CubicQuant: Parametric Non-Uniform Codebooks for High-Throughput LLM Inference with 1-8-Bit Weights

# Parametric Non-Uniform Codebooks for High-Throughput LLM Inference with 1–8-Bit Weights
Source: [https://arxiv.org/html/2608.06763](https://arxiv.org/html/2608.06763)
\(Technical Report · August 2026\)

###### Abstract

Weight quantization for large\-language\-model inference must reconcile two competing goals: reconstruction levels should adapt to the strongly varying local statistics of model weights, yet their representation should remain compact and regular enough for direct GPU execution\. Uniform integers favor regularity but constrain every group to a linear grid\. Low\-bit floating\-point formats redistribute levels through a fixed exponent–mantissa structure, whereas learned codebooks gain flexibility at the cost of less regular decoding and additional metadata\.

We introduce CubicQuant, a parametric non\-uniform scalar format that preserves a dense integer code stream while adapting its reconstruction levels independently within each weight group\. A monotonic cubic curve, specified by two shape parameters and one scale, maps uniformly spaced magnitude codes to non\-uniform levels\. The family spans 1–8\-bit weight payloads, contains symmetric uniform integer quantization as an exact special case, and has effective widthB\+64/GB\+64/Gbits per weight for payload widthBBand group sizeGG\. We derive its population distortion under Uniform, Gaussian, and Laplace distributions, formulate both continuous and Dynamic\-A8\-carrier\-aware fitting objectives, and describe direct packed\-weight GPU execution with model\-dtype or dynamically quantized INT8 activations\.

In a finite\-group G128 experiment with 15,360 samples per distribution, W4 CubicQuant reduced reconstruction RMSE relative to optimally clipped four\-bit uniform integer quantization by 3\.90% on Uniform, 13\.49% on Gaussian, and 28\.14% on Laplace samples\. Relative to the best enumerated four\-bit finite floating\-point format, the reductions were 3\.90%, 9\.44%, and 6\.27%\. Preliminary H200 kernel measurements further reveal a workload\-dependent crossover: model\-dtype execution is faster for narrow GEMV, while Dynamic A8 becomes favorable as row count grows\. Together, the results establish the representational promise and executable character of the format, while leaving downstream model quality and cross\-device end\-to\-end performance as open evaluation questions\.

## 1Introduction

Autoregressive inference repeatedly streams large weight matrices through GPU memory\. Reducing their stored width directly relieves capacity and bandwidth pressure, particularly in decode\-like regimes where weight reuse across tokens is limited\. At low precision, however, storage width alone is not sufficient: the placement of the few available reconstruction levels determines whether a format preserves the mass near zero, the tails, or neither\. A useful format must therefore balance statistical adaptability against the regularity required by high\-throughput matrix kernels\.

The established scalar choices occupy distinct points on this trade\-off\. Uniform integers offer dense packing and simple arithmetic, but every group is restricted to an affine level grid\. Floating\-point formats devote bits to a fixed exponent–mantissa hierarchy and thereby favor a predetermined dynamic range\. Free or learned codebooks can adapt all levels, yet their lookup and metadata structure is less naturally aligned with conventional GPU dot\-product paths\. Vector and additive codebooks enlarge the design space further, but also change the decoding problem from scalar reconstruction to structured lookup\.

CubicQuant occupies the middle ground between a fixed grid and a free codebook\. It retains scalar integer codes and a regular packed bitstream, but makes the represented level positions parametric\. Within each group, a normalized monotonic cubic maps uniformly spaced magnitude indices to non\-uniform reconstruction levels\. Two shape coefficients control the distribution of interior levels and a scale fixes the endpoint\. The resulting codebook is adaptive, yet sufficiently structured to be regenerated from compact metadata inside the consuming GPU tile\.

Four requirements shape the format:

1. 1\.Compactness\.Weight codes remain densely packed from one to eight bits, and metadata cost is explicit rather than hidden in a model\-level average\.
2. 2\.A stable numerical contract\.Zero and both signed endpoints are exact; the scale is persisted and applied in FP32; the codebook is monotonic\.
3. 3\.Estimator independence\.The representation does not prescribe how its codes and parameters are estimated; data\-free, activation\-aware, and second\-order objectives can share the same format\.
4. 4\.Direct GPU execution\.Cubic weights can be consumed directly by GPU kernels with either BF16/FP16 activations or dynamically quantized INT8 activations\. The packed weights are reconstructed within each compute tile rather than materialized as a full\-precision tensor, while the kernel strategy may vary with tensor shape and GPU architecture without changing the numerical format\.

This report develops that design through five contributions:

- •a symmetric, groupwise, two\-parameter family of scalar non\-uniform codebooks that contains uniform signed integer quantization as an exact special case;
- •a packed checkpoint format spanning one\- to eight\-bit weights, with FP32 scaling and a closed\-form storage overhead;
- •a groupwise reference fitting procedure and a Dynamic\-A8\-carrier\-aware objective, both separable from richer data\-driven error estimators;
- •GPU execution paths for model\-dtype and Dynamic\-A8 activations, with device\- and tensor\-shape\-aware calibration; and
- •analytical memory and operation models together with preliminary numerical and Hopper kernel evidence that delimit the present claims\.

CubicQuant is developed within the QuantTrio open\-source project\. Its reference implementation uses vLLM as a systems substrate, while the contribution studied here is confined to the weight representation, fitting objectives, packed operators, and their evaluation\. Scheduler, attention, and memory\-management behavior are not attributed to the format\[[24](https://arxiv.org/html/2608.06763#bib.bib24)\]\.

The remainder of the report follows the format from definition to evidence\. Section 2 specifies the code space, level function, and metadata\. Section 3 derives population distortion and connects it to finite\-group behavior\. Section 4 formulates offline fitting, including the deterministic Dynamic\-A8 carrier projection\. Section 5 describes packed GPU realization, and Section 6 separates analytical properties from the presently available numerical and kernel evidence\. Sections 7–9 position the work, state its limitations, and conclude\.

## 2CubicQuant representation

Throughout this report,Wb​AaW\_\{b\}A\_\{a\}denotes abb\-bit weight payload and anaa\-bit activation precision; W4A8 therefore means four\-bit weights with eight\-bit activations\. After this definition, W1–W8 is used as a range notation for the eight supported weight payload widths, not as the name of a single format\.

### 2\.1Signed codes

LetBBbe the payload width\. ForB\>1B\>1, define

M=2B−1−1,k∈\{−M,…,0,…,M\}\.M=2^\{B\-1\}\-1,\\qquad k\\in\\\{\-M,\\ldots,0,\\ldots,M\\\}\.\(1\)
The signed code range has2B−12^\{B\}\-1valid values\. The remaining two’s\-complement bit pattern,−2B−1\-2^\{B\-1\}, is reserved and decodes deterministically to zero\. This choice gives an exact zero while preserving symmetric positive and negative endpoints\. W1 is a separate binary case with codes in\{−1,\+1\}\\\{\-1,\+1\\\}and no zero code\.

Packed codes are stored as a continuous bitstream in row\-major logical order\. At widths that do not divide eight, such as W3, W5, W6, and W7, an individual code may span two adjacent bytes\. The format therefore specifies both cross\-byte decoding and the treatment of unused bits at the end of each logical row\.

### 2\.2Parametric level function

ForB\>1B\>1, reconstruction is defined by

tk\\displaystyle t\_\{k\}=\|k\|M,\\displaystyle=\\frac\{\|k\|\}\{M\},\(2\)c\\displaystyle c=1−a−b,\\displaystyle=1\-a\-b,q​\(t\)\\displaystyle q\(t\)=t​\[a\+t​\(b\+c​t\)\],\\displaystyle=t\\left\[a\+t\\left\(b\+ct\\right\)\\right\],w^​\(k\)\\displaystyle\\widehat\{w\}\(k\)=sign⁡\(k\)​s​q​\(tk\)\.\\displaystyle=\\operatorname\{sign\}\(k\)\\,s\\,q\(t\_\{k\}\)\.
The reconstruction combines a stored integer code with group\-local scale and shape parameters\. The notation and the role of each quantity are summarized below\.

The factorttgivesq​\(0\)=0q\(0\)=0, whilec=1−a−bc=1\-a\-bgivesq​\(1\)=1q\(1\)=1\. Thus zero and both group endpoints are exact by construction\. The two stored shape coefficients jointly redistribute the interior reconstruction levels\. Candidate shape parameters must also satisfy

q′​\(t\)=a\+2​b​t\+3​\(1−a−b\)​t2\>0,t∈\[0,1\]\.q^\{\\prime\}\(t\)=a\+2bt\+3\(1\-a\-b\)t^\{2\}\>0,\\qquad t\\in\[0,1\]\.\(3\)
which prevents level inversions\. Strict monotonicity can be checked cheaply because the derivative is quadratic\.

Uniform symmetric integer quantization is not merely a baseline adjacent to the format\. It is a point inside the family:

a=1,b=0,c=0⟹q​\(t\)=t\.a=1,\\qquad b=0,\\qquad c=0\\quad\\Longrightarrow\\quad q\(t\)=t\.\(4\)
Including this point in every fitting search ensures that, before metadata rounding, a Cubic candidate need not be worse than the corresponding optimally clipped uniform\-integer candidate under the same objective\.

Figure[1](https://arxiv.org/html/2608.06763#S2.F1)illustrates how the two shape coefficients redistribute levels without changing either endpoint\. Curves below the linear grid assign smaller reconstructed magnitudes to a given interior code index, thereby concentrating more discrete levels near zero while preserving the group scale att=1t=1\.

00\.10\.10\.20\.20\.30\.30\.40\.40\.50\.50\.60\.60\.70\.70\.80\.80\.90\.91100\.20\.20\.40\.40\.60\.60\.80\.811ttq​\(t\)q\(t\)a=1\.0,b=0\.0a=1\.0,\\ b=0\.0\(INT\)a=0\.8,b=−0\.4a=0\.8,\\ b=\-0\.4a=0\.5,b=−0\.3a=0\.5,\\ b=\-0\.3Figure 1:Illustrative members of the normalized Cubic level family\. All curves satisfyq​\(0\)=0q\(0\)=0andq​\(1\)=1q\(1\)=1; the shape parameters alter only the interior level placement\.
### 2\.3Metadata and effective width

Let a Linear weight matrix have logical shapeN×KN\\times K\. CubicQuant uses a\[1,G\]\[1,G\]group shape: each output row is divided independently into contiguous groups ofGGinput\-channel weights\. Group boundaries are identical across rows, but the fitted metadata is not shared\. The three metadata tensors therefore have logical shapeN×\(K/G\)N\\times\(K/G\)whenKKis divisible byGG\.

Each group storesG​BGBpayload bits, one FP32 scaless, and two FP16 shape coefficientsaaandbb\.

The metadata is therefore eight bytes per group\. Ignoring row\-tail padding and container metadata, its contribution is

N​\(K/G\)​64N​K=64Gbits per weight,\\frac\{N\(K/G\)\\,64\}\{NK\}=\\frac\{64\}\{G\}\\qquad\\text\{bits per weight\},\(5\)
and the effective stored width is

Beff=B\+64Gbits per weight\.B\_\{\\mathrm\{eff\}\}=B\+\\frac\{64\}\{G\}\\qquad\\text\{bits per weight\}\.\(6\)
The following comparison places this layout beside two representative weight scaling schemes\. It describes both metadata geometry and effective stored width, not a universal rule for all integer or floating\-point formats\.

The\[1,G\]\[1,G\]integer layout is the common groupwise weight\-only case; asymmetric variants additionally store a zero point\. The FP8 row represents two\-dimensional\[G,G\]\[G,G\]weight scaling; activation scaling may instead use\[1,G\]\[1,G\]tiles\. Its smaller scale overhead is not a like\-for\-like representation comparison: each FP8 payload already carries exponent and mantissa fields, whereas INT and Cubic payloads rely more directly on group\-local metadata\.

At the same value ofGG, the effective widths are:

GGSymmetric INT,\[1,G\]\[1,G\]↓\\downarrowCubicQuant,\[1,G\]\[1,G\]↓\\downarrowFP8,\[G,G\]\[G,G\]↓\\downarrow128B\+0\.12500B\+0\.12500B\+0\.500B\+0\.5008\+0\.0019538\+0\.001953256B\+0\.06250B\+0\.06250B\+0\.250B\+0\.2508\+0\.0004888\+0\.000488512B\+0\.03125B\+0\.03125B\+0\.125B\+0\.1258\+0\.0001228\+0\.000122Larger groups amortize metadata and codebook\-generation work, but provide one curve for a wider region of the tensor\. Whether the additional shape freedom is enough to compensate for that coarser granularity is an empirical question; it is tested rather than assumed\.

The shape parameters use FP16, whereas the scale remains FP32\. This asymmetry keeps the dominant magnitude in higher precision while allowing the two dimensionless shape coefficients to contribute only four metadata bytes per group\.

## 3Quantization distortion under reference distributions

This section gives a population reference for the finite\-group experiment in Section 3\.6\. LetXXbe a zero\-mean, unit\-variance symmetric random variable with densityf​\(x\)f\(x\), and letQQbe a scalar quantizer\. Its mean\-squared distortion is

D​\(Q\)=𝔼​\[\(X−Q​\(X\)\)2\]\.D\(Q\)=\\mathbb\{E\}\\\!\\left\[\(X\-Q\(X\)\)^\{2\}\\right\]\.\(7\)
Throughout this section,QQreconstructs the continuous Cubic levelss​q​\(tk\)sq\(t\_\{k\}\)\. Accordingly,DCD\_\{\\rm C\}and every numerical value reported below are pre\-carrier population references\. They characterize the representation itself before the additional INT8 grid projection used by Dynamic\-A8 execution\. Section 4\.3 defines the joint continuous/carrier objective adopted for dual\-path representation fitting; its numerical optimum is a different quantity and must not be inferred from the tables in this section\.

We first derive an exact expression in which the Cubic shape parameters are explicit\. We then bound the best Cubic quantizer between a free Lloyd–Max codebook and the uniform\-integer member of the Cubic family\.

### 3\.1Exact distortion and explicit shape dependence

ForB\>1B\>1, writeM=2B−1−1M=2^\{B\-1\}\-1,ti=i/Mt\_\{i\}=i/M, and define three fixed basis terms

ri=ti3,ui=ti−ti3,vi=ti2−ti3\.r\_\{i\}=t\_\{i\}^\{3\},\\qquad u\_\{i\}=t\_\{i\}\-t\_\{i\}^\{3\},\\qquad v\_\{i\}=t\_\{i\}^\{2\}\-t\_\{i\}^\{3\}\.\(8\)
Substitutingc=1−a−bc=1\-a\-binto the level function and collecting the coefficients ofaaandbbgives

q​\(t\)=a​t\+b​t2\+\(1−a−b\)​t3=t3\+a​\(t−t3\)\+b​\(t2−t3\)\.q\(t\)=at\+bt^\{2\}\+\(1\-a\-b\)t^\{3\}=t^\{3\}\+a\(t\-t^\{3\}\)\+b\(t^\{2\}\-t^\{3\}\)\.\(9\)
Evaluating this identity attit\_\{i\}makes each stored level affine in the two shape parameters:

yi=s​q​\(ti\)=s​\(ri\+a​ui\+b​vi\)\.y\_\{i\}=s\\,q\(t\_\{i\}\)=s\\left\(r\_\{i\}\+au\_\{i\}\+bv\_\{i\}\\right\)\.\(10\)
The nearest\-neighbor boundary between adjacent levels is therefore

hi=s2​\[ri\+ri\+1\+a​\(ui\+ui\+1\)\+b​\(vi\+vi\+1\)\]\.h\_\{i\}=\\frac\{s\}\{2\}\\left\[r\_\{i\}\+r\_\{i\+1\}\+a\(u\_\{i\}\+u\_\{i\+1\}\)\+b\(v\_\{i\}\+v\_\{i\+1\}\)\\right\]\.\(11\)
Equations \([10](https://arxiv.org/html/2608.06763#S3.E10)\) and \([11](https://arxiv.org/html/2608.06763#S3.E11)\) show precisely whereaaandbbenter the distortion: they move both the reconstruction values and their Voronoi boundaries\. Withh−1=0h\_\{\-1\}=0andhM=∞h\_\{M\}=\\infty, symmetry gives

D​\(s,a,b\)=2​∑i=0M∫hi−1hi\(x−yi\)2​f​\(x\)​dx\.D\(s,a,b\)=2\\sum\_\{i=0\}^\{M\}\\int\_\{h\_\{i\-1\}\}^\{h\_\{i\}\}\(x\-y\_\{i\}\)^\{2\}f\(x\)\\,\\mathrm\{d\}x\.\(12\)
For each quantization cell, define its order\-rrcell moment

μr​\(ℓ,u\)=∫ℓuxr​f​\(x\)​dx,r∈\{0,1,2\}\.\\mu\_\{r\}\(\\ell,u\)=\\int\_\{\\ell\}^\{u\}x^\{r\}f\(x\)\\,\\mathrm\{d\}x,\\qquad r\\in\\\{0,1,2\\\}\.\(13\)
Expanding the square turns each cell integral into

μ2​\(hi−1,hi\)−2​yi​μ1​\(hi−1,hi\)\+yi2​μ0​\(hi−1,hi\)\.\\mu\_\{2\}\(h\_\{i\-1\},h\_\{i\}\)\-2y\_\{i\}\\mu\_\{1\}\(h\_\{i\-1\},h\_\{i\}\)\+y\_\{i\}^\{2\}\\mu\_\{0\}\(h\_\{i\-1\},h\_\{i\}\)\.\(14\)
For compactness, putRi​\(a,b\)=ri\+a​ui\+b​viR\_\{i\}\(a,b\)=r\_\{i\}\+au\_\{i\}\+bv\_\{i\}\. Substitutingyi=s​Riy\_\{i\}=sR\_\{i\}andhi=s​\(Ri\+Ri\+1\)/2h\_\{i\}=s\(R\_\{i\}\+R\_\{i\+1\}\)/2into the preceding equation gives the complete three\-parameter objective

DB​\(s,a,b\)=2​∑i=0M\[μ2−2​s​Ri​\(a,b\)​μ1\+s2​Ri​\(a,b\)2​μ0\]\(hi−1,hi\)\\boxed\{D\_\{B\}\(s,a,b\)=2\\sum\_\{i=0\}^\{M\}\\left\[\\mu\_\{2\}\-2sR\_\{i\}\(a,b\)\\mu\_\{1\}\+s^\{2\}R\_\{i\}\(a,b\)^\{2\}\\mu\_\{0\}\\right\]\_\{\(h\_\{i\-1\},h\_\{i\}\)\}\}\(15\)
where\[μr\]\(hi−1,hi\)\[\\mu\_\{r\}\]\_\{\(h\_\{i\-1\},h\_\{i\}\)\}means that every cell moment is evaluated on that cell’s two boundaries\. Thusaaandbbare present both explicitly inRiR\_\{i\}and implicitly through the limitshi−1,hih\_\{i\-1\},h\_\{i\}; treating the boundaries as fixed while changing the curve would give the wrong distortion\.

For the unit\-variance reference distributions, the required positive\-half moments are closed form\. LetA=3A=\\sqrt\{3\},β=1/2\\beta=1/\\sqrt\{2\},Ex=exp⁡\(−x/β\)E\_\{x\}=\\exp\(\-x/\\beta\), and letϕ\\phiandΦ\\Phidenote the standard\-normal PDF and CDF\. For Uniform, replace\[ℓ,u\]\[\\ell,u\]by its intersection with the support\[0,A\]\[0,A\]\. Then

Uniform:μr​\(ℓ,u\)\\displaystyle\\text\{Uniform:\}\\quad\\mu\_\{r\}\(\\ell,u\)=ur\+1−ℓr\+12​A​\(r\+1\),\\displaystyle=\\frac\{u^\{r\+1\}\-\\ell^\{r\+1\}\}\{2A\(r\+1\)\},\(16\)Gaussian:μ0\\displaystyle\\text\{Gaussian:\}\\quad\\mu\_\{0\}=Φ​\(u\)−Φ​\(ℓ\),\\displaystyle=\\Phi\(u\)\-\\Phi\(\\ell\),\(17\)μ1\\displaystyle\\mu\_\{1\}=ϕ​\(ℓ\)−ϕ​\(u\),\\displaystyle=\\phi\(\\ell\)\-\\phi\(u\),\(18\)μ2\\displaystyle\\mu\_\{2\}=μ0\+ℓ​ϕ​\(ℓ\)−u​ϕ​\(u\),\\displaystyle=\\mu\_\{0\}\+\\ell\\phi\(\\ell\)\-u\\phi\(u\),\(19\)Laplace:μ0\\displaystyle\\text\{Laplace:\}\\quad\\mu\_\{0\}=12​\(Eℓ−Eu\),\\displaystyle=\\tfrac\{1\}\{2\}\(E\_\{\\ell\}\-E\_\{u\}\),\(20\)μ1\\displaystyle\\mu\_\{1\}=12​\[\(ℓ\+β\)​Eℓ−\(u\+β\)​Eu\],\\displaystyle=\\tfrac\{1\}\{2\}\[\(\\ell\+\\beta\)E\_\{\\ell\}\-\(u\+\\beta\)E\_\{u\}\],\(21\)μ2\\displaystyle\\mu\_\{2\}=12​\[\(ℓ2\+2​β​ℓ\+2​β2\)​Eℓ−\(u2\+2​β​u\+2​β2\)​Eu\],\\displaystyle=\\tfrac\{1\}\{2\}\[\(\\ell^\{2\}\+2\\beta\\ell\+2\\beta^\{2\}\)E\_\{\\ell\}\-\(u^\{2\}\+2\\beta u\+2\\beta^\{2\}\)E\_\{u\}\],\(22\)
For the unbounded final cell, the upper\-endpoint terms are evaluated in the limitu→∞u\\to\\inftyand vanish\. Substitution into \([14](https://arxiv.org/html/2608.06763#S3.E14)\) then gives an exact evaluation ofD​\(s,a,b\)D\(s,a,b\)\.

These expressions follow directly from elementary antiderivatives\. Uniform uses∫xr​dx=xr\+1/\(r\+1\)\\int x^\{r\}\\,\\mathrm\{d\}x=x^\{r\+1\}/\(r\+1\)\. For Gaussian,ϕ′​\(x\)=−x​ϕ​\(x\)\\phi^\{\\prime\}\(x\)=\-x\\phi\(x\)gives

∫x​ϕ​\(x\)​dx=−ϕ​\(x\),∫x2​ϕ​\(x\)​dx=Φ​\(x\)−x​ϕ​\(x\)\.\\int x\\phi\(x\)\\,\\mathrm\{d\}x=\-\\phi\(x\),\\qquad\\int x^\{2\}\\phi\(x\)\\,\\mathrm\{d\}x=\\Phi\(x\)\-x\\phi\(x\)\.\(23\)
For the positive half of the unit\-variance Laplace density,f​\(x\)=Ex/\(2​β\)f\(x\)=E\_\{x\}/\(2\\beta\); integratingxr​Exx^\{r\}E\_\{x\}forr=0,1,2r=0,1,2produces the three endpoint differences in \([22](https://arxiv.org/html/2608.06763#S3.E22)\)\. No distribution tail is discarded: the last cell is evaluated with upper limitu→∞u\\to\\infty\.

The monotonicity condition also gives an explicit feasible domain for the shape search\. Putc=1−a−bc=1\-a\-b\. Sinceq′​\(t\)=a\+2​b​t\+3​c​t2q^\{\\prime\}\(t\)=a\+2bt\+3ct^\{2\}, its minimum over\[0,1\]\[0,1\]is attained at an endpoint or, when it lies inside the interval, att∗=−b/\(3​c\)t\_\{\*\}=\-b/\(3c\)\. The endpoint values areq′​\(0\)=aq^\{\\prime\}\(0\)=aandq′​\(1\)=3−2​a−bq^\{\\prime\}\(1\)=3\-2a\-b\. Ifc≤0c\\leq 0, the quadratic is concave and its minimum on a closed interval is an endpoint\. Ifc\>0c\>0, an interior minimum exists exactly when0<t∗<10<t\_\{\*\}<1, equivalently−3​c<b<0\-3c<b<0, and substitution givesq′​\(t∗\)=a−b2/\(3​c\)q^\{\\prime\}\(t\_\{\*\}\)=a\-b^\{2\}/\(3c\)\. Hence

m​\(a,b\)=\{min⁡\{a,3−2​a−b,a−b23​c\},c\>0​and−3​c<b<0,min⁡\{a,3−2​a−b\},otherwise,m\(a,b\)=\\begin\{cases\}\\min\\\!\\left\\\{a,\\,3\-2a\-b,\\,a\-\\dfrac\{b^\{2\}\}\{3c\}\\right\\\},&c\>0\\ \\text\{and\}\\ \-3c<b<0,\\\\\[5\.0pt\] \\min\\\!\\left\\\{a,\\,3\-2a\-b\\right\\\},&\\text\{otherwise\},\\end\{cases\}\(24\)
The admissible closure is𝒜=\{\(a,b\):m​\(a,b\)≥0\}\\mathcal\{A\}=\\\{\(a,b\):m\(a,b\)\\geq 0\\\}\. Together, Equations \([10](https://arxiv.org/html/2608.06763#S3.E10)\), \([14](https://arxiv.org/html/2608.06763#S3.E14)\), and \([24](https://arxiv.org/html/2608.06763#S3.E24)\) reduce population fitting to a constrained optimization over only three variables: the scalessand the two stored shape parameters\(a,b\)\(a,b\)\. The next subsection compares the optimum of this constrained family with free and uniform codebooks\.

### 3\.2Cubic lower and upper bounds

Let𝒴B\\mathcal\{Y\}\_\{B\}contain every ordered, symmetric scalar codebook with2B−12^\{B\}\-1reconstruction levels and an exact zero\. Let𝒞B\\mathcal\{C\}\_\{B\}contain only the codebooks generated by \([10](https://arxiv.org/html/2608.06763#S3.E10)\) withs\>0s\>0and\(a,b\)∈𝒜\(a,b\)\\in\\mathcal\{A\}, and letℐB\\mathcal\{I\}\_\{B\}be the scaled uniform integer codebooks obtained at\(a,b\)=\(1,0\)\(a,b\)=\(1,0\)\. These are nested:

ℐB⊆𝒞B⊆𝒴B\.\\mathcal\{I\}\_\{B\}\\subseteq\\mathcal\{C\}\_\{B\}\\subseteq\\mathcal\{Y\}\_\{B\}\.\(25\)
The first inclusion follows by direct substitution: at\(a,b\)=\(1,0\)\(a,b\)=\(1,0\), Equation \([9](https://arxiv.org/html/2608.06763#S3.E9)\) reduces toq​\(t\)=tq\(t\)=t, so every scaled uniform\-integer level set\{0,s/M,2​s/M,…,s\}\\\{0,s/M,2s/M,\\ldots,s\\\}is a member of𝒞B\\mathcal\{C\}\_\{B\}\. The second inclusion follows fromm​\(a,b\)≥0m\(a,b\)\\geq 0: the Cubic levels are ordered, symmetric, contain zero, and number2B−12^\{B\}\-1, which are precisely the defining requirements of𝒴B\\mathcal\{Y\}\_\{B\};𝒴B\\mathcal\{Y\}\_\{B\}simply removes the cubic\-curve restriction\.

For any objectiveDDand feasible sets𝒮1⊆𝒮2\\mathcal\{S\}\_\{1\}\\subseteq\\mathcal\{S\}\_\{2\},

infz∈𝒮2D​\(z\)≤infz∈𝒮1D​\(z\),\\inf\_\{z\\in\\mathcal\{S\}\_\{2\}\}D\(z\)\\leq\\inf\_\{z\\in\\mathcal\{S\}\_\{1\}\}D\(z\),\(26\)
because the minimization on the left may choose every candidate available on the right and possibly more\. Applying \([26](https://arxiv.org/html/2608.06763#S3.E26)\) first to𝒞B⊆𝒴B\\mathcal\{C\}\_\{B\}\\subseteq\\mathcal\{Y\}\_\{B\}and then toℐB⊆𝒞B\\mathcal\{I\}\_\{B\}\\subseteq\\mathcal\{C\}\_\{B\}proves

DLM​\(B\)⏟minY∈𝒴B⁡D​\(Y\)≤DC​\(B\)⏟mins\>0,\(a,b\)∈𝒜⁡D​\(s,a,b\)≤DINT​\(B\)⏟mins\>0⁡D​\(s,1,0\)\.\\underbrace\{D\_\{\\mathrm\{LM\}\}\(B\)\}\_\{\\displaystyle\\min\_\{Y\\in\\mathcal\{Y\}\_\{B\}\}D\(Y\)\}\\;\\leq\\;\\underbrace\{D\_\{\\mathrm\{C\}\}\(B\)\}\_\{\\displaystyle\\min\_\{s\>0,\(a,b\)\\in\\mathcal\{A\}\}D\(s,a,b\)\}\\;\\leq\\;\\underbrace\{D\_\{\\mathrm\{INT\}\}\(B\)\}\_\{\\displaystyle\\min\_\{s\>0\}D\(s,1,0\)\}\.\(27\)
A finite floating\-point family does not extend this chain\. For anE​e​M​m\\mathrm\{E\}e\\mathrm\{M\}mformat with one sign bit,e\+m=B−1e\+m=B\-1, let all exponent/mantissa patterns denote finite values\. With exponent fieldEE, mantissa fieldJJ, and biasγe=2e−1−1\\gamma\_\{e\}=2^\{e\-1\}\-1, its nonnegative raw levels are

ge,m​\(E,J\)=\{J​2−m​21−γe,E=0,\(1\+J​2−m\)​2E−γe,E\>0\.g\_\{e,m\}\(E,J\)=\\begin\{cases\}J\\,2^\{\-m\}\\,2^\{1\-\\gamma\_\{e\}\},&E=0,\\\\ \(1\+J\\,2^\{\-m\}\)\\,2^\{E\-\\gamma\_\{e\}\},&E\>0\.\\end\{cases\}\(28\)
Sort the distinct nonnegative values and normalize by the largest one:0=z0<z1<⋯<zM=10=z\_\{0\}<z\_\{1\}<\\cdots<z\_\{M\}=1, whereM=2B−1−1M=2^\{B\-1\}\-1\. With an external scaless, the positive reconstruction levels and their nearest\-neighbor boundaries are

yiFP=s​zi,κi​\(s\)=s2​\(zi\+zi\+1\),κ−1=0,κM=∞\.y\_\{i\}^\{\\mathrm\{FP\}\}=sz\_\{i\},\\qquad\\kappa\_\{i\}\(s\)=\\frac\{s\}\{2\}\(z\_\{i\}\+z\_\{i\+1\}\),\\qquad\\kappa\_\{\-1\}=0,\\quad\\kappa\_\{M\}=\\infty\.\(29\)
Substituting these quantities into the same cell\-moment expansion used for Cubic gives the complete FP objective

De,m​\(s\)=2​∑i=0M\[μ2​\(κi−1,κi\)−2​s​zi​μ1​\(κi−1,κi\)\+s2​zi2​μ0​\(κi−1,κi\)\]\.D\_\{e,m\}\(s\)=2\\sum\_\{i=0\}^\{M\}\\left\[\\mu\_\{2\}\(\\kappa\_\{i\-1\},\\kappa\_\{i\}\)\-2sz\_\{i\}\\mu\_\{1\}\(\\kappa\_\{i\-1\},\\kappa\_\{i\}\)\+s^\{2\}z\_\{i\}^\{2\}\\mu\_\{0\}\(\\kappa\_\{i\-1\},\\kappa\_\{i\}\)\\right\]\.\(30\)
The best finite\-FP reference at widthBBis therefore

DFP​\(B\)=mine\+m=B−1⁡mins\>0⁡De,m​\(s\)\.D\_\{\\mathrm\{FP\}\}\(B\)=\\min\_\{e\+m=B\-1\}\\;\\min\_\{s\>0\}D\_\{e,m\}\(s\)\.\(31\)
It obeys the separate boundDLM​\(B\)≤DFP​\(B\)D\_\{\\mathrm\{LM\}\}\(B\)\\leq D\_\{\\mathrm\{FP\}\}\(B\)because every finite\-FP codebook is also a member of the unrestricted set𝒴B\\mathcal\{Y\}\_\{B\}\. The finite\-FP and Cubic families do not contain one another, so there is no distribution\-free ordering between them; their comparison must be evaluated directly\.

It remains to characterize the free\-codebook lower bound\. For fixed adjacent levelsyi<yi\+1y\_\{i\}<y\_\{i\+1\}, nearest\-neighbor assignment places their boundary where their squared errors are equal:

\(hi−yi\)2=\(hi−yi\+1\)2⟹hi=yi\+yi\+12\.\(h\_\{i\}\-y\_\{i\}\)^\{2\}=\(h\_\{i\}\-y\_\{i\+1\}\)^\{2\}\\quad\\Longrightarrow\\quad h\_\{i\}=\\frac\{y\_\{i\}\+y\_\{i\+1\}\}\{2\}\.\(32\)
For fixed boundaries, differentiating the distortion of celliiwith respect to its reconstruction value gives

∂∂yi​∫hi−1hi\(x−yi\)2​f​\(x\)​dx=2​\[yi​μ0​\(hi−1,hi\)−μ1​\(hi−1,hi\)\]\.\\frac\{\\partial\}\{\\partial y\_\{i\}\}\\int\_\{h\_\{i\-1\}\}^\{h\_\{i\}\}\(x\-y\_\{i\}\)^\{2\}f\(x\)\\,\\mathrm\{d\}x=2\\left\[y\_\{i\}\\mu\_\{0\}\(h\_\{i\-1\},h\_\{i\}\)\-\\mu\_\{1\}\(h\_\{i\-1\},h\_\{i\}\)\\right\]\.\(33\)
Setting this derivative to zero yields the conditional mean\. Alternating the two necessary conditions gives the Lloyd–Max fixed point\[[1](https://arxiv.org/html/2608.06763#bib.bib1),[2](https://arxiv.org/html/2608.06763#bib.bib2)\]:

hi\\displaystyle h\_\{i\}=yi\+yi\+12,\\displaystyle=\\frac\{y\_\{i\}\+y\_\{i\+1\}\}\{2\},\(34\)yi\\displaystyle y\_\{i\}=𝔼​\[X∣hi−1≤X<hi\]\.\\displaystyle=\\mathbb\{E\}\[X\\mid h\_\{i\-1\}\\leq X<h\_\{i\}\]\.\(35\)
The resulting codebook belongs to the largest feasible set𝒴B\\mathcal\{Y\}\_\{B\}; its distortion is therefore the lower term in \([27](https://arxiv.org/html/2608.06763#S3.E27)\)\.

ThusDC−DLMD\_\{\\mathrm\{C\}\}\-D\_\{\\mathrm\{LM\}\}measures the restriction imposed by the two\-parameter cubic curve, whileDINT−DCD\_\{\\mathrm\{INT\}\}\-D\_\{\\mathrm\{C\}\}measures the benefit of those two parameters over a linear grid\. We report the fraction of the available gap closed by Cubic as

ηB=DINT−DCDINT−DLM\.\\eta\_\{B\}=\\frac\{D\_\{\\mathrm\{INT\}\}\-D\_\{\\mathrm\{C\}\}\}\{D\_\{\\mathrm\{INT\}\}\-D\_\{\\mathrm\{LM\}\}\}\.\(36\)
Equation \([27](https://arxiv.org/html/2608.06763#S3.E27)\) ensures0≤ηB≤10\\leq\\eta\_\{B\}\\leq 1whenever the denominator is nonzero\. This is a bound on the*optimized Cubic family*, not a claim that every arbitrary\(a,b\)\(a,b\)choice is better than INT\.

### 3\.3Population solutions

ForX∼Uniform​\[−3,3\]X\\sim\\mathrm\{Uniform\}\[\-\\sqrt\{3\},\\sqrt\{3\}\], the optimal oddL=2B−1L=2^\{B\}\-1level codebook has equal\-width cells of widthΔ=2​3/L\\Delta=2\\sqrt\{3\}/L\. The outer positive reconstruction point is the center of the last cell,s∗=3−Δ/2=3​\(1−1/L\)s^\{\*\}=\\sqrt\{3\}\-\\Delta/2=\\sqrt\{3\}\(1\-1/L\)\. Because the density is1/\(2​3\)1/\(2\\sqrt\{3\}\), translating every cell to\[−Δ/2,Δ/2\]\[\-\\Delta/2,\\Delta/2\]gives

D∗=L2​3​∫−Δ/2Δ/2x2​dx=L​Δ324​3=1L2\.D^\{\*\}=\\frac\{L\}\{2\\sqrt\{3\}\}\\int\_\{\-\\Delta/2\}^\{\\Delta/2\}x^\{2\}\\,\\mathrm\{d\}x=\\frac\{L\\Delta^\{3\}\}\{24\\sqrt\{3\}\}=\\frac\{1\}\{L^\{2\}\}\.\(37\)
The cell centers are uniformly spaced, so the corresponding Cubic parameters area∗=1,b∗=0a^\{\*\}=1,b^\{\*\}=0\. Collecting the result,

s∗=3​\(1−1L\),a∗=1,b∗=0,D∗=1L2\.s^\{\*\}=\\sqrt\{3\}\\left\(1\-\\frac\{1\}\{L\}\\right\),\\qquad a^\{\*\}=1,\\quad b^\{\*\}=0,\\qquad D^\{\*\}=\\frac\{1\}\{L^\{2\}\}\.\(38\)
Therefore all three terms in \([27](https://arxiv.org/html/2608.06763#S3.E27)\) are equal for Uniform at W2–W8\. Cubic cannot improve the population optimum there; any finite\-group improvement comes from adapting to sample fluctuations within individual groups\.

W3 provides a second result that follows from the parameter count rather than from a numerical coincidence\. HereM=3M=3and a free symmetric codebook has three positive levels0<y1<y2<y30<y\_\{1\}<y\_\{2\}<y\_\{3\}\. Setp1=y1/y3p\_\{1\}=y\_\{1\}/y\_\{3\},p2=y2/y3p\_\{2\}=y\_\{2\}/y\_\{3\}, ands=y3s=y\_\{3\}\. Evaluating \([9](https://arxiv.org/html/2608.06763#S3.E9)\) att=1/3t=1/3andt=2/3t=2/3gives

27​p1=1\+8​a\+2​b,27​p2=8\+10​a\+4​b\.27p\_\{1\}=1\+8a\+2b,\\qquad 27p\_\{2\}=8\+10a\+4b\.\(39\)
Solving this two\-by\-two system yields

a=1\+9​p1−92​p2,b=18​p2−452​p1−92\.a=1\+9p\_\{1\}\-\\frac\{9\}\{2\}p\_\{2\},\\qquad b=18p\_\{2\}\-\\frac\{45\}\{2\}p\_\{1\}\-\\frac\{9\}\{2\}\.\(40\)
Thus every W3 free\-codebook solution whose recovered\(a,b\)\(a,b\)lies in𝒜\\mathcal\{A\}is represented exactly by Cubic\. The Gaussian and Laplace Lloyd–Max solutions below satisfy this feasibility check, so equality at W3 is explained by \([40](https://arxiv.org/html/2608.06763#S3.E40)\); it is not inferred from rounded table entries\.

For Gaussian and Laplace, we evaluated the cell moments in \([22](https://arxiv.org/html/2608.06763#S3.E22)\), solved Lloyd–Max to convergence, minimized the Cubic objective overssand the feasible\(a,b\)\(a,b\)domain, independently optimized the INT scale, and enumerated all finite\-FP splits in \([31](https://arxiv.org/html/2608.06763#S3.E31)\)\. All distributions have unit variance; values are MSE\. Concretely, every candidate split uses levels from \([28](https://arxiv.org/html/2608.06763#S3.E28)\), evaluates \([30](https://arxiv.org/html/2608.06763#S3.E30)\), minimizes its single scale variable, and competes in the outer minimum of \([31](https://arxiv.org/html/2608.06763#S3.E31)\)\. The best finite\-FP reference table reports that outer minimum and its winning split\.

The finite\-FP comparison uses a different denominator from \([36](https://arxiv.org/html/2608.06763#S3.E36)\)\. Its final column is the direct relative MSE reduction

ρBFP=DFP−DCDFP\.\\rho\_\{B\}^\{\\mathrm\{FP\}\}=\\frac\{D\_\{\\mathrm\{FP\}\}\-D\_\{\\mathrm\{C\}\}\}\{D\_\{\\mathrm\{FP\}\}\}\.\(41\)
Gaussian

Laplace

Best finite\-FP reference

At W2 there is no interior positive level, so\(a,b\)\(a,b\)is not identifiable; the table reports the canonical INT point\. At W3,s,a,bs,a,bprovide exactly three degrees of freedom for the three positive levels, and the optimized Cubic codebook reaches the Lloyd–Max lower bound for both tested unbounded distributions\. From W4 onward,DC\>DLMD\_\{\\mathrm\{C\}\}\>D\_\{\\mathrm\{LM\}\}exposes the cost of describing more than three positive levels with onlys,a,bs,a,b\. Even so, Cubic closes 84\.8–96\.1% of the Gaussian gap and 91\.4–96\.9% of the Laplace gap over W4–W8\. The decreasinga∗a^\{\*\}and increasingly negativeb∗b^\{\*\}move interior levels toward the high\-density center while preserving a more distant endpoint for the tails\.

For Uniform, the finite\-FP split E1M\(B−2\)\(B\-2\)is the same linear grid as INT and therefore also attains1/\(2B−1\)21/\(2^\{B\}\-1\)^\{2\}\. For Gaussian and Laplace, Cubic is better than the best enumerated finite\-FP split at every width above W2 in this population comparison, but this is a numerical result for the stated distributions rather than a consequence of set inclusion\.

### 3\.4Numerical verification of analytic distortion

An independent Monte Carlo estimate checks the analytic distortion in the preceding subsection\. For each distribution, 2,000,000 samples are assigned to their nearest reconstruction levels and their empirical squared error is compared with the closed\-form cell\-moment result\. Across all 21 distribution–bit\-width combinations, the largest discrepancy is 2\.10 estimated standard errors\. HereDanalyticD\_\{\\rm analytic\}is the optimized Cubic population valueDCD\_\{\\rm C\}and SE is the estimated standard error of the Monte Carlo mean\.

Uniform

Gaussian

Laplace

The larger relative percentages occur only where W8 MSE is very small and tail events dominate sampling variance; the standardized discrepancies remain within 2\.10 standard errors\.

### 3\.5From population optima to finite\-group fitting

The preceding results optimize one continuous\-level quantizer for an entire reference distribution\. A groupwise continuous fitter instead selects a separate scale and shape for every finite group of weights\. These are different optimization problems, even when the weights in a group are independent samples from the same distribution\.

LetSG=\(X1,…,XG\)S\_\{G\}=\(X\_\{1\},\\ldots,X\_\{G\}\)denote a group ofGGvalues, letθ=\(s,a,b\)\\theta=\(s,a,b\)collect the continuous Cubic parameters, and letℓC​\(X,θ\)\\ell\_\{\\rm C\}\(X,\\theta\)include assignment to the nearest continuous level generated byθ\\theta\. Define the empirical distortion of one group as

D^GC​\(θ;SG\)=1G​∑j=1GℓC​\(Xj,θ\),θ^​\(SG\)∈arg⁡minθ⁡D^GC​\(θ;SG\)\.\\widehat\{D\}\_\{G\}^\{\\rm C\}\(\\theta;S\_\{G\}\)=\\frac\{1\}\{G\}\\sum\_\{j=1\}^\{G\}\\ell\_\{\\rm C\}\(X\_\{j\},\\theta\),\\qquad\\widehat\{\\theta\}\(S\_\{G\}\)\\in\\arg\\min\_\{\\theta\}\\widehat\{D\}\_\{G\}^\{\\rm C\}\(\\theta;S\_\{G\}\)\.\(42\)
The expected distortion after independently fitting every group is therefore

RGC=𝔼SG​\[minθ⁡D^GC​\(θ;SG\)\]\.R\_\{G\}^\{\\rm C\}=\\mathbb\{E\}\_\{S\_\{G\}\}\\\!\\left\[\\min\_\{\\theta\}\\widehat\{D\}\_\{G\}^\{\\rm C\}\(\\theta;S\_\{G\}\)\\right\]\.\(43\)
By contrast, Section 3\.3 reports the population optimum

Dpop,C∗=minθ⁡𝔼X​\[ℓC​\(X,θ\)\]\.D\_\{\{\\rm pop\},\\rm C\}^\{\*\}=\\min\_\{\\theta\}\\mathbb\{E\}\_\{X\}\[\\ell\_\{\\rm C\}\(X,\\theta\)\]\.\(44\)
Ifθ∗\\theta^\{\*\}is a population minimizer, then for every observed groupSGS\_\{G\},

minθ⁡D^GC​\(θ;SG\)≤D^GC​\(θ∗;SG\)\.\\min\_\{\\theta\}\\widehat\{D\}\_\{G\}^\{\\rm C\}\(\\theta;S\_\{G\}\)\\leq\\widehat\{D\}\_\{G\}^\{\\rm C\}\(\\theta^\{\*\};S\_\{G\}\)\.\(45\)
Taking expectations on both sides gives

RGC≤𝔼SG​\[D^GC​\(θ∗;SG\)\]=𝔼X​\[ℓC​\(X,θ∗\)\]=Dpop,C∗\.R\_\{G\}^\{\\rm C\}\\leq\\mathbb\{E\}\_\{S\_\{G\}\}\[\\widehat\{D\}\_\{G\}^\{\\rm C\}\(\\theta^\{\*\};S\_\{G\}\)\]=\\mathbb\{E\}\_\{X\}\[\\ell\_\{\\rm C\}\(X,\\theta^\{\*\}\)\]=D\_\{\{\\rm pop\},\\rm C\}^\{\*\}\.\(46\)
This inequality is an in\-group reconstruction statement for static weights, not a claim about out\-of\-sample generalization\. It says that a fitter allowed to adapt\(s,a,b\)\(s,a,b\)to each realized group cannot have larger expected reconstruction error than a single population\-wide parameter choice\. Under the usual law\-of\-large\-numbers and regularity conditions, the empirical objective approaches the population objective asGGgrows, so this finite\-group adaptation advantage diminishes\.

This distinction resolves an otherwise apparent contradiction in the Uniform result\. Section 3\.3 shows that the linear INT grid is population optimal for a perfectly Uniform distribution\. A finite group, however, is not exactly Uniform: its empirical histogram, extrema, and spacing fluctuate\. Cubic can adapt its two shape parameters to those fluctuations and may obtain a small groupwise improvement even though it has no population\-level advantage for Uniform data\.

Group size consequently controls two opposing effects\. Smaller groups permit more local adaptation but require more scale and shape metadata per weight; larger groups amortize metadata and codebook\-generation cost but force one curve to represent a broader region of the tensor\. Real weight groups are also neither independent nor stationary, so the size of this tradeoff cannot be inferred from the reference distributions alone\. Finite\-GGsimulation and tensor\-level experiments are therefore used to measure this finite\-group effect, while the population analysis supplies the distribution\-controlled reference point\. The same empirical\-versus\-population argument applies when the loss is replaced consistently by the joint carrier\-aware loss in Section 4\.3, but the resulting optimum and decision cells are not the continuous quantities tabulated in this section\.

### 3\.6Finite\-group reconstruction experiment

The population results above use one parameter set for an entire distribution\. To measure the distinct finite\-group setting of Section 3\.5, we drew 15,360 zero\-centered values from each reference distribution with seed 42 and divided them into groups ofG=128G=128\. Every group was fitted independently\. Cubic used joint scale/shape fitting with optimized clipping; symmetric INT independently optimized its clipping scale; and the finite\-FP reference enumerated every splite\+m=B−1e\+m=B\-1, optimized the scale of each candidate, and retained the lowest\-error split\.

Because every candidate optimizes an external scale, the experiment is invariant to a common positive rescaling of the source values\. The table therefore reports the scale\-free quantity

NRMSE=N−1​∑j=1N\(xj−x^j\)2σX,\\operatorname\{NRMSE\}=\\frac\{\\sqrt\{N^\{\-1\}\\sum\_\{j=1\}^\{N\}\(x\_\{j\}\-\\widehat\{x\}\_\{j\}\)^\{2\}\}\}\{\\sigma\_\{X\}\},\(47\)
whereN=15,360N=15\{,\}360and eachx^j\\widehat\{x\}\_\{j\}is reconstructed with the parameters fitted to its own group\. Lower is better; boldface marks the unique minimum in each row, while exact ties are left unbolded\. The experiment records W2–W6 and W8; the absent W1 and W7 rows are not interpolated\. As elsewhere in Section 3, these are continuous\-level reconstructions before Dynamic\-A8 carrier projection\.

W2 has no movable interior positive level, so all three families coincide in this experiment\. From W3 upward, per\-group shape adaptation consistently reduces reconstruction error\. The improvement over clipped INT is modest for bounded Uniform samples and larger for Gaussian and Laplace samples\. At W4, the Cubic NRMSE reductions are 3\.90%, 13\.49%, and 28\.14%, respectively\. Against the best finite\-FP candidate, the corresponding reductions are 3\.90%, 9\.44%, and 6\.27%\.

Unlike the population result, the small Uniform gain does not imply that a nonlinear grid is optimal for a Uniform law\. It is the finite\-sample adaptation effect derived in Section 3\.5: each realized group departs slightly from the population distribution, and Cubic can fit those local fluctuations\. The experiment therefore supplies a finite\-GGcounterpart to the population tables, not evidence about CUDA throughput or model\-level task quality\. Since onlyG=128G=128is measured here, it also does not determine the best group size\.

## 4Groupwise parameter fitting

### 4\.1Continuous reference objective

The representation becomes a quantizer only after codes and group parameters have been selected\. The reference estimator operates independently on each static weight group and minimizes original\-domain squared reconstruction error over codes, scale, and admissible shape parameters:

mins,a,b,\{kj\}​∑j\[wj−sign⁡\(kj\)​s​q​\(\|kj\|M\)\]2\.\\min\_\{s,a,b,\\\{k\_\{j\}\\\}\}\\sum\_\{j\}\\left\[w\_\{j\}\-\\operatorname\{sign\}\(k\_\{j\}\)sq\\\!\\left\(\\frac\{\|k\_\{j\}\|\}\{M\}\\right\)\\right\]^\{2\}\.\(48\)
Heressis both the endpoint and the clipping threshold; it is not constrained to equal the largest magnitude in the group\. Values beyondsssaturate at the signed endpoint\. For fixed levels, each value is assigned to its nearest reconstruction\. The reference search traverses a deterministic set of monotonic\(a,b\)\(a,b\)candidates, jointly refines scale and clipping, and always includes the uniform\-integer point\. The objective is reevaluated after mappingss,aa, andbbto their serialized precisions, so parameter rounding is part of the fitted representation rather than a post hoc perturbation\.

The fitting loss is MSE/SSE\. Its aggregate, scale\-free reporting form is

NRMSEC=∑j\(wj−w^j\)2∑jwj2,\\operatorname\{NRMSE\}\_\{\\rm C\}=\\sqrt\{\\frac\{\\sum\_\{j\}\(w\_\{j\}\-\\widehat\{w\}\_\{j\}\)^\{2\}\}\{\\sum\_\{j\}w\_\{j\}^\{2\}\}\},\(49\)
which is scale\-free across tensors and bit widths\. Taking the square root is a reporting transformation; it does not alter the candidate selected by MSE\.

This continuous objective is the finite\-group counterpart of the population analysis in Section 3 and one component of the dual\-path objective introduced in Section 4\.3\. It depends only on the weights themselves and therefore serves as a controlled reference estimator\. That choice does not imply that every coordinate has equal functional importance; Section 4\.2 separates the format from richer estimators that incorporate model sensitivity\.

### 4\.2Compatibility with data\-driven estimation

The weight\-only fitter is a reference estimator, not a restriction of the Cubic format\. LetE=W−W^E=W\-\\widehat\{W\}be the weight error of a linear operator andxxits input\. Unweighted weight MSE minimizes∥E∥F2\\lVert E\\rVert\_\{F\}^\{2\}\. If representative activations are available, the expected output error is instead

𝔼​‖E​x‖22=tr⁡\(E​Σx​E⊤\),Σx=𝔼​\[x​x⊤\]\.\\mathbb\{E\}\\\|Ex\\\|\_\{2\}^\{2\}=\\operatorname\{tr\}\(E\\Sigma\_\{x\}E^\{\\top\}\),\\qquad\\Sigma\_\{x\}=\\mathbb\{E\}\[xx^\{\\top\}\]\.\(50\)
A diagonal covariance becomes a coordinate\-weighted reconstruction objective\. Block\-diagonal, low\-rank, or full covariance models retain progressively more cross\-coordinate structure at progressively greater fitting cost\. Approximate Hessians and other sensitivity estimates can play the same role\.

Changing the estimator changes which codes and\(s,a,b\)\(s,a,b\)are chosen; it does not change the packed bitstream, metadata schema, level function, or runtime kernel\. CubicQuant can therefore be combined with data\-driven methods such as second\-order or activation\-aware fitting\. The experiments in this report evaluate only the data\-free reference path and make no claim that it dominates such methods\.

### 4\.3Dynamic\-A8 carrier\-aware fitting

The model\-dtype execution path can reconstruct the continuous Cubic value\. The Dynamic\-A8 path instead maps the normalized Cubic level to a signed INT8 carrier:

r​\(k\)=round⁡\[127​sign⁡\(k\)​q​\(\|k\|M\)\]\.r\(k\)=\\operatorname\{round\}\\\!\\left\[127\\,\\operatorname\{sign\}\(k\)q\\\!\\left\(\\frac\{\|k\|\}\{M\}\\right\)\\right\]\.\(51\)
The deployed normalized level is consequently

q~​\(t\)=round⁡\(127​q​\(t\)\)127\.\\widetilde\{q\}\(t\)=\\frac\{\\operatorname\{round\}\(127q\(t\)\)\}\{127\}\.\(52\)
For a weightwjw\_\{j\}assigned to codekjk\_\{j\}, define the two reconstructions

w^jC=sign⁡\(kj\)​s​q​\(tkj\),w^jA8=s127​r​\(kj\)=sign⁡\(kj\)​s​q~​\(tkj\)\.\\widehat\{w\}\_\{j\}^\{\\rm C\}=\\operatorname\{sign\}\(k\_\{j\}\)sq\(t\_\{k\_\{j\}\}\),\\qquad\\widehat\{w\}\_\{j\}^\{\\rm A8\}=\\frac\{s\}\{127\}r\(k\_\{j\}\)=\\operatorname\{sign\}\(k\_\{j\}\)s\\widetilde\{q\}\(t\_\{k\_\{j\}\}\)\.\(53\)
The first is consumed by the model\-dtype path; the second is the effective weight represented by the signed INT8 carrier in the Dynamic\-A8 path\. Their groupwise squared errors are

ℒC=∑j\(wj−w^jC\)2,ℒA8=∑j\(wj−w^jA8\)2\.\\mathcal\{L\}\_\{\\rm C\}=\\sum\_\{j\}\(w\_\{j\}\-\\widehat\{w\}\_\{j\}^\{\\rm C\}\)^\{2\},\\qquad\\mathcal\{L\}\_\{\\rm A8\}=\\sum\_\{j\}\(w\_\{j\}\-\\widehat\{w\}\_\{j\}^\{\\rm A8\}\)^\{2\}\.\(54\)
The dual\-path fitting policy studied here gives the two execution paths equal weight:

ℒjoint=12​ℒC\+12​ℒA8\.\\mathcal\{L\}\_\{\\rm joint\}=\\frac\{1\}\{2\}\\mathcal\{L\}\_\{\\rm C\}\+\\frac\{1\}\{2\}\\mathcal\{L\}\_\{\\rm A8\}\.\(55\)
This objective is distinct from the continuous distortion analyzed in Section 3\. To state the corresponding population quantity, putzi=q​\(ti\)z\_\{i\}=q\(t\_\{i\}\)andz~i=q~​\(ti\)\\widetilde\{z\}\_\{i\}=\\widetilde\{q\}\(t\_\{i\}\)\. On the positive half\-line, the joint distortion is

DBjoint​\(s,a,b\)=2​∑i=0M∫hi−1jointhijoint\(x−s​zi\)2\+\(x−s​z~i\)22​f​\(x\)​dx\.D\_\{B\}^\{\\rm joint\}\(s,a,b\)=2\\sum\_\{i=0\}^\{M\}\\int\_\{h\_\{i\-1\}^\{\\rm joint\}\}^\{h\_\{i\}^\{\\rm joint\}\}\\frac\{\(x\-sz\_\{i\}\)^\{2\}\+\(x\-s\\widetilde\{z\}\_\{i\}\)^\{2\}\}\{2\}f\(x\)\\,\\mathrm\{d\}x\.\(56\)
Equating the joint costs of adjacent codes gives their decision boundary

hijoint=s2​\(zi\+12\+z~i\+12\)−\(zi2\+z~i2\)\(zi\+1\+z~i\+1\)−\(zi\+z~i\)\.h\_\{i\}^\{\\rm joint\}=\\frac\{s\}\{2\}\\frac\{\(z\_\{i\+1\}^\{2\}\+\\widetilde\{z\}\_\{i\+1\}^\{2\}\)\-\(z\_\{i\}^\{2\}\+\\widetilde\{z\}\_\{i\}^\{2\}\)\}\{\(z\_\{i\+1\}\+\\widetilde\{z\}\_\{i\+1\}\)\-\(z\_\{i\}\+\\widetilde\{z\}\_\{i\}\)\}\.\(57\)
Equation[56](https://arxiv.org/html/2608.06763#S4.E56)can be evaluated with the same cell moments as Equation[14](https://arxiv.org/html/2608.06763#S3.E14), but its boundaries are not the midpoints in Equation[11](https://arxiv.org/html/2608.06763#S3.E11)\. Moreover, the rounding operation makes it piecewise non\-smooth in\(a,b\)\(a,b\)\. The continuous population values and bounds in Section 3 therefore remain representation references; they are not numerical evaluations of the joint fitting policy\.

Thus the carrier correction is not an additional diagnostic applied after a continuous\-only fit: it participates in the selection of codes, scale, and shape parameters\. The implementation minimizesℒC\+ℒA8\\mathcal\{L\}\_\{\\rm C\}\+\\mathcal\{L\}\_\{\\rm A8\}, omitting the common factor1/21/2because it cannot change the minimizer\. For reporting, it restores the mean and emits

NRMSEjoint=ℒC\+ℒA82​∑jwj2\\operatorname\{NRMSE\}\_\{\\rm joint\}=\\sqrt\{\\frac\{\\mathcal\{L\}\_\{\\rm C\}\+\\mathcal\{L\}\_\{\\rm A8\}\}\{2\\sum\_\{j\}w\_\{j\}^\{2\}\}\}\(58\)
together with the A8\-only diagnosticℒA8/∑jwj2\\sqrt\{\\mathcal\{L\}\_\{\\rm A8\}/\\sum\_\{j\}w\_\{j\}^\{2\}\}\.

The joint objective is used throughout the deterministic candidate search\. For fixed codes and shape parameters, letuj=\|wj\|u\_\{j\}=\|w\_\{j\}\|and usezj=q​\(tkj\)z\_\{j\}=q\(t\_\{k\_\{j\}\}\),z~j=q~​\(tkj\)\\widetilde\{z\}\_\{j\}=\\widetilde\{q\}\(t\_\{k\_\{j\}\}\)\. The shared scale that minimizes Equation[55](https://arxiv.org/html/2608.06763#S4.E55)has the closed form

s∗=∑juj​\(zj\+z~j\)∑j\(zj2\+z~j2\)\.s^\{\*\}=\\frac\{\\sum\_\{j\}u\_\{j\}\(z\_\{j\}\+\\widetilde\{z\}\_\{j\}\)\}\{\\sum\_\{j\}\(z\_\{j\}^\{2\}\+\\widetilde\{z\}\_\{j\}^\{2\}\)\}\.\(59\)
Conversely, for fixed\(s,a,b\)\(s,a,b\), each code is assigned by minimizing

\(uj−s​q​\(tk\)\)2\+\(uj−s​q~​\(tk\)\)2\(u\_\{j\}\-sq\(t\_\{k\}\)\)^\{2\}\+\(u\_\{j\}\-s\\widetilde\{q\}\(t\_\{k\}\)\)^\{2\}\(60\)
over the admissible magnitudes\. Assignment and scale updates alternate while the estimator traverses the same monotonic shape and soft\-clipping candidates as the continuous fitter\. After convertingaaandbbto FP16 and retainingssin FP32, the final code assignment is repeated under the same joint objective\. The evaluated object is therefore the serialized representation rather than an unrounded optimization intermediate\.

Equal weighting is a representation\-design choice, not an assumption about the frequency with which either execution path is selected, and no averaging occurs during inference\. A model\-dtype kernel reconstructsw^C\\widehat\{w\}^\{\\rm C\}, whereas a Dynamic\-A8 kernel uses the carrier represented byr​\(k\)r\(k\)\. For W1 and W2 the available normalized levels already lie exactly on the INT8 carrier grid, soℒC=ℒA8\\mathcal\{L\}\_\{\\rm C\}=\\mathcal\{L\}\_\{\\rm A8\}and the correction has no effect\.

Optimizing only the continuous curve at W3–W8 can choose parameters that look favorable before the second grid projection but lose that advantage in the actual A8 kernel\. The joint correction remains data\-free because it models deterministic runtime arithmetic rather than activation statistics\. It changes neither code packing nor stored metadata and adds no online fitting work; the same checkpoint can therefore serve both model\-dtype and Dynamic\-A8 execution\. In particular, the correction covers only the weight\-side projection onto the INT8 carrier grid\. It does not model dynamic activation quantization or its propagation into operator outputs; doing so requires an activation distribution or sensitivity estimator of the kind discussed in Section 4\.2\.

## 5GPU realization

### 5\.1Fused decode\-and\-compute principle

Cubic codes are not native Tensor Core operands\. Expanding an entire matrix to BF16 or INT8 before multiplication would write and reread a conventional weight tensor, eliminating much of the bandwidth benefit of the packed representation\. CubicQuant therefore couples reconstruction to consumption: each output tile loads only its packed payload and group metadata, generates a temporary model\-dtype or INT8 operand, and immediately consumes that operand in the dot\-product main loop\. The expanded tile remains local to the kernel\.

Two complementary kernel families instantiate this principle\. Generic kernels cover W1–W8 for model\-dtype and Dynamic\-A8 activations, with separate narrow\-row GEMV and two\-dimensional dense organizations\. Native CUDA kernels add packed\-word extraction and DP4A\-style signed INT8 dot products for selected Dynamic\-A8 regimes\. These families are alternative realizations of the same operator semantics; the numerical format does not depend on which realization is selected\.

### 5\.2Packed extraction and level reconstruction

Letβp\\beta\_\{p\}denote packed bytepp\. For logical coordinatejj, definepj=⌊j​B/8⌋p\_\{j\}=\\lfloor jB/8\\rfloorandδj=\(j​B\)mod8\\delta\_\{j\}=\(jB\)\\bmod 8\. The unsigned payload is recovered by

uj=\[\(βpj≫δj\)\|\(βpj\+1≪\(8−δj\)\)\]&\(2B−1\),u\_\{j\}=\\left\[\(\\beta\_\{p\_\{j\}\}\\\!\\gg\\\!\\delta\_\{j\}\)\\;\\mathbin\{\|\}\\;\(\\beta\_\{p\_\{j\}\+1\}\\\!\\ll\\\!\(8\-\\delta\_\{j\}\)\)\\right\]\\mathbin\{\\&\}\(2^\{B\}\-1\),\(61\)
where the second\-byte contribution is omitted when the field does not cross a byte boundary\. W1 maps its single payload bit directly to\{−1,\+1\}\\\{\-1,\+1\\\}; wider codes are sign\-extended, and the reserved most\-negative pattern is mapped to zero as required by the format\. BecauseG​BGBis byte aligned, a weight group does not share a partial byte with the next group’s metadata domain\.

The group index isg=⌊j/G⌋g=\\lfloor j/G\\rfloor\. A tile loadssgs\_\{g\}and, where needed,\(ag,bg\)\(a\_\{g\},b\_\{g\}\)once for the output channels it covers\. Three reconstruction strategies are used:

1. 1\.W1 and W2 admit direct algebraic mappings and need no curve table\.
2. 2\.A nonnegative Cubic table contains2B−12^\{B\-1\}entries, including zero; its signed values are obtained by applying the decoded sign\.
3. 3\.Direct evaluation uses Horner\-form fused multiply\-adds,

q​\(t\)=t​fma⁡\(t,fma⁡\(t,c,b\),a\)\.q\(t\)=t\\,\\operatorname\{fma\}\\\!\\left\(t,\\operatorname\{fma\}\(t,c,b\),a\\right\)\.\(62\)
For some routed W3/A8 kernels, the two interior positive carrier levels are precomputed from\(a,b\)\(a,b\)after loading and reused as compact INT8 metadata\. For W4–W8, eligible CUDA kernels construct a group\-local carrier table in shared memory, while generic kernels may construct levels through static evaluation or evaluate the curve directly\. A simple arithmetic comparison isO​\(2B−1\)O\(2^\{B\-1\}\)work to construct the positive table versusO​\(G\)O\(G\)curve evaluations to decode a group\. This is only a cost\-model term, not a dispatch law: packed\-load pattern, table reuse, register pressure, shared\-memory capacity, output shape, and GPU architecture determine the measured crossover\.

### 5\.3Model\-dtype execution

The model\-dtype path accepts FP16 or BF16 activations and implements the continuous reconstruction in Equation[2](https://arxiv.org/html/2608.06763#S2.E2)\. Its dense and narrow\-row tactics have different, explicit rounding points\. In a dense Triton tile,sg​q​\(tk\)s\_\{g\}q\(t\_\{k\}\)is formed using FP32 metadata arithmetic, converted to the activation dtype, and supplied with the activation tile totl\.dot; the dot accumulator is FP32\. In the narrow\-row GEMV path, activations and reconstructed normalized weights are multiplied in FP32 andsgs\_\{g\}is applied to the group reduction in FP32\. Both store the requested model dtype, but they need not be bitwise identical because the dense path rounds the temporary weight operand before the dot\.

The continuous formula is therefore the semantic reference for this execution mode\. No individual model\-dtype kernel is treated as an oracle for Dynamic A8; both modes are compared independently with the mathematical reconstruction defined by their respective equations\.

### 5\.4Dynamic\-A8 execution

For an activation rowxx, the canonical path computes

x¯=max⁡\(maxj⁡\|xj\|,ϵ\),dx=x¯127,\\bar\{x\}=\\max\\\!\\left\(\\max\_\{j\}\|x\_\{j\}\|,\\epsilon\\right\),\\qquad d\_\{x\}=\\frac\{\\bar\{x\}\}\{127\},\(63\)
and

x8,j=clip⁡\(roundaway⁡\(xj/dx\),−127,127\)\.x\_\{8,j\}=\\operatorname\{clip\}\\\!\\left\(\\operatorname\{round\}\_\{\\rm away\}\(x\_\{j\}/d\_\{x\}\),\-127,127\\right\)\.\(64\)
Hereroundaway\\operatorname\{round\}\_\{\\rm away\}denotes nearest\-integer rounding with halfway cases away from zero, matching the activation\-quantization semantics\. This is distinct from the round\-to\-nearest\-even operation used to construct the weight carrierr​\(k\)r\(k\)in Section 4\.3\. The distinction only affects exact halfway cases but is part of the executable contract\. An all\-zero row uses the positive safeguardϵ\\epsilonand produces zero activation carriers\.

For weight groupgg, the signed integer partial and reconstructed output are

Pg\\displaystyle P\_\{g\}=∑j∈gx8,j​rg​\(kj\),\\displaystyle=\\sum\_\{j\\in g\}x\_\{8,j\}r\_\{g\}\(k\_\{j\}\),\(65\)y\\displaystyle y=∑gdx​sg127​Pg\.\\displaystyle=\\sum\_\{g\}\\frac\{d\_\{x\}s\_\{g\}\}\{127\}P\_\{g\}\.\(66\)
Integer products accumulate into INT32 partials\. With the supportedG≤512G\\leq 512, a full\-group worst\-case magnitude is bounded by

\|Pg\|≤G​1272≤512×1272=8,258,048<231−1\.\|P\_\{g\}\|\\leq G\\,127^\{2\}\\leq 512\\times 127^\{2\}=8\{,\}258\{,\}048<2^\{31\}\-1\.\(67\)
Dense kernels may form still smallerKK\-tile partials\. Each completed partial is converted to FP32 before multiplication bydx​sg/127d\_\{x\}s\_\{g\}/127; cross\-group accumulation is also FP32, followed by the requested output cast\. The activation scale is generated at runtime and is neither stored in the checkpoint nor estimated from a calibration corpus\.

After carrier construction, W1–W8 all perform one signed INT8 product per logical weight\. This equality concerns the main dot\-product order only\. Their total costs remain different because packed traffic, cross\-byte extraction, carrier\-table construction, direct curve evaluation, and achievable occupancy all depend onBBandGG\.

Dynamic A8 is an operator\-local compute mode, not a persistent integer graph\. An ordinary operator returns model\-dtype output, and a later operator quantizes its own input when invoked\. Selected fused expert paths can combine activation and quantization adjacent to the down projection, but this optimization does not change the general boundary or the checkpoint format\.

### 5\.5Workload regimes and routed execution

For an ordinary Linear operator,MMis the number of activation rows\. SmallMMprovides too little row parallelism for a conventional matrix tile, so the narrow path assigns parallel work across output channels and consumes one or a few rows as GEMV\. AsMMgrows, a two\-dimensional tile can reuse each decoded weight across several activation rows and can make effective use of native dot instructions\. Thus decode\-like and prefill\-like shapes have different crossovers even at identical\(B,G,N,K\)\(B,G,N,K\)\.

For a routed operator, the relevant work is the number and distribution of active token–matrix routes rather than the input\-token count alone\. Routes can be compacted and grouped to reuse a decoded tile, processed independently with route\-major outputs, or reduced within the kernel when the layout permits it\. Dense tiles remain useful when route density is high\. These choices alter work organization but not the Cubic reconstruction or route weights\.

Routing probabilities, expert selection, and expert placement remain outside the Cubic format\. The kernels consume the route map supplied by the serving system and support both local and expert\-parallel layouts without assigning a special meaning to expert numbering\.

### 5\.6Shape\- and device\-dependent kernel selection

Performance depends on\(B,G,M,N,K\)\(B,G,M,N,K\), activation dtype, route topology, GPU architecture, compiler stack, and kernel organization\. No single threshold inMMorBBreliably determines whether a narrow, dense, native, or generic kernel is fastest across this space\. CubicQuant therefore treats kernel choice as a shape\- and device\-dependent optimization problem rather than a property of the numerical format\.

For every admissible operator signature, candidate families are first filtered by payload width, alignment, dtype, and architectural constraints\. Numerically valid candidates are then compared on representative row\-count regimes, with choices including GEMV versus dense tiling, block height, route grouping, route\-level parallelism, and native CUDA versus generic kernels\. The selected strategy is a deterministic function of the measured execution environment and operator signature\.

This empirical selection changes neither codes nor group parameters\. It is also distinct from quantizer fitting: synthetic operands are sufficient because the quantity being estimated is kernel latency, not an activation distribution or model\-sensitivity statistic\. A conservative implementation remains available whenever no specialized candidate satisfies the numerical and structural constraints\.

## 6Experimental evidence

The evidence is organized in increasing order of system dependence\. Section 3 establishes distribution\-controlled representation error; this section first quantifies storage and computational work, then reports isolated kernel measurements and one negative fusion result\. End\-to\-end serving behavior is discussed only as an attribution boundary because it also depends on attention, communication, scheduling, and memory management\.

### 6\.1Experimental scope and portability

The implementation can be compiled natively for several GPU generations, but binary compatibility and measured performance are distinct claims:

GPU generationNative targetLatency evidence in this reportAmpere data centerSM80Not measuredAmpere consumerSM86Not measuredAda consumer/data centerSM89Not measuredHopperSM90 and SM90aSingle\-H200 kernel measurementsBlackwell data centerSM100Not measuredBlackwell consumerSM120Not measuredThe isolated measurements were collected on one NVIDIA H200 with 143,771 MiB of memory and CUDA compute capability 9\.0, using the SM90a target\. The storage and work model in Section 6\.2 is architecture independent; every measured latency in this report is Hopper\-specific\.

Native targets were verified from the produced code objects rather than inferred solely from compiler flags\[[25](https://arxiv.org/html/2608.06763#bib.bib25)\]\. Physical\-device evidence remains limited to Hopper, so the other rows establish portability of the implementation, not performance equivalence across architectures\.

### 6\.2Computational work and memory traffic

Consider a matrix multiplication withMMinput rows,NNoutput columns, input dimensionKK, weight widthBB, and weight group sizeGG\. Ignoring row\-tail padding, the packed Cubic matrix occupies

TW=N​K​\(B8\+8G\)​bytes\.T\_\{\\rm W\}=NK\\left\(\\frac\{B\}\{8\}\+\\frac\{8\}\{G\}\\right\)\\ \\text\{bytes\}\.\(68\)
The first term is the packed payload\. The second is one FP32 scale and two FP16 shape coefficients per group\. Relative to a two\-byte BF16 matrix, the ideal resident\-weight compression factor is therefore

CBF16/Cubic=2B/8\+8/G\.C\_\{\\rm BF16/Cubic\}=\\frac\{2\}\{B/8\+8/G\}\.\(69\)
These are representation ratios, not measured kernel speedups\. A tile\-local kernel reads the compressed payload and metadata without writing a complete BF16 weight tensor, but it must generate or look up Cubic levels and perform the same number of logical dot\-product terms as an uncompressed matrix multiplication\. The leading work therefore remainsO​\(M​N​K\)O\(MNK\)\.

ComponentLeading workIdeal full\-matrix traffic or statePacked weights and metadataDecode/lookup per visited tileN​K​\(B/8\+8/G\)NK\(B/8\+8/G\)bytesMain dot productM​N​KMNKproducts and accumulationsOutput\-tile dependentGroup rescalingM​N​K/GMNK/Ggroup contributionsFP32 scale/shape metadata included aboveDynamic\-A8 activation encodingM​KMKreduction/rounding workBF16 source plus anM​KMK\-byte INT8 carrierDynamic\-A8 activation scalesMMvalues for per\-token A84​M4Mbytes under the operator semanticsThe resident Dynamic\-A8 operand is approximatelyM​K\+4​MMK\+4Mbytes rather than2​M​K2MKbytes for BF16, but producing it also reads the BF16 source and writes the carrier\. Consequently, activation compression becomes a traffic reduction only when the carrier is fused with, or reused by, enough downstream work\. A16 and Dynamic\-A8 also have different constant factors: A16 reconstructs levels into floating\-point dot operands, whereas Dynamic\-A8 maps them to INT8 carriers, uses integer dot products with INT32 partial sums, and performs FP32 group rescaling\. Direct maps, Horner evaluation, and shared\-memory lookup tables alter the decode constant but not the asymptotic complexity\. This is why latency must be measured as a function of\(M,N,K,B,G\)\(M,N,K,B,G\)rather than inferred from storage width alone\.

### 6\.3H200 kernel crossover: model dtype versus Dynamic A8

The kernel experiment compares the model\-dtype path, labeled A16, with the operator\-local Dynamic\-A8 path on two large linear shapes at G512:\(N,K\)=\(3584,3072\)\(N,K\)=\(3584,3072\)and\(6144,3584\)\(6144,3584\)\. These dimensions are drawn from actual projection shapes in a widely used, large\-scale open model and serve here as representative large\-model workloads; neither the kernels nor their dispatch rules are specialized to that model\. Both paths consume the same packed Cubic representation and produce the same output dtype\. The ratio below istA16/tA8t\_\{\\rm A16\}/t\_\{\\rm A8\}, so values above one favor Dynamic A8\. Measurements are available for representative W2, W5, and W8 cases and do not constitute a complete W1–W8 performance sweep\.

Weight widthM=1M=1M=16M=16M=64M=64M=256M=256W20\.40–0\.59×\\times1\.00–1\.60×\\times1\.74–2\.03×\\times2\.21–2\.23×\\timesW50\.78–0\.87×\\times1\.62–2\.60×\\times2\.94–3\.16×\\times3\.35–3\.93×\\timesW80\.35–0\.42×\\times1\.62–2\.40×\\times2\.37–4\.19×\\times4\.31–4\.46×\\timesFor one of those shapes,\(N,K\)=\(3584,3072\)\(N,K\)=\(3584,3072\), the underlying measured latencies were as follows; each cell is A16/A8 in milliseconds\.

The measurements show a crossover rather than a universal A8 advantage\. AtM=1M=1, activation encoding and integer\-path setup dominate, and the mature A16 GEMV is faster\. AsMMgrows, those fixed costs are amortized and the integer main loop becomes favorable\. The crossover also depends on width and shape, supporting the shape\- and device\-dependent selection developed in Section 5\.6\. The measurements are preliminary Hopper evidence rather than a general throughput claim\.

### 6\.4Persistent activation carriers: a negative result

An experimental path persisted groupwise INT8 activation carriers between adjacent MLP operators to avoid an intermediate BF16 tensor\. Although this reduces the nominal intermediate width, it did not improve either isolated or end\-to\-end performance and is therefore not part of CubicQuant’s retained execution design\.

On a representative W1/activation\-to\-W2 boundary with G512 weights and G256 activation groups, the online boundary was slower than the established non\-online Dynamic\-A8 path:

Routed rowsNon\-online boundary \(ms\)↓\\downarrowOnline boundary \(ms\)↓\\downarrowNon\-online / online160\.10920\.14080\.776×\\times640\.36510\.39450\.926×\\times1280\.71120\.75760\.939×\\timesThe intended bandwidth saving was real, but the producer needed groupwise reductions, synchronization, scale generation, and an additional carrier write\. At the tested boundary those costs exceeded the saved BF16 traffic\. This negative result narrows the contribution: Dynamic A8 is an operator\-local compute mode, not a persistent all\-integer activation graph\. A future design would need producer\-consumer fusion that keeps the intermediate in registers or shared memory rather than merely changing its global\-memory dtype\.

### 6\.5Interpretation of system\-level evidence

Large routed operators and complete multi\-GPU serving workloads have been executed with both activation modes, but those observations combine Cubic kernels with attention, collective communication, scheduling, and memory management\. They cannot isolate the causal contribution of the weight format\. Accordingly, this report makes no aggregate serving\-speedup, power, or energy claim\.

The isolated latency results support a narrower conclusion: CubicQuant is directly executable from its packed representation, and Dynamic A8 has a measured crossover with model\-dtype execution as row count grows\. Establishing an end\-to\-end advantage requires matched experiments that hold the model, parallel decomposition, attention and KV representation, request distribution, and hardware operating point fixed\. Those controls are necessary because the full system difference cannot be attributed to the weight format alone\.

## 7Related work

### 7\.1Uniform and second\-order post\-training quantization

GPTQ uses approximate second\-order information to minimize layer\-wise error during one\-shot weight quantization\[[3](https://arxiv.org/html/2608.06763#bib.bib3)\]\. AWQ uses activation statistics to identify salient channels and applies an equivalent scaling transformation while retaining a hardware\-friendly low\-bit format\[[4](https://arxiv.org/html/2608.06763#bib.bib4)\]\. SmoothQuant migrates activation outlier difficulty into weights through an equivalent transformation to enable W8A8 execution\[[5](https://arxiv.org/html/2608.06763#bib.bib5)\]\. OmniQuant learns clipping and equivalent transformations through block\-wise calibration\[[6](https://arxiv.org/html/2608.06763#bib.bib6)\]\. These methods motivate the distinction made above: the stored representation and the estimator used to select it are separable\. CubicQuant’s present fitter is data\-free, but its codes and shape parameters can be selected with activation\-aware or second\-order objectives\.

### 7\.2Distribution\-matched and adaptive scalar formats

QLoRA introduced NormalFloat4, a fixed non\-uniform codebook constructed for normally distributed weights, together with double quantization for training memory reduction\[[7](https://arxiv.org/html/2608.06763#bib.bib7)\]\. Subsequent analysis showed that absmax normalization changes the blockwise source distribution and therefore the codebook favored by a reconstruction objective\[[8](https://arxiv.org/html/2608.06763#bib.bib8)\]\. BOF4 formalizes that dependence by optimizing a blockwise codebook for a specified source model and loss\[[9](https://arxiv.org/html/2608.06763#bib.bib9)\]\. Standard finite floating\-point formats instead place levels through a fixed exponent–mantissa split; E4M3 and E5M2 are representative FP8 instances\[[13](https://arxiv.org/html/2608.06763#bib.bib13)\]\.

Adaptive Block\-Scaled Data Types introduce IF4, which evaluates native FP4 and scaled INT4 for each group and records the lower\-MSE choice\[[10](https://arxiv.org/html/2608.06763#bib.bib10)\]\. IF4 therefore adapts the local scalar grid through a discrete choice between two fixed formats; it does not learn the reconstruction levels or continuously deform the selected grid\.

LO\-BCQ clusters blocks according to their local statistics and designs a separate quantization codebook for each cluster\[[11](https://arxiv.org/html/2608.06763#bib.bib11)\]\. A block selector then identifies the shared cluster codebook used for reconstruction\. This provides finer adaptation than a single global grid, at the cost of a codebook library and explicit selection metadata\.

AAAC learns two activation\-aware scalar codebooks per layer and lets each weight group select between them\[[12](https://arxiv.org/html/2608.06763#bib.bib12)\]\. CubicQuant makes a different locality–metadata trade\-off: every group receives its own codebook, but that codebook is restricted to a monotonic cubic family represented by two shape coefficients\. Thus AAAC offers two freer layer\-level alternatives with a one\-bit group selector, whereas CubicQuant offers continuously varying group\-local shapes with explicit parameter metadata\.

SqueezeLLM combines sensitivity\-based non\-uniform quantization with a sparse representation for outliers\[[14](https://arxiv.org/html/2608.06763#bib.bib14)\], while SpQR isolates sensitive outliers into a higher\-precision sparse component\[[15](https://arxiv.org/html/2608.06763#bib.bib15)\]\. CubicQuant instead retains a homogeneous packed scalar stream and has no separate sparse outlier matrix\.

### 7\.3Vector and additive codebooks

AQLM represents weight vectors through sums of learned codebook entries and optimizes layer/block output reconstruction\[[16](https://arxiv.org/html/2608.06763#bib.bib16)\]\. QuIP\# combines randomized Hadamard incoherence processing with lattice codebooks\[[17](https://arxiv.org/html/2608.06763#bib.bib17)\]\. Such vector representations can express correlations unavailable to a scalar curve and have shown strong extreme\-compression accuracy\. CubicQuant instead restricts itself to a scalar monotonic codebook to retain simple bit packing, compact metadata, and a direct tile\-local decode path\. The approaches therefore occupy different points in the accuracy, fitting\-cost, metadata, and kernel\-complexity space\.

### 7\.4Packed low\-bit GPU execution

LUT\-GEMM replaces repeated scalar dequantization with lookup\-based products for sub\-four\-bit weight\-only inference\[[18](https://arxiv.org/html/2608.06763#bib.bib18)\]\. FLUTE develops this line for non\-uniform LUT formats and explicitly addresses irregular widths such as W3 through offline restructuring, vectorized lookup, and fused matrix multiplication\[[19](https://arxiv.org/html/2608.06763#bib.bib19)\]\. MARLIN instead targets mixed\-precision packed weights and shows that the balance between memory traffic, dequantization, and matrix work changes with batch size\[[20](https://arxiv.org/html/2608.06763#bib.bib20)\]\. These systems motivate CubicQuant’s tile\-local reconstruction and shape\-dependent tactic selection, but do not directly execute its parametric level function\.

### 7\.5Dynamic weight\-activation execution

LLM\.int8\(\) exposes the importance of activation outliers and uses a mixed\-precision decomposition around an INT8 matrix product\[[21](https://arxiv.org/html/2608.06763#bib.bib21)\]\. ZeroQuant and SmoothQuant demonstrate that backend integration, equivalent transformations, and fusion are central to realizing W8A8 benefits\[[22](https://arxiv.org/html/2608.06763#bib.bib22),[5](https://arxiv.org/html/2608.06763#bib.bib5)\]\. QServe extends the systems argument to W4A8 serving and identifies dequantization overhead as a first\-order concern at larger batches\[[23](https://arxiv.org/html/2608.06763#bib.bib23)\]\. CubicQuant’s Dynamic\-A8 path follows the same principle—temporary activation quantization and weight reconstruction must be consumed inside or adjacent to the matrix kernel—but differs in how compressed weights are mapped to the INT8 operand\. Its carrier rounding is exposed in the offline objective rather than treated as an invisible implementation detail\.

## 8Limitations

Downstream task quality remains unmeasured\.Distributional distortion and finite\-group NRMSE characterize scalar reconstruction; they do not establish perplexity, reasoning, instruction following, long\-context behavior, or task accuracy\.

Finite\-group numerical coverage is incomplete\.The reported experiment contains one group size and omits W1 and W7\. It therefore demonstrates the observed distributional trend but cannot identify the best group size or provide a complete 1–8\-bit comparison\.

The systems evidence does not establish a general speedup\.The H200 measurements cover representative widths and shapes rather than a matched W1–W8 sweep across GPU generations\. They establish a workload\-dependent crossover, not a universal ordering between model\-dtype and Dynamic\-A8 execution\.

CubicQuant is not a native Tensor Core type\.Packed codes must be reconstructed into model\-dtype values or an INT8 carrier\. Any bandwidth benefit therefore depends on coupling reconstruction to consumption and avoiding full\-matrix expansion\.

Metadata and locality trade against one another\.Scale and shape parameters contribute64/G64/Gbits per weight\. Smaller groups increase local adaptability but also metadata cost; larger groups amortize metadata while forcing one curve to describe a broader tensor region\.

Weight\-only MSE is not universally optimal\.The reference estimator gives no special weight to directions emphasized by real activations\. Covariance\- or Hessian\-weighted fitting may preserve operator behavior more faithfully, but requires representative information beyond the static weights\.

Physical performance validation is Hopper\-centric\.Native compilation targets several architectures, but the latency evidence is drawn from H200\. Compilation coverage does not substitute for numerical and performance evaluation on Ampere, Ada, and Blackwell devices\.

Persistent activations and KV\-cache formats are outside the retained scope\.The measured persistent\-activation design was unfavorable, and attention\-state quantization poses a separate numerical and architectural problem\.

## 9Conclusion

CubicQuant separates two properties that fixed scalar formats bind together: the payload remains a regular integer bitstream, while group\-local parameters adapt the values represented by its magnitude codes\. The normalized cubic guarantees exact zero and endpoints, preserves ordering through an explicit monotonicity constraint, and contains symmetric uniform integer quantization as an exact special case\. Its additional flexibility costs two shape coefficients per group, giving a transparent effective width ofB\+64/GB\+64/Gbits per weight\.

The distributional analysis explains where that flexibility is useful\. The linear grid is already population\-optimal for a Uniform source, whereas Gaussian and Laplace distributions benefit from moving interior levels toward their high\-density center while retaining distant endpoints\. Across W4–W8, the optimized cubic family recovers most of the gap between uniform integers and a free Lloyd–Max codebook for the two unbounded reference distributions\. Finite groups add a second adaptation effect because each realized group may depart from its population law\.

The same representation supports two execution semantics\. Model\-dtype kernels consume continuous Cubic levels, while Dynamic\-A8 kernels consume the deterministic INT8 projection of those levels\. Incorporating both reconstructions into the fitting objective allows one serialized weight format to serve either path\. Tile\-local reconstruction preserves compressed weight traffic, and empirical kernel selection accommodates the fact that narrow GEMV and denser row regimes favor different realizations\.

The present evidence establishes a parametric scalar format with explicit distortion theory, finite\-group reconstruction gains, and direct packed GPU execution\. It does not yet establish downstream model quality or a universal end\-to\-end speedup\. Those questions require broader numerical, architectural, and model\-level evaluation, but they do not alter the central result: a compact parametric curve can recover much of the adaptability of a free codebook without abandoning a regular scalar bitstream\.

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## Appendix ANormative format properties

1. 1\.B∈\{1,…,8\}B\\in\\\{1,\\ldots,8\\\}andG​BGBis byte\-aligned\.
2. 2\.W1 uses binary−1/\+1\-1/\+1codes; W2–W8 reserve the most\-negative signed pattern and decode it to zero\.
3. 3\.The scalessis FP32 and positive for nonzero groups;aaandbbare FP16\.
4. 4\.The coefficientc=1−a−bc=1\-a\-bis derived and is not stored\.
5. 5\.Every accepted curve satisfiesq​\(0\)=0q\(0\)=0,q​\(1\)=1q\(1\)=1, and strict monotonicity\.
6. 6\.Packing order and logical tensor shape determine all tail decoding\.
7. 7\.Runtime A8 carrier generation uses round\-to\-nearest\-even, matching the reference framework behavior\.

## Appendix BReference decoding pseudocode

defcubic\_levels\(bits, scale, a, b\):ifbits==1:return\(\-scale, scale\)maximum=\(1<<\(bits\-1\)\)\-1c=1\.0\-a\-blevels=\[\]forindexinrange\(maximum\+1\):t=index/maximumq=t\*\(a\+t\*\(b\+t\*c\)\)levels\.append\(scale\*q\)returnlevelsdefdecode\(code, levels\):returnsign\(code\)\*levels\[abs\(code\)\]

## Appendix CExperimental reporting variables

Reproducing the numerical experiments requires the random seed, source distribution, sample count, group size, payload width, parameter precisions, clipping search, baseline scale optimization, and finite\-format enumeration rule\. Both the objective used for candidate selection and the statistic used for reporting must be stated explicitly\.

Reproducing kernel measurements additionally requires the GPU architecture and operating policy, driver and CUDA versions, compiler stack, tensor shape, activation mode, kernel family, correctness tolerance, repetition count, and reported latency statistic\. System\-level measurements require all non\-Cubic components that can affect the observation—including attention, KV state, parallel decomposition, request distribution, and concurrency—to be held fixed or reported separately\.