−𝒬​\(w\)\|\|w\-\\mathcal\{Q\}\(w\)\|, which can be at odds\.

Taken together, Tables[2](https://arxiv.org/html/2607.08779#S5.T2)–[4](https://arxiv.org/html/2607.08779#S5.T4)support a consistent interpretation\. The sign of the scale is a local degree of freedom with measurable model\-level effects when clipping is material\. Direct casting isolates this effect: signed absmax consistently improves over the conventional strictly positive absmax grid\. In practical PTQ pipelines, signed symmetric quantization decouples the accuracy and systems axes: it deploys through exactly the symmetric runtime path, while its accuracy moves toward that of the asymmetric format\. On Llama3 8B, it reaches the asymmetric accuracy regime at 4 bits, closes most of the recovery gap at 3 bits, and substantially improves overQ2\_0at 2 bits, with jointly learned grids and rotations even exceedingQ2\_1in aggregate few\-shot accuracy recovery despite worse perplexity\. Thus, the sign remains a useful degree of freedom even when rotations and quantizer parameters are learned from data, at no extra inference cost relative to the conventional symmetric format\.

## 6Discussion and Conclusion

Standard symmetric quantization constrains the scale factor to be positive by convention\. We showed that allowing signed scales exposes a zero\-overhead degree of freedom\. The signed absmax grid uses this freedom to place the extra signed\-integer endpoint on the dominant\-outlier tail\. Our analysis separates the role of scale magnitude, which controls rounding error, from the role of scale sign, which controls clipping\. Theoretically, this separation yields a worst\-caseℓ2\\ell\_\{2\}bound under which signed absmax is conditionally bound\-optimal\. Empirically, this degree of freedom consistently improves conventional symmetric quantization at few bits and composes with rotations, Qronos, learned grid parameters, and dynamic activation quantization\.

#### Limitations and future work\.

Our analysis focuses mainly on data\-free, per\-group weight quantization\. This is intentionally narrow: it isolates the degree of freedom and yields a closed\-form rule that already captures much of the benefit observed in our PTQ pipelines\. Learned rotations and grid parameters can provide additional gains, suggesting that a fully data\-aware treatment of signed scales is a promising extension rather than a prerequisite for the method\. More broadly, future work should evaluate signed scales across additional model families, operator types such as convolutions, quantization granularities, and deployment backends\. Our results nevertheless suggest that the sign of the scale should be treated as a quantization parameter rather than fixed by convention\.

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## Appendix AProofs

Throughout this appendix, we adopt the assumptions stated in Section[4](https://arxiv.org/html/2607.08779#S4): the signed integer alphabet𝒜q:=\{−2q−1,…,2q−1−1\}\\mathcal\{A\}\_\{q\}:=\\\{\-2^\{q\-1\},\\ldots,2^\{q\-1\}\-1\\\}, and the no\-midpoint conditionwi/α∉ℤ\+12w\_\{i\}/\\alpha\\notin\\mathbb\{Z\}\+\\tfrac\{1\}\{2\}for alli∈\[d\]i\\in\[d\]\(a measure\-zero condition under any continuous weight distribution\), under which rounding to nearest is unique and satisfies⌊−x⌉=−⌊x⌉\\lfloor\-x\\rceil=\-\\lfloor x\\rceilwheneverx∉ℤ\+12x\\notin\\mathbb\{Z\}\+\\tfrac\{1\}\{2\}\.

### A\.1Proof of Lemma[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)

See[4\.2](https://arxiv.org/html/2607.08779#S4.Thmtheorem2)

###### Proof\.

The mapRRdoes not depend onγ\\gamma, soE​\(w\)E\(w\)isγ\\gamma\-invariant by construction\. Setui:=γ​wiu\_\{i\}:=\\gamma w\_\{i\}\. Since⌊−x⌉=−⌊x⌉\\lfloor\-x\\rceil=\-\\lfloor x\\rceilunder the no\-midpoint assumption,⌊wi/\(γα\)⌉=⌊ui/α⌉\\lfloor w\_\{i\}/\(\\gamma\\alpha\)\\rceil=\\lfloor u\_\{i\}/\\alpha\\rceil, so

𝒬γ\(wi\)=γαclip\(⌊ui/α⌉;𝒜q\),R\(wi\)=γα⌊ui/α⌉\.\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)=\\gamma\\alpha\\,\\textrm\{clip\}\\bigl\(\\lfloor u\_\{i\}/\\alpha\\rceil;\\,\\mathcal\{A\}\_\{q\}\\bigr\),\\qquad R\(w\_\{i\}\)=\\gamma\\alpha\\,\\lfloor u\_\{i\}/\\alpha\\rceil\.
*Step 1:𝒬γ​\(wi\)=R​\(wi\)\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)=R\(w\_\{i\}\)fori∉Cγi\\notin C\_\{\\gamma\}\.*By Definition[4\.1](https://arxiv.org/html/2607.08779#S4.Thmtheorem1),i∉Cγi\\notin C\_\{\\gamma\}means⌊ui/α⌉∈𝒜q\\lfloor u\_\{i\}/\\alpha\\rceil\\in\\mathcal\{A\}\_\{q\}, so the clip is inactive\.

*Step 2:\(wi−𝒬γ​\(wi\)\)2−\(wi−R​\(wi\)\)2=2​α​\(γ​wi−T\)\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\)^\{2\}\-\(w\_\{i\}\-R\(w\_\{i\}\)\)^\{2\}=2\\alpha\(\\gamma w\_\{i\}\-T\)fori∈Cγi\\in C\_\{\\gamma\}\.*Fori∈Cγi\\in C\_\{\\gamma\},ui∈\(M−α/2,M\]u\_\{i\}\\in\(M\-\\alpha/2,M\], so⌊ui/α⌉=2q−1\\lfloor u\_\{i\}/\\alpha\\rceil=2^\{q\-1\}and the clip caps it at2q−1−12^\{q\-1\}\-1, givingR​\(wi\)=γ​MR\(w\_\{i\}\)=\\gamma Mand𝒬γ​\(wi\)=γ​\(M−α\)\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)=\\gamma\(M\-\\alpha\)\. Since\(wi−γ​c\)2=\(ui−c\)2\(w\_\{i\}\-\\gamma c\)^\{2\}=\(u\_\{i\}\-c\)^\{2\}for any constantcc,

\(wi−𝒬γ​\(wi\)\)2−\(wi−R​\(wi\)\)2=\(ui−\(M−α\)\)2−\(ui−M\)2=2​α​\(ui−T\)\.\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\)^\{2\}\-\(w\_\{i\}\-R\(w\_\{i\}\)\)^\{2\}=\(u\_\{i\}\-\(M\-\\alpha\)\)^\{2\}\-\(u\_\{i\}\-M\)^\{2\}=2\\alpha\(u\_\{i\}\-T\)\.
Splitting the sum atCγC\_\{\\gamma\}and applying Steps 1–2,

‖w−𝒬γ​\(w\)‖22\\displaystyle\\\|w\-\\mathcal\{Q\}\_\{\\gamma\}\(w\)\\\|\_\{2\}^\{2\}=∑i=1d\(wi−R​\(wi\)\)2\+∑i∈Cγ\[\(wi−𝒬γ​\(wi\)\)2−\(wi−R​\(wi\)\)2\]\\displaystyle=\\sum\_\{i=1\}^\{d\}\\bigl\(w\_\{i\}\-R\(w\_\{i\}\)\\bigr\)^\{2\}\+\\sum\_\{i\\in C\_\{\\gamma\}\}\\Bigl\[\\bigl\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\\bigr\)^\{2\}\-\\bigl\(w\_\{i\}\-R\(w\_\{i\}\)\\bigr\)^\{2\}\\Bigr\]=E​\(w\)\+2​α​∑i∈Cγ\(γ​wi−T\)\.∎\\displaystyle=E\(w\)\+2\\alpha\\sum\_\{i\\in C\_\{\\gamma\}\}\(\\gamma w\_\{i\}\-T\)\.\\qed

### A\.2Proof of Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)

See[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3)

###### Proof\.

Setui:=γ​wiu\_\{i\}:=\\gamma w\_\{i\}\. Sinceγ∈\{−1,\+1\}\\gamma\\in\\\{\-1,\+1\\\}, multiplication byγ\\gammapreserves absolute error:

\|wi−𝒬γ​\(wi\)\|=\|ui−𝒬\+​\(ui\)\|,\|w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\|=\|u\_\{i\}\-\\mathcal\{Q\}\_\{\+\}\(u\_\{i\}\)\|,where𝒬\+\\mathcal\{Q\}\_\{\+\}denotes positive\-scale symmetric quantization with stepα\\alphaon the alphabet𝒜q\\mathcal\{A\}\_\{q\}\.

Ifi∉Cγi\\notin C\_\{\\gamma\}, then no clipping occurs and𝒬\+\(ui\)=α⌊ui/α⌉\\mathcal\{Q\}\_\{\+\}\(u\_\{i\}\)=\\alpha\\,\\lfloor u\_\{i\}/\\alpha\\rceil, so round\-to\-nearest gives\|wi−𝒬γ​\(wi\)\|≤α/2\|w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\|\\leq\\alpha/2\.

Ifi∈Cγi\\in C\_\{\\gamma\}, then Definition[4\.1](https://arxiv.org/html/2607.08779#S4.Thmtheorem1)givesui∈\(T,M\]u\_\{i\}\\in\(T,M\], whereT=M−α/2T=M\-\\alpha/2\. Clipping mapsuiu\_\{i\}to the largest representable pointM−αM\-\\alpha, so\|wi−𝒬γ​\(wi\)\|=ui−\(M−α\)∈\(α/2,α\]\|w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\|=u\_\{i\}\-\(M\-\\alpha\)\\in\(\\alpha/2,\\alpha\]\.

Therefore

\(wi−𝒬γ​\(wi\)\)2≤α24\+3​α24​1\{i∈Cγ\},\\bigl\(w\_\{i\}\-\\mathcal\{Q\}\_\{\\gamma\}\(w\_\{i\}\)\\bigr\)^\{2\}\\leq\\frac\{\\alpha^\{2\}\}\{4\}\+\\frac\{3\\alpha^\{2\}\}\{4\}\\,\\mathbf\{1\}\_\{\\\{i\\in C\_\{\\gamma\}\\\}\},and summing overiiyields‖w−𝒬γ​\(w\)‖22≤α24​\(d\+3​\|Cγ\|\)\\\|w\-\\mathcal\{Q\}\_\{\\gamma\}\(w\)\\\|\_\{2\}^\{2\}\\leq\\tfrac\{\\alpha^\{2\}\}\{4\}\\bigl\(d\+3\|C\_\{\\gamma\}\|\\bigr\)\. Substitutingα=M/2q−1\\alpha=M/2^\{q\-1\}givesα24=2−2​q​M2\\tfrac\{\\alpha^\{2\}\}\{4\}=2^\{\-2q\}\\,M^\{2\}, and writing\|Cγ\|=ργ​d\|C\_\{\\gamma\}\|=\\rho\_\{\\gamma\}dyields

‖w−𝒬γ​\(w\)‖22≤2−2​q​M2​d​\(1\+3​ργ\)\.∎\\\|w\-\\mathcal\{Q\}\_\{\\gamma\}\(w\)\\\|\_\{2\}^\{2\}\\;\\leq\\;2^\{\-2q\}\\,M^\{2\}\\,d\\,\(1\+3\\rho\_\{\\gamma\}\)\.\\qed

### A\.3Proof of Corollary[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)

See[4\.4](https://arxiv.org/html/2607.08779#S4.Thmtheorem4)

###### Proof\.

By Theorem[4\.3](https://arxiv.org/html/2607.08779#S4.Thmtheorem3), the worst\-case bound2−2​q​‖w‖∞2​d​\(1\+3​ργ\)2^\{\-2q\}\\,\\\|w\\\|\_\{\\infty\}^\{2\}\\,d\\,\(1\+3\\rho\_\{\\gamma\}\)has a strictly positive prefactor independent ofγ\\gammaand is strictly increasing inργ=\|Cγ\|/d\\rho\_\{\\gamma\}=\|C\_\{\\gamma\}\|/d, so its minimizers overγ∈\{−1,\+1\}\\gamma\\in\\\{\-1,\+1\\\}are exactlyarg​minγ∈\{−1,\+1\}⁡\|Cγ\|\\operatorname\*\{arg\\,min\}\_\{\\gamma\\in\\\{\-1,\+1\\\}\}\|C\_\{\\gamma\}\|\. Sinceγ\\gammatakes only the two values±1\\pm 1, the condition\|Cγ⋆\|≤\|C−γ⋆\|\|C\_\{\\gamma^\{\\star\}\}\|\\leq\|C\_\{\-\\gamma^\{\\star\}\}\|implies thatγ⋆\\gamma^\{\\star\}minimizes\|Cγ\|\|C\_\{\\gamma\}\|, and therefore minimizes the bound\. ∎

### A\.4Proof of Theorem[4\.6](https://arxiv.org/html/2607.08779#S4.Thmtheorem6)

See[4\.6](https://arxiv.org/html/2607.08779#S4.Thmtheorem6)

###### Proof\.

For fixed\(s,z\)\(s,z\), the quantizer𝒬​\(⋅;s,z\)\\mathcal\{Q\}\(\\cdot;s,z\)returns the unique nearest point in the grids​\(𝒜q−z\)s\(\\mathcal\{A\}\_\{q\}\-z\)whenever midpoint ties are excluded\. Hence, the representable grid of𝒬​\(⋅;−α,0\)\\mathcal\{Q\}\(\\cdot;\-\\alpha,0\)is−α​𝒜q\-\\alpha\\,\\mathcal\{A\}\_\{q\}, while the representable grid of𝒬​\(⋅;α,−1\)\\mathcal\{Q\}\(\\cdot;\\alpha,\-1\)isα​\(𝒜q\+1\)\\alpha\(\\mathcal\{A\}\_\{q\}\+1\)\. Since𝒜q=\{−2q−1,…,2q−1−1\}\\mathcal\{A\}\_\{q\}=\\\{\-2^\{q\-1\},\\ldots,2^\{q\-1\}\-1\\\}, these grids coincide:

−α​𝒜q=α​\(𝒜q\+1\)=α⋅\{−\(2q−1−1\),…,2q−1\}\.\-\\alpha\\,\\mathcal\{A\}\_\{q\}\\;=\\;\\alpha\(\\mathcal\{A\}\_\{q\}\+1\)\\;=\\;\\alpha\\cdot\\\{\-\(2^\{q\-1\}\-1\),\\ldots,2^\{q\-1\}\\\}\.Therefore,𝒬​\(⋅;−α,0\)\\mathcal\{Q\}\(\\cdot;\-\\alpha,0\)and𝒬​\(⋅;α,−1\)\\mathcal\{Q\}\(\\cdot;\\alpha,\-1\)have the same representable grid, and under the no\-midpoint\-tie assumption they agree pointwise:

𝒬​\(w;−α,0\)=𝒬​\(w;α,−1\)for all​w∈ℝ​with​w/α∉ℤ\+12\.∎\\mathcal\{Q\}\(w;\-\\alpha,0\)=\\mathcal\{Q\}\(w;\\alpha,\-1\)\\quad\\text\{for all \}w\\in\\mathbb\{R\}\\text\{ with \}w/\\alpha\\notin\\mathbb\{Z\}\+\\tfrac\{1\}\{2\}\.\\qed

## Appendix BImplementation Details forllama\.cpp

This appendix expands on the inference measurements in Table[1](https://arxiv.org/html/2607.08779#S3.T1)\. The goal is to quantify the systems cost of the deployed quantization format, independent of our sign selection rule \(Section[3](https://arxiv.org/html/2607.08779#S3)\)\. In particular, the measurements compare symmetric and asymmetric GGUF formats with matched group size and bit width\. Since signed symmetric quantization uses the same deployed format as the standard symmetric quantization, these measurements apply to both; the sign rule changes scale selection offline, not the runtime kernel path\.

We consider two standard 4\-bit GGUF formats:Q4\_1\(asymmetric\) andQ4\_0\(symmetric\)\. We also contribute two new 2\-bit GGUF formats,Q2\_1\(asymmetric\) andQ2\_0\(symmetric\), enabling direct comparison of signed symmetric and asymmetric quantization at INT2 precision\. We benchmark CPU inference throughput using thellama\.cppframework\[[12](https://arxiv.org/html/2607.08779#bib.bib47)\]on an AMD EPYC™ “Turin” 128\-core processor with the ZenDNN\(L\) backend v1\.0\.0\.

### B\.1Formats and block layout

Consistent with theQ4\_0andQ4\_1standard, all formats use a block size of 32 weights\. For Q2, each block packs 32 two\-bit indices into 8 bytes, with four indices per byte\. For Q4, each block packs 32 four\-bit indices into 16 bytes, with two indices per byte\.

Q4\_0andQ2\_0store a single scales=γ​αs=\\gamma\\alpha, which is strictly positive by convention but may be negative under our implementation\.Q4\_1andQ2\_1also store a scalessand minimummm\.

### B\.2Measurement setup

#### Hardware\.

CPU experiments run on an AMD EPYC™ 9755 \(Zen 5, “Turin”\) system with 128 physical cores per socket \(256 logical\), two NUMA nodes, and 1\.5 TB DDR5\. The CPU supports AVX\-512 \(F/BW/DQ/VL\), AVX\-512VNNI, AVXVNNIINT8, AVX2, and FMA\.

#### Software and builds\.

All experiments usellama\.cppat commitb1be68e8\(master, ggml\-org/llama\.cpp\), compiled with GCC 14\.3\.0 and CMake 4\.2\.1 on Ubuntu 22\.04 \(Linux 5\.15\) inReleasemode withGGML\_NATIVE=ONandGGML\_OPENMP=ON, enabling\-march=nativeand all ISA extensions supported by the target CPU\. We evaluate two configurations from this same commit: \(i\) an*unmodified upstream*baseline build, supporting BF16,Q4\_0, andQ4\_1with no custom kernels or code modifications, and \(ii\) a*modified build*that addsQ2\_0/Q2\_1quantization support and AVX\-512–optimized dot product kernels for all quantized types\. In the baseline build,Q4\_0/Q4\_1dot product kernels do not have explicit AVX\-512 paths and execute via 256\-bit AVX2 VNNI instructions, while other operations \(e\.g\., attention, RMSNorm, SiLU, softmax\) use existing 512\-bit AVX\-512 kernels\. Models are converted from HuggingFace safetensors to BF16 GGUF and then quantized toQ4\_0,Q4\_1,Q2\_0, andQ2\_1\. However, we note that one could similarly export Brevitas\-quantized models that are compatible with thellama\.cppruntime\.

#### Benchmarking and metrics\.

Throughput is measured usingllama\-bench\. We evaluate a 512\-token prefill followed by 512\-token autoregressive decode, with prefill batch size 512 and 128 threads\. All CPU runs are pinned to a single NUMA node usingnumactl \-\-cpunodebind=0 \-\-membind=0, mapping one thread per physical core and avoiding cross\-NUMA traffic\. We report tokens/s for: \(i\)*prefill \(pp512\)*, a compute\-bound phase dominated by matrix multiplications, and \(ii\)*decode \(tg512\)*, a memory\-bandwidth\-bound phase\. Each experiment performs one warmup iteration followed by 20 timed repetitions; reported results are the mean over repetitions\. Across all reported configurations, over 90% exhibit a coefficient of variation \(standard deviation divided by mean\) below 6%\.

### B\.3Kernel\-level cost gap

The throughput comparison is between deployed formats, not between sign selection policies\. Unsigned symmetric and signed symmetric quantization both deploy as a symmetric format withz=0z=0; the only difference is the sign assigned to each per\-group scale\. When scales are stored as signed real values, the sign introduces no additional metadata\. Therefore, Table[1](https://arxiv.org/html/2607.08779#S3.T1)characterizes the runtime profile inherited by signed symmetric quantization from the symmetric family\.

#### Unpacking overhead\.

After loading 8 bytes of packed 2\-bit indices from a Q2 format, the kernel must expand them to 32 individual bytes in element order\. This requires broadcasting the 8\-byte value to a 256\-bit register, extracting four shift\-and groups, and interleaving the results with twounpacklo\_epi8and oneunpacklo\_epi16, approximately 10 SIMD instructions per block\. By comparison, the 4\-bit nibble unpack \(bytes\_from\_nibbles\_32\) needs only a 128\-bit load, one shift, and one mask: approximately 3 instructions\. This3×3\\timesdifference in unpack cost is a fixed overhead shared by bothQ2\_0andQ2\_1, and it is the primary reason the symmetric\-over\-asymmetric throughput advantage is smaller at 2 bits than at 4 bits \(Table[1](https://arxiv.org/html/2607.08779#S3.T1)\)\. On the AVX\-512 path, a dual\-block unpack primitive loads two blocks’qsfields into a single 128\-bit register and produces a 512\-bit result \(64 bytes\), halving the number of unpack calls per iteration\.

#### Zero\-point cost in quantized GEMM\.

The asymmetric format also adds work in the GEMM itself\[[17](https://arxiv.org/html/2607.08779#bib.bib8),[19](https://arxiv.org/html/2607.08779#bib.bib49)\]\. Consider one output elementYi​k=∑jXi​j​Wk​jY\_\{ik\}=\\sum\_\{j\}X\_\{ij\}W\_\{kj\}withWWquantized in groups of sizegg, so that within groupbb,Wk​j≈sk​b​\(Wq,k​j−zk​b\)W\_\{kj\}\\approx s\_\{kb\}\(W\_\{q,kj\}\-z\_\{kb\}\)\. Expanding,

Yi​k≈∑bsk​b​∑j∈bXi​j​Wq,k​j⏟integer GEMM−∑bsk​b​zk​b​∑j∈bXi​j⏟row\-sum correction\.Y\_\{ik\}\\;\\approx\\;\\sum\_\{b\}s\_\{kb\}\\underbrace\{\\sum\_\{j\\in b\}X\_\{ij\}\\,W\_\{q,kj\}\}\_\{\\text\{integer GEMM\}\}\\;\-\\;\\sum\_\{b\}s\_\{kb\}z\_\{kb\}\\underbrace\{\\sum\_\{j\\in b\}X\_\{ij\}\}\_\{\\text\{row\-sum correction\}\}\.Both formats compute the integer GEMM term\. The asymmetric format \(z≠0z\\neq 0\) additionally accumulates a per\-block row\-sum ofXXscaled bys​zszper output, while the symmetric format \(z=0z=0\) skips this term entirely\.

## Appendix CAdditional Experiment Details

This appendix provides implementation details for the experiments in Section[5](https://arxiv.org/html/2607.08779#S5)\. We separate the quantization graph from the quantization pipeline\. The graph specifies where rotations and quantizers are inserted into the model, while the pipeline specifies how grid parameters, rotations, and rounding decisions are chosen\.

Across all experiments, we quantize linear projection weights using group size 32, matching the GGUF block convention used byllama\.cpp\. Unless otherwise stated \(i\.e\., Table[2](https://arxiv.org/html/2607.08779#S5.T2)\), we evaluate models with dynamicQ8\_0activation quantization\. Weight grids are calculated \(or learned\) per group, while activation grids are computed dynamically from the corresponding \(potentially transformed\) activations per\-token during each forward pass\.

### C\.1Direct casting

The direct\-casting experiments in Table[2](https://arxiv.org/html/2607.08779#S5.T2)isolate the effect of the scale\-sign rule\. We quantize all linear projection weights groupwise with group size 32 and do not use rotations, error correction, learned grid parameters, calibration data, or activation quantization\. The conventional absmax baseline uses the signed integer alphabet with zero pointz=0z=0, strictly positive scales=α\>0s=\\alpha\>0, andα=‖w‖∞/2q−1\\alpha=\\\|w\\\|\_\{\\infty\}/2^\{q\-1\}per group\. The signed absmax variant uses the same alphabet and zero point, but setss=γ​αs=\\gamma\\alpha, whereγ∈\{\+1,−1\}\\gamma\\in\\\{\+1,\-1\\\}is chosen by Equation[5](https://arxiv.org/html/2607.08779#S3.E5)\. All values are rounded to nearest and clipped to the target integer alphabet\.

### C\.2Algorithm details

![Refer to caption](https://arxiv.org/html/2607.08779v1/x2.png)Figure 2:We illustrated the mergeable rotation graph used by the PTQ pipelines behind Tables[3](https://arxiv.org/html/2607.08779#S5.T3)and[4](https://arxiv.org/html/2607.08779#S5.T4), with full pipeline details in Appendices[C\.2](https://arxiv.org/html/2607.08779#A3.SS2)and[C\.3](https://arxiv.org/html/2607.08779#A3.SS3)\. Normalized Hadamard rotations are inserted only at rotation\-invariant points where they fold into adjacent linear projections at deployment, leaving the graph architecture, and therefore thellama\.cppruntime contract, unchanged\. This figure is adapted from Liu et al\.\[[20](https://arxiv.org/html/2607.08779#bib.bib17)\]The experiments in Table[3](https://arxiv.org/html/2607.08779#S5.T3)evaluate whether the signed grid remains useful when composed with a practical PTQ pipeline\. We first insert normalized Hadamard rotations into rotation\-invariant regions of the model only where rotations can be fully merged, as shown in Figure[2](https://arxiv.org/html/2607.08779#A3.F2)\. Note that thellama\.cppruntime assumes a fixed contract on the model \(i\.e\., no changes to the graph architecture and therefore no online rotations can be added\), which is a notable departure from QuaRot\[[2](https://arxiv.org/html/2607.08779#bib.bib16)\]and SpinQuant\[[20](https://arxiv.org/html/2607.08779#bib.bib17)\]\. These rotations are merged before grid calculation, so weight grids are calculated based on the rotated weights\. Activations are quantized dynamically withQ8\_0, with activation quantization grids computed from the rotated activations\. Inllama\.cpp\[[12](https://arxiv.org/html/2607.08779#bib.bib47)\],Q8\_0on activations assumes a “narrow range” quantizer, where activations are scaled to be symmetric such that𝒜q=\{−K,…,K\}\\mathcal\{A\}\_\{q\}=\\\{\-K,\\ldots,K\\\}forK=2q−1−1K=2^\{q\-1\}\-1\. We emulate this quantizer within Brevitas during calibration\.

After the rotations and grids are fixed, we apply Qronos as the adaptive rounding method\[[33](https://arxiv.org/html/2607.08779#bib.bib18)\]\. Qronos is run with the same transformed graph and dynamic activation\-quantized execution path used for evaluation, so that the rounding decisions account for the mismatch between the quantized model and its full\-precision counterpart underQ8\_0activation quantization\.

For the studied formats,Q4\_0uses the conventional symmetric grid withz=0z=0ands\>0s\>0\.Q4\_0suses the same signed alphabet andz=0z=0, but allowss=γ​αs=\\gamma\\alphawithγ∈\{\+1,−1\}\\gamma\\in\\\{\+1,\-1\\\}\.Q4\_1uses the corresponding asymmetric min\-max grid\. We use analogous definitions for Q3 and Q2\.

For Qronos, we use 128 calibration sequences of length 2048\. The covariance matrix is damped as in Qronos, with damping parameterλ=10−4​σ1\\lambda=10^\{\-4\}\\,\\sigma\_\{1\}whereσ1\\sigma\_\{1\}is the largest singular value of the covariance matrix\. We quantize weights in descending order of the diagonal of the covariance matrix\[[16](https://arxiv.org/html/2607.08779#bib.bib56)\]\. All Qronos hyperparameters are held fixed acrossQ4\_1,Q4\_0s, andQ4\_0at the same bit width\.

### C\.3Jointly learned rotations and grid parameters

Table[4](https://arxiv.org/html/2607.08779#S5.T4)evaluates a data\-driven pipeline in which rotations and grid parameters are learned jointly before applying Qronos\. The goal is to test whether the signed scale remains useful when the quantization pipeline is allowed to adapt from data, rather than relying only on the closed\-form signed absmax rule\.

We initialize rotations and grid parameters from the fixed\-rotation pipeline in Table[3](https://arxiv.org/html/2607.08779#S5.T3)\. Starting from this initialization, we jointly optimize the rotations and grid parameters for 100 steps on FineWeb\[[22](https://arxiv.org/html/2607.08779#bib.bib54)\], using batches of 8 sequences\. Rotations are optimized on the Stiefel manifold using Cayley SGD, following SpinQuant\[[20](https://arxiv.org/html/2607.08779#bib.bib17)\]\. Quantized weights and activations are simulated during optimization using the straight\-through estimator\[[3](https://arxiv.org/html/2607.08779#bib.bib55),[8](https://arxiv.org/html/2607.08779#bib.bib15)\]\.

The three grid families differ only in their parameter constraints\. ForQ4\_1, we learn the scale and zero\-point offset jointly, as in LSQ\+\[[4](https://arxiv.org/html/2607.08779#bib.bib46)\]\. We follow LSQ\[[9](https://arxiv.org/html/2607.08779#bib.bib48)\]when learning scales forQ4\_0andQ4\_0s, constraining the scale to be strictly positive \(s\>0s\>0\) forQ4\_0and non\-zero \(s≠0s\\neq 0\) forQ4\_0s, with the latter initialized from the signed absmax grid\. We use analogous parameterizations for Q3 and Q2\. After the 100\-step joint optimization, the learned rotations are merged and the grid parameters are frozen, and Qronos is applied using the same adaptive rounding setup described in Appendix[C\.2](https://arxiv.org/html/2607.08779#A3.SS2)\.

### C\.4Implementation notes

All quantization experiments are implemented using Brevitas\[[29](https://arxiv.org/html/2607.08779#bib.bib45)\]and run on an AMD Instinct™ MI325X GPU with 256 GB of GPU memory\. The learned grid experiments use the same quantization graph as the fixed\-rotation experiments; only the procedure for choosing rotations and grid parameters changes\. In particular,Q4\_0sandQ4\_0share the same deployed symmetric inference path inllama\.cpp\. The signed variant changes the sign of the stored scale but does not introduce a zero point or any additional runtime arithmetic\.