On Explicit Super-Expressive Approximation for Neural Networks

# 1 Introduction
Source: [https://arxiv.org/html/2607.06781](https://arxiv.org/html/2607.06781)
On Explicit Super\-Expressive Approximation for Neural Networks

Feng\-Lei Fan1, Ze\-Yu Li2, Chen\-Yu Wang2, and Jian\-Jun Wang3,∗

1Department of Data Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong\. fenglfan@cityu\.edu\.hk

2Department of Mathematics, Chinese University of Hong Kong, University Avenue, Shatin, N\.T\., Hong Kong\. 1155184076@link\.cuhk\.edu\.hk;WangChenyuCUHK@outlook\.com

3School of Civil Engineering and Architecture, Guangxi Minzu University, Nanning 530006, China\. 20250017@gxmzu\.edu\.cn

∗Corresponding author\.

###### Abstract

In this work, we investigate the fixed\-architecture neural network approximation with explicit parameter bounds and elementary activations\. While prior work demonstrated super\-expressive approximation using fixed\-size networks, they lack quantitative and non\-asymptotic characterizations of parameter magnitude with respect to the approximation error\. We resolve this issue by introducing the Chinese Remainder Theorem as a constructive encoding mechanism\. For Lipschitz continuous functions on\[0,1\]D\[0,1\]^\{D\}, we construct a width\-max⁡\{D,4\}\\max\\\{D,4\\\}, depth\-55network with explicit parameter\-error trade\-offs\. For Hölder\-smooth functions inCAr,γ​\(\[0,1\]D\)C^\{r,\\gamma\}\_\{A\}\\left\(\[0,1\]^\{D\}\\right\), our fixed network of widthmax⁡\{2​D,D\+5​N\+1\}\\max\\\{2D,\\ D\+5N\+1\\\}and depthr\+9r\+9achieves the parameter magnitude𝒫\\mathcal\{P\}bounded bylog2⁡𝒫=𝒪​\(ε−2​D/\(r\+γ\)​log⁡\(1/ε\)\)\\log\_\{2\}\\mathcal\{P\}=\\mathcal\{O\}\\bigl\(\\varepsilon^\{\-2D/\(r\+\\gamma\)\}\\log\(1/\\varepsilon\)\\bigr\)\. This is the dual result compared to those in the parameter\-bounded and architecture\-unbounded paradigm\.

Keywords:Neural network approximation, fixed\-architecture neural networks, elementary super\-expressive activation, Hölder\-smooth functions

### 1\.1Background and Our Result

Over the past few years, deep neural networks have achieved tremendous success in artificial intelligence\(Babaiee et al\.,,[2025](https://arxiv.org/html/2607.06781#bib.bib1); Esser et al\.,,[2024](https://arxiv.org/html/2607.06781#bib.bib9)\)and data science\(Wu and Yao,,[2025](https://arxiv.org/html/2607.06781#bib.bib42); Gicic et al\.,,[2024](https://arxiv.org/html/2607.06781#bib.bib10)\), motivating extensive research to establish the theoretical foundation of deep learning\. In this context, a substantial body of research explores the approximation capacity of deep neural networks to reveal the strong power of deep learning\. These works investigate how well a network can express the target function via the composition of affine linear transforms and nonlinear activations\. Typically, the approximation error is characterized in terms of the total number of parameters, the width and depth, or the number of neurons in a neural network\. In particular, the approximation of several important basic function spaces has been well studied,e\.g\., continuous function space\(Yarotsky,,[2018](https://arxiv.org/html/2607.06781#bib.bib45); Shen et al\.,,[2020](https://arxiv.org/html/2607.06781#bib.bib35)\),CsC^\{s\}function space\(Yarotsky and Zhevnerchuk,,[2020](https://arxiv.org/html/2607.06781#bib.bib47); Lu et al\.,,[2021](https://arxiv.org/html/2607.06781#bib.bib29)\), Sobolev space\(Yarotsky,,[2017](https://arxiv.org/html/2607.06781#bib.bib44); Hon and Yang,,[2021](https://arxiv.org/html/2607.06781#bib.bib15)\), Besov space\(Siegel,,[2023](https://arxiv.org/html/2607.06781#bib.bib39)\), and Korobov space\(Blanchard and Bennouna,,[2022](https://arxiv.org/html/2607.06781#bib.bib4); Yang and Lu,,[2024](https://arxiv.org/html/2607.06781#bib.bib43)\)\. Although these studies have reached \(nearly\) optimal approximation rates, the size of the network typically grows exponentially as the error goes small\.

One special direction of approximation results utilizes super\-expressive activation functions, which allow to approximate a class of functions by a fixed\-architecture network independent of the function and the error\. That is, without enlarging the size of the network, one can achieve arbitrary accuracy by merely adjusting the weights, where multiple activations can be applied\. The first fixed\-architecture approximation result was given by\(Maiorov and Pinkus,,[1999](https://arxiv.org/html/2607.06781#bib.bib30)\), which established the existence of super\-expressive activations\. The key idea is to construct a dense countable subset inC​\(\[0,1\]\)C\(\[0,1\]\)and then reduce the multivariate case to the univariate one via the Kolmogorov Superposition Theorem \(KST\)\(Kolmogorov,,[1957](https://arxiv.org/html/2607.06781#bib.bib20)\)\. Nevertheless, the super\-expressive activation here is quite complex and has no closed\-form formulation\. Following the idea of KST, this computational intractability was solved by\(Yarotsky,,[2021](https://arxiv.org/html/2607.06781#bib.bib46)\)\. In\(Yarotsky,,[2021](https://arxiv.org/html/2607.06781#bib.bib46)\), it was shown that anyC​\(\(\[0,1\]d\)\)C\\left\(\(\[0,1\]^\{d\}\\right\)\)function can be approximated by a fixed\-size network with multiple elementary activation functions\. In\(Zhang et al\.,,[2022](https://arxiv.org/html/2607.06781#bib.bib50)\), the authors utilized the elementary universal activation function \(EUAF\) and constructed a fixed network to approximate any continuous functions\. This work explicitly showed that an EUAF network requires only𝒪​\(1\)\\mathcal\{O\}\(1\)depth and𝒪​\(d2\)\\mathcal\{O\}\(d^\{2\}\)width to achieve super\-expressiveness, using irrational winding in Diophantine approximations and KST\. Furthermore,Wang et al\., \([2025](https://arxiv.org/html/2607.06781#bib.bib41)\)proposed a parametric EUAF \(PEUAF\) based fixed\-architecture network with a similar super\-expressive result\. They also showed that such super\-expressive networks are trainable in practice\.

However, achieving super\-expressiveness does come with a cost\. Once the width and depth are fixed, the approximation burden is transferred to the magnitude of the parameters\. Hence, establishing a quantitative and non\-asymptotic parameter boundw\.r\.t\.the approximation error is crucial in fixed\-architecture approximation\.Beknazaryan, \([2022](https://arxiv.org/html/2607.06781#bib.bib3)\)gives an explicit super\-expressive approximation of Hölder continuous functions by a refined Kronecker’s Theorem\. Although the network size and parameter magnitude are explicit, the activation is recalibrated as the error tolerance changes\. That is,Beknazaryan, \([2022](https://arxiv.org/html/2607.06781#bib.bib3)\)trades the growth of architecture for growth of activation complexity\. To the best of our knowledge, existing work on fixed\-architecture network approximation lacks quantitative and non\-asymptotic bounds on parameter magnitudesw\.r\.t\.approximation error, and those that do provide relevant results rely either on error\-dependent activation functions or on prior encoding of the target function in the initial input\. The main reason is that they utilized the Diophantine approximation or the KST, which contains unknown prefactors hard to be explicitly formulated\.

In this work, we propose a quantitative and non\-asymptotic fixed\-architecture approximation of Lipschitz continuous and Hölder\-smooth functions with the parameter magnitude explicitly characterized by the approximation accuracy\. Our central mechanism is the introduction of the Chinese Remainder Theorem \(CRT\)\. Given a partition of the input domain indexed by coprime integers and the quantization of the function value indexed by integers, we leverage CRT to explicitly associate each cell to a quantized value\. Then a fixed\-architecture network with multiple elementary super\-expressive activation functions to reconstruct the target\. Our main results are as follows:

∙\\bulletApproximation of Lipschitz functions\.For any Lipschitz function in\[0,1\]D\[0,1\]^\{D\}, we construct a network of widthmax⁡\{D,4\}\\max\\\{D,4\\\}and depth55to not only prove the existence of such a fixed\-size network but also provide explicit parameter magnitudes𝒫\\mathcal\{P\}satisfyinglog2⁡𝒫≤O​\(ε−2​D​log⁡\(1/ε\)\)\\log\_\{2\}\\mathcal\{P\}\\leq O\\bigl\(\\varepsilon^\{\-2D\}\\log\(1/\\varepsilon\)\\bigr\)that relate the target accuracyε\\varepsilonto the parameter magnitude of the network\. To the best of our knowledge, this is the first explicit relation made in the area of super\-expressive approximation\.

∙\\bulletApproximation of Hölder\-smooth functions\.For any Hölder\-smooth functionf∈CAr,γ​\(\[0,1\]D\)f\\in C^\{r,\\gamma\}\_\{A\}\(\[0,1\]^\{D\}\), we construct a rational gridwise polynomial surrogate via a local Taylor expansion\. This approximation is achieved using a fixed\-architecture network with widthmax⁡\{2​D,D\+5​N\+1\}\\max\\\{2D,\\ D\+5N\+1\\\}and depthr\+9r\+9\. Crucially, higher target smoothness mitigates the growth of the required parameter magnitude, which we bound bylog2⁡𝒫≤O​\(ε−2​D/\(r\+γ\)​log⁡\(1/ε\)\)\\log\_\{2\}\\mathcal\{P\}\\leq O\\bigl\(\\varepsilon^\{\-2D/\(r\+\\gamma\)\}\\log\(1/\\varepsilon\)\\bigr\)\. This demonstrates that the smoothness of the target function improves the scaling law, reducing the complexity exponent from2​D2Dto2​D/\(r\+γ\)2D/\(r\+\\gamma\)\.

Traditional approximation results provide the change in architecture with respect to the error while the parameter magnitude is fixed\. Our result can be regarded as dual to the traditional paradigm, which is instrumental in completing the picture of super\-expressive approximation\.

### 1\.2Related Work

Classical neural network approximation\. The approximation capacity of neural networks has been a central theme in deep learning theory and has seen a flourishing body of work in recent years\. Early works in universal approximation\(Cybenko,,[1989](https://arxiv.org/html/2607.06781#bib.bib8); Hornik et al\.,,[1989](https://arxiv.org/html/2607.06781#bib.bib17); Hornik,,[1991](https://arxiv.org/html/2607.06781#bib.bib16); Leshno et al\.,,[1993](https://arxiv.org/html/2607.06781#bib.bib23); Barron,,[1993](https://arxiv.org/html/2607.06781#bib.bib2)\)showed that sufficiently wide or deep fully\-connected neural networks can approximate any continuous function on compact sets without characterizing the approximation rate\. The universal approximation theorem of other networks is also widely studied\. For example,[Zhou, 2020b](https://arxiv.org/html/2607.06781#bib.bib52);[Zhou, 2020a](https://arxiv.org/html/2607.06781#bib.bib51); Yu and Zhou, \([2023](https://arxiv.org/html/2607.06781#bib.bib48)\); Li et al\., \([2025](https://arxiv.org/html/2607.06781#bib.bib26)\)demonstrated that deep convolutional networks can approximate arbitrary continuous functions on compact sets with sufficient depth, showing that locality and weight sharing preserve universality\. In addition,\(Yun et al\.,,[2020](https://arxiv.org/html/2607.06781#bib.bib49)\)proved that transformers universally approximate continuous permutation\-equivariant sequence\-to\-sequence maps, andLin and Jegelka, \([2018](https://arxiv.org/html/2607.06781#bib.bib28)\)showed that the ReLU activated ResNet with one neuron in each layer is a universal approximator\. Besides, the universal approximation of ResNet can also be established through the dynamic system\(Li et al\.,,[2023](https://arxiv.org/html/2607.06781#bib.bib25),[2019](https://arxiv.org/html/2607.06781#bib.bib27)\)\.

While the classical universal approximation theorems mainly focus on the existence of neural network approximants, a more quantitative line of research has subsequently emerged, aiming to characterize the approximation error in terms of network width, depth, parameter count, and the number of neurons\. The error estimate using the total number of parameters in ReLU networks has been well studied and has reached \(nearly\) optimal approximation rate for several basic function spaces,e\.g\., continuous functions\(Yarotsky,,[2018](https://arxiv.org/html/2607.06781#bib.bib45)\), smooth functions\(Yarotsky and Zhevnerchuk,,[2020](https://arxiv.org/html/2607.06781#bib.bib47); Petersen and Voigtlaender,,[2018](https://arxiv.org/html/2607.06781#bib.bib31)\), and Sobolev space\(Hon and Yang,,[2021](https://arxiv.org/html/2607.06781#bib.bib15); Yarotsky,,[2017](https://arxiv.org/html/2607.06781#bib.bib44)\)\. Errors in terms of width and depth further motivated the works on the non\-asymptotic approximation,e\.g\.,continuous functions\(Shen et al\.,,[2020](https://arxiv.org/html/2607.06781#bib.bib35),[2022](https://arxiv.org/html/2607.06781#bib.bib38)\),CsC^\{s\}functions\(Lu et al\.,,[2021](https://arxiv.org/html/2607.06781#bib.bib29)\), and functions in Korobov space\(Blanchard and Bennouna,,[2022](https://arxiv.org/html/2607.06781#bib.bib4); Yang and Lu,,[2024](https://arxiv.org/html/2607.06781#bib.bib43)\), which not only achieve \(nearly\) optimal approximation error, but also are explicit without unknown prefactors\. Although the errors are \(nearly\) optimal in classical network approximation, the size of the networks increases exponentially as the error goes small\.

Super\-expressive activation and fixed\-architecture approximation\. Designing super\-expressive activations is one of the directions to improve the approximation rate, where a class of target functions can be approximated by a neural network using a finite family of activation functions and fixed architecture only depending on the input dimension\. The initial idea of super\-expressive approximation comes from the KST\(Kolmogorov,,[1957](https://arxiv.org/html/2607.06781#bib.bib20)\), which shows that any continuous functionf∈C​\(\[0,1\]d\)f\\in C\\left\(\[0,1\]^\{d\}\\right\)can be represented asf​\(𝐱\)=∑i=02​dgi​\(∑j=1dhi,j​\(xj\)\)f\(\\mathbf\{x\}\)=\\sum\_\{i=0\}^\{2d\}g\_\{i\}\\left\(\\sum\_\{j=1\}^\{d\}h\_\{i,j\}\(x\_\{j\}\)\\right\)\. Note that the KST can be viewed as the network composition with activations depending on the target function\. See\(Igelnik and Parikh,,[2003](https://arxiv.org/html/2607.06781#bib.bib18); Kůrková,,[1991](https://arxiv.org/html/2607.06781#bib.bib21),[1992](https://arxiv.org/html/2607.06781#bib.bib22); Maiorov and Pinkus,,[1999](https://arxiv.org/html/2607.06781#bib.bib30)\)for further studies on the connection of KST and neural networks\. Especially, in\(Maiorov and Pinkus,,[1999](https://arxiv.org/html/2607.06781#bib.bib30)\), the authors used the KST to design a special activation and construct a fixed\-size network with𝒪​\(d\)\\mathcal\{O\}\(d\)neurons, which can approximate any continuous function on\[−1,1\]d\[\-1,1\]^\{d\}with an arbitrary error\. Since this activation is complicated and has no closed form, this network cannot be applied in practice\. Later, several works\(Guliyev and Ismailov,,[2016](https://arxiv.org/html/2607.06781#bib.bib12);[Guliyev and Ismailov, 2018b,](https://arxiv.org/html/2607.06781#bib.bib14); Ismailov,,[2014](https://arxiv.org/html/2607.06781#bib.bib19);[Guliyev and Ismailov, 2018a,](https://arxiv.org/html/2607.06781#bib.bib13)\)further studied similar activations\. For a long time, the computational intractability in super\-expressive approximation remains a hard issue\. Another line of research explored weaker forms of super\-expressiveness with elementary activations, where the network size has to grow for higher accuracy, but much slower than the power laws in the classical neural network approximation theory\. For example,Yarotsky and Zhevnerchuk, \([2020](https://arxiv.org/html/2607.06781#bib.bib47)\)showed that a network with ReLU andsin\\operatorname\{sin\}activations can approximate Lipschitz functions with exponential error ratew\.r\.t\.the number of parameters\. Besides,[Shen et al\., 2021b](https://arxiv.org/html/2607.06781#bib.bib37)also demonstrated that a three\-layer network with⌊⋅⌋\\lfloor\\cdot\\rfloor,2x2^\{x\}, and𝟏x≥0\\mathbf\{1\}\_\{x\\geq 0\}as activations can approximate Lipschitz functions with a similar error\.

The computational intractability of super\-expressive approximation was solved in\(Yarotsky,,[2021](https://arxiv.org/html/2607.06781#bib.bib46)\), where a fixed\-size network was constructed with multiple elementary activation functions,e\.g\.,sin\\operatorname\{sin\}&arcsin\\operatorname\{arcsin\},⌊⋅⌋\\lfloor\\cdot\\rfloor& a non\-polynomial analytic function, or a specificC1C^\{1\}piecewise elementary function, to approximate anyC​\(\[0,1\]d\)C\\left\(\[0,1\]^\{d\}\\right\)functions within arbitrarily small error\. It was also shown inYarotsky, \([2021](https://arxiv.org/html/2607.06781#bib.bib46)\)that most of the practically used activations are not super\-expressive\. However, this result does not give explicit characterization of the size dependence ondd\. To make the dependence clear,Zhang et al\., \([2022](https://arxiv.org/html/2607.06781#bib.bib50)\)showed that a EUAF network of width36​d​\(2​d\+1\)36d\(2d\+1\)and depth1111can approximate any continuous function on add\-dimensional closed cube\. Later,\(Wang et al\.,,[2025](https://arxiv.org/html/2607.06781#bib.bib41)\)showed that a fixed\-architecture network with with parametric EUAF is super\-expressive are trainable in practice\. To further give the parameter magnitude in terms of approximation accuracy,Beknazaryan, \([2022](https://arxiv.org/html/2607.06781#bib.bib3)\)utilized piecewise exponential and⌊⋅⌋\\lfloor\\cdot\\rflooractivations to give a super\-expressive approximation of Hölder continuous functions with explicit integer weights, where the activation is not fixed in the usual sense and varies with the target accuracy\. In addition,Bournez et al\., \([2025](https://arxiv.org/html/2607.06781#bib.bib6)\)constructed a fixed ReLU network to approximate arbitrary continuous functions, but it requires the encoding of the target function to be given in the initial input\.

## 2Preliminaries

We collect notations and definitions used throughout the paper\. We first introduce the basic function spaces, including Lipschitz continuous functions, Hölder\-smooth functions, and gridwise polynomials\.

###### Definition 1\(Lipschitz Continuous Function\)\.

LetA\>0A\>0\. Define

LipA⁡\(Ω\):=\{f:Ω→ℝ:\|f​\(𝐱\)−f​\(𝐲\)\|≤A​‖𝐱−𝐲‖∞\}\.\\operatorname\{Lip\}\_\{A\}\(\\Omega\):=\\bigl\\\{f:\\Omega\\to\\mathbb\{R\}:\\ \|f\(\\mathbf\{x\}\)\-f\(\\mathbf\{y\}\)\|\\leq A\\\|\\mathbf\{x\}\-\\mathbf\{y\}\\\|\_\{\\infty\}\\bigr\\\}\.\(1\)We sayf∈LipA⁡\(Ω\)f\\in\\operatorname\{Lip\}\_\{A\}\(\\Omega\)is Lipschitz continuous onΩ\\Omega\.

###### Definition 2\(Hölder\-Smooth Function\)\.

LetA\>0A\>0,r∈ℕr\\in\\mathbb\{N\},γ∈\(0,1\]\\gamma\\in\(0,1\]\. Define

CAr,γ​\(Ω\):=\{f:Ω→ℝ:supg∈∂rf𝐱≠𝐲\|g​\(𝐱\)−g​\(𝐲\)\|‖𝐱−𝐲‖∞γ≤A\},C^\{r,\\gamma\}\_\{A\}\(\\Omega\):=\\biggl\\\{f:\\Omega\\to\\mathbb\{R\}:\\ \\sup\_\{\\begin\{subarray\}\{c\}g\\in\\partial^\{r\}f\\\\ \\mathbf\{x\}\\neq\\mathbf\{y\}\\end\{subarray\}\}\\frac\{\\bigl\|g\(\\mathbf\{x\}\)\-g\(\\mathbf\{y\}\)\\bigr\|\}\{\\\|\\mathbf\{x\}\-\\mathbf\{y\}\\\|\_\{\\infty\}^\{\\gamma\}\}\\leq A\\biggr\\\},\(2\)whereffhas continuous partial derivatives up to orderrrand∂rf\\partial^\{r\}fis the set of up to order\-rrpartial derivatives\. We sayf∈CAr,γ​\(Ω\)f\\in C^\{r,\\gamma\}\_\{A\}\(\\Omega\)is Hölder\-smooth onΩ\\Omega\. In particularCA0,1​\(Ω\)=LipA⁡\(Ω\)C^\{0,1\}\_\{A\}\(\\Omega\)=\\operatorname\{Lip\}\_\{A\}\(\\Omega\)\.

###### Definition 3\(Gridwise Polynomials with Rational Coefficients\)\.

LetPolyD,r​\(ℚ\)\\mathrm\{Poly\}\_\{D,r\}\(\\mathbb\{Q\}\)denote the polynomials at degreerrwith rational coefficients:

PolyD,r​\(ℚ\):=\{∑\|𝜶\|≤rc𝜶​𝐱𝜶\|c𝜶∈ℚ\},\\mathrm\{Poly\}\_\{D,r\}\(\\mathbb\{Q\}\):=\\bigg\\\{\\sum\_\{\\begin\{subarray\}\{c\}\\\\ \|\\bm\{\\alpha\}\|\\leq r\\end\{subarray\}\}c\_\{\\bm\{\\alpha\}\}\\mathbf\{x\}^\{\\bm\{\\alpha\}\}\\;\\bigg\|\\;c\_\{\\bm\{\\alpha\}\}\\in\\mathbb\{Q\}\\bigg\\\},\(3\)where𝜶:=\(α1,α2,…,αD\)∈ℕ0D\\bm\{\\alpha\}:=\(\\alpha\_\{1\},\\alpha\_\{2\},\.\.\.,\\alpha\_\{D\}\)\\in\\mathbb\{N\}\_\{0\}^\{D\}and𝐱𝜶:=∏d=1Dxdαd\\mathbf\{x\}^\{\\bm\{\\alpha\}\}:=\\prod\_\{d=1\}^\{D\}x\_\{d\}^\{\\alpha\_\{d\}\}\.

Consider the uniform partition of the domainΩ=\[0,1\]D\\Omega=\[0,1\]^\{D\}intoMDM^\{D\}grids indexed by𝐦∈\{0,…,M−1\}D\\mathbf\{m\}\\in\\\{0,\\dots,M\-1\\\}^\{D\}:

Ω𝐦:=∏d=1D\[mdM,md\+1M\),\\Omega\_\{\\mathbf\{m\}\}:=\\prod\_\{d=1\}^\{D\}\\left\[\\frac\{m\_\{d\}\}\{M\},\\frac\{m\_\{d\}\+1\}\{M\}\\right\),\(4\)with the convention that the right endpoint is included whenmd=M−1m\_\{d\}=M\-1\. A functionf:Ω→ℝf:\\Omega\\to\\mathbb\{R\}is called a gridwise polynomial with rational coefficients if, for every gridΩ𝐦\\Omega\_\{\\mathbf\{m\}\}, there exists a polynomialP∈PolyD,r​\(ℚ\)P\\in\\mathrm\{Poly\}\_\{D,r\}\(\\mathbb\{Q\}\)such thatf​\(𝐱\)=P​\(M​𝐱−𝐦\)f\(\\mathbf\{x\}\)=P\(M\\mathbf\{x\}\-\\mathbf\{m\}\)for all𝐱∈Ω𝐦\\mathbf\{x\}\\in\\Omega\_\{\\mathbf\{m\}\}\.

Gridwise polynomials form an important class of functions\. They are capable of capturing local variations, boundary layers, spikes, and piecewise regular structures, while avoiding the rigidity of a single global polynomial\. Moreover, they have a natural connection to scientific computing: splines and finite element functions are precisely built from local polynomial pieces over grids\. In fact, gridwise polynomials are the Cartesian\-grid analogue of the local polynomial spaces used in finite element discretizations\(Ciarlet,,[2002](https://arxiv.org/html/2607.06781#bib.bib7)\), which are fundamental in modeling a wide range of physical phenomena, including elasticity, heat transfer, and fluid flow\. On structured Cartesian or hexahedral meshes, this is exactly the type of polynomial structure captured by the gridwise class\(Reddy and Gartling,,[2010](https://arxiv.org/html/2607.06781#bib.bib32)\)\.

We introduce the bit length to measure the bits required to encode a rational number, which follows the standard convention in algorithmic information theory and computational complexity\.

###### Definition 4\(Bit Length of a Rational Number\(Schrijver,,[1986](https://arxiv.org/html/2607.06781#bib.bib33)\)\)\.

Forq∈ℚq\\in\\mathbb\{Q\}, writeqqin reduced form:

q=s​\(q\)g​\(q\),q=\\frac\{s\(q\)\}\{g\(q\)\},\(5\)wheres​\(q\)∈ℤs\(q\)\\in\\mathbb\{Z\},g​\(q\)∈ℕ\+g\(q\)\\in\\mathbb\{N\}\_\{\+\}, andgcd⁡\(\|s​\(q\)\|,g​\(q\)\)=1\\gcd\(\|s\(q\)\|,g\(q\)\)=1\. Forq=0q=0, we use the conventions​\(q\)=0s\(q\)=0andg​\(q\)=1g\(q\)=1\. Define the rational bit length by

𝚋𝚒𝚝𝚜​\(q\):=⌈log2⁡\(\|s​\(q\)\|\+2\)⌉\+⌈log2⁡\(g​\(q\)\+1\)⌉\.\\mathtt\{bits\}\(q\):=\\bigl\\lceil\\log\_\{2\}\(\|s\(q\)\|\+2\)\\bigr\\rceil\+\\bigl\\lceil\\log\_\{2\}\(g\(q\)\+1\)\\bigr\\rceil\.\(6\)

We then introduce the activation functions used in this work, which are all simple and have been widely used in literature\. See Figure[1](https://arxiv.org/html/2607.06781#S2.F1)for visualization\.

###### Definition 5\(Floor Activation\([Shen et al\., 2021a,](https://arxiv.org/html/2607.06781#bib.bib36)\)\)\.

The floor activation is given by

ρfloor:ℝ→ℤ,ρfloor​\(t\):=⌊t⌋,\\rho\_\{\\mathrm\{floor\}\}:\\mathbb\{R\}\\to\\mathbb\{Z\},\\qquad\\rho\_\{\\mathrm\{floor\}\}\(t\):=\\lfloor t\\rfloor,\(7\)where⌊t⌋\\lfloor t\\rfloordenotes the largest integer not exceedingtt\. Equivalently,ρfloor​\(t\)=k\\rho\_\{\\mathrm\{floor\}\}\(t\)=kwhenevert∈\[k,k\+1\)t\\in\[k,k\+1\)for somek∈ℤk\\in\\mathbb\{Z\}\. We also writeρfrac​\(t\)=t−⌊t⌋\\rho\_\{\\mathrm\{frac\}\}\(t\)=t\-\\lfloor t\\rflooras a derived notation

###### Definition 6\(RePU Activation,\(Shen et al\.,,[2023](https://arxiv.org/html/2607.06781#bib.bib34)\)\)\.

For an integers∈ℕ\+s\\in\\mathbb\{N\}\_\{\+\}, the*rectified power unit*RePUs\\mathrm\{RePU\}\_\{s\}is defined as

ρs​\(t\)=\{ts,t≥0,0,t<0\.\\rho\_\{s\}\(t\)=\\begin\{cases\}t^\{s\},&t\\geq 0,\\\\ 0,&t<0\.\\end\{cases\}\(8\)In particular, whens=1s=1,ρs\\rho\_\{s\}is the usual ReLU activation; whens=2s=2, it is often called the ReQU activation\. We can use RePU activations as exact algebraic primitives for multiplication\. More precisely, we define

𝙼𝚞𝚕𝚝​\(u,v\):=\(ρ2​\(u\+v\)\+ρ2​\(−u−v\)−ρ2​\(u−v\)−ρ2​\(v−u\)\)/4\.\\mathtt\{Mult\}\(u,v\):=\\big\(\\rho\_\{2\}\(u\+v\)\+\\rho\_\{2\}\(\-u\-v\)\-\\rho\_\{2\}\(u\-v\)\-\\rho\_\{2\}\(v\-u\)\\big\)/4\.\(9\)

###### Definition 7\(Reciprocal Activation\(Boullé et al\.,,[2020](https://arxiv.org/html/2607.06781#bib.bib5)\)\)\.

Defineρinv:ℝ→ℝ\\rho\_\{\\rm\\mathrm\{inv\}\}:\\mathbb\{R\}\\to\\mathbb\{R\}by

ρinv​\(t\)=\{1/t,t≥1,2−t,t<1,\\rho\_\{\\rm\\mathrm\{inv\}\}\(t\)=\\begin\{cases\}1/t,&t\\geq 1,\\\\ 2\-t,&t<1,\\end\{cases\}\(10\)which, together withρfloor\\rho\_\{\\mathrm\{floor\}\}, enables the exact modular operation

Mmodp=M−p​ρfloor​\(M​ρinv​\(p\)\),M\\bmod p=M\-p\\rho\_\{\\mathrm\{floor\}\}\\bigl\(M\\rho\_\{\\mathrm\{inv\}\}\(p\)\\bigr\),for everyM∈ℤM\\in\\mathbb\{Z\}andp∈ℕ\+p\\in\\mathbb\{N\}^\{\+\}\.

![Refer to caption](https://arxiv.org/html/2607.06781v1/x1.png)Figure 1:Visualization of the floor activationρfloor​\(t\)\\rho\_\{\\mathrm\{floor\}\}\(t\), the RePU activationρ1​\(t\),ρ2​\(t\)\\rho\_\{1\}\(t\),\\rho\_\{2\}\(t\), and the reciprocal activationρinv\\rho\_\{\\mathrm\{inv\}\}\.
## 3Approximation of Lipschitz Functions

In this section, we propose a fixed\-architecture construction for approximating Lipschitz continuous functions with explicit parameter growth\. This bridges a critical gap in super\-expressive approximation: previous works either established mere existence without quantifying parameter scales\(Yarotsky,,[2021](https://arxiv.org/html/2607.06781#bib.bib46); Zhang et al\.,,[2022](https://arxiv.org/html/2607.06781#bib.bib50)\)\. Unlike their work that used irrational winding and KST, we apply CRT to give an explicit construction\. For the first time, our theorem provides an explicit bound on the parameter magnitude in terms of the approximation error\.

###### Theorem 1\(Lipschitz Continuous Function\)\.

LetΩ=\[0,1\]D\\Omega=\[0,1\]^\{D\}andf∈LipA⁡\(Ω\)f\\in\\operatorname\{Lip\}\_\{A\}\(\\Omega\)\. Then for arbitraryε\>0\\varepsilon\>0, there exists a networkΦ\\Phiof widthmax⁡\{D,4\}\\max\\\{D,4\\\}and depth55withρfloor,ρ1,ρ2\\rho\_\{\\mathrm\{floor\}\},\\rho\_\{1\},\\rho\_\{2\}, andρinv\\rho\_\{\\mathrm\{inv\}\}as activations, such that

\|Φ​\(𝐱\)−f​\(𝐱\)\|≤ε\.\\bigl\|\\Phi\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\)\\bigr\|\\ \\leq\\varepsilon\.\(11\)Moreover, the parameter magnitude satisfies

𝒫​\(Φ\)≤∏i=1MεD\(1\+i​\(Jε\+1\)​\(MεD\)\!\),\\mathcal\{P\}\(\\Phi\)\\;\\leq\\;\\prod\_\{i=1\}^\{M\_\{\\varepsilon\}^\{D\}\}\\Bigl\(1\+i\\,\(J\_\{\\varepsilon\}\+1\)\\,\(M\_\{\\varepsilon\}^\{D\}\)\!\\Bigr\),\(12\)whereB=‖f‖L∞​\(Ω\)B=\\\|f\\\|\_\{L^\{\\infty\}\(\\Omega\)\},Mε:=⌈2​A/ε⌉M\_\{\\varepsilon\}:=\\left\\lceil 2A/\\varepsilon\\right\\rceil, andJε:=⌈4​B/ε⌉J\_\{\\varepsilon\}:=\\left\\lceil 4B/\\varepsilon\\right\\rceil\.

### 3\.1Idea of the Construction in Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1)

Here, we show the idea of proving Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1)step by step\. The construction follows a fixed\-architecture finite\-fitting strategy, which has been proposed in\([Shen et al\., 2021b,](https://arxiv.org/html/2607.06781#bib.bib37)\)for a weaker form of super\-expressiveness\. Our two novel main ideas are: 1\) First, a continuous function is simplified to a finite number of representative values through quantization\. 2\) Then, unique pairwise coprime integers label the domain grids, enabling CRT to bridge the domain and function value\. We take the following four steps:

*1\. Divide and reduce the target to finitely many codes\.*ChooseM∈ℕ\+M\\in\\mathbb\{N\}\_\{\+\}, we use𝐦=\(m1,…,mD\)∈\{0,…,M−1\}D\\mathbf\{m\}=\(m\_\{1\},\\ldots,m\_\{D\}\)\\in\\\{0,\\ldots,M\-1\\\}^\{D\}as the coordinate of a grid\. For each𝐦\\mathbf\{m\}, define

Q𝐦:=∏d=1D\[mdM,md\+1M\),Q\_\{\\mathbf\{m\}\}:=\\prod\_\{d=1\}^\{D\}\\left\[\\frac\{m\_\{d\}\}\{M\},\\frac\{m\_\{d\}\+1\}\{M\}\\right\),\(13\)with the convention that an interval whose right endpoint is11is closed, and otherwise is half\-open\. Then⋃𝐦∈\{0,…,M−1\}DQ𝐦\\bigcup\_\{\\mathbf\{m\}\\in\\\{0,\\ldots,M\-1\\\}^\{D\}\}Q\_\{\\mathbf\{m\}\}forms a partition ofΩ=\[0,1\]d\\Omega=\[0,1\]^\{d\}\. Let𝐱𝐦:=𝐦/M\\mathbf\{x\}\_\{\\mathbf\{m\}\}:=\\mathbf\{m\}/Mbe the corner ofQ𝐦Q\_\{\\mathbf\{m\}\}\.

Now, we quantize the function value offf\. Recall that‖f‖∞=B\\\|f\\\|\_\{\\infty\}=B\. Then we chooseJ∈ℕ\+J\\in\\mathbb\{N\}\_\{\+\}and setδ:=2​B/J\\delta:=2B/J\. Thus, we haveJJintervals\[j⋅δ−B,\(j\+1\)⋅δ−B\],j=0,1,⋯,J−1\[j\\cdot\\delta\-B,\(j\+1\)\\cdot\\delta\-B\],j=0,1,\\cdots,J\-1\. Due to the Lipschitz continuity offf, for anyJJ, as long asMMis sufficient large, there always exists

−B\+j​δ≤f​\(𝐱𝐦\)≤−B\+\(j\+1\)​δ,\-B\+j\\delta\\leq f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\leq\-B\+\(j\+1\)\\delta,\(14\)for somejj\. Thus, we establish the one\-to\-one mapping between the grid𝐦\\mathbf\{m\}and thejj\-th quantization level\. For convenience, we refer to it asj𝐦j\_\{\\mathbf\{m\}\}\. Our approximation is to use a piecewise constant functionf¯​\(𝐱\)\\bar\{f\}\(\\mathbf\{x\}\)overQ𝐦Q\_\{\\mathbf\{m\}\}to approximate the target functionff\. In each gridQ𝐦Q\_\{\\mathbf\{m\}\}, the constant is the bottom value of the quantization interval\. Mathematically, we have

f¯​\(𝐱\):=−B\+j𝐦​δ\\bar\{f\}\(\\mathbf\{x\}\):=\-B\+j\_\{\\mathbf\{m\}\}\\delta\(15\)for∀𝐱∈Q𝐦\\forall\\mathbf\{x\}\\in Q\_\{\\mathbf\{m\}\}\. Then, for everyx∈Q𝐦x\\in Q\_\{\\mathbf\{m\}\},

\|f¯​\(𝐱\)−f​\(𝐱\)\|≤\|f¯​\(𝐱\)−f​\(𝐱𝐦\)\|\+\|f​\(𝐱𝐦\)−f​\(𝐱\)\|≤δ\+A/M\.\|\\bar\{f\}\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\)\|\\leq\|\\bar\{f\}\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\|\+\|f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\-f\(\\mathbf\{x\}\)\|\\leq\\delta\+A/M\.\(16\)See Figure[2](https://arxiv.org/html/2607.06781#S3.F2)for our partition of the input domain and the quantization for the function value offf\.

![Refer to caption](https://arxiv.org/html/2607.06781v1/x2.png)Figure 2:Illustration of Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1)pipeline\.*2\. Extract the active grid index off the grid boundary\.*For𝐱∈Ω\\mathbf\{x\}\\in\\Omega, each coordinatexdx\_\{d\}lies strictly inside one of the intervals\[md/M,\(md\+1\)/M\]\[m\_\{d\}/M,\(m\_\{d\}\+1\)/M\]\. We derive the index of the grid where𝐱\\mathbf\{x\}belongs by the following equation:

𝐦=\[ρfloor​\(M​x1\),ρfloor​\(M​x2\),⋯,ρfloor​\(M​xD\)\]∈\{0,1,⋯,M−1\}D\.\\mathbf\{m\}=\[\\rho\_\{\\mathrm\{floor\}\}\(Mx\_\{1\}\),\\rho\_\{\\mathrm\{floor\}\}\(Mx\_\{2\}\),\\cdots,\\rho\_\{\\mathrm\{floor\}\}\(Mx\_\{D\}\)\]\\in\\\{0,1,\\cdots,M\-1\\\}^\{D\}\.\(17\)Then, we generate a unique annotation for each grid based on its associated index:

Φ1​\(𝐱\):=1\+∑d=1DMd−1​ρfloor​\(M​xd\),\\Phi\_\{1\}\(\\mathbf\{x\}\):=1\+\\sum\_\{d=1\}^\{D\}M^\{d\-1\}\\rho\_\{\\mathrm\{floor\}\}\(Mx\_\{d\}\),\(18\)which is exactly the base\-MMencoding of the active grid coordinate\. The first sub\-network realizesΦ1\\Phi\_\{1\}\. In the activation model containingρfloor\\rho\_\{\\mathrm\{floor\}\}, this sub\-network is implemented by applyingDDfloor units toM​x1,…,M​xDMx\_\{1\},\\ldots,Mx\_\{D\}, followed by one affine combination\.

*3\. Construct a group of modulus by CRT\.*Now, we assign a modulo for each grid based on its unique annotation\. Set

c:=\(J\+1\)​MD\!,p​\(𝐱\):=1\+Φ1​\(𝐱\)​c,c:=\(J\+1\)M^\{D\}\!,\\qquad p\(\\mathbf\{x\}\):=1\+\\Phi\_\{1\}\(\\mathbf\{x\}\)c,\(19\)thenp​\(𝐱\)\>Jp\(\\mathbf\{x\}\)\>J, and the integersp​\(𝐱\)p\(\\mathbf\{x\}\)andp​\(𝐱′\)p\(\\mathbf\{x\}^\{\\prime\}\)are pairwise coprime if𝐱\\mathbf\{x\}and𝐱′\\mathbf\{x\}^\{\\prime\}are from two different grids\. The coprimality ofp​\(𝐱\)p\(\\mathbf\{x\}\)is formally confirmed by Lemma[8](https://arxiv.org/html/2607.06781#Thmtheorem8)\. Then by CRT, for𝐱\\mathbf\{x\}located atQ𝐦Q\_\{\\mathbf\{m\}\}, there exists an integerKKsuch that

K≡j𝐦\(modp​\(𝐱\)\),K\\equiv j\_\{\\mathbf\{m\}\}\\pmod\{p\(\\mathbf\{x\}\)\},\(20\)Since0≤j𝐦<J<p​\(𝐱\)0\\leq j\_\{\\mathbf\{m\}\}<J<p\(\\mathbf\{x\}\), the codej𝐦j\_\{\\mathbf\{m\}\}is exactly the standard residue ofKmodp​\(𝐱\)K\\mod p\(\\mathbf\{x\}\)\. Thus

j𝐦=Kmodp​\(𝐱\)=K−p​\(𝐱\)​⌊Kp​\(𝐱\)⌋\.j\_\{\\mathbf\{m\}\}=K\\bmod p\(\\mathbf\{x\}\)=K\-p\(\\mathbf\{x\}\)\\left\\lfloor\\frac\{K\}\{p\(\\mathbf\{x\}\)\}\\right\\rfloor\.\(21\)
The quotient term in \([21](https://arxiv.org/html/2607.06781#S3.E21)\) is realized by

q​\(𝐱\):=ρfloor​\(K​ρinv​\(p​\(𝐱\)\)\)\.q\(\\mathbf\{x\}\):=\\rho\_\{\\mathrm\{floor\}\}\\\!\\left\(K\\rho\_\{\\mathrm\{inv\}\}\(p\(\\mathbf\{x\}\)\)\\right\)\.\(22\)SinceKKis a fixed target\-dependent parameter, the factorK​ρinv​\(p​\(𝐱\)\)K\\rho\_\{\\mathrm\{inv\}\}\(p\(\\mathbf\{x\}\)\)is obtained by affine scaling after the inverse activation\. Henceq​\(𝐱\)=⌊Kp​\(𝐱\)⌋\.q\(\\mathbf\{x\}\)=\\left\\lfloor\\frac\{K\}\{p\(\\mathbf\{x\}\)\}\\right\\rfloor\.

It remains to realize the productp​\(𝐱\)​q​\(𝐱\)p\(\\mathbf\{x\}\)q\(\\mathbf\{x\}\)\. This product can not be produced by an affine layer because both factors are hidden\-channel quantities\. With the ReQU multiplication module𝙼𝚞𝚕𝚝\\mathtt\{Mult\}, the CRT module is

Φ2​\(𝐱\):=K−𝙼𝚞𝚕𝚝​\(p​\(𝐱\),q​\(𝐱\)\)\.\\Phi\_\{2\}\(\\mathbf\{x\}\):=K\-\\mathtt\{Mult\}\(p\(\\mathbf\{x\}\),q\(\\mathbf\{x\}\)\)\.\(23\)
*4\. Reconstruct the value and control the error\.*Finally, we use a network to represent the following function

Φ3​\(j𝐦\):=−B\+j𝐦​δ,\\Phi\_\{3\}\(j\_\{\\mathbf\{m\}\}\):=\-B\+j\_\{\\mathbf\{m\}\}\\delta,\(24\)which reconstructs the function value from the quantization level\. Then

f¯​\(𝐱\)=Φ3∘Φ2∘Φ1,\\bar\{f\}\(\\mathbf\{x\}\)=\\Phi\_\{3\}\\circ\\Phi\_\{2\}\\circ\\Phi\_\{1\},\(25\)concludes our construction\. See Figure[2](https://arxiv.org/html/2607.06781#S3.F2)for an illustration of the target network\.

Based on the aforementioned construction, we can give the width and depth of the network and estimate the parameter bounds based on the error\.

### 3\.2Proof of Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1)

In our proof idea, the modulip​\(𝐱\)p\(\\mathbf\{x\}\)must satisfy two requirements such that CRT can be applied to have a fixed\-architecture network to realize an arbitrarily small error\. First, each modulusp​\(𝐱\)p\(\\mathbf\{x\}\)must be greater than the number of intervalsJJ\. Second,p​\(𝐱\)p\(\\mathbf\{x\}\)andp​\(𝐱′\)p\(\\mathbf\{x\}^\{\\prime\}\)must be pairwise coprime for𝐱\\mathbf\{x\}and𝐱′\\mathbf\{x\}^\{\\prime\}are from different grids\. Gödel’sβ\\beta\-function provides a classical method for forming coprime numbers\(Gödel,,[1931](https://arxiv.org/html/2607.06781#bib.bib11); Smith,,[2013](https://arxiv.org/html/2607.06781#bib.bib40)\)\.

###### Lemma 8\(Pairwise coprimality ofpip\_\{i\}\)\.

GivenJ,M,D∈ℕ\+J,M,D\\in\\mathbb\{N\}\_\{\+\}, and setc:=\(J\+1\)​\(MD\)\!\.c:=\(J\+1\)\(M^\{D\}\)\!\.Define

pi:=1\+i​c,i=1,…,MD\.p\_\{i\}:=1\+ic,\\qquad i=1,\\ldots,M^\{D\}\.\(26\)Thenp1,…,pMDp\_\{1\},\\ldots,p\_\{M^\{D\}\}are pairwise coprime\. Moreover,pi\>Jp\_\{i\}\>Jfor everyi=1,…,MDi=1,\\ldots,M^\{D\}\.

###### Proof\.

i\) First, sincec=\(J\+1\)​\(MD\)\!≥J\+1c=\(J\+1\)\(M^\{D\}\)\!\\geq J\+1, we havepi=1\+i​c\>J,p\_\{i\}=1\+ic\>J,fori=1,…,MDi=1,\\ldots,M^\{D\}\. ii\) Second, fix1≤a<b≤MD1\\leq a<b\\leq M^\{D\}, and letg∈ℕ\+g\\in\\mathbb\{N\}\_\{\+\}be a common divisor ofpap\_\{a\}andpbp\_\{b\}, which meansg∣pag\\mid p\_\{a\}andg∣pb\.g\\mid p\_\{b\}\.Thenggalso divides their difference:g∣\(pb−pa\)\.g\\mid\(p\_\{b\}\-p\_\{a\}\)\.Sincepb−pa=\(1\+b​c\)−\(1\+a​c\)=\(b−a\)​c,p\_\{b\}\-p\_\{a\}=\(1\+bc\)\-\(1\+ac\)=\(b\-a\)c,we getg∣\(b−a\)​c\.g\\mid\(b\-a\)c\.Sincepa=1\+a​cp\_\{a\}=1\+ac, any common divisor ofpap\_\{a\}andccmust dividepa−a​c=1\.p\_\{a\}\-ac=1\.Hencegcd⁡\(pa,c\)=1\\gcd\(p\_\{a\},c\)=1\. Becauseg∣pag\\mid p\_\{a\}, it follows thatgcd⁡\(g,c\)=1\.\\gcd\(g,c\)=1\.Together withg∣\(b−a\)​cg\\mid\(b\-a\)c, this impliesg∣\(b−a\)\.g\\mid\(b\-a\)\.Since1≤b−a≤MD−11\\leq b\-a\\leq M^\{D\}\-1, the integerb−ab\-adivides\(MD\)\!\(M^\{D\}\)\!\. So we have\(b−a\)∣c\.\(b\-a\)\\mid c\.Thusg∣cg\\mid c\. Together withg∣pag\\mid p\_\{a\}, this givesg∣\(pa−a​c\)\.g\\mid\(p\_\{a\}\-ac\)\.Thereforeg=1g=1\. Hencegcd⁡\(pa,pb\)=1\\gcd\(p\_\{a\},p\_\{b\}\)=1\. Sincea<ba<bis arbitrary,p1,…,pMDp\_\{1\},\\ldots,p\_\{M^\{D\}\}are pairwise coprime\. ∎

###### Lemma 9\(Chinese Remainder Theorem \(CRT\)\)\.

Letp1,…,pNp\_\{1\},\\dots,p\_\{N\}be pairwise coprime integers, and letb1,…,bN∈ℤb\_\{1\},\\dots,b\_\{N\}\\in\\mathbb\{Z\}\. Then the system of congruences

K≡bj​mod​pj,j=1,…,N,K\\equiv b\_\{j\}~~\\mathrm\{mod\}~\{p\_\{j\}\},\\qquad j=1,\\dots,N,\(27\)has a solutionK∈ℤK\\in\\mathbb\{Z\}\. Moreover, the solution is unique modulo∏j=1Npj\.\\prod\_\{j=1\}^\{N\}p\_\{j\}\.Equivalently, there exists a unique residue class\[K\]∈ℤ/P​ℤ\[K\]\\in\\mathbb\{Z\}/P\\mathbb\{Z\}satisfying all the congruences, and one may choose its representative so that0≤K<∏j=1Npj\.0\\leq K<\\prod\_\{j=1\}^\{N\}p\_\{j\}\.

###### Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1)\.

Combining Lemma[8](https://arxiv.org/html/2607.06781#Thmtheorem8)and Proposition[9](https://arxiv.org/html/2607.06781#Thmtheorem9), it can be seen that Eq\. \([21](https://arxiv.org/html/2607.06781#S3.E21)\) holds true\. The modulip1,…,pMDp\_\{1\},\\ldots,p\_\{M^\{D\}\}are pairwise coprime\. Hence, there exists an integerKKsuch thatK≡j𝐦\(modp𝐦\)\.K\\equiv j\_\{\\mathbf\{m\}\}\\pmod\{p\_\{\\mathbf\{m\}\}\}\.Sincep𝐦\>Jp\_\{\\mathbf\{m\}\}\>Jandj𝐦∈\{0,…,J−1\}j\_\{\\mathbf\{m\}\}\\in\\\{0,\\ldots,J\-1\\\}, we have0≤j𝐦<J<p𝐦\.0\\leq j\_\{\\mathbf\{m\}\}<J<p\_\{\\mathbf\{m\}\}\.Therefore reducingKKmodulop𝐦p\_\{\\mathbf\{m\}\}recoversj𝐦j\_\{\\mathbf\{m\}\}exactly:j𝐦=Kmodp𝐦\.j\_\{\\mathbf\{m\}\}=K\\bmod p\_\{\\mathbf\{m\}\}\.Becausep​\(𝐱\)=p𝐦p\(\\mathbf\{x\}\)=p\_\{\\mathbf\{m\}\}onQ𝐦Q\_\{\\mathbf\{m\}\}, the sub\-networkΦ2\\Phi\_\{2\}generatesj𝐦j\_\{\\mathbf\{m\}\}exactly\. Then throughΦ3\\Phi\_\{3\}, we can reconstruct the function valueΦ3∘Φ2∘Φ1​\(𝐱\)=f¯​\(𝐱\)\\Phi\_\{3\}\\circ\\Phi\_\{2\}\\circ\\Phi\_\{1\}\(\\mathbf\{x\}\)=\\bar\{f\}\(\\mathbf\{x\}\)from the quantization level\.

Now, we estimate the relationship between the error and the magnitude of the parameters\. Since there are parameters in allΦ1,Φ2,Φ3\\Phi\_\{1\},\\Phi\_\{2\},\\Phi\_\{3\}dependent onϵ\\epsilon, we provide their relationship, respectively\.

Parameters inΦ1\\Phi\_\{1\}: By the local reconstruction estimate Eq\. \([16](https://arxiv.org/html/2607.06781#S3.E16)\),

\|f¯​\(𝐱\)−f​\(𝐱\)\|≤δ\+A/M,𝐱∈Ω\.\|\\bar\{f\}\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\)\|\\leq\\delta\+A/M,\\qquad\\mathbf\{x\}\\in\\Omega\.\(28\)Sincef∈LipA⁡\(Ω\)f\\in\\operatorname\{Lip\}\_\{A\}\(\\Omega\), choosingM≥2​A/εM\\geq 2A/\\varepsilonandJ≥4​B/εJ\\geq 4B/\\varepsilongivesδ=2​B/J≤ε/2\\delta=2B/J\\leq\\varepsilon/2\. Hence,

\|f¯​\(𝐱\)−f​\(𝐱\)\|≤A/M\+δ≤ε,𝐱∈Ω\.\|\\bar\{f\}\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\)\|\\leq A/M\+\\delta\\leq\\varepsilon,\\qquad\\mathbf\{x\}\\in\\Omega\.\(29\)FixMε=⌈2​A/ε⌉M\_\{\\varepsilon\}=\\lceil 2A/\\varepsilon\\rceilandJε=⌈4​B/ε⌉J\_\{\\varepsilon\}=\\lceil 4B/\\varepsilon\\rceil\. The moduleΦ1\\Phi\_\{1\}uses the grid scaleMεM\_\{\\varepsilon\}and the weights1,Mε,…,MεD−11,M\_\{\\varepsilon\},\\ldots,M\_\{\\varepsilon\}^\{D\-1\}, so

𝒫​\(Φ1\)≤MεD\.\\mathcal\{P\}\(\\Phi\_\{1\}\)\\leq M\_\{\\varepsilon\}^\{D\}\.\(30\)
Parameters inΦ2\\Phi\_\{2\}: The parameters ofΦ2\\Phi\_\{2\}areccin Eq\. \([26](https://arxiv.org/html/2607.06781#S3.E26)\) and the CRT integerKK\. The dominant one isKK, bounded by the modulus product: the least nonnegative solution satisfies

0≤K≤∏i=1MεDpi−1=∏i=1MεD\(1\+i​\(Jε\+1\)​\(MεD\)\!\)−1\.0\\leq K\\leq\\prod\_\{i=1\}^\{M\_\{\\varepsilon\}^\{D\}\}p\_\{i\}\-1=\\prod\_\{i=1\}^\{M\_\{\\varepsilon\}^\{D\}\}\\Bigl\(1\+i\\,\(J\_\{\\varepsilon\}\+1\)\(M\_\{\\varepsilon\}^\{D\}\)\!\\Bigr\)\-1\.\(31\)Thus

𝒫​\(Φ2\)≤∏i=1MεD\(1\+i​\(Jε\+1\)​\(MεD\)\!\)\.\\mathcal\{P\}\(\\Phi\_\{2\}\)\\leq\\prod\_\{i=1\}^\{M\_\{\\varepsilon\}^\{D\}\}\\Bigl\(1\+i\\,\(J\_\{\\varepsilon\}\+1\)\(M\_\{\\varepsilon\}^\{D\}\)\!\\Bigr\)\.\(32\)
We count the width and depth layer\-wise over the entire composed network\. In each layer, the width equals the number of neurons, including identity channels that carry a value forward for use in a later layer\. Following the standard approximation literature\(Yarotsky,,[2017](https://arxiv.org/html/2607.06781#bib.bib44);[Shen et al\., 2021a,](https://arxiv.org/html/2607.06781#bib.bib36)\), Each hidden layer consisting of affine transformations and element\-wise activations is counted as one layer\. The detailed calculation appears in Table[1](https://arxiv.org/html/2607.06781#S3.T1)\.

Width:InΦ1\\Phi\_\{1\}, the first layer’s affine map scales the input toM​𝐱M\\mathbf\{x\}, and its activation appliesρfloor\\rho\_\{\\mathrm\{floor\}\}to produce⌊M​xd⌋\\lfloor Mx\_\{d\}\\rfloor\(d=1,…,Dd=1,\\dots,D\)\. Since no later module requires the original coordinatesxdx\_\{d\}, no carry channels are needed, giving this layer a width ofDD\.

InΦ2\(1\)\\Phi\_\{2\}^\{\(1\)\}andΦ2\(2\)\\Phi\_\{2\}^\{\(2\)\}, the second layer’s affine map linearly combines theDDfloor outputs to directly compute the modulusp​\(𝐱\)=1\+c​Φ1​\(𝐱\)p\(\\mathbf\{x\}\)=1\+c\\,\\Phi\_\{1\}\(\\mathbf\{x\}\)\. Then, its activation appliesρinv\\rho\_\{\\mathrm\{inv\}\}to produce1/p​\(𝐱\)1/p\(\\mathbf\{x\}\), while applyingρ1\\rho\_\{1\}to carryp​\(𝐱\)p\(\\mathbf\{x\}\)forward\. This layer has the width of22\.

InΦ2\(3\)\\Phi\_\{2\}^\{\(3\)\}, the third layer’s affine map scales1/p​\(𝐱\)1/p\(\\mathbf\{x\}\)by the fixed target\-dependent constantKKto obtainK/p​\(𝐱\)K/p\(\\mathbf\{x\}\)while still carryingp​\(𝐱\)p\(\\mathbf\{x\}\)\. Its activation appliesρfloor\\rho\_\{\\mathrm\{floor\}\}to produceq​\(𝐱\)=⌊K/p​\(𝐱\)⌋q\(\\mathbf\{x\}\)=\\lfloor K/p\(\\mathbf\{x\}\)\\rfloorandρ1\\rho\_\{1\}to carryp​\(𝐱\)p\(\\mathbf\{x\}\)\. This layer has the width of22\.

InΦ2\(4\)\\Phi\_\{2\}^\{\(4\)\}, the fourth layer constructs the components for𝙼𝚞𝚕𝚝​\(p​\(𝐱\),q​\(𝐱\)\)\\mathtt\{Mult\}\(p\(\\mathbf\{x\}\),q\(\\mathbf\{x\}\)\)\. Its affine map forms the four linear combinationsp\+qp\+q,−p−q\-p\-q,p−qp\-q,q−pq\-p\(width44\)\. Then, its activation appliesρ2\\rho\_\{2\}\(RePU\) to each component\. Hence, this layer has width44\.

InΦ3\\Phi\_\{3\}, the fifth \(output\) layer is a pure affine map\. It first combines theρ2\\rho\_\{2\}outputs to evaluateΦ2​\(𝐱\)=K−𝙼𝚞𝚕𝚝​\(p,q\)\\Phi\_\{2\}\(\\mathbf\{x\}\)=K\-\\mathtt\{Mult\}\(p,q\), and simultaneously scales it byδ\\deltaand subtractsBBto reconstruct−B\+j𝐦​δ\-B\+j\_\{\\mathbf\{m\}\}\\delta\. This outputs a scalar, giving a width of11\.

The maximum width across all layers is thereforemax⁡\{D,4\}\\max\\\{D,\\,4\\\}\.

Depth:By strictly grouping one affine map and one activation into a single layer, the network comprises11layer forΦ1\\Phi\_\{1\},33layers forΦ2\\Phi\_\{2\}, and11output layer forΦ3\\Phi\_\{3\}\(which absorbs the final affine summation ofΦ2\\Phi\_\{2\}\), giving a total depth of55\.

∎

Table 1:Layer\-wise parameter magnitude and width\-depth accounting for the continuous target construction\.

## 4Approximation of Hölder\-Smooth Functions

In this part, we give a super\-expressive approximation of Hölder\-smooth targets, which demonstrates that smoother targets make parameters grow slower as the approximation error decreases\. The key of the construction is that any Hölder\-smooth targets can be approximated by a gridwise Taylor polynomial function of rational coefficients\.

###### Theorem 2\(Hölder\-smooth function\)\.

Letf∈CAr,γ​\(Ω\)f\\in C^\{r,\\gamma\}\_\{A\}\(\\Omega\)withr∈ℕr\\in\\mathbb\{N\}andγ∈\(0,1\]\\gamma\\in\(0,1\]\. For everyε∈\(0,1\)\\varepsilon\\in\(0,1\), there exists a networkΨε\\Psi\_\{\\varepsilon\}with activationρfloor,ρ1,ρ2\\rho\_\{\\mathrm\{floor\}\},\\rho\_\{1\},\\rho\_\{2\}andρinv\\rho\_\{\\mathrm\{inv\}\}as activation functions such that

\|Ψε​\(𝐱\)−f​\(𝐱\)\|≤ε\(𝐱∈Ω\),\\bigl\|\\Psi\_\{\\varepsilon\}\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\)\\bigr\|\\leq\\varepsilon\\qquad\(\\mathbf\{x\}\\in\\Omega\),\(33\)where the width ismax⁡\{2​D,D\+5​N\+1\}\\max\\\{2D,\\,D\+5N\+1\\\}and the depth isr\+9r\+9\. Furthermore, the parameter magnitude ofΨε\\Psi\_\{\\varepsilon\}satisfies

log2⁡𝒫​\(Ψε\)≤CD,r,γ,A,B​ε−2​D/\(r\+γ\)​log2⁡\(1/ε\),\\log\_\{2\}\\mathcal\{P\}\(\\Psi\_\{\\varepsilon\}\)\\leq C\_\{D,r,\\gamma,A,B\}\\,\\varepsilon^\{\-2D/\(r\+\\gamma\)\}\\log\_\{2\}\(1/\\varepsilon\),\(34\)with leading coefficient

CD,r,γ,A,B=2​\(1\+κ\)2​D​\[N​\(2\+2​log2⁡\(2​N\)\+log2⁡\(B\+1\)\)\+Dr\+γ\+D​log2⁡\(1\+κ\)\],C\_\{D,r,\\gamma,A,B\}=2\(1\+\\kappa\)^\{2D\}\\Bigl\[N\\bigl\(2\+2\\log\_\{2\}\(2N\)\+\\log\_\{2\}\(B\+1\)\\bigr\)\+\\tfrac\{D\}\{r\+\\gamma\}\+D\\log\_\{2\}\(1\+\\kappa\)\\Bigr\],where

N\\displaystyle N:=\(D\+rr\),κ:=\(2​Dr​Γ​\(γ\+1\)Γ​\(r\+γ\+1\)​A\)1/\(r\+γ\),and\\displaystyle=\\binom\{D\+r\}\{r\},\\qquad\\kappa=\\left\(\\frac\{2D^\{r\}\\Gamma\(\\gamma\+1\)\}\{\\Gamma\(r\+\\gamma\+1\)\}A\\right\)^\{1/\(r\+\\gamma\)\},\\text\{and\}B\\displaystyle B:=max0≤k≤r​sup𝐱∈Ω,g∈∂kf\|g​\(𝐱\)\|\.\\displaystyle=\\max\_\{0\\leq k\\leq r\}\\ \\sup\_\{\\mathbf\{x\}\\in\\Omega,\\ g\\in\\partial^\{k\}f\}\|g\(\\mathbf\{x\}\)\|\.

To give the proof, we first introduce two auxiliary lemmas\. Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)quantifies how the CRT parameter must grow as the accuracy increases, and Lemma[11](https://arxiv.org/html/2607.06781#Thmtheorem11)guarantees that the Taylor coefficients can be rationalized to finite bit length without introducing approximation error\.

###### Lemma 10\(Gridwise polynomial with rational coefficients\)\.

Let⋃m=0MD−1Ω𝐦\\bigcup\_\{m=0\}^\{M^\{D\}\-1\}\\Omega\_\{\\mathbf\{m\}\}be a uniform partition of the domain\[0,1\]D\[0,1\]^\{D\}, whereMMis the number of intervals along each coordinate\. Supposef:Ω→ℝf:\\Omega\\to\\mathbb\{R\}is a gridwise polynomial with rational coefficients of degreerr, then there exists a networkΨ\\Psiwith activationρfloor\\rho\_\{\\mathrm\{floor\}\},ρ1\\rho\_\{1\},ρ2\\rho\_\{2\}andρinv\\rho\_\{\\mathrm\{inv\}\}, where the width ismax⁡\{2​D,D\+5​N\+1\}\\max\\\{2D,\\,D\+5N\+1\\\}and the depth isr\+9r\+9, such that

Ψ​\(𝐱\)=f​\(𝐱\)\.\\Psi\(\\mathbf\{x\}\)=f\(\\mathbf\{x\}\)\.\(35\)Furthermore, the parameter magnitude ofΨ\\Psisatisfies

log2⁡𝒫​\(Ψ\)≤MD​\[\(MD\+1\)​D​log2⁡M\+F​\(MD​N\+1\)\+3\]\+1,\\log\_\{2\}\\mathcal\{P\}\(\\Psi\)\\leq M^\{D\}\[\(M^\{D\}\+1\)\\,D\\log\_\{2\}M\\;\+\\;F\\,\(M^\{D\}N\+1\)\\;\+\\;3\]\+1,\(36\)whereFFis the maximum bit length of coefficients offfin lowest terms, andN:=\(D\+rr\)N:=\\binom\{D\+r\}\{r\}\.

The detailed proof of Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)can be found in Section[5](https://arxiv.org/html/2607.06781#S5)\. Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)shows that rational gridwise polynomials can be exactly realized in a fixed architecture\. This is significant for two reasons\. First, the CRT provides an exact integer encoding of all gridwise data, with rationality ensuring that polynomial coefficients can be represented as integers\. Second, the network recovers a full local polynomial on each grid, not just a constant label\. This is achieved without growing the architecture with the target information stored entirely in the CRT congruence data\.

###### Lemma 11\(Uniform low\-bit rational approximation\)\.

For everyc^∈ℝ\\hat\{c\}\\in\\mathbb\{R\}with\|c^\|≤L\|\\hat\{c\}\|\\leq Land∀0<η<12\\forall 0<\\eta<\\frac\{1\}\{2\}, there exists a rational numberc=s/gc=s/gin lowest terms with\|c^−c\|≤η\|\\hat\{c\}\-c\|\\leq\\eta, satisfying

bits⁡\(c\)≤2​⌈log2⁡1η⌉\+log2⁡\(L\+2\)\+2\.\\operatorname\{bits\}\(c\)\\leq 2\\left\\lceil\\log\_\{2\}\\frac\{1\}\{\\eta\}\\right\\rceil\+\\log\_\{2\}\(L\+2\)\+2\.\(37\)

###### Proof\.

Setq:=⌈log2⁡1η⌉q:=\\left\\lceil\\log\_\{2\}\\frac\{1\}\{\\eta\}\\right\\rceiland define the dyadic roundingc:=2−q​⌊2q​c^\+12⌋∈2−q​ℤ\.c:=2^\{\-q\}\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor\\in 2^\{\-q\}\\mathbb\{Z\}\.Since⌊2q​c^\+12⌋\\lfloor 2^\{q\}\\hat\{c\}\+\\tfrac\{1\}\{2\}\\rflooris a nearest integer to2q​c^2^\{q\}\\hat\{c\}, we have\|⌊2q​c^\+12⌋−2q​c^\|≤12\.\\left\|\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor\-2^\{q\}\\hat\{c\}\\right\|\\leq\\frac\{1\}\{2\}\.Therefore\|c−c^\|≤2−q−1≤η\.\|c\-\\hat\{c\}\|\\leq 2^\{\-q\-1\}\\leq\\eta\.

If⌊2q​c^\+12⌋=0\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor=0, thenc=2−q​⌊2q​c^\+12⌋=0c=2^\{\-q\}\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor=0, and the claimed bounds are immediate\. If⌊2q​c^\+12⌋≠0\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor\\neq 0, since every divisor of2q2^\{q\}is a power of22, we may writegcd⁡\(\|⌊2q​c^\+12⌋\|,2q\)=2e\\gcd\(\|\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor\|,2^\{q\}\)=2^\{e\}, where0≤e≤q0\\leq e\\leq q\. Hence, in lowest terms,

c=sg,s=⌊2q​c^\+12⌋2e,g=2q−e\.c=\\frac\{s\}\{g\},s=\\frac\{\\left\\lfloor 2^\{q\}\\hat\{c\}\+\\frac\{1\}\{2\}\\right\\rfloor\}\{2^\{e\}\},g=2^\{q\-e\}\.\(38\)In particular,g≤2q\.g\\leq 2^\{q\}\.Moreover, using\|c^\|≤L\|\\hat\{c\}\|\\leq Land the rounding estimate above,

\|s\|=\|c\|​g≤\(\|c^\|\+2−q−1\)​2q≤\(L\+12\)​2q≤\(L\+1\)​2q\.\|s\|=\|c\|g\\leq\\bigl\(\|\\hat\{c\}\|\+2^\{\-q\-1\}\\bigr\)2^\{q\}\\leq\\left\(L\+\\frac\{1\}\{2\}\\right\)2^\{q\}\\leq\(L\+1\)2^\{q\}\.\(39\)Therefore,

\|s\|\+2≤\(L\+1\)​2q\+2≤\(L\+2\)​2q,\|s\|\+2\\leq\(L\+1\)2^\{q\}\+2\\leq\(L\+2\)2^\{q\},\(40\)where we useq≥1q\\geq 1\. Also,

g\+1≤2q\+1≤2q\+1\.g\+1\\leq 2^\{q\}\+1\\leq 2^\{q\+1\}\.\(41\)By the definition of bit length,⌈log2⁡\(\|s\|\+2\)⌉≤log2⁡\(L\+2\)\+q\+1,\\left\\lceil\\log\_\{2\}\(\|s\|\+2\)\\right\\rceil\\leq\\log\_\{2\}\(L\+2\)\+q\+1,and⌈log2⁡\(g\+1\)⌉≤q\+1\.\\left\\lceil\\log\_\{2\}\(g\+1\)\\right\\rceil\\leq q\+1\.Adding two estimates gives

bits⁡\(c\)≤2​q\+log2⁡\(L\+2\)\+2=2​⌈log2⁡1η⌉\+log2⁡\(L\+2\)\+2\.\\operatorname\{bits\}\(c\)\\leq 2q\+\\log\_\{2\}\(L\+2\)\+2=2\\left\\lceil\\log\_\{2\}\\frac\{1\}\{\\eta\}\\right\\rceil\+\\log\_\{2\}\(L\+2\)\+2\.\(42\)This bound depends only onLLandη\\etainstead of the particular value ofc^\\hat\{c\}\. ∎

###### Proof of Theorem[2](https://arxiv.org/html/2607.06781#Thmmaintheorem2)\.

Fixε∈\(0,1\)\\varepsilon\\in\(0,1\)and an integerM≥1M\\geq 1to be chosen in Step 1 in Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1), and partitionΩ=\[0,1\]D\\Omega=\[0,1\]^\{D\}intoMDM^\{D\}gridsΩ𝐦\\Omega\_\{\\mathbf\{m\}\}with corners𝐱𝐦=𝐦/M\\mathbf\{x\}\_\{\\mathbf\{m\}\}=\\mathbf\{m\}/M, so that onΩ𝐦\\Omega\_\{\\mathbf\{m\}\}the local coordinate isM​\(𝐱−𝐱𝐦\)∈\[0,1\]DM\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\in\[0,1\]^\{D\}\. We construct a rational gridwise polynomialf~\\tilde\{f\}of degreerrwith‖f−f~‖L∞​\(Ω\)≤ε\\\|f\-\\tilde\{f\}\\\|\_\{L^\{\\infty\}\(\\Omega\)\}\\leq\\varepsilonand realize it exactly via Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)\.

Taylor error estimate\.Let

T𝐦​\(𝐱\):=∑\|𝜶\|≤r∂𝜶f​\(𝐱𝐦\)𝜶\!​\(𝐱−𝐱𝐦\)𝜶=∑i≤Nc^𝐦,i​\(M​\(𝐱−𝐱𝐦\)\)𝜶\(i\),T\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\):=\\sum\_\{\|\\bm\{\\alpha\}\|\\leq r\}\\frac\{\\partial^\{\\bm\{\\alpha\}\}f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\}\{\\bm\{\\alpha\}\!\}\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}\}=\\sum\_\{i\\leq N\}\\hat\{c\}\_\{\\mathbf\{m\},i\}\\,\\bigl\(M\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\bigr\)^\{\\bm\{\\alpha\}^\{\(i\)\}\},\(43\)and

c𝐦,i:=∂𝜶\(i\)f​\(𝐱𝐦\)𝜶\(i\)\!​M\|𝜶\(i\)\|c\_\{\\mathbf\{m\},i\}:=\\frac\{\\partial^\{\\bm\{\\alpha\}^\{\(i\)\}\}f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\}\{\\bm\{\\alpha\}^\{\(i\)\}\!\\,M^\{\|\\bm\{\\alpha\}^\{\(i\)\}\|\}\}\(44\)be the degree\-rrTaylor polynomial offfat𝐱𝐦\\mathbf\{x\}\_\{\\mathbf\{m\}\}, using\(𝐱−𝐱𝐦\)𝜶=M−\|𝜶\|​\(M​\(𝐱−𝐱𝐦\)\)𝜶\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}\}=M^\{\-\|\\bm\{\\alpha\}\|\}\\bigl\(M\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\bigr\)^\{\\bm\{\\alpha\}\}\. Forr≥1r\\geq 1, putϕ​\(t\):=f​\(𝐱𝐦\+t​\(𝐱−𝐱𝐦\)\)\\phi\(t\):=f\\bigl\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\+t\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\bigr\)\. Taylor’s theorem with integral remainder, with the order\-rrterm moved intoT𝐦T\_\{\\mathbf\{m\}\}via1\(r−1\)\!​∫01\(1−t\)r−1​𝑑t=1r\!\\tfrac\{1\}\{\(r\-1\)\!\}\\int\_\{0\}^\{1\}\(1\-t\)^\{r\-1\}\\,dt=\\tfrac\{1\}\{r\!\}, gives

f​\(𝐱\)−T𝐦​\(𝐱\)=1\(r−1\)\!​∫01\(1−t\)r−1​\(ϕ\(r\)​\(t\)−ϕ\(r\)​\(0\)\)​𝑑t\.f\(\\mathbf\{x\}\)\-T\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\)=\\frac\{1\}\{\(r\-1\)\!\}\\int\_\{0\}^\{1\}\(1\-t\)^\{r\-1\}\\bigl\(\\phi^\{\(r\)\}\(t\)\-\\phi^\{\(r\)\}\(0\)\\bigr\)\\,dt\.\(45\)Sinceϕ\(r\)​\(t\)=∑\|𝜶\|=rr\!𝜶\!​\(𝐱−𝐱𝐦\)𝜶​∂𝜶f​\(𝐱𝐦\+t​\(𝐱−𝐱𝐦\)\)\\phi^\{\(r\)\}\(t\)=\\sum\_\{\|\\bm\{\\alpha\}\|=r\}\\tfrac\{r\!\}\{\\bm\{\\alpha\}\!\}\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}\}\\,\\partial^\{\\bm\{\\alpha\}\}f\\bigl\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\+t\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\bigr\)and‖𝐱−𝐱𝐦‖∞≤1/M\\\|\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\\\|\_\{\\infty\}\\leq 1/M,

\|ϕ\(r\)​\(t\)−ϕ\(r\)​\(0\)\|≤r\!​A​tγ​‖𝐱−𝐱𝐦‖∞γ​∑\|𝜶\|=r\|\(𝐱−𝐱𝐦\)𝜶\|𝜶\!≤r\!​A​tγ​M−γ⋅\(D/M\)rr\!,\\bigl\|\\phi^\{\(r\)\}\(t\)\-\\phi^\{\(r\)\}\(0\)\\bigr\|\\leq r\!\\,A\\,t^\{\\gamma\}\\\|\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\\\|\_\{\\infty\}^\{\\gamma\}\\sum\_\{\|\\bm\{\\alpha\}\|=r\}\\frac\{\|\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}\}\|\}\{\\bm\{\\alpha\}\!\}\\leq r\!\\,A\\,t^\{\\gamma\}M^\{\-\\gamma\}\\cdot\\frac\{\(D/M\)^\{r\}\}\{r\!\},\(46\)where the last step uses∑\|𝜶\|=r\|\(𝐱−𝐱𝐦\)𝜶\|𝜶\!=1r\!​\(∑d\|xd−x𝐦,d\|\)r≤\(D/M\)rr\!\\sum\_\{\|\\bm\{\\alpha\}\|=r\}\\tfrac\{\|\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}\}\|\}\{\\bm\{\\alpha\}\!\}=\\tfrac\{1\}\{r\!\}\\bigl\(\\sum\_\{d\}\|x\_\{d\}\-x\_\{\\mathbf\{m\},d\}\|\\bigr\)^\{r\}\\leq\\tfrac\{\(D/M\)^\{r\}\}\{r\!\}\. With∫01\(1−t\)r−1​tγ​𝑑t=Γ​\(γ\+1\)​Γ​\(r\)Γ​\(r\+γ\+1\)\\int\_\{0\}^\{1\}\(1\-t\)^\{r\-1\}t^\{\\gamma\}\\,dt=\\tfrac\{\\Gamma\(\\gamma\+1\)\\Gamma\(r\)\}\{\\Gamma\(r\+\\gamma\+1\)\},

\|f​\(𝐱\)−T𝐦​\(𝐱\)\|≤Dr​Γ​\(γ\+1\)Γ​\(r\+γ\+1\)​A​M−\(r\+γ\)=CD,r,γ​A​M−\(r\+γ\)\.\|f\(\\mathbf\{x\}\)\-T\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\)\|\\leq\\frac\{D^\{r\}\\Gamma\(\\gamma\+1\)\}\{\\Gamma\(r\+\\gamma\+1\)\}\\,A\\,M^\{\-\(r\+\\gamma\)\}=C\_\{D,r,\\gamma\}\\,A\\,M^\{\-\(r\+\\gamma\)\}\.\(47\)Forr=0r=0, this is the direct Hölder bound\|f​\(𝐱\)−f​\(𝐱𝐦\)\|≤A​‖𝐱−𝐱𝐦‖∞γ≤A​M−γ\|f\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\|\\leq A\\\|\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\\\|\_\{\\infty\}^\{\\gamma\}\\leq AM^\{\-\\gamma\}, again withCD,0,γ=1C\_\{D,0,\\gamma\}=1\. Choosing

M=Mε:=⌈\(2​CD,r,γ​A\)1/\(r\+γ\)​ε−1/\(r\+γ\)⌉M=M\_\{\\varepsilon\}:=\\Bigl\\lceil\(2C\_\{D,r,\\gamma\}A\)^\{1/\(r\+\\gamma\)\}\\,\\varepsilon^\{\-1/\(r\+\\gamma\)\}\\Bigr\\rceil\(48\)yields\|f−T𝐦\|≤ε/2\|f\-T\_\{\\mathbf\{m\}\}\|\\leq\\varepsilon/2on every grid\.

Rationalization\.Since\|∂𝜶\(i\)f​\(𝐱𝐦\)\|≤B\|\\partial^\{\\bm\{\\alpha\}^\{\(i\)\}\}f\(\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\|\\leq Band𝜶\(i\)\!,M\|𝜶\(i\)\|≥1\\bm\{\\alpha\}^\{\(i\)\}\!,M^\{\|\\bm\{\\alpha\}^\{\(i\)\}\|\}\\geq 1, we have\|c^𝐦,i\|≤B\|\\hat\{c\}\_\{\\mathbf\{m\},i\}\|\\leq B\. Apply Lemma[11](https://arxiv.org/html/2607.06781#Thmtheorem11)withL=BL=Bandη:=ε/\(2​N\)<12\\eta:=\\varepsilon/\(2N\)<\\tfrac\{1\}\{2\}\(valid asε<1≤N\\varepsilon<1\\leq N\): eachc^𝐦,i\\hat\{c\}\_\{\\mathbf\{m\},i\}admits a rationalc𝐦,i=s𝐦,i/g𝐦,ic\_\{\\mathbf\{m\},i\}=s\_\{\\mathbf\{m\},i\}/g\_\{\\mathbf\{m\},i\}in lowest terms with\|c𝐦,i−c^𝐦,i\|≤η\|c\_\{\\mathbf\{m\},i\}\-\\hat\{c\}\_\{\\mathbf\{m\},i\}\|\\leq\\etaand

𝚋𝚒𝚝𝚜\(c𝐦,i\)≤2⌈log2\(2N/ε\)⌉\+log2\(B\+2\)\+2=:Fε\.\\mathtt\{bits\}\(c\_\{\\mathbf\{m\},i\}\)\\leq 2\\bigl\\lceil\\log\_\{2\}\(2N/\\varepsilon\)\\bigr\\rceil\+\\log\_\{2\}\(B\+2\)\+2=:F\_\{\\varepsilon\}\.\(49\)Definef~​\(𝐱\):=∑i≤Nc𝐦,i​\(M​\(𝐱−𝐱𝐦\)\)𝜶\(i\)\\tilde\{f\}\(\\mathbf\{x\}\):=\\sum\_\{i\\leq N\}c\_\{\\mathbf\{m\},i\}\\,\\bigl\(M\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\bigr\)^\{\\bm\{\\alpha\}^\{\(i\)\}\}onΩ𝐦\\Omega\_\{\\mathbf\{m\}\}\. As0≤\(M​\(𝐱−𝐱𝐦\)\)𝜶\(i\)≤10\\leq\\bigl\(M\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\bigr\)^\{\\bm\{\\alpha\}^\{\(i\)\}\}\\leq 1,

\|T𝐦​\(𝐱\)−f~​\(𝐱\)\|≤∑i≤N\|c^𝐦,i−c𝐦,i\|≤N​η=ε/2,\|T\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\)\-\\tilde\{f\}\(\\mathbf\{x\}\)\|\\leq\\sum\_\{i\\leq N\}\|\\hat\{c\}\_\{\\mathbf\{m\},i\}\-c\_\{\\mathbf\{m\},i\}\|\\leq N\\eta=\\varepsilon/2,\(50\)hence\|f​\(𝐱\)−f~​\(𝐱\)\|≤ε\|f\(\\mathbf\{x\}\)\-\\tilde\{f\}\(\\mathbf\{x\}\)\|\\leq\\varepsilonfor all𝐱∈Ω\\mathbf\{x\}\\in\\Omega\.

Realization by a fixed\-size network\.f~\\tilde\{f\}is a gridwise polynomial with rational coefficients of degreerron the uniformMεDM\_\{\\varepsilon\}^\{D\}\-partition, so Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)furnishes a networkΨε\\Psi\_\{\\varepsilon\}of the same activations, widthmax⁡\{2​D,D\+5​N\+1\}\\max\\\{2D,\\,D\+5N\+1\\\}, and depthr\+9r\+9withΨε=f~\\Psi\_\{\\varepsilon\}=\\tilde\{f\}\. The architecture depends only onD,rD,randNN; hence, it is independent ofε\\varepsilonandff, and\|Ψε​\(𝐱\)−f​\(𝐱\)\|≤ε\|\\Psi\_\{\\varepsilon\}\(\\mathbf\{x\}\)\-f\(\\mathbf\{x\}\)\|\\leq\\varepsilon\.

Parameter magnitude\.By Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)withM=MεM=M\_\{\\varepsilon\}and bit lengthF=FεF=F\_\{\\varepsilon\}\(forr≥1r\\geq 1,r=0r=0follows from the exact bound of Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)withN=1N=1\),

log2⁡𝒫​\(Ψε\)≤2​Mε2​D​\(D​log2⁡Mε\+N​Fε\)\.\\log\_\{2\}\\mathcal\{P\}\(\\Psi\_\{\\varepsilon\}\)\\leq 2M\_\{\\varepsilon\}^\{2D\}\\bigl\(D\\log\_\{2\}M\_\{\\varepsilon\}\+NF\_\{\\varepsilon\}\\bigr\)\.\(51\)Writeκ:=\(2​Dr​Γ​\(γ\+1\)Γ​\(r\+γ\+1\)​A\)1/\(r\+γ\)\\kappa:=\\bigl\(\\tfrac\{2D^\{r\}\\Gamma\(\\gamma\+1\)\}\{\\Gamma\(r\+\\gamma\+1\)\}A\\bigr\)^\{1/\(r\+\\gamma\)\}\. The choice ofMεM\_\{\\varepsilon\}andε<1\\varepsilon<1give

Mε2​D≤\(1\+κ\)2​D​ε−2​D/\(r\+γ\),log2⁡Mε≤log2⁡\(1\+κ\)\+1r\+γ​log2⁡\(1/ε\),M\_\{\\varepsilon\}^\{2D\}\\leq\(1\+\\kappa\)^\{2D\}\\varepsilon^\{\-2D/\(r\+\\gamma\)\},\\qquad\\log\_\{2\}M\_\{\\varepsilon\}\\leq\\log\_\{2\}\(1\+\\kappa\)\+\\tfrac\{1\}\{r\+\\gamma\}\\log\_\{2\}\(1/\\varepsilon\),\(52\)whileFε≤2​log2⁡\(1/ε\)\+2​log2⁡\(2​N\)\+log2⁡\(B\+1\)\+O​\(1\)F\_\{\\varepsilon\}\\leq 2\\log\_\{2\}\(1/\\varepsilon\)\+2\\log\_\{2\}\(2N\)\+\\log\_\{2\}\(B\+1\)\+O\(1\)\. Substituting and collecting the coefficient ofε−2​D/\(r\+γ\)​log2⁡\(1/ε\)\\varepsilon^\{\-2D/\(r\+\\gamma\)\}\\log\_\{2\}\(1/\\varepsilon\)yields

log2⁡𝒫​\(Ψε\)≤CD,r,γ,A,B​ε−2​D/\(r\+γ\)​log2⁡\(1/ε\),\\log\_\{2\}\\mathcal\{P\}\(\\Psi\_\{\\varepsilon\}\)\\leq C\_\{D,r,\\gamma,A,B\}\\,\\varepsilon^\{\-2D/\(r\+\\gamma\)\}\\log\_\{2\}\(1/\\varepsilon\),\(53\)with

CD,r,γ,A,B=2​\(1\+κ\)2​D​\[N​\(2\+2​log2⁡\(2​N\)\+log2⁡\(B\+1\)\)\+Dr\+γ\+D​log2⁡\(1\+κ\)\],C\_\{D,r,\\gamma,A,B\}=2\(1\+\\kappa\)^\{2D\}\\Bigl\[N\\bigl\(2\+2\\log\_\{2\}\(2N\)\+\\log\_\{2\}\(B\+1\)\\bigr\)\+\\tfrac\{D\}\{r\+\\gamma\}\+D\\log\_\{2\}\(1\+\\kappa\)\\Bigr\],\(54\)which is the constant in the statement\. ∎

![Refer to caption](https://arxiv.org/html/2607.06781v1/x3.png)Figure 3:Scaling law of the parameter magnitudelog2⁡𝒫​\(Ψε\)\\log\_\{2\}\\mathcal\{P\}\(\\Psi\_\{\\varepsilon\}\)with respect to the target accuracylog2⁡\(1/ε\)\\log\_\{2\}\(1/\\varepsilon\)\.
## 5Proof of Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)

Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)shows an exact realization for gridwise polynomials with rational coefficients\. Exact realization phenomena already appear in several related settings, but with different scopes\. The exact result of\(Zhang et al\.,,[2022](https://arxiv.org/html/2607.06781#bib.bib50)\)concerns finite\-valued classification functions on pairwise disjoint bounded closed sets\. Its key mechanism is to directly leverage the distance function

f​\(𝐱\):=dist⁡\(𝐱,B\)dist⁡\(𝐱,A\)\+dist⁡\(𝐱,B\)for any​𝐱∈ℝD,f\(\\mathbf\{x\}\):=\\frac\{\\operatorname\{dist\}\(\\mathbf\{x\},B\)\}\{\\operatorname\{dist\}\(\\mathbf\{x\},A\)\+\\operatorname\{dist\}\(\\mathbf\{x\},B\)\}\\qquad\\text\{for any \}\\mathbf\{x\}\\in\\mathbb\{R\}^\{D\},\(55\)whereAAandBBare two disjoint bounded closed sets, and eliminate the error at the margin: the labels are rescaled to separated odd integers, a continuous lifting is approximated within less than half the label gap, and a final correction map sends the approximate value exactly back to the intended label\. But in\(Zhang et al\.,,[2022](https://arxiv.org/html/2607.06781#bib.bib50)\), the distance function is not represented by a network; therefore, its form is implicit\. The ReQU\-based exact algebraic result of\(Li et al\.,,[2020](https://arxiv.org/html/2607.06781#bib.bib24)\)is best understood as an exact polynomial\-evaluation theorem\. Since a ReQU activationρs​\(t\)\\rho\_\{s\}\(t\)directly generates power functions, products and monomials can be built exactly by algebraic identities and network composition\. Linear combinations of these monomials then represent a polynomial with no approximation error\. In this sense, once the polynomial coefficients are known, ReQU networks provide a natural exact evaluator for local polynomial rules\. The scope of this exactness is algebraic: it applies to polynomials and to smooth functions only after polynomial approximants have first replaced them\.

### 5\.1Proof Idea of Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)

The proof is parallel to that of[Theorem1](https://arxiv.org/html/2607.06781#Thmmaintheorem1), with one important difference\. The recovered gridwise object is a quantized constant, whereas here the recovered object is a local polynomial\. The exact construction is feasible because, on each grid, the local polynomial is exactly determined by finitely many rational coefficients, and these coefficients can be represented exactly by integers\.

*1\. Calculation of the local coordinates\.*For gridwise polynomials, we do not need to partition their input domain, as they are already defined on uniform grids\. Thus, as in Steps 1–2 of Section[3\.1](https://arxiv.org/html/2607.06781#S3.SS1), a single floor layer computesρfloor​\(M​xd\)\\rho\_\{\\mathrm\{floor\}\}\(Mx\_\{d\}\),d=1,…,Dd=1,\\dots,D, from which we read off two quantities\. The first is the grid annotation

Ψ1\(1\)​\(𝐱\):=1\+∑d=1DMd−1​ρfloor​\(M​xd\)∈\{1,…,MD\},\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\):=1\+\\sum\_\{d=1\}^\{D\}M^\{d\-1\}\\rho\_\{\\mathrm\{floor\}\}\(Mx\_\{d\}\)\\in\\\{1,\\dots,M^\{D\}\\\},\(56\)the base\-MMencoding of the active grid\. Because here our goal is an exact representation of local polynomials, we do not need to quantizeff\. Instead, we compute the local coordinate,

Ψ1\(2\)​\(𝐱\):=M​\(𝐱−𝐱𝐦\)=M​𝐱−ρfloor​\(M​𝐱\)∈\[0,1\)D,\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\):=M\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)=M\\mathbf\{x\}\-\\rho\_\{\\mathrm\{floor\}\}\(M\\mathbf\{x\}\)\\in\[0,1\)^\{D\},\(57\)where𝐱𝐦:=𝐦/M\\mathbf\{x\}\_\{\\mathbf\{m\}\}:=\\mathbf\{m\}/Mis the corner ofΩ𝐦\\Omega\_\{\\mathbf\{m\}\}\. Sharing the sameDDfloor units and carrying𝐱\\mathbf\{x\}forward, the sub\-network is

Ψ1​\(𝐱\):=\(Ψ1\(1\)​\(𝐱\),Ψ1\(2\)​\(𝐱\)\)\.\\Psi\_\{1\}\(\\mathbf\{x\}\):=\\big\(\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\),\\,\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\big\)\.\(58\)
*2\. Exact representation by coordinate\-wise CRT\.*SettingN:=\(D\+rr\)N:=\\binom\{D\+r\}\{r\}, we fix the ordering of the polynomial exponents\{𝜶∈ℕD:0≤\|𝜶\|≤r\}=\{𝜶\(1\),…,𝜶\(N\)\}\.\\\{\\bm\{\\alpha\}\\in\\mathbb\{N\}^\{D\}:0\\leq\|\\bm\{\\alpha\}\|\\leq r\\\}=\\\{\\bm\{\\alpha\}^\{\(1\)\},\\dots,\\bm\{\\alpha\}^\{\(N\)\}\\\}\.Each gridΩ𝐦\\Omega\_\{\\mathbf\{m\}\}carries a rational polynomial that, in the local coordinateM​\(𝐱−𝐱𝐦\)∈\[0,1\]DM\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\in\[0,1\]^\{D\}, reads

Y𝐦​\(𝐱−𝐱𝐦\)=∑i≤Nc𝐦,i​M\|𝜶\(i\)\|​\(𝐱−𝐱𝐦\)𝜶\(i\),c𝐦,i∈ℚ\.Y\_\{\\mathbf\{m\}\}\\\!\\big\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\\big\)=\\sum\_\{i\\leq N\}c\_\{\\mathbf\{m\},i\}\\,M^\{\|\\bm\{\\alpha\}^\{\(i\)\}\|\}\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}^\{\(i\)\}\},\\qquad c\_\{\\mathbf\{m\},i\}\\in\\mathbb\{Q\}\.\(59\)Since the index set\{\(𝐦,i\):1≤\|𝐦\|≤MD,1≤i≤N\}\\\{\(\\mathbf\{m\},i\):1\\leq\|\\mathbf\{m\}\|\\leq M^\{D\},\\ 1\\leq i\\leq N\\\}is finite, so is the set\{c𝐦,i\}\\\{c\_\{\\mathbf\{m\},i\}\\\}\. Since everyc𝐦,ic\_\{\\mathbf\{m\},i\}is a rational, we can multiply everyc𝐦,ic\_\{\\mathbf\{m\},i\}with a common integer such that we havec𝐦,i=s𝐦,i/Bfc\_\{\\mathbf\{m\},i\}=s\_\{\\mathbf\{m\},i\}/B\_\{f\}, ands𝐦,i∈ℤs\_\{\\mathbf\{m\},i\}\\in\\mathbb\{Z\}\. Thus, all our later constructions can be built on integers\. WithCf:=max𝐦,i⁡\|s𝐦,i\|C\_\{f\}:=\\max\_\{\\mathbf\{m\},i\}\|s\_\{\\mathbf\{m\},i\}\|, we define

Γf:=\(1\+2​Cf\)​\(MD\)\!\.\\Gamma\_\{f\}:=\(1\+2C\_\{f\}\)\\,\(M^\{D\}\)\!\.\(60\)The sub\-networkΨ2\(1\)\\Psi^\{\(1\)\}\_\{2\}computes the modulus by one affine layer,

Ψ2\(1\)​\(𝐱\):=1\+Ψ1\(1\)​\(𝐱\)​Γf=p𝐦\.\\Psi^\{\(1\)\}\_\{2\}\(\\mathbf\{x\}\):=1\+\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\)\\Gamma\_\{f\}=p\_\{\\mathbf\{m\}\}\.\(61\)Because we use the same construction, by Lemma[8](https://arxiv.org/html/2607.06781#Thmtheorem8),p𝐦\>Γfp\_\{\\mathbf\{m\}\}\>\\Gamma\_\{f\}and\{p𝐦\}\|𝐦\|=1MD\\\{p\_\{\\mathbf\{m\}\}\\\}\_\{\|\\mathbf\{m\}\|=1\}^\{M^\{D\}\}are pairwise coprime\. For each monomial indexi≤Ni\\leq N, Proposition[9](https://arxiv.org/html/2607.06781#Thmtheorem9)gives an integerKiK\_\{i\}with

Ki≡s𝐦,i\(modp𝐦\),m=1,…,MD\.K\_\{i\}\\equiv s\_\{\\mathbf\{m\},i\}\\pmod\{p\_\{\\mathbf\{m\}\}\},\\qquad m=1,\\dots,M^\{D\}\.\(62\)Sinces𝐦,i∈\[−Cf,Cf\]s\_\{\\mathbf\{m\},i\}\\in\[\-C\_\{f\},C\_\{f\}\], settingE𝐦,i:=Kimodp𝐦E\_\{\\mathbf\{m\},i\}:=K\_\{i\}\\bmod p\_\{\\mathbf\{m\}\}, Lemma[12](https://arxiv.org/html/2607.06781#Thmtheorem12)gives the signed representative

s𝐦,i=\{E𝐦,i,0≤E𝐦,i≤Cf,E𝐦,i−p𝐦,p𝐦−Cf≤E𝐦,i<p𝐦\.s\_\{\\mathbf\{m\},i\}=\\begin\{cases\}E\_\{\\mathbf\{m\},i\},&0\\leq E\_\{\\mathbf\{m\},i\}\\leq C\_\{f\},\\\\\[2\.0pt\] E\_\{\\mathbf\{m\},i\}\-p\_\{\\mathbf\{m\}\},&p\_\{\\mathbf\{m\}\}\-C\_\{f\}\\leq E\_\{\\mathbf\{m\},i\}<p\_\{\\mathbf\{m\}\}\.\\end\{cases\}\(63\)The residueE𝐦,iE\_\{\\mathbf\{m\},i\}is realized byΨ2,i\(2\)\\Psi^\{\(2\)\}\_\{2,i\}:

Ψ2,i\(2\)​\(𝐱\):=E𝐦,i=Ki−Ψ2\(1\)​\(𝐱\)​ρfloor​\(Ki​ρinv​\(Ψ2\(1\)​\(𝐱\)\)\)\.\\Psi^\{\(2\)\}\_\{2,i\}\(\\mathbf\{x\}\):=E\_\{\\mathbf\{m\},i\}=K\_\{i\}\-\\Psi^\{\(1\)\}\_\{2\}\(\\mathbf\{x\}\)\\,\\rho\_\{\\mathrm\{floor\}\}\\\!\\big\(K\_\{i\}~\\rho\_\{\\mathrm\{inv\}\}\(\\Psi^\{\(1\)\}\_\{2\}\(\\mathbf\{x\}\)\)\\big\)\.\(64\)By Lemma[13](https://arxiv.org/html/2607.06781#Thmtheorem13), the signed values𝐦,is\_\{\\mathbf\{m\},i\}can be recovered fromΨ2,i\(3\)\\Psi^\{\(3\)\}\_\{2,i\}by oneρinv\\rho\_\{\\mathrm\{inv\}\}, oneρfloor\\rho\_\{\\mathrm\{floor\}\}, and the multiplication module𝙼𝚞𝚕𝚝​\(u,v\):=\(ρ2​\(u\+v\)\+ρ2​\(−u−v\)−ρ2​\(u−v\)−ρ2​\(v−u\)\)/4\\mathtt\{Mult\}\(u,v\):=\\big\(\\rho\_\{2\}\(u\+v\)\+\\rho\_\{2\}\(\-u\-v\)\-\\rho\_\{2\}\(u\-v\)\-\\rho\_\{2\}\(v\-u\)\\big\)/4:

Ψ2,i\(3\)​\(𝐱\):=s𝐦,i=Ψ2,i\(2\)​\(𝐱\)−𝙼𝚞𝚕𝚝​\(Ψ2\(1\)​\(𝐱\),ρfloor​\(𝙼𝚞𝚕𝚝​\(Ψ2,i\(2\)​\(𝐱\),ρinv​\(Ψ2\(1\)​\(𝐱\)−Cf\)\)\)\),\\Psi^\{\(3\)\}\_\{2,i\}\(\\mathbf\{x\}\):=s\_\{\\mathbf\{m\},i\}=\\Psi^\{\(2\)\}\_\{2,i\}\(\\mathbf\{x\}\)\-\\mathtt\{Mult\}\\\!\\Big\(\\Psi^\{\(1\)\}\_\{2\}\(\\mathbf\{x\}\),\\,\\rho\_\{\\mathrm\{floor\}\}\\big\(\\mathtt\{Mult\}\(\\Psi^\{\(2\)\}\_\{2,i\}\(\\mathbf\{x\}\),\\rho\_\{\\mathrm\{inv\}\}\(\\Psi^\{\(1\)\}\_\{2\}\(\\mathbf\{x\}\)\-C\_\{f\}\)\)\\big\)\\Big\),\(65\)so thatc𝐦,i=Bf−1​s𝐦,i=Bf−1​Ψ2,i\(3\)​\(𝐱\)c\_\{\\mathbf\{m\},i\}=B^\{\-1\}\_\{f\}s\_\{\\mathbf\{m\},i\}=B^\{\-1\}\_\{f\}\\Psi^\{\(3\)\}\_\{2,i\}\(\\mathbf\{x\}\)\.

For each fixedii, the coefficientc𝐦,ic\_\{\\mathbf\{m\},i\}is produced by the serial chainΨ2,i\(3\)∘Ψ2,i\(2\)\\Psi^\{\(3\)\}\_\{2,i\}\\circ\\Psi^\{\(2\)\}\_\{2,i\}built on the shared modulusΨ2\(1\)\\Psi^\{\(1\)\}\_\{2\}\. TheseNNchains run over the single modulus channelΨ2\(1\)​\(𝐱\)\\Psi^\{\(1\)\}\_\{2\}\(\\mathbf\{x\}\)in parallel, while the local coordinateΨ1\(2\)​\(𝐱\)\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)from Step 1 is fed into later layers unchanged for Step 3\. Collecting the outputs,Ψ2\\Psi\_\{2\}sends the pair\(Ψ1\(1\)​\(𝐱\),Ψ1\(2\)​\(𝐱\)\)\\big\(\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\),\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\big\)to

Ψ2​\(𝐱\):=\(Bf−1​Ψ2,i\(3\)​\(𝐱\)i≤N,Ψ1\(2\)​\(𝐱\)\)=\(𝐜𝐦,Ψ1\(2\)​\(𝐱\)\),𝐜𝐦:=\(c𝐦,1,…,c𝐦,N\)\.\\Psi\_\{2\}\(\\mathbf\{x\}\):=\\Big\(B^\{\-1\}\_\{f\}\\Psi^\{\(3\)\}\_\{2,i\}\(\\mathbf\{x\}\)\_\{i\\leq N\},\\ \\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\,\\Big\)=\\big\(\\mathbf\{c\}\_\{\\mathbf\{m\}\},\\ \\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\big\),\\mathbf\{c\}\_\{\\mathbf\{m\}\}:=\(c\_\{\\mathbf\{m\},1\},\\dots,c\_\{\\mathbf\{m\},N\}\)\.\(66\)
*3\. Exact polynomial computation\.*From the local coordinateΨ1\(2\)​\(𝐱\)∈\[0,1\]D\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\in\[0,1\]^\{D\}, define for eachi≤Ni\\leq Nthe monomial

Ψ3,i\(1\)​\(𝐱\):=\[Ψ1\(2\)​\(𝐱\)\]𝜶\(i\)=∏d=1D\(Ψ1,d\(2\)​\(𝐱\)\)αd\(i\)\.\\Psi^\{\(1\)\}\_\{3,i\}\(\\mathbf\{x\}\):=\[\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\]^\{\\bm\{\\alpha\}^\{\(i\)\}\}=\\prod\_\{d=1\}^\{D\}\\big\(\\Psi^\{\(2\)\}\_\{1,d\}\(\\mathbf\{x\}\)\\big\)^\{\\alpha^\{\(i\)\}\_\{d\}\}\.\(67\)By Lemma[14](https://arxiv.org/html/2607.06781#Thmtheorem14), each monomial\[Ψ1\(2\)​\(𝐱\)\]𝜶\(i\)\[\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\]^\{\\bm\{\\alpha\}^\{\(i\)\}\}is computed exactly\. Running theNNblocks in parallel on disjoint channels \(sharing the inputΨ1\(2\)​\(𝐱\)\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)and carrying the coefficientsc𝐦,ic\_\{\\mathbf\{m\},i\}\), and aligning every chain to terminate at layerrr, yields all monomials simultaneously\. We writeΨ3\(1\):=\(Ψ3,1\(1\),…,Ψ3,N\(1\)\)\\Psi^\{\(1\)\}\_\{3\}:=\(\\Psi^\{\(1\)\}\_\{3,1\},\\dots,\\Psi^\{\(1\)\}\_\{3,N\}\)\. The moduleΨ3\(2\)\\Psi^\{\(2\)\}\_\{3\}then scales each monomial by its recovered coefficient via𝙼𝚞𝚕𝚝\\mathtt\{Mult\}and sums by one affine layer:

Ψ3\(2\)​\(𝐜𝐦,Ψ3\(1\)​\(𝐱\)\):=∑i≤N𝙼𝚞𝚕𝚝​\(c𝐦,i,Ψ1\(2\)​\(𝐱\)𝜶\(i\)\)=Y𝐦​\(Ψ1\(2\)​\(𝐱\)\)\.\\Psi^\{\(2\)\}\_\{3\}\\\!\\big\(\\mathbf\{c\}\_\{\\mathbf\{m\}\},\\Psi^\{\(1\)\}\_\{3\}\(\\mathbf\{x\}\)\\big\):=\\sum\_\{i\\leq N\}\\mathtt\{Mult\}\\\!\\big\(c\_\{\\mathbf\{m\},i\},\\,\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)^\{\\bm\{\\alpha\}^\{\(i\)\}\}\\big\)=Y\_\{\\mathbf\{m\}\}\\\!\\big\(\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\big\)\.\(68\)and we setΨ3​\(𝐜𝐦,Ψ1\(2\)​\(𝐱\)\):=Ψ3\(2\)​\(𝐜𝐦,Ψ3\(1\)​\(Ψ1\(2\)​\(𝐱\)\)\)\\Psi\_\{3\}\\big\(\\mathbf\{c\}\_\{\\mathbf\{m\}\},\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\big\):=\\Psi^\{\(2\)\}\_\{3\}\\big\(\\mathbf\{c\}\_\{\\mathbf\{m\}\},\\Psi^\{\(1\)\}\_\{3\}\(\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\)\\big\)\.

Each module is thus a map on the copied tuple:

Ψ1:𝐱↦\(Ψ1\(1\)​\(𝐱\),Ψ1\(2\)​\(𝐱\)\);Ψ2:\(Ψ1\(1\),Ψ1\(2\)\)↦\(𝐜𝐦,Ψ1\(2\)\);Ψ3:\(𝐜𝐦,Ψ1\(2\)\)↦Y𝐦​\(Ψ1\(2\)\),\\Psi\_\{1\}:\\mathbf\{x\}\\mapsto\\big\(\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\),\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\\big\);\\Psi\_\{2\}:\\big\(\\Psi^\{\(1\)\}\_\{1\},\\Psi^\{\(2\)\}\_\{1\}\\big\)\\mapsto\\big\(\\mathbf\{c\}\_\{\\mathbf\{m\}\},\\Psi^\{\(2\)\}\_\{1\}\\big\);\\Psi\_\{3\}:\\big\(\\mathbf\{c\}\_\{\\mathbf\{m\}\},\\Psi^\{\(2\)\}\_\{1\}\\big\)\\mapsto Y\_\{\\mathbf\{m\}\}\\\!\\big\(\\Psi^\{\(2\)\}\_\{1\}\\big\),\(69\)so the three compose serially\. Sincef​\(𝐱\)=Y𝐦​\(𝐱−𝐱𝐦\)f\(\\mathbf\{x\}\)=Y\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)onΩ𝐦\\Omega\_\{\\mathbf\{m\}\},

Ψ​\(𝐱\):=Ψ3∘Ψ2∘Ψ1​\(𝐱\)=Y𝐦​\(𝐱−𝐱𝐦\)=f​\(𝐱\)\.\\Psi\(\\mathbf\{x\}\):=\\Psi\_\{3\}\\circ\\Psi\_\{2\}\\circ\\Psi\_\{1\}\(\\mathbf\{x\}\)=Y\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)=f\(\\mathbf\{x\}\)\.\(70\)
The block diagram of the networkΨ=Ψ3∘Ψ2∘Ψ1​\(𝐱\)\\Psi=\\Psi\_\{3\}\\circ\\Psi\_\{2\}\\circ\\Psi\_\{1\}\(\\mathbf\{x\}\)is visualized in Figure[4](https://arxiv.org/html/2607.06781#S5.F4)\.

![Refer to caption](https://arxiv.org/html/2607.06781v1/x4.png)Figure 4:Architecture of the target network realizingΨ\\Psi\.
### 5\.2Proof of Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)

###### Lemma 12\(Range of the canonical residue\)\.

LetC,p∈ℕ\+C,p\\in\\mathbb\{N\}\_\{\+\}withp\>2​Cp\>2C, and lets∈\[−C,C\]∩ℤs\\in\[\-C,C\]\\cap\\mathbb\{Z\}andK∈ℤK\\in\\mathbb\{Z\}satisfyK≡s\(modp\)K\\equiv s\\pmod\{p\}\. ThenKmodpK\\bmod plies in one of the disjoint intervals\[0,C\]\[0,C\]and\[p−C,p\)\[p\-C,p\)\.

###### Proof\.

SinceK≡s\(modp\)K\\equiv s\\pmod\{p\}is the unique element which is congruent tossmodulopp\.

1. i\)Ifs≥0s\\geq 0, then0≤s≤C<p0\\leq s\\leq C<p, sossis already in\{0,…,p−1\}\\\{0,\\dots,p\-1\\\}ands=Kmodp∈\[0,C\]s=K\\bmod p\\in\[0,C\]\.
2. ii\)Ifs<0s<0, then−C≤s<0\-C\\leq s<0, sop−C≤p\+s<pp\-C\\leq p\+s<pwithp\+s∈\{0,…,p−1\}p\+s\\in\\\{0,\\dots,p\-1\\\}\. HenceKmodp=p\+s∈\[p−C,p\)K\\bmod p=p\+s\\in\[p\-C,p\)ands=Kmodp−ps=K\\bmod p\-p\.

Becausep\>2​Cp\>2CgivesC<p−CC<p\-C, the intervals\[0,C\]\[0,C\]and\[p−C,p\)\[p\-C,p\)are disjoint, andKmodpK\\bmod pfalls in one of them\. ∎

###### Lemma 13\(Recoveringssviaρinv\\rho\_\{\\mathrm\{inv\}\},ρfloor\\rho\_\{\\mathrm\{floor\}\}andρ2\\rho\_\{2\}\)\.

Givens:\(\[0,C\]∪\[p−C,p\)\)×ℕ\+→ℤs:\\big\(\[0,C\]\\cup\[p\-C,p\)\\big\)\\times\\mathbb\{N\}\_\{\+\}\\to\\mathbb\{Z\}by

s​\(E,p\)=\{E,0≤E≤C,E−p,p−C≤E<p,s\(E,p\)=\\begin\{cases\}E,&0\\leq E\\leq C,\\\\\[2\.0pt\] E\-p,&p\-C\\leq E<p,\\end\{cases\}\(71\)whereC,p∈ℕ\+C,p\\in\\mathbb\{N\}\_\{\+\}withp\>2​Cp\>2C, so that the two intervals\[0,C\]\[0,C\]and\[p−C,p\)\[p\-C,p\)are disjoint, there is a network with activationρfloor\\rho\_\{\\mathrm\{floor\}\},ρinv\\rho\_\{\\mathrm\{inv\}\}andρ2\\rho\_\{2\}to realize it\.

###### Proof\.

There are two cases:

1. i\)IfE∈\[0,C\]E\\in\[0,C\], then0≤Ep−C<10\\leq\\frac\{E\}\{p\-C\}<1\. Thereforeρfloor​\(Ep−C\)=0\\rho\_\{\\mathrm\{floor\}\}\\left\(\\frac\{E\}\{p\-C\}\\right\)=0\.
2. ii\)IfE∈\[p−C,p\)E\\in\[p\-C,p\), then1≤Ep−C<pp−C<21\\leq\\frac\{E\}\{p\-C\}<\\frac\{p\}\{p\-C\}<2\. Thereforeρfloor​\(Ep−C\)=1\\rho\_\{\\mathrm\{floor\}\}\\left\(\\frac\{E\}\{p\-C\}\\right\)=1\.

Sos=E−p​ρfloor​\(Ep−C\)\.s=E\-p~\\rho\_\{\\mathrm\{floor\}\}\\left\(\\frac\{E\}\{p\-C\}\\right\)\.The quantity\(p−C\)−1\(p\-C\)^\{\-1\}can be produced byρinv\\rho\_\{\\mathrm\{inv\}\}, the product can be produced by𝙼𝚞𝚕𝚝\\mathtt\{Mult\}based onρ2\\rho\_\{2\}, and cases are distinguished withρfloor\\rho\_\{\\mathrm\{floor\}\}\. Hences​\(E,p\)s\(E,p\)is realized exactly on\[0,C\]∪\[p−C,p\)\[0,C\]\\cup\[p\-C,p\)\. ∎

###### Lemma 14\(ReQU realization of a monomial\)\.

Given a monomial

𝐱𝜶=∏d=1Dxdαd,𝐱∈\[0,1\]D,\\mathbf\{x\}^\{\\bm\{\\alpha\}\}=\\prod\_\{d=1\}^\{D\}x\_\{d\}^\{\\alpha\_\{d\}\},\\qquad\\mathbf\{x\}\\in\[0,1\]^\{D\},\(72\)where𝛂=\(α1,…,αD\)∈ℕD\\bm\{\\alpha\}=\(\\alpha\_\{1\},\\dots,\\alpha\_\{D\}\)\\in\\mathbb\{N\}^\{D\}is a multi\-index of degree\|𝛂\|=∑d=1Dαd≤r\|\\bm\{\\alpha\}\|=\\sum\_\{d=1\}^\{D\}\\alpha\_\{d\}\\leq r, there exists a network with activationρ2\\rho\_\{2\}\(through𝙼𝚞𝚕𝚝\\mathtt\{Mult\}\) that computes𝐱𝛂\\mathbf\{x\}^\{\\bm\{\\alpha\}\}exactly on\[0,1\]D\[0,1\]^\{D\}, with width44and depth\|𝛂\|\|\\bm\{\\alpha\}\|\.

###### Proof\.

By Eq\. \([9](https://arxiv.org/html/2607.06781#S2.E9)\),𝙼𝚞𝚕𝚝​\(u,v\)=u​v\\mathtt\{Mult\}\(u,v\)=uv, using four ReQU units in a single layer\. Each coordinatexdx\_\{d\}is repeatedαd\\alpha\_\{d\}times, writing as𝐱𝜶\\mathbf\{x\}^\{\\bm\{\\alpha\}\}\. Accumulate one factor per layer, as:

𝐱𝜶=𝙼𝚞𝚕𝚝​\(⋯​𝙼𝚞𝚕𝚝​\(𝙼𝚞𝚕𝚝​\(1,x1,…,x1⏟α1\),x2,…,x2⏟α2\),…,xD,…,xD⏟αD\)\.\\mathbf\{x\}^\{\\bm\{\\alpha\}\}=\\mathtt\{Mult\}\\Bigl\(\\cdots\\mathtt\{Mult\}\\bigl\(\\mathtt\{Mult\}\(1,\\underbrace\{x\_\{1\},\\dots,x\_\{1\}\}\_\{\\alpha\_\{1\}\}\),\\,\\underbrace\{x\_\{2\},\\dots,x\_\{2\}\}\_\{\\alpha\_\{2\}\}\\bigr\),\\dots,\\,\\underbrace\{x\_\{D\},\\dots,x\_\{D\}\}\_\{\\alpha\_\{D\}\}\\Bigr\)\.\(73\)Each𝙼𝚞𝚕𝚝\\mathtt\{Mult\}is one layer of four ReQU units, so the width is44; there are\|𝜶\|\|\\bm\{\\alpha\}\|such layers, so the depth is\|𝜶\|\|\\bm\{\\alpha\}\|\. ∎

###### Proof of Lemma[10](https://arxiv.org/html/2607.06781#Thmtheorem10)\.

The construction of Steps 1–3 producesΨ=Ψ3∘Ψ2∘Ψ1\\Psi=\\Psi\_\{3\}\\circ\\Psi\_\{2\}\\circ\\Psi\_\{1\}\. Specially,Ψ1\\Psi\_\{1\}outputs the grid indexΨ1\(1\)​\(𝐱\)\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\)together with the local coordinateM​\(𝐱−𝐱𝐦\)∈\[0,1\)DM\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\\in\[0,1\)^\{D\}\. Next, by Lemma[8](https://arxiv.org/html/2607.06781#Thmtheorem8)the moduli\{p𝐦\}\\\{p\_\{\\mathbf\{m\}\}\\\}over theMDM^\{D\}grids are pairwise coprime, so Proposition[9](https://arxiv.org/html/2607.06781#Thmtheorem9)furnishes integersKiK\_\{i\}withKi≡s𝐦,i\(modp𝐦\)K\_\{i\}\\equiv s\_\{\\mathbf\{m\},i\}\\pmod\{p\_\{\\mathbf\{m\}\}\}for every grid\. Then Lemma[12](https://arxiv.org/html/2607.06781#Thmtheorem12)givesE𝐦,i∈\[0,Cf\]∪\[p𝐦−Cf,p𝐦\)E\_\{\\mathbf\{m\},i\}\\in\[0,C\_\{f\}\]\\cup\[p\_\{\\mathbf\{m\}\}\-C\_\{f\},p\_\{\\mathbf\{m\}\}\), and Lemma[13](https://arxiv.org/html/2607.06781#Thmtheorem13)recoverss𝐦,is\_\{\\mathbf\{m\},i\}fromE𝐦,iE\_\{\\mathbf\{m\},i\}exactly throughρinv,ρfloor,𝙼𝚞𝚕𝚝\\rho\_\{\\mathrm\{inv\}\},\\rho\_\{\\mathrm\{floor\}\},\\mathtt\{Mult\}\. Thus,c𝐦,i=Bf−1​s𝐦,ic\_\{\\mathbf\{m\},i\}=B\_\{f\}^\{\-1\}s\_\{\\mathbf\{m\},i\}\. So thatΨ2\\Psi\_\{2\}recovers the exact rational coefficients\(c𝐦,i\)i≤N\(c\_\{\\mathbf\{m\},i\}\)\_\{i\\leq N\}of the polynomialY𝐦Y\_\{\\mathbf\{m\}\}on the active gridΩ𝐦\\Omega\_\{\\mathbf\{m\}\}\. Finally,Ψ3\\Psi\_\{3\}constructs the target polynomial: by Lemma[14](https://arxiv.org/html/2607.06781#Thmtheorem14), each monomial\(𝐱−𝐱𝐦\)𝜶\(i\)\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}^\{\(i\)\}\}is realized exactly, so is∑i≤Nc𝐦,i​\(𝐱−𝐱𝐦\)𝜶\(i\)\\sum\_\{i\\leq N\}c\_\{\\mathbf\{m\},i\}\\,\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)^\{\\bm\{\\alpha\}^\{\(i\)\}\}\. Since our network can exactly represent every polynomial over each grid, our network can represent exactly the whole polynomial\.

It remains to bound the width, depth, and parameter magnitudes, withN:=\(D\+rr\)N:=\\binom\{D\+r\}\{r\}andMDM^\{D\}grids\. As Table[2](https://arxiv.org/html/2607.06781#S5.T2)shows, our counting follows the standard approximation literature: the width of a layer is its number of neurons, including identity channels copied forward; each hidden layer consisting of an affine map followed by element\-wise activations counts as one layer\.

Width and depth ofΨ1\\Psi\_\{1\}\.InΨ1\\Psi\_\{1\}, it first appliesρfloor\\rho\_\{\\mathrm\{floor\}\}to eachM​xdMx\_\{d\}and carries𝐱\\mathbf\{x\}\(needed to form the local coordinate\) as the input\. Then, the affine combination for the indexΨ1\(1\)​\(𝐱\)\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\)together withΨ1\(2\)​\(𝐱\)=M​𝐱−ρfloor​\(M​𝐱\)\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)=M\\mathbf\{x\}\-\\rho\_\{\\mathrm\{floor\}\}\(M\\mathbf\{x\}\)\. Hence, forΦ1\\Phi\_\{1\}, we haveW1=2​DW\_\{1\}=2DandL1=1L\_\{1\}=1\.

Width and depth ofΨ2\\Psi\_\{2\}\.InΨ2\\Psi\_\{2\},DDchannels inΨ1\(2\)​\(𝐱\)\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)are copied until being passed intoΨ3\\Psi\_\{3\}\. Recovering coefficients uses two reciprocals:1/p𝐦1/p\_\{\\mathbf\{m\}\}for the residueE𝐦,i=Ki−p𝐦​⌊Ki/p𝐦⌋E\_\{\\mathbf\{m\},i\}=K\_\{i\}\-p\_\{\\mathbf\{m\}\}\\lfloor K\_\{i\}/p\_\{\\mathbf\{m\}\}\\rfloor, and1/\(p𝐦−Cf\)1/\(p\_\{\\mathbf\{m\}\}\-C\_\{f\}\)for the signed values𝐦,i=E𝐦,i−p𝐦​⌊E𝐦,i/\(p𝐦−Cf\)⌋s\_\{\\mathbf\{m\},i\}=E\_\{\\mathbf\{m\},i\}\-p\_\{\\mathbf\{m\}\}\\lfloor E\_\{\\mathbf\{m\},i\}/\(p\_\{\\mathbf\{m\}\}\-C\_\{f\}\)\\rfloor\. Formingp𝐦p\_\{\\mathbf\{m\}\}andp𝐦−Cfp\_\{\\mathbf\{m\}\}\-C\_\{f\}together, a singleρinv\\rho\_\{\\mathrm\{inv\}\}layer produces both\. This yields Layers 2\-7, as seen in Table[2](https://arxiv.org/html/2607.06781#S5.T2), which results in a depth of 6\. The widest layer is𝙼𝚞𝚕𝚝\\mathtt\{Mult\}, which forms𝙼𝚞𝚕𝚝​\(E​𝐦,i,1/\(p𝐦−Cf\)\)\\mathtt\{Mult\}\(E\{\\mathbf\{m\},i\},1/\(p\_\{\\mathbf\{m\}\}\-C\_\{f\}\)\)for alli≤Ni\\leq N, with a width ofD\+5​N\+1D\+5N\+1\.

Width and depth ofΨ3\\Psi\_\{3\}\.By Lemma[14](https://arxiv.org/html/2607.06781#Thmtheorem14), each monomial\[Ψ1\(2\)​\(𝐱\)\]𝜶\(i\)\[\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\)\]^\{\\bm\{\\alpha\}^\{\(i\)\}\}is a sequential product of\|𝜶\(i\)\|≤r\|\\bm\{\\alpha\}^\{\(i\)\}\|\\leq rfactors, one𝙼𝚞𝚕𝚝\\mathtt\{Mult\}per Layer\. RunningNNaccumulators in parallel and padding shorter chains with𝙼𝚞𝚕𝚝​\(⋅,1\)\\mathtt\{Mult\}\(\\cdot,1\), all monomials are obtained after multiplying the entries ofΨ1\(2\)​\(𝐱\)\\Psi^\{\(2\)\}\_\{1\}\(\\mathbf\{x\}\), with the coefficient extractionc𝐦,ic\_\{\\mathbf\{m\},i\}sharing the first of these layers\. One further𝙼𝚞𝚕𝚝\\mathtt\{Mult\}layer scales each monomial by its coefficient, and one affine map sums them toY𝐦​\(𝐱−𝐱𝐦\)Y\_\{\\mathbf\{m\}\}\(\\mathbf\{x\}\-\\mathbf\{x\}\_\{\\mathbf\{m\}\}\)\. The widest layer runsNNmultiplications, which needs4​N4NReQU units withD\+ND\+Ncopied channels\. Hence, forΨ3\\Psi\_\{3\}, we haveW3=D\+5​NW\_\{3\}=D\+5NandL3=r\+2L\_\{3\}=r\+2\.

Now we conclude the total width and depth based on the width and depth of three modules:

\{W​\(Ψ\)=max⁡\{W1,W2,W3\}=max⁡\{2​D,D\+5​N\+1\},L​\(Ψ\)=L1\+L2\+L3=r\+9,\\begin\{cases\}&W\(\\Psi\)=\\max\\\{W\_\{1\},W\_\{2\},W\_\{3\}\\\}=\\max\\\{2D,\\ D\+5N\+1\\\},\\\\ &L\(\\Psi\)=L\_\{1\}\+L\_\{2\}\+L\_\{3\}=r\+9,\\end\{cases\}\(74\)whereW​\(Ψ\)=D\+5​N\+1W\(\\Psi\)=D\+5N\+1forr≠0r\\neq 0\.

Parameter magnitudes\.We now estimate the largest parameter magnitude𝒫​\(Ψ\)\\mathcal\{P\}\(\\Psi\)\. The parameters ofΨ1\\Psi\_\{1\}areM,M2,…,MD−1M,M^\{2\},\\dots,M^\{D\-1\}; those ofΨ2\\Psi\_\{2\}areΓf\\Gamma\_\{f\},CfC\_\{f\},Bf−1B\_\{f\}^\{\-1\},KiK\_\{i\}, and fixed𝙼𝚞𝚕𝚝\\mathtt\{Mult\}weights of absolute value≤1\\leq 1; and those ofΨ3\\Psi\_\{3\}are padding constants11and𝙼𝚞𝚕𝚝\\mathtt\{Mult\}weights≤1\\leq 1\. The dominant parameters are the CRT integersKiK\_\{i\}, controlled by the modulus product\. Each modulusp𝐦=1\+Ψ1\(1\)​\(𝐱\)​Γfp\_\{\\mathbf\{m\}\}=1\+\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\)\\Gamma\_\{f\}withΨ1\(1\)​\(𝐱\)≤MD\\Psi^\{\(1\)\}\_\{1\}\(\\mathbf\{x\}\)\\leq M^\{D\}satisfiesp𝐦≤pmax:=1\+MD​Γfp\_\{\\mathbf\{m\}\}\\leq p\_\{\\max\}:=1\+M^\{D\}\\Gamma\_\{f\}andΓf=\(1\+2​Cf\)​\(MD\)\!\\Gamma\_\{f\}=\(1\+2C\_\{f\}\)\(M^\{D\}\)\!\. Since the least nonnegative CRT solutions obey0≤Ki<∏𝐦p𝐦≤pmaxMD0\\leq K\_\{i\}<\\prod\_\{\\mathbf\{m\}\}p\_\{\\mathbf\{m\}\}\\leq p\_\{\\max\}^\{M^\{D\}\}, and all other parameters are at mostCf≤pmaxMDC\_\{f\}\\leq p\_\{\\max\}^\{M^\{D\}\}, we have

𝒫​\(Ψ\)≤pmaxMD\+Cf≤2​\(1\+MD​\(1\+2​Cf\)​\(MD\)\!\)MD,\\mathcal\{P\}\(\\Psi\)\\leq p\_\{\\max\}^\{M^\{D\}\}\+C\_\{f\}\\leq 2\\bigl\(1\+M^\{D\}\(1\+2C\_\{f\}\)\(M^\{D\}\)\!\\bigr\)^\{M^\{D\}\},\(75\)which is an explicit bound withM,D,NM,D,Nand the integer scaleCfC\_\{f\}of the rational coefficients\.

The bound in Eq\. \([75](https://arxiv.org/html/2607.06781#S5.E75)\) depends onffonly through the integer scaleCfC\_\{f\}of its coefficients\. For any fixed gridwise polynomial with rational coefficients,CfC\_\{f\}is finite and determined byff\. Remember we write eachc𝐦,i=s𝐦,i/g𝐦,ic\_\{\\mathbf\{m\},i\}=s\_\{\\mathbf\{m\},i\}/g\_\{\\mathbf\{m\},i\}in lowest terms, let

F:=max𝐦,i⁡𝚋𝚒𝚝𝚜​\(c𝐦,i\),F:=\\max\_\{\\mathbf\{m\},i\}\\mathtt\{bits\}\(c\_\{\\mathbf\{m\},i\}\),\(76\)and the coefficient bit length carried byff\. Then\|s𝐦,i\|<2F\|s\_\{\\mathbf\{m\},i\}\|<2^\{F\}andg𝐦,i<2Fg\_\{\\mathbf\{m\},i\}<2^\{F\}, soBf≤2F​MD​NB\_\{f\}\\leq 2^\{FM^\{D\}N\}andCf≤2F​Bf≤2F​\(MD​N\+1\)C\_\{f\}\\leq 2^\{F\}B\_\{f\}\\leq 2^\{F\(M^\{D\}N\+1\)\}\. Substituting Eq\. \([76](https://arxiv.org/html/2607.06781#S5.E76)\) into Eq\. \([75](https://arxiv.org/html/2607.06781#S5.E75)\) and combining\(MD\)\!≤\(MD\)MD\(M^\{D\}\)\!\\leq\(M^\{D\}\)^\{M^\{D\}\},1\+2​Cf≤2F​\(MD​N\+1\)\+21\+2C\_\{f\}\\leq 2^\{F\(M^\{D\}N\+1\)\+2\}, and1\+MD​Γf≤2​MD​Γf1\+M^\{D\}\\Gamma\_\{f\}\\leq 2M^\{D\}\\Gamma\_\{f\}give

log2⁡pmax≤D​\(MD\+1\)​log2⁡M\+F​\(MD​N\+1\)\+3\.\\log\_\{2\}p\_\{\\max\}\\leq D\(M^\{D\}\+1\)\\log\_\{2\}M\+F\(M^\{D\}N\+1\)\+3\.\(77\)Therefore

𝒫​\(Ψ\)≤2MD​\[\(MD\+1\)​D​log2⁡M\+F​\(MD​N\+1\)\+3\]\+1,\\mathcal\{P\}\(\\Psi\)\\leq 2^\{\\,M^\{D\}\[\(M^\{D\}\+1\)\\,D\\log\_\{2\}M\\;\+\\;F\\,\(M^\{D\}N\+1\)\\;\+\\;3\]\+1\},\(78\)with𝒫​\(Ψ\)≤22​M2​D​\(D​log2⁡M\+F​N\)\\mathcal\{P\}\(\\Psi\)\\leq 2^\{2M^\{2D\}\(D\\log\_\{2\}M\+FN\)\}forr≠0r\\neq 0\.

∎

Table 2:Architecture and parameter magnitudes for the construction of the gridwise polynomial with rational coefficients\.

## 6Conclusion

This paper has developed a novel fixed\-architecture approximation framework based on the Chinese Remainder Theorem, which moves one step further in the domain of super\-expressiveness\. For Lipschitz targets \(Theorem[1](https://arxiv.org/html/2607.06781#Thmmaintheorem1)\), our construction for the first time enjoys an explicit relation between parameter magnitude and the approximation error\. For Hölder\-smooth targets \(Theorem[2](https://arxiv.org/html/2607.06781#Thmmaintheorem2)\), our construction also realizes an explicit representation, which shows a smoothness scaling law\. The key to our two constructions is that the CRT ensures the accurate estimation of integer magnitude\. There is still room for improvement in our constructions\. First, the CRT parameter can be enormous\. Though it is achievable, there might be redundancy in the construction\. Second, like other super\-expressive constructions, our result uses nonstandard activations, which narrows the scope of our theory\. Our future work will attempt to address these limitations\.

### Acknowledgments

This work is supported in part by the Lagrange Mathematics and Computing Research Center, Huawei Technologies France, and the Startup Fund of the City University of Hong Kong\.

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