Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials

# Completeness Theory for Graph Neural Network Interatomic Potentials
Source: [https://arxiv.org/html/2609.00528](https://arxiv.org/html/2609.00528)
## Why Multi\-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials

Pingbing Ming††thanks:SKLMS, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, No\.˜55, East Road Zhong\-Guan\-Cun, Beijing 100190, China \(\)\.Han Wang††thanks:National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Fenghao East Road 2, Beijing 100094, P\.R\.˜China, and HEDPS, CAPT, College of Engineering, Peking University, Beijing 100871, P\.R\.˜China \(\)\.Email:[wang\_han@iapcm\.ac\.cn](mailto:[email protected])

###### Abstract

We prove that the Hypergraph Neural Network, an invariant architecture with 3\-body message passing, is a universal approximator for potential energy surfaces\. Our main contribution is a*multi\-layer completeness theory*\. We show thatLLlayers of message passing on sparse, cutoff\-based graphs achieve the same representational power as having access to the fullLL\-hop neighborhood, provided the configurations are generic, satisfy an overlap condition and a connectivity condition\. This provides the first rigorous justification for the common practice of using multi\-layer message passing with a per\-layer cutoff smaller than the physical interaction range, the setting used by virtually all practical graph neural network based machine\-learned interatomic potentials\. As immediate consequences, we show that both DPA3 and CHGNet architectures inherit universal approximation\.

###### keywords

machine\-learned interatomic potentials, graph neural networks, hypergraph neural networks, universal approximation

††runningheads:Completeness Theory for GNN Interatomic Potentials / P\. Ming and H\. Wang###### MSC

68T07, 68R10, 41A65, 92E10

## 1Introduction

Machine\-learned interatomic potentials \(MLIPs\)\[[6](https://arxiv.org/html/2609.00528#bib.bib6),[2](https://arxiv.org/html/2609.00528#bib.bib2),[42](https://arxiv.org/html/2609.00528#bib.bib42)\]have become an essential tool in computational chemistry and materials science, enabling molecular dynamics simulations that achieve near\-quantum\-mechanical accuracy at a fraction of the computational cost\. Applications range from drug discovery\[[37](https://arxiv.org/html/2609.00528#bib.bib37)\]and catalysis\[[21](https://arxiv.org/html/2609.00528#bib.bib21)\]to materials design\[[9](https://arxiv.org/html/2609.00528#bib.bib9)\]and phase diagram exploration\[[38](https://arxiv.org/html/2609.00528#bib.bib38)\]\. Recent advances in atomistic foundation models\[[41](https://arxiv.org/html/2609.00528#bib.bib41),[3](https://arxiv.org/html/2609.00528#bib.bib3),[24](https://arxiv.org/html/2609.00528#bib.bib24),[10](https://arxiv.org/html/2609.00528#bib.bib10)\]have further expanded the scope of MLIPs to universal potentials covering much of the periodic table\. This raises a fundamental question: can a multi\-layer GNN\-based MLIP approximate an arbitrary potential energy surface \(PES\) while rigorously preserving the required physical symmetries and continuity constraints? The answer to this question is of central importance, as it determines whether the predictive accuracy of such models is inherently limited by the architectural design, or whether it ultimately depends only on the training data and optimization procedure\.

A closely related concept is*completeness*\. A model is said to be complete if it produces distinct outputs for any two local environments that are not related by symmetry, that is, two environments yield the same output only if they are related by a rotation or reflection combined with a permutation of atoms of the same species\. Completeness is a necessary condition for universal approximation, yet whether it is also sufficient is a separate question that we shall address in this work\. The concept was introduced by Bartók et al\.\[[1](https://arxiv.org/html/2609.00528#bib.bib1)\]in the context of invariant representations, and brought into sharp focus by Pozdnyakov et al\.\[[29](https://arxiv.org/html/2609.00528#bib.bib29)\], who demonstrated that the widely used 3\-body representations, such as SOAP power spectra, are*incomplete*, in that there exist geometrically distinct environments that share identical sets of pairwise distances and triplet angles\. Incompleteness thus imposes a fundamental ceiling on approximation accuracy that cannot be overcome, regardless of the training data\.

Further incompleteness results have since revealed the scope of the problem\. Pozdnyakov and Ceriotti\[[27](https://arxiv.org/html/2609.00528#bib.bib27)\]showed that distance\-only GNNs, such as SchNet\[[30](https://arxiv.org/html/2609.00528#bib.bib30)\], are incomplete for 3D point clouds even with arbitrary depth and nonlinearity, so angular information is not merely helpful but*necessary*\. Cen et al\.\[[8](https://arxiv.org/html/2609.00528#bib.bib8)\]showed that equivariant GNNs with fixed tensor degreelldegenerate to zero on symmetric structures for specific values ofll, e\.g\.,l=1l=1fails on all symmetric graphs, demonstrating that low\-degree equivariant features are also insufficient\.

Several approaches have been developed to achieve completeness\. The Atomic Cluster Expansion \(ACE\)\[[14](https://arxiv.org/html/2609.00528#bib.bib14),[15](https://arxiv.org/html/2609.00528#bib.bib15)\]and Moment Tensor Potentials \(MTP\)\[[32](https://arxiv.org/html/2609.00528#bib.bib32)\]construct systematically improvable polynomial bases for invariant functions\. These are*linear*models for which basis density directly yields universal approximation\. Equivariant message\-passing architectures such as Tensor Field Networks\[[34](https://arxiv.org/html/2609.00528#bib.bib34)\], NequIP\[[5](https://arxiv.org/html/2609.00528#bib.bib5)\], PaiNN\[[31](https://arxiv.org/html/2609.00528#bib.bib31)\], SEGNN\[[7](https://arxiv.org/html/2609.00528#bib.bib7)\], and Equivariant Transformers\[[33](https://arxiv.org/html/2609.00528#bib.bib33)\]useE⁡\(3\)E\(3\)\-equivariant convolutions to propagate vector and tensor features, achieving high data efficiency but without a formal completeness guarantee\. Building on this line of work, MACE\[[4](https://arxiv.org/html/2609.00528#bib.bib4)\]achieves completeness by constructing ACE basis functions within the GNN framework: tensor products of 2\-body equivariant features are symmetrized via Clebsch–Gordan coefficients to build\(ν\+1\)\(\\nu\{\+\}1\)\-body features that are mathematically equivalent to the ACE basis, so completeness follows directly from ACE theory\. A different approach processes*invariant*scalar features, namely distances and bond angles, with nonlinear networks\. Models in this class include DimeNet\[[17](https://arxiv.org/html/2609.00528#bib.bib17)\], GemNet\[[16](https://arxiv.org/html/2609.00528#bib.bib16)\], SphereNet\[[23](https://arxiv.org/html/2609.00528#bib.bib23)\], ALIGNN\[[11](https://arxiv.org/html/2609.00528#bib.bib11)\], M3GNet\[[9](https://arxiv.org/html/2609.00528#bib.bib9)\], CHGNet\[[13](https://arxiv.org/html/2609.00528#bib.bib13)\], and DPA3\[[41](https://arxiv.org/html/2609.00528#bib.bib41)\]\. Finally, coordinate\-frame methods construct local reference frames from pairs of neighbors and apply nonlinear functions before symmetrization: Nigam et al\.\[[25](https://arxiv.org/html/2609.00528#bib.bib25)\]proposed 3\-center\-1\-neighbor representations, and Pozdnyakov and Ceriotti\[[28](https://arxiv.org/html/2609.00528#bib.bib28)\]introduced the Equivariant Coordinate System Ensemble \(ECSE\)\.

On the theoretical side, several foundational results underpin the analysis of model expressiveness\. The DeepSets universality theorem\[[40](https://arxiv.org/html/2609.00528#bib.bib40),[36](https://arxiv.org/html/2609.00528#bib.bib36),[18](https://arxiv.org/html/2609.00528#bib.bib18)\]provides established thatϕ∘\(∑jg\)\\phi\\circ\(\\sum\_\{j\}g\)is a universal approximator for permutation\-invariant functions\. Villar et al\.\[[35](https://arxiv.org/html/2609.00528#bib.bib35)\]proved that any function invariant under translations, rotations, reflections, and permutations can be expressed as a permutation\-invariant function of the Gram matrix of displacement vectors, a result known as the “scalars are universal” theorem\. Li et al\.\[[22](https://arxiv.org/html/2609.00528#bib.bib22)\]demonstrated that several invariant GNN architectures, such as DimeNet, SphereNet and GemNet, achieveE⁡\(3\)E\(3\)\-completeness on fully connected graphs\. Joshi et al\.\[[20](https://arxiv.org/html/2609.00528#bib.bib20)\]introduced the Geometric Weisfeiler–Leman test, covering both invariant and equivariant models, and proved an equivalence between pairwise discrimination within a model class and universal approximation\.

Despite this progress, significant gaps remain\. The completeness results of Villar et al\.\[[35](https://arxiv.org/html/2609.00528#bib.bib35)\]and Li et al\.\[[22](https://arxiv.org/html/2609.00528#bib.bib22)\]are established only for*fully connected*graphs; for the sparse, cutoff\-based graphs commonly employed in practice, completeness remains an open question\. Villar et al\. characterize*what*form invariant functions must take, but do not show any specific architecture can actually approximate them\. Nigam et al\.\[[25](https://arxiv.org/html/2609.00528#bib.bib25)\]identify the correct geometric insight, namely that three\-center features with nonlinearity applied before summation achieve completeness, yet their argument is presented as a proof sketch rather than a rigorous derivation\. The discrimination–UAT equivalence of Joshi et al\.\[[20](https://arxiv.org/html/2609.00528#bib.bib20)\]operates on the*function class*, requiring only that for each pair of environments some function in the class separates them\. This is conceptually different from completeness of a*single*model with fixed weights\. Moreover, to the best of our knowledge, all existing theoretical analyses operate within a single cutoff neighborhood\. In practice, however, multi\-layer message passing extends the receptive field from the per\-layer cutoffrcr\_\{c\}to a larger physical interaction rangeRc\>rcR\_\{c\}\>r\_\{c\}, and the impact of this extended range on completeness has not yet been addressed\.

We close these gaps by introducing the Hypergraph Neural Network \(HGNN\), an invariant 3\-body architecture, and proving the following multi\-layer completeness theorem for HGNN:

Main result \(see[theorem26](https://arxiv.org/html/2609.00528#Thmtheorem26)\)\.*On generic configurations satisfying an overlap and a connectivity condition, and forLLlarge enough that theLL\-hop neighborhood covers the physical interaction rangeRcR\_\{c\}, theLL\-layer HGNN is a universal approximator for the PES at rangeRcR\_\{c\}\.*

The above result extends completeness from the per\-layer cutoffrcr\_\{c\}to the physical rangeRcR\_\{c\}via multi\-layer message passing, a setting not addressed in prior work\. The proof operates on*sparse*cutoff\-based graphs, provides a rigorous continuous approximation guarantee for the HGNN, and characterizes a*single*fixed\-weight representation rather than a function class\. Beyond this, the HGNN serves as a*reference theoretical architecture*: any practical architecture that can simulate the HGNN inherits its universal approximation property\. We demonstrate this implication explicitly with DPA3 and CHGNet, i\.e\.,[corollaries27](https://arxiv.org/html/2609.00528#Thmtheorem27)and[29](https://arxiv.org/html/2609.00528#Thmtheorem29)\. The proofs further reveal one key architectural requirement, that message functions must be Multi\-Layer Perceptrons \(MLPs\)\[[12](https://arxiv.org/html/2609.00528#bib.bib12),[19](https://arxiv.org/html/2609.00528#bib.bib19)\], rather than single linear layers, to ensure the desired expressiveness\.

The paper is organized as follows\.[section2](https://arxiv.org/html/2609.00528#S2)introduces the PES properties, symmetry framework, and the GNN\-based interatomic potential architectures considered in this work, namely, HGNN, ALIGNN, CHGNet, DPA3\.[section3](https://arxiv.org/html/2609.00528#S3)proves the completeness–UAT equivalence and establishes the existence of complete representations via Gram matrix reconstruction\.[section4](https://arxiv.org/html/2609.00528#S4)proves that the HGNN can approximate complete representations\.[section5](https://arxiv.org/html/2609.00528#S5)states the UAT and derives corollaries for DPA3 and CHGNet, with the simulation proofs given in the supplementary materials\.[section6](https://arxiv.org/html/2609.00528#S6)discusses implications, assumptions, limitations, and open problems\.

### 2\.1Potential energy surface and target function class

Consider a system ofNNatoms with typeszi∈𝒵=\{1,…,T\}z\_\{i\}\\in\\mathcal\{Z\}=\\\{1,\\ldots,T\\\}and pairwise distinct positions𝒓i∈ℝ3\\bm\{r\}\_\{i\}\\in\\mathbb\{R\}^\{3\}\. WriteBr=\{𝒓∈ℝ3:\|𝒓\|<r\}B\_\{r\}=\\\{\\bm\{r\}\\in\\mathbb\{R\}^\{3\}:\|\\bm\{r\}\|<r\\\}for the open ball of radiusrr\. We assume the density is bounded, in that the number of atoms in any ball of fixed radius admits a uniform bound\. We writeMMfor such a bound at the largest cutoff appearing below\.

###### Definition 1\(Local environment\)\.

The*local environment*of atomjjwithin cutoffrris

𝒟j\(r\)=\(zj,\{\(Δ𝒓j​k,zk\):k≠j,Δ𝒓j​k∈Br\}\),Δ𝒓j​k=𝒓k−𝒓j,\\mathcal\{D\}\_\{j\}^\{\(r\)\}=\\bigl\(z\_\{j\},\\;\\\{\(\\Delta\\bm\{r\}\_\{jk\},\\,z\_\{k\}\):k\\neq j,\\;\\Delta\\bm\{r\}\_\{jk\}\\in B\_\{r\}\\\}\\bigr\),\\qquad\\Delta\\bm\{r\}\_\{jk\}=\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\},\(1\)where the neighbor set is unordered \(a multiset\)\. That is,𝒟j\(r\)\\mathcal\{D\}\_\{j\}^\{\(r\)\}consists of the center typezjz\_\{j\}together with the displacement vectors and types of all neighbors withinrr\. We write𝒟j≡𝒟j\(rc\)\\mathcal\{D\}\_\{j\}\\equiv\\mathcal\{D\}\_\{j\}^\{\(r\_\{c\}\)\}when the cutoff is the defaultrcr\_\{c\}\.

A PES is a mapE:⋃N=1∞\(ℝ3×𝒵\)N→ℝE:\\bigcup\_\{N=1\}^\{\\infty\}\(\\mathbb\{R\}^\{3\}\\times\\mathcal\{Z\}\)^\{N\}\\to\\mathbb\{R\}subject to four structural properties:

#### \(P1\) Symmetry

EEis invariant under translations, rotations/reflections, and permutation of same\-type atoms:

E⁡\(\{𝒓i\+𝒕,zi\}\)\\displaystyle E\(\\\{\\bm\{r\}\_\{i\}\+\\bm\{t\},\\,z\_\{i\}\\\}\)=E⁡\(\{𝒓i,zi\}\)\\displaystyle=E\(\\\{\\bm\{r\}\_\{i\},\\,z\_\{i\}\\\}\)∀𝒕∈ℝ3,\\displaystyle\\forall\\,\\bm\{t\}\\in\\mathbb\{R\}^\{3\},\(2\)E⁡\(\{R​𝒓i,zi\}\)\\displaystyle E\(\\\{R\\bm\{r\}\_\{i\},\\,z\_\{i\}\\\}\)=E⁡\(\{𝒓i,zi\}\)\\displaystyle=E\(\\\{\\bm\{r\}\_\{i\},\\,z\_\{i\}\\\}\)∀R∈O⁡\(3\),\\displaystyle\\forall\\,R\\in O\(3\),\(3\)E⁡\(\{𝒓π⁡\(i\),zi\}\)\\displaystyle E\(\\\{\\bm\{r\}\_\{\\pi\(i\)\},\\,z\_\{i\}\\\}\)=E⁡\(\{𝒓i,zi\}\)\\displaystyle=E\(\\\{\\bm\{r\}\_\{i\},\\,z\_\{i\}\\\}\)∀π∈SN1×⋯×SNT,\\displaystyle\\forall\\,\\pi\\in S\_\{N\_\{1\}\}\\times\\cdots\\times S\_\{N\_\{T\}\},\(4\)whereNt=\|\{i:zi=t\}\|N\_\{t\}=\|\\\{i:z\_\{i\}=t\\\}\|,SnS\_\{n\}denotes the symmetric group onnnelements, andπ\\pipermutes only positions of same\-type atoms\. Each atom’s type is an intrinsic attribute that travels with it under permutation\. Swapping two atoms exchanges their positions while each retains its type, so only same\-type swaps leave the configuration unchanged\.

#### \(P2\) Extensivity

EEdecomposes as a sum of atomic contributions:

E⁡\(\{𝒓i,zi\}i=1N\)=∑i=1Nεi\.E\(\\\{\\bm\{r\}\_\{i\},\\,z\_\{i\}\\\}\_\{i=1\}^\{N\}\)=\\sum\_\{i=1\}^\{N\}\\varepsilon\_\{i\}\.\(5\)

#### \(P3\) Locality

There exists a physical interaction rangeRc\>0R\_\{c\}\>0such thatεi\\varepsilon\_\{i\}depends only on atoms withinRcR\_\{c\}of atomii:εi=ε⁡\(𝒟i\(Rc\)\)\\varepsilon\_\{i\}=\\varepsilon\(\\mathcal\{D\}\_\{i\}^\{\(R\_\{c\}\)\}\)\.

#### \(P4\) Smoothness

ε\\varepsilonis at leastCkC^\{k\}\(k≥2k\\geq 2\) with respect to the neighbor positions, including at the interaction\-range boundary, where neighbors enter and leave the environment\. Sinceε\\varepsilonisCkC^\{k\}, the forces areCk−1C^\{k\-1\}and the Hessian isCk−2C^\{k\-2\}\. Requiringk≥2k\\geq 2therefore makes all three continuous\.

The extensivity and locality properties reduce the problem from a function on a3​N3N\-dimensional configuration space to a function on the space of local environments, which has bounded dimensionality\. We now formalize this space\.

###### Definition 2\(Environment space\)\.

For a fixed cutoffr\>0r\>0and a type signature\(z0,𝐳\)\(z\_\{0\},\\bm\{z\}\)withz0∈𝒵z\_\{0\}\\in\\mathcal\{Z\},𝐳∈𝒵≤n=\{\(z1,…,zn\)∈𝒵n:z1≤⋯≤zn\}\\bm\{z\}\\in\\mathcal\{Z\}^\{n\}\_\{\\leq\}=\\\{\(z\_\{1\},\\ldots,z\_\{n\}\)\\in\\mathcal\{Z\}^\{n\}:z\_\{1\}\\leq\\cdots\\leq z\_\{n\}\\\}, andn≤Mn\\leq M, the*stratum*ℰz0,𝐳\(r\)\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\)\}is the set of ordered lists of the neighbors of a local environment within cutoffrr\([definition1](https://arxiv.org/html/2609.00528#Thmtheorem1)\) whose center has typez0z\_\{0\}, arranged so that thekk\-th neighbor has typezkz\_\{k\}\. Since the types are fixed by the signature, such a list is determined by its tuple of displacement vectors, written

𝒟=\(Δ​𝒓1,…,Δ​𝒓n\)∈\(Br∖\{0\}\)n⊂ℝ3​n\.\\mathcal\{D\}=\(\\Delta\\bm\{r\}\_\{1\},\\ldots,\\Delta\\bm\{r\}\_\{n\}\)\\in\\bigl\(B\_\{r\}\\setminus\\\{0\\\}\\bigr\)^\{n\}\\;\\subset\\;\\mathbb\{R\}^\{3n\}\.\(6\)Every such tuple occurs, so the stratum is an open subset ofℝ3​n\\mathbb\{R\}^\{3n\}and inherits its Euclidean metric, standard topology, andC∞C^\{\\infty\}differentiable structure\. The type signature\(z0,𝐳\)\(z\_\{0\},\\bm\{z\}\)is index data rather than set data, so strata with different signatures are disjoint even when they have equal dimension\. The*environment space*with cutoffrris the disjoint union over all center types and neighbor compositions:

ℰ\(r\)=⨆z0∈𝒵⨆n=0M⨆𝒛∈𝒵≤nℰz0,𝒛\(r\)\.\\mathcal\{E\}^\{\(r\)\}=\\bigsqcup\_\{z\_\{0\}\\in\\mathcal\{Z\}\}\\;\\bigsqcup\_\{n=0\}^\{M\}\\;\\bigsqcup\_\{\\bm\{z\}\\in\\mathcal\{Z\}^\{n\}\_\{\\leq\}\}\\;\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\)\}\\,\.\(7\)We writeℰ≡ℰ\(rc\)\\mathcal\{E\}\\equiv\\mathcal\{E\}^\{\(r\_\{c\}\)\}andℰz0,𝐳≡ℰz0,𝐳\(rc\)\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}\\equiv\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\_\{c\}\)\}when the cutoff is the defaultrcr\_\{c\}\.

The Euclidean metric on each stratum is

d⁡\(𝒟,𝒟′\)=\(∑k=1n‖Δ​𝒓k−Δ​𝒓k′‖2\)1/2,𝒟,𝒟′∈ℰz0,𝒛\(r\)\.d\(\\mathcal\{D\},\\mathcal\{D\}^\{\\prime\}\)=\\Bigl\(\\sum\_\{k=1\}^\{n\}\\\|\\Delta\\bm\{r\}\_\{k\}\-\\Delta\\bm\{r\}\_\{k\}^\{\\prime\}\\\|^\{2\}\\Bigr\)^\{1/2\},\\qquad\\mathcal\{D\},\\mathcal\{D\}^\{\\prime\}\\in\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\)\}\.\(8\)Two tuples in the same stratum list the same local environment precisely when they differ by a type\-preserving permutation of the neighbors, which is part of the symmetry group of[definition3](https://arxiv.org/html/2609.00528#Thmtheorem3)\. We refer to elements ofℰ\(r\)\\mathcal\{E\}^\{\(r\)\}simply as local environments\. Since the type tuple is nondecreasing, every local environment has all of its representatives in a single stratum\. The mathematical structure ofℰ\\mathcal\{E\}is stratum\-wise, in that continuity, compactness, andCkC^\{k\}\-smoothness are defined with respect to the Euclidean metric and differentiable structure on each stratumℰz0,𝒛\(r\)⊂ℝ3​n\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\)\}\\subset\\mathbb\{R\}^\{3n\}\. Globally,ℰ\(r\)\\mathcal\{E\}^\{\(r\)\}is a disjoint union of strata of different dimensions, while it is not a linear space\.

The*symmetry group*acting on a stratum ofℰ\\mathcal\{E\}withnnneighbors of types𝒛\\bm\{z\}is

G=O\(3\)×\(Sn1×⋯×SnT\),G=O\(3\)\\times\\bigl\(S\_\{n\_\{1\}\}\\times\\cdots\\times S\_\{n\_\{T\}\}\\bigr\),\(9\)whereSntS\_\{n\_\{t\}\}permutes thentn\_\{t\}neighbors of typettwith∑tnt=n\\sum\_\{t\}n\_\{t\}=n\. We writeΓ=Sn1×⋯×SnT\\Gamma=S\_\{n\_\{1\}\}\\times\\cdots\\times S\_\{n\_\{T\}\}for the permutation part\. Only type\-preserving permutations appear, since atom types are intrinsic attributes, cf\. \(P1\)\. The action of\(R,π\)∈G\(R,\\pi\)\\in Gon a stratum element𝒟=\(Δ​𝒓1,…,Δ​𝒓n\)∈ℰz0,𝒛\\mathcal\{D\}=\(\\Delta\\bm\{r\}\_\{1\},\\ldots,\\Delta\\bm\{r\}\_\{n\}\)\\in\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}is

\(R,π\)⋅\(Δ​𝒓1,…,Δ​𝒓n\)=\(R​Δ​𝒓π−1​\(1\),…,R​Δ​𝒓π−1​\(n\)\)\.\(R,\\pi\)\\cdot\(\\Delta\\bm\{r\}\_\{1\},\\ldots,\\Delta\\bm\{r\}\_\{n\}\)=\(R\\,\\Delta\\bm\{r\}\_\{\\pi^\{\-1\}\(1\)\},\\ldots,R\\,\\Delta\\bm\{r\}\_\{\\pi^\{\-1\}\(n\)\}\)\.\(10\)
###### Definition 3\(Symmetry equivalence\)\.

Two environments𝒟,𝒟′\\mathcal\{D\},\\mathcal\{D\}^\{\\prime\}, possibly of different atoms or different configurations, are*equivalent*, written𝒟∼𝒟′\\mathcal\{D\}\\sim\\mathcal\{D\}^\{\\prime\}, if they lie in the same stratum and there exists\(R,π\)∈G\(R,\\pi\)\\in Gsuch that\(R,π\)⋅𝒟=𝒟′\(R,\\pi\)\\cdot\\mathcal\{D\}=\\mathcal\{D\}^\{\\prime\}, that is,

R​Δ​𝒓k=Δ​𝒓π⁡\(k\)′for all​k\.R\\,\\Delta\\bm\{r\}\_\{k\}=\\Delta\\bm\{r\}\_\{\\pi\(k\)\}^\{\\prime\}\\qquad\\text\{for all \}k\.\(11\)

Writedj​k=\|𝒓k−𝒓j\|d\_\{jk\}=\|\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\}\|for the distance between atomsjjandkk, andθi​j​k\\theta\_\{ijk\}for the bond angle atjjbetween neighborsiiandkk\.

Rotations and reflections change the displacement vectors but leave their mutual inner products unchanged\. Collecting those inner products therefore retains exactly theO⁡\(3\)O\(3\)invariants of the geometry\. Two environments have the same inner products if and only if they lie in the sameO⁡\(3\)O\(3\)orbit\.

###### Definition 4\(Gram matrix\)\.

The Gram matrix of a local environment is

Gi​k=Δ𝒓i⋅Δ𝒓k=didkcosθi​j​k,i,k=1,…,n,G\_\{ik\}=\\Delta\\bm\{r\}\_\{i\}\\cdot\\Delta\\bm\{r\}\_\{k\}=d\_\{i\}\\,d\_\{k\}\\,\\cos\\theta\_\{ijk\},\\qquad i,k=1,\\ldots,n,\(12\)wheredi=\|Δ​𝐫i\|d\_\{i\}=\|\\Delta\\bm\{r\}\_\{i\}\|andθi​j​k\\theta\_\{ijk\}is the angle betweenΔ​𝐫i\\Delta\\bm\{r\}\_\{i\}andΔ​𝐫k\\Delta\\bm\{r\}\_\{k\}\.

Within a stratumℰz0,𝒛\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}, the type tuple𝒛\\bm\{z\}is fixed, so the Gram matrixGGalone encodes all geometric information\. The type\-preserving permutation groupΓ\\Gammais determined by the stratum\.

For any atomkk, we writeG\(k\)G^\{\(k\)\}for the Gram matrix of atomkk’s own local environment\. Its entries areGa​b\(k\)=Δ​𝒓a\(k\)⋅Δ​𝒓b\(k\)G^\{\(k\)\}\_\{ab\}=\\Delta\\bm\{r\}\_\{a\}^\{\(k\)\}\\cdot\\Delta\\bm\{r\}\_\{b\}^\{\(k\)\}withΔ​𝒓a\(k\)=𝒓a−𝒓k\\Delta\\bm\{r\}\_\{a\}^\{\(k\)\}=\\bm\{r\}\_\{a\}\-\\bm\{r\}\_\{k\}\.

###### Definition 5\(Invariant function\)\.

A functionε:ℰ\(r\)→ℝ\\varepsilon:\\mathcal\{E\}^\{\(r\)\}\\to\\mathbb\{R\}is*invariant*under the symmetry groupGGifε⁡\(𝒟\)=ε⁡\(𝒟′\)\\varepsilon\(\\mathcal\{D\}\)=\\varepsilon\(\\mathcal\{D\}^\{\\prime\}\)whenever𝒟∼𝒟′\\mathcal\{D\}\\sim\\mathcal\{D\}^\{\\prime\}\.

###### Definition 6\(Target function space\)\.

Denote byℱRck\\mathcal\{F\}\_\{R\_\{c\}\}^\{k\}the set of functionsε:ℰ\(Rc\)→ℝ\\varepsilon:\\mathcal\{E\}^\{\(R\_\{c\}\)\}\\to\\mathbb\{R\}satisfying:

1. 1\.GG\-invariance;
2. 2\.CkC^\{k\}\-smoothness on each stratum: the restriction ofε\\varepsilonto eachℰz0,𝒛\(Rc\)⊂ℝ3​n\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(R\_\{c\}\)\}\\subset\\mathbb\{R\}^\{3n\}is aCkC^\{k\}function with respect to the Euclidean differentiable structure;
3. 3\.cross\-stratumCkC^\{k\}matching: if𝒟=\(Δ​𝒓1,…,Δ​𝒓n\)\\mathcal\{D\}=\(\\Delta\\bm\{r\}\_\{1\},\\ldots,\\Delta\\bm\{r\}\_\{n\}\)lies inℰz0,𝒛\(Rc\)\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(R\_\{c\}\)\}and𝒟−\\mathcal\{D\}^\{\-\}denotes the environment with themm\-th neighbor deleted, thenε⁡\(𝒟\)→ε⁡\(𝒟−\)\\varepsilon\(\\mathcal\{D\}\)\\to\\varepsilon\(\\mathcal\{D\}^\{\-\}\)together with all partial derivatives of order≤k\\leq kin the retained coordinates, while all derivatives involvingΔ​𝒓m\\Delta\\bm\{r\}\_\{m\}tend to zero, as\|Δ​𝒓m\|→Rc\|\\Delta\\bm\{r\}\_\{m\}\|\\to R\_\{c\}\.

Condition 3 of[definition6](https://arxiv.org/html/2609.00528#Thmtheorem6)ensures thatε\\varepsilonis well defined and isCkC^\{k\}across the strata of different dimension thatℰ\(Rc\)\\mathcal\{E\}^\{\(R\_\{c\}\)\}comprises\. It is met, for instance, byε⁡\(𝒟\)=∑i=1ns⁡\(\|Δ​𝒓i\|\)​u​\(z0,zi,\|Δ​𝒓i\|\)\\varepsilon\(\\mathcal\{D\}\)=\\sum\_\{i=1\}^\{n\}s\(\|\\Delta\\bm\{r\}\_\{i\}\|\)\\,u\(z\_\{0\},z\_\{i\},\|\\Delta\\bm\{r\}\_\{i\}\|\)withuuaCkC^\{k\}function ands:\[0,Rc\]→\[0,1\]s:\[0,R\_\{c\}\]\\to\[0,1\]satisfyings\(m\)​\(Rc\)=0s^\{\(m\)\}\(R\_\{c\}\)=0for0≤m≤k0\\leq m\\leq k\.

### 2\.2Machine learning interatomic potentials

An MLIP approximates the local energyε⁡\(𝒟j\(Rc\)\)\\varepsilon\(\\mathcal\{D\}\_\{j\}^\{\(R\_\{c\}\)\}\)by

εθ​\(𝒟j\(Rc\)\)=fθ​\(Φθ​\(𝒟j\(Rc\)\)\),\\varepsilon\_\{\\theta\}\(\\mathcal\{D\}\_\{j\}^\{\(R\_\{c\}\)\}\)=f\_\{\\theta\}\\bigl\(\\Phi\_\{\\theta\}\(\\mathcal\{D\}\_\{j\}^\{\(R\_\{c\}\)\}\)\\bigr\),\(13\)whereΦθ:ℰ\(Rc\)→ℝd\\Phi\_\{\\theta\}:\\mathcal\{E\}^\{\(R\_\{c\}\)\}\\to\\mathbb\{R\}^\{d\}is a*representation*that maps the local environment to add\-dimensional feature vector, andfθ:ℝd→ℝf\_\{\\theta\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}is a readout MLP that maps the representation to an atomic energy contribution\. Both carry learnable parameters, collectively denotedθ\\theta, and the total energy is

Eθ​\(\{𝒓i,zi\}i=1N\)=∑i=1Nεθ​\(𝒟i\(Rc\)\)\.E\_\{\\theta\}\(\\\{\\bm\{r\}\_\{i\},\\,z\_\{i\}\\\}\_\{i=1\}^\{N\}\)=\\sum\_\{i=1\}^\{N\}\\varepsilon\_\{\\theta\}\(\\mathcal\{D\}\_\{i\}^\{\(R\_\{c\}\)\}\)\.\(14\)The ansatz builds \(P2\) and \(P3\) into the architecture\. The total energy is a sum of atomic contributions, each depending only on theRcR\_\{c\}\-environment\. The representation must beGG\-invariant so that \(P1\) holds\. Smoothness ofΦθ\\Phi\_\{\\theta\}andfθf\_\{\\theta\}, including at the cutoff boundary, yields \(P4\)\. The quality of the MLIP hinges on the expressiveness of the representation\.

Two architectural families instantiate this ansatz\. For*linear*models such as ACE\[[14](https://arxiv.org/html/2609.00528#bib.bib14),[15](https://arxiv.org/html/2609.00528#bib.bib15)\]and MTP\[[32](https://arxiv.org/html/2609.00528#bib.bib32)\],Φθ\\Phi\_\{\\theta\}provides a complete polynomial basis\{Bα\}\\\{B\_\{\\alpha\}\\\}andfθ=∑αcα​Bαf\_\{\\theta\}=\\sum\_\{\\alpha\}c\_\{\\alpha\}B\_\{\\alpha\}is a linear combination\. Universal approximation then follows from the density of the basis\. For*nonlinear*models,fθf\_\{\\theta\}is an MLP and the relevant criterion is not basis density but*injectivity*ofΦθ\\Phi\_\{\\theta\}on symmetry orbits\.[section2\.3](https://arxiv.org/html/2609.00528#S2.SS3)describes specific GNN\-based realizations ofΦθ\\Phi\_\{\\theta\}\.

Figure 1:Graph structures for atomistic modeling\. \(a\) A graph with 2\-body edges, in which each edgeej​ie\_\{ji\}connects a pair of atoms and carries the interatomic distance\. \(b\) A hypergraph with 3\-body hyperedges, in which each hyperedge𝒂i​j​k\\bm\{a\}\_\{ijk\}\(shaded triangle\) connects a triplet\(j,i,k\)\(j,i,k\)and carries the distances and bond angle\. The three hyperedges are shown in orange, blue, and green\. Dashed lines indicate the outer edge of each triangle\. \(c\) The line graph transformℒ\\mathcal\{L\}, under which each edge in the atom graphGGbecomes a node \(square\) inℒ⁡\(G\)\\mathcal\{L\}\(G\), and two nodes inℒ⁡\(G\)\\mathcal\{L\}\(G\)are connected if the corresponding edges share an atom\. The edges inℒ⁡\(G\)\\mathcal\{L\}\(G\)use the same colors as the hyperedges in \(b\), showing the correspondence between 3\-body hyperedges and line graph edges\. ALIGNN, CHGNet, and DPA3 use this two\-graph structure to implement 3\-body message passing\.
### 2\.3Graph neural networks for atomistic systems

Fix a center atomjjwith neighbor set𝒩\(j\)=\{k:k≠j,\|𝒓k−𝒓j\|<rc\}\\mathcal\{N\}\(j\)=\\\{k:k\\neq j,\\;\|\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\}\|<r\_\{c\}\\\}, wherercr\_\{c\}is the per\-layer cutoff\. Each atomkkcarries a node feature vector𝒏k∈ℝd\\bm\{n\}\_\{k\}\\in\\mathbb\{R\}^\{d\}\.

An atomic system is naturally represented as a*graph*, see[fig\.1](https://arxiv.org/html/2609.00528#S2.F1)\(a\), in which atoms are nodes and pairs within cutoffrcr\_\{c\}are connected by edges\. A GNN updates node features by*message passing*\. At each layer, every nodejjcollects information from its neighbors, aggregates it, and updates its own feature\. A single message\-passing layer has the general form

𝒏j\(l\+1\)=ψ⁡\(𝒏j\(l\),⨁k∈𝒩⁡\(j\)sa​\(dj​k\)​ϕ​\(𝒏j\(l\),𝒏k\(l\),dj​k\)\),\\bm\{n\}\_\{j\}^\{\(l\+1\)\}=\\psi\\Bigl\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\;\\bigoplus\_\{k\\in\\mathcal\{N\}\(j\)\}s\_\{a\}\(d\_\{jk\}\)\\,\\phi\\bigl\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\,\\bm\{n\}\_\{k\}^\{\(l\)\},\\,d\_\{jk\}\\bigr\)\\Bigr\),\(15\)whereϕ\\phiis a message function,⨁\\bigoplusis a permutation\-invariant aggregation, typically a summation,ψ\\psiis an update function, andsa:\[0,rc\]→\[0,1\]s\_\{a\}\\colon\[0,r\_\{c\}\]\\to\[0,1\]is a smooth switch function, positive on\[0,rc\)\[0,r\_\{c\}\)withsa​\(rc\)=sa′​\(rc\)=0s\_\{a\}\(r\_\{c\}\)=s\_\{a\}^\{\\prime\}\(r\_\{c\}\)=0, so that a neighbor’s contribution leaves the energy and the forces continuous as it crosses the cutoff boundary\. Bothϕ\\phiandψ\\psiare parametrized by MLPs\. The representation must be invariant under bothO⁡\(3\)O\(3\)andΓ\\Gamma\. In \([15](https://arxiv.org/html/2609.00528#S2.E15)\),O⁡\(3\)O\(3\)\-invariance is achieved by using only the scalar distancedj​kd\_\{jk\}as geometric input, andΓ\\Gamma\-invariance by the permutation\-invariant aggregation\.

In the context of MLIPs, GNNs serve as representations\. The output feature𝒏j\(L\)\\bm\{n\}\_\{j\}^\{\(L\)\}afterLLlayers serves as the representation, writtenΦ\(L\)\\Phi^\{\(L\)\}\. The key advantage of stacking layers is that the receptive field grows\. AfterLLlayers, the feature of atomjjdepends on all atoms reachable byLLhops, each shorter thanrcr\_\{c\}\. This allows the model to capture interactions at rangeRc\>rcR\_\{c\}\>r\_\{c\}without using a large single\-layer cutoff, which would be computationally expensive\.

However, the framework \([15](https://arxiv.org/html/2609.00528#S2.E15)\) uses only pairwise geometric information: the scalar distancedj​kd\_\{jk\}along each edge\. Pozdnyakov et al\.\[[29](https://arxiv.org/html/2609.00528#bib.bib29)\]showed that such two\-body representations are incomplete, and that even adding three\-body invariant correlations does not fully resolve the problem, since certain configurations with identical distance and angle distributions are geometrically distinct\.

### 2\.4Hypergraph neural networks

A*hypergraph*, see[fig\.1](https://arxiv.org/html/2609.00528#S2.F1)\(b\), generalizes a graph by allowing*hyperedges*that connect more than two nodes simultaneously\. An HGNN passes messages along such hyperedges\. In the architecture considered here, each hyperedge contains a center atomjjand two of its neighborsiiandkk\. It carries the 3\-body information, namely the bond\-angle cosinecos⁡θi​j​k\\cos\\theta\_\{ijk\}in addition to the distancesdj​id\_\{ji\}anddj​kd\_\{jk\}\. The message\-passing update aggregates over all hyperedges incident to a given node, analogous to \([15](https://arxiv.org/html/2609.00528#S2.E15)\) but with each message depending on a triplet rather than a pair\. This gives the HGNN access to angular information that standard two\-body GNNs lack\. Unlike the GNN framework of[section2\.3](https://arxiv.org/html/2609.00528#S2.SS3), which maintains only node and edge features, the HGNN computes three\-body angle features at each layer\. However, only node features are carried across layers, and the angle features are recomputed from scratch at each layer\. The HGNN layer comprises an angle\-feature calculation and an aggregation operation, defined as follows\.

#### Angle features

For each ordered pairi,k∈𝒩⁡\(j\)i,k\\in\\mathcal\{N\}\(j\)withi≠ki\\neq k:

𝒂j;i,k=ρ3​\(𝒏j\(in\),𝒏i\(in\),𝒏k\(in\),dj​i,dj​k,cos⁡θi​j​k\)∈ℝda,\\bm\{a\}\_\{j;\\,i,k\}=\\rho\_\{3\}\\Bigl\(\\bm\{n\}\_\{j\}^\{\(\\mathrm\{in\}\)\},\\;\\bm\{n\}\_\{i\}^\{\(\\mathrm\{in\}\)\},\\;\\bm\{n\}\_\{k\}^\{\(\\mathrm\{in\}\)\},\\;d\_\{ji\},\\;d\_\{jk\},\\;\\cos\\theta\_\{ijk\}\\Bigr\)\\in\\mathbb\{R\}^\{d\_\{a\}\},\(16\)whereρ3\\rho\_\{3\}is an MLP anddad\_\{a\}is the angle\-feature dimension\.

#### Aggregation

𝒏j\(out\)=ψ3​\(∑i,k∈𝒩⁡\(j\)i≠ksa​\(dj​i\)​sa​\(dj​k\)​𝒂j;i,k\),\\bm\{n\}\_\{j\}^\{\(\\mathrm\{out\}\)\}=\\psi\_\{3\}\\Bigl\(\\sum\_\{\\begin\{subarray\}\{c\}i,k\\in\\mathcal\{N\}\(j\)\\\\ i\\neq k\\end\{subarray\}\}s\_\{a\}\(d\_\{ji\}\)\\,s\_\{a\}\(d\_\{jk\}\)\\,\\bm\{a\}\_\{j;\\,i,k\}\\Bigr\),\(17\)whereψ3\\psi\_\{3\}is an MLP andsas\_\{a\}is the switch function of \([15](https://arxiv.org/html/2609.00528#S2.E15)\)\.

#### Multi\-layer stacking

AnLL\-layer HGNN stacksLLsuch layers\. The input node features for layer 1 are type embeddings:𝒏k\(0\)=type​\_​embed​\(zk\)\\bm\{n\}\_\{k\}^\{\(0\)\}=\\mathrm\{type\\\_embed\}\(z\_\{k\}\)\. The output of layerllprovides the input for layerl\+1l\+1, in that𝒏k\(l\)≡𝒏k\(out,l\)\\bm\{n\}\_\{k\}^\{\(l\)\}\\equiv\\bm\{n\}\_\{k\}^\{\(\\mathrm\{out\},l\)\}serves as𝒏k\(in\)\\bm\{n\}\_\{k\}^\{\(\\mathrm\{in\}\)\}in \([16](https://arxiv.org/html/2609.00528#S2.E16)\) for the next layer\. The*LL\-layer HGNN representation*Φ\(L\)\\Phi^\{\(L\)\}maps theLL\-hop surroundings of atomjjto𝒏j\(L\)\\bm\{n\}\_\{j\}^\{\(L\)\}\. Its domain is formalized in[definition18](https://arxiv.org/html/2609.00528#Thmtheorem18)\. The total energy isEθ=∑jfθ​\(𝒏j\(L\)\)E\_\{\\theta\}=\\sum\_\{j\}f\_\{\\theta\}\(\\bm\{n\}\_\{j\}^\{\(L\)\}\), wherefθf\_\{\\theta\}is a readout MLP\.

### 2\.5DPA3: GNN on a line graph series

Three architectures used in practice fall within the scope of our results: ALIGNN\[[11](https://arxiv.org/html/2609.00528#bib.bib11)\], CHGNet\[[13](https://arxiv.org/html/2609.00528#bib.bib13)\], and DPA3\[[41](https://arxiv.org/html/2609.00528#bib.bib41)\]\. All three maintain node, edge, and angle features and update them in the order angle→\\toedge→\\tonode\. They differ, however, in the specific form of the message functions and in how angular information is propagated to the nodes\. Among these differences, the last one is particularly consequential\. As we show in[section5](https://arxiv.org/html/2609.00528#S5), it determines which of the three architectures inherits universal approximation\. Specifically, DPA3 and CHGNet build their messages from MLPs, whereas ALIGNN relies on single gated linear layers within its edge\-gated convolutions\. We present DPA3 in detail here, as it contains the core ingredients\. CHGNet and ALIGNN are described in[sections2](https://arxiv.org/html/2609.00528#S2a)and[3](https://arxiv.org/html/2609.00528#S3a)of the supplementary materials, respectively\.

DPA3\[[41](https://arxiv.org/html/2609.00528#bib.bib41)\]adopts the same angle→\\toedge→\\tonode aggregation pattern as ALIGNN and CHGNet, formalizing it through a*line graph series*\(LiGS\)\. Starting from the atomic graphG\(1\)≡GG^\{\(1\)\}\\equiv G, one recursively applies the line graph transform to obtain a seriesG\(1\),G\(2\),G\(3\),…G^\{\(1\)\},G^\{\(2\)\},G^\{\(3\)\},\\ldots, whereG\(k\)=ℒ⁡\(G\(k−1\)\)G^\{\(k\)\}=\\mathcal\{L\}\(G^\{\(k\-1\)\}\)\. In particular,G\(2\)=ℒ⁡\(G\)G^\{\(2\)\}=\\mathcal\{L\}\(G\)\. The vertices and edges of successive graphs correspond to progressively higher\-order geometric entities:G\(1\)G^\{\(1\)\}has atoms/bonds,G\(2\)G^\{\(2\)\}has bonds/angles,G\(3\)G^\{\(3\)\}has angles/dihedrals, and so forth\. The LiGS thus naturally extends to arbitrary orderKK\. In practice, DPA3 usesK=2K=2, limiting itself to 3\-body angular features, althoughK=3K=3with 4\-body dihedral features has also been explored in\[[41](https://arxiv.org/html/2609.00528#bib.bib41)\]\.

DPA3 uses*directed*edges, so each bond\{j,i\}\\\{j,i\\\}gives rise to two directed edges\(j,i\)\(j,i\)and\(i,j\)\(i,j\), which carry independent features\. DPA3 maintains three feature types: node features𝒏j\(l\)∈ℝd\\bm\{n\}\_\{j\}^\{\(l\)\}\\in\\mathbb\{R\}^\{d\}, edge features𝒆j​i\(l\)∈ℝd\\bm\{e\}\_\{ji\}^\{\(l\)\}\\in\\mathbb\{R\}^\{d\}\(which serve as vertex features of the graphG\(2\)G^\{\(2\)\}\), and angle features𝒂j;i,k\(l\)∈ℝd\\bm\{a\}\_\{j;i,k\}^\{\(l\)\}\\in\\mathbb\{R\}^\{d\}\(which serve as edge features of the graphG\(2\)G^\{\(2\)\}\)\. The initial representations are constructed from type embeddings for nodes, distance embeddings for edges, and angle embeddings for angles\.

Each DPA3 layer applies four updates in sequence, each consuming the most recent value of its inputs\.111This ordering is not described in\[[41](https://arxiv.org/html/2609.00528#bib.bib41)\]\. It is implemented in the DeePMD\-kit source as thesequential\_updateoption of the DPA3 descriptor, enabled by settingsequential\_update = True\. The default path instead computes all four updates from the layer\-llfeatures and combines them additively\.

*1: Edge self\-message\.*The edge features first load the node features of both endpoints:

𝒆j​i\(l\+12\)=𝒆j​i\(l\)\+δs\(1\)​ϕs\(1\)​\(𝒏j\(l\),𝒏i\(l\),𝒆j​i\(l\)\)\.\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}=\\bm\{e\}\_\{ji\}^\{\(l\)\}\+\\delta\_\{s\}^\{\(1\)\}\\,\\phi\_\{s\}^\{\(1\)\}\\bigl\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\,\\bm\{n\}\_\{i\}^\{\(l\)\},\\,\\bm\{e\}\_\{ji\}^\{\(l\)\}\\bigr\)\.\(18\)
*2: Angle self\-message\.*The angle features are updated from the two node\-enriched edges that span them:

𝒂j;i,k\(l\+1\)=𝒂j;i,k\(l\)\+δs\(2\)​ϕs\(2\)​\(𝒆j​i\(l\+12\),𝒆j​k\(l\+12\),𝒂j;i,k\(l\)\)\.\\bm\{a\}\_\{j;i,k\}^\{\(l\+1\)\}=\\bm\{a\}\_\{j;i,k\}^\{\(l\)\}\+\\delta\_\{s\}^\{\(2\)\}\\,\\phi\_\{s\}^\{\(2\)\}\\bigl\(\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{e\}\_\{jk\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{a\}\_\{j;i,k\}^\{\(l\)\}\\bigr\)\.\(19\)
*3: Angle→\\toedge\.*Each edge aggregates over the angles centered at its tail atom:

𝒆j​i\(l\+1\)=𝒆j​i\(l\+12\)\+δu\(2\)​ϕu\(2\)​\(∑k∈𝒩\(3\)​\(j\)∖\{i\}wj​i​k​ϕc\(2\)​\(𝒆j​i\(l\+12\),𝒆j​k\(l\+12\),𝒂j;i,k\(l\+1\)\)\),\\bm\{e\}\_\{ji\}^\{\(l\+1\)\}=\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}\+\\delta\_\{u\}^\{\(2\)\}\\,\\phi\_\{u\}^\{\(2\)\}\\Bigl\(\\sum\_\{k\\in\\mathcal\{N\}^\{\(3\)\}\(j\)\\setminus\\\{i\\\}\}w\_\{jik\}\\,\\phi\_\{c\}^\{\(2\)\}\\bigl\(\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{e\}\_\{jk\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{a\}\_\{j;i,k\}^\{\(l\+1\)\}\\bigr\)\\Bigr\),\(20\)where𝒩\(3\)​\(j\)\\mathcal\{N\}^\{\(3\)\}\(j\)denotes the angle\-cutoff neighbors ofjj\(dj​k<rc\(3\)d\_\{jk\}<r\_\{c\}^\{\(3\)\}\),wj​i​kw\_\{jik\}is a switch prefactor, andδu\(2\)\\delta\_\{u\}^\{\(2\)\}is a trainable step size\.

*4: Edge→\\tonode\.*Each node aggregates over its edges:

𝒏j\(l\+1\)=𝒏j\(l\)\+δu\(1\)​ϕu\(1\)​\(∑i∈𝒩\(2\)​\(j\)wj​i​ϕc\(1\)​\(𝒏j\(l\),𝒏i\(l\),𝒆j​i\(l\+1\)\)\),\\bm\{n\}\_\{j\}^\{\(l\+1\)\}=\\bm\{n\}\_\{j\}^\{\(l\)\}\+\\delta\_\{u\}^\{\(1\)\}\\,\\phi\_\{u\}^\{\(1\)\}\\Bigl\(\\sum\_\{i\\in\\mathcal\{N\}^\{\(2\)\}\(j\)\}w\_\{ji\}\\,\\phi\_\{c\}^\{\(1\)\}\\bigl\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\,\\bm\{n\}\_\{i\}^\{\(l\)\},\\,\\bm\{e\}\_\{ji\}^\{\(l\+1\)\}\\bigr\)\\Bigr\),\(21\)where𝒩\(2\)​\(j\)\\mathcal\{N\}^\{\(2\)\}\(j\)denotes the edge\-cutoff neighbors ofjj\(dj​i<rc\(2\)d\_\{ji\}<r\_\{c\}^\{\(2\)\}\)\. DPA3 uses two cutoff radii:rc\(3\)≤rc\(2\)r\_\{c\}^\{\(3\)\}\\leq r\_\{c\}^\{\(2\)\}\. Allϕ\\phifunctions \(ϕs\(1\)\\phi\_\{s\}^\{\(1\)\},ϕs\(2\)\\phi\_\{s\}^\{\(2\)\},ϕc\(2\)\\phi\_\{c\}^\{\(2\)\},ϕu\(2\)\\phi\_\{u\}^\{\(2\)\},ϕc\(1\)\\phi\_\{c\}^\{\(1\)\},ϕu\(1\)\\phi\_\{u\}^\{\(1\)\}\) are MLPs\. The prefactors are products of switch\-function values,wj​i​k=sa\(3\)​\(dj​i\)​sa\(3\)​\(dj​k\)w\_\{jik\}=s\_\{a\}^\{\(3\)\}\(d\_\{ji\}\)\\,s\_\{a\}^\{\(3\)\}\(d\_\{jk\}\)andwj​i=sa\(2\)​\(dj​i\)w\_\{ji\}=s\_\{a\}^\{\(2\)\}\(d\_\{ji\}\), wheresa\(2\)s\_\{a\}^\{\(2\)\}andsa\(3\)s\_\{a\}^\{\(3\)\}are switch functions as in \([15](https://arxiv.org/html/2609.00528#S2.E15)\) for the cutoffsrc\(2\)r\_\{c\}^\{\(2\)\}andrc\(3\)r\_\{c\}^\{\(3\)\}\. Theδ\\deltastep sizes are trainable scalars\. The total energy isEθ=∑jMLP⁡\(𝒏j\(L\)\)E\_\{\\theta\}=\\sum\_\{j\}\\mathrm\{MLP\}\(\\bm\{n\}\_\{j\}^\{\(L\)\}\)\.

As noted above, the LiGS extends to arbitrary orderKK\. Empirically, the DPA3 paper found thatK=2K=2yields optimal performance, whileK=3K=3offers no further improvement and may even degrade accuracy\. Our completeness theory provides an explanation for this observation\. On generic configurations, 3\-body information already suffices for universal approximation, so higher body\-order features are theoretically unnecessary\.

## 3Representation Completeness

### 3\.1Completeness and universal approximation

As described in[section2\.2](https://arxiv.org/html/2609.00528#S2.SS2), most MLIP architectures decompose the local energy asε⁡\(𝒟\)=f⁡\(Φ⁡\(𝒟\)\)\\varepsilon\(\\mathcal\{D\}\)=f\(\\Phi\(\\mathcal\{D\}\)\), whereΦ:ℰ\(Rc\)→ℝd\\Phi:\\mathcal\{E\}^\{\(R\_\{c\}\)\}\\to\\mathbb\{R\}^\{d\}is aGG\-invariant representation andf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}is a readout network\.

###### Definition 7\(Completeness\)\.

A*complete*representation at cutoffrris a continuousGG\-invariant mapΦ:ℰ\(r\)→ℝd\\Phi:\\mathcal\{E\}^\{\(r\)\}\\to\\mathbb\{R\}^\{d\}that separates orbits: for any two local environments𝒟1,𝒟2∈ℰ\(r\)\\mathcal\{D\}\_\{1\},\\mathcal\{D\}\_\{2\}\\in\\mathcal\{E\}^\{\(r\)\},

Φ⁡\(𝒟1\)=Φ⁡\(𝒟2\)implies𝒟1∼𝒟2\.\\Phi\(\\mathcal\{D\}\_\{1\}\)=\\Phi\(\\mathcal\{D\}\_\{2\}\)\\quad\\text\{implies\}\\quad\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}\.\(22\)

Since∼\\simincludes both rotations/reflections and type\-preserving permutations, completeness requires separating orbits under the full symmetry groupGG\.

###### Definition 8\(Universal approximation\)\.

A parametrized family\{εθ\}\\\{\\varepsilon\_\{\\theta\}\\\}of continuous functions onℰ\(Rc\)\\mathcal\{E\}^\{\(R\_\{c\}\)\}is a*universal approximator*forℱRc0\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}on a subsetA⊆ℰ\(Rc\)A\\subseteq\\mathcal\{E\}^\{\(R\_\{c\}\)\}if for everyε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}, every compactK⊆AK\\subseteq A, and everyδ\>0\\delta\>0there is a parameterθ\\thetawith‖εθ−ε‖C0​\(K\)<δ\\\|\\varepsilon\_\{\\theta\}\-\\varepsilon\\\|\_\{C^\{0\}\(K\)\}<\\delta\. We omit the subset whenA=ℰ\(Rc\)A=\\mathcal\{E\}^\{\(R\_\{c\}\)\}\.

Completeness constrains the representation alone, whereas universal approximation constrains the architecturefθ∘Φf\_\{\\theta\}\\circ\\Phias a whole\. That an incomplete representation caps the achievable accuracy is immediate\. The substance of the next result is the converse, that completeness suffices\.

###### Theorem 9\(Completeness⇔\\LeftrightarrowUAT\)\.

LetΦ:ℰ\(Rc\)→ℝd\\Phi:\\mathcal\{E\}^\{\(R\_\{c\}\)\}\\to\\mathbb\{R\}^\{d\}be a continuousGG\-invariant representation, and let\{fθ\}\\\{f\_\{\\theta\}\\\}be a family of MLPs, which by\[[12](https://arxiv.org/html/2609.00528#bib.bib12),[19](https://arxiv.org/html/2609.00528#bib.bib19)\]approximate every continuous function onℝd\\mathbb\{R\}^\{d\}uniformly on compact sets\. Then the architectureεθ=fθ∘Φ\\varepsilon\_\{\\theta\}=f\_\{\\theta\}\\circ\\Phiis a universal approximator forℱRc0\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}in the sense of[definition8](https://arxiv.org/html/2609.00528#Thmtheorem8)if and only ifΦ\\Phiis complete\.

Continuity ofΦ\\Phiand compactness of test sets are understood with respect to the Euclidean metric on each stratum\.

###### Proof\.

*Completeness⇒\\RightarrowUAT\.*Letε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}andK⊂ℰ\(Rc\)K\\subset\\mathcal\{E\}^\{\(R\_\{c\}\)\}be compact\. Sinceε\\varepsilonisGG\-invariant andΦ\\Phiis complete,Φ⁡\(𝒟1\)=Φ⁡\(𝒟2\)\\Phi\(\\mathcal\{D\}\_\{1\}\)=\\Phi\(\\mathcal\{D\}\_\{2\}\)forces𝒟1∼𝒟2\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}and henceε⁡\(𝒟1\)=ε⁡\(𝒟2\)\\varepsilon\(\\mathcal\{D\}\_\{1\}\)=\\varepsilon\(\\mathcal\{D\}\_\{2\}\), sof⁡\(Φ⁡\(𝒟\)\)=ε⁡\(𝒟\)f\(\\Phi\(\\mathcal\{D\}\)\)=\\varepsilon\(\\mathcal\{D\}\)defines a functionffon the compact setΦ⁡\(K\)\\Phi\(K\)\. The restrictionΦ\|K\\Phi\|\_\{K\}is a continuous surjection from a compact space onto a Hausdorff one, hence a quotient map\. The compositionf∘Φ\|K=ε\|Kf\\circ\\Phi\|\_\{K\}=\\varepsilon\|\_\{K\}is continuous, soffis continuous\. By the Tietze extension theorem,ffextends to a continuous function onℝd\\mathbb\{R\}^\{d\}, and by MLP universality, for anyδ\>0\\delta\>0, there existsfθf\_\{\\theta\}that approximates that extension onΦ⁡\(K\)\\Phi\(K\)with‖fθ∘Φ−ε‖C0​\(K\)<δ\\\|f\_\{\\theta\}\\circ\\Phi\-\\varepsilon\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

*Converse \(incompleteness⇒\\Rightarrownot UAT\)\.*IfΦ\\Phiis incomplete, there exist𝒟1≁𝒟2\\mathcal\{D\}\_\{1\}\\not\\sim\\mathcal\{D\}\_\{2\}withΦ⁡\(𝒟1\)=Φ⁡\(𝒟2\)\\Phi\(\\mathcal\{D\}\_\{1\}\)=\\Phi\(\\mathcal\{D\}\_\{2\}\)\. For any readout networkff,f⁡\(Φ⁡\(𝒟1\)\)=f⁡\(Φ⁡\(𝒟2\)\)f\(\\Phi\(\\mathcal\{D\}\_\{1\}\)\)=f\(\\Phi\(\\mathcal\{D\}\_\{2\}\)\), so the architecture cannot distinguish𝒟1\\mathcal\{D\}\_\{1\}and𝒟2\\mathcal\{D\}\_\{2\}\. It remains to exhibit a targetε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}withε⁡\(𝒟1\)≠ε⁡\(𝒟2\)\\varepsilon\(\\mathcal\{D\}\_\{1\}\)\\neq\\varepsilon\(\\mathcal\{D\}\_\{2\}\)\.

LetΦ∗\\Phi^\{\*\}be a complete representation whose components lie inℱRc0\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}\. Such a representation is constructed in[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13)via the switched Gram matrix, without relying on the present theorem, so no circularity arises\. Setε⁡\(𝒟\)=min⁡\{1,‖Φ∗​\(𝒟\)−Φ∗​\(𝒟2\)‖\}\\varepsilon\(\\mathcal\{D\}\)=\\min\\bigl\\\{1,\\\|\\Phi^\{\*\}\(\\mathcal\{D\}\)\-\\Phi^\{\*\}\(\\mathcal\{D\}\_\{2\}\)\\\|\\bigr\\\}\. Thenε\\varepsilonisGG\-invariant and continuous becauseΦ∗\\Phi^\{\*\}is, and it satisfies the cross\-stratum matching of[definition6](https://arxiv.org/html/2609.00528#Thmtheorem6)because the components ofΦ∗\\Phi^\{\*\}do, soε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}\. Moreoverε⁡\(𝒟2\)=0\\varepsilon\(\\mathcal\{D\}\_\{2\}\)=0, whileε⁡\(𝒟1\)\>0\\varepsilon\(\\mathcal\{D\}\_\{1\}\)\>0, because completeness and𝒟1≁𝒟2\\mathcal\{D\}\_\{1\}\\not\\sim\\mathcal\{D\}\_\{2\}giveΦ∗​\(𝒟1\)≠Φ∗​\(𝒟2\)\\Phi^\{\*\}\(\\mathcal\{D\}\_\{1\}\)\\neq\\Phi^\{\*\}\(\\mathcal\{D\}\_\{2\}\)\.

TakingK=\{𝒟1,𝒟2\}K=\\\{\\mathcal\{D\}\_\{1\},\\mathcal\{D\}\_\{2\}\\\}\(compact\) andδ=ε⁡\(𝒟1\)/2\\delta=\\varepsilon\(\\mathcal\{D\}\_\{1\}\)/2, every modelfθ∘Φf\_\{\\theta\}\\circ\\Phisatisfies‖fθ∘Φ−ε‖C0​\(K\)≥ε⁡\(𝒟1\)/2\>0\\\|f\_\{\\theta\}\\circ\\Phi\-\\varepsilon\\\|\_\{C^\{0\}\(K\)\}\\geq\\varepsilon\(\\mathcal\{D\}\_\{1\}\)/2\>0, so the architecture is not a universal approximator\.

### 3\.2Generic configurations and Gram matrix reconstruction

This subsection is purely information\-theoretic\. We show that complete representations exist, without reference to any neural network architecture\. Consider a local environment𝒟\\mathcal\{D\}with center typez0z\_\{0\}andnnneighbors at displacementsΔ​𝒓1,…,Δ​𝒓n∈Brc∖\{0\}\\Delta\\bm\{r\}\_\{1\},\\ldots,\\Delta\\bm\{r\}\_\{n\}\\in B\_\{r\_\{c\}\}\\setminus\\\{0\\\}with typesz1,…,znz\_\{1\},\\ldots,z\_\{n\}\.

###### Definition 10\(Generic configuration\)\.

A local environment𝒟∈ℰ\(r\)\\mathcal\{D\}\\in\\mathcal\{E\}^\{\(r\)\}is*generic*if no two same\-type neighbors are equidistant from the center, that is, ifzi=zkz\_\{i\}=z\_\{k\}andi≠ki\\neq kimply\|Δ​𝐫i\|≠\|Δ​𝐫k\|\|\\Delta\\bm\{r\}\_\{i\}\|\\neq\|\\Delta\\bm\{r\}\_\{k\}\|\. We writeℰ∗⊂ℰ\(r\)\\mathcal\{E\}^\{\*\}\\subset\\mathcal\{E\}^\{\(r\)\}for the set of generic environments\.

On each stratum, the complement ofℰ∗\\mathcal\{E\}^\{\*\}is the union, over the finitely many same\-type pairsi≠ki\\neq k, of the zero sets of the nonzero polynomials\|Δ​𝒓i\|2−\|Δ​𝒓k\|2\|\\Delta\\bm\{r\}\_\{i\}\|^\{2\}\-\|\\Delta\\bm\{r\}\_\{k\}\|^\{2\}\. The zero set of a polynomial that does not vanish identically is closed and has zero Lebesgue measure in\(ℝ3\)n\(\\mathbb\{R\}^\{3\}\)^\{n\}\. The complement ofℰ∗\\mathcal\{E\}^\{\*\}is a finite union of such sets, so it is closed and has zero Lebesgue measure\. Henceℰ∗\\mathcal\{E\}^\{\*\}is open and dense inℰ\(r\)\\mathcal\{E\}^\{\(r\)\}\.

A complete representation has two requirements, invariance underGGand separation of distinctGG\-orbits\. The Gram matrix meets the first for theO⁡\(3\)O\(3\)factor, and the second amounts to the converse, that it determines the displacement vectors up to a single orthogonal transformation\. That is the content of[lemma11](https://arxiv.org/html/2609.00528#Thmtheorem11)\. The permutation factorΓ\\Gammais handled separately in[lemma12](https://arxiv.org/html/2609.00528#Thmtheorem12)\.

###### Lemma 11\(Gram matrix reconstruction\)\.

If𝐀,𝐁∈ℝn×3\\bm\{A\},\\bm\{B\}\\in\\mathbb\{R\}^\{n\\times 3\}satisfy𝐀​𝐀T=𝐁​𝐁T\\bm\{A\}\\bm\{A\}^\{T\}=\\bm\{B\}\\bm\{B\}^\{T\}, then there existsR∈O⁡\(3\)R\\in O\(3\)such that𝐁=𝐀​R\\bm\{B\}=\\bm\{A\}R\.

###### Proof\.

It follows from the thin SVD and𝑨​𝑨T=𝑩​𝑩T\\bm\{A\}\\bm\{A\}^\{T\}=\\bm\{B\}\\bm\{B\}^\{T\}that

𝑨=P​U𝑩=P​V\\bm\{A\}=PU\\qquad\\bm\{B\}=PVwithP=\(𝑨​𝑨T\)1/2P=\(\\bm\{A\}\\bm\{A\}^\{T\}\)^\{1/2\}andUT​U=VT​V=I3U^\{T\}U=V^\{T\}V=I\_\{3\}\. Whenrank​𝑨<3\\mathrm\{rank\}\\,\\bm\{A\}<3, the factors are not unique\. The column spacecol​U\\text\{col\}\\,UofUUis a 3\-dimensional subspace containingcol​𝑨\\text\{col\}\\,\\bm\{A\}, determined by an arbitrary orthonormal completion in the thin SVD, and likewise forVV\. Sincecol​𝑨=col​P=col​𝑩\\text\{col\}\\,\\bm\{A\}=\\text\{col\}\\,P=\\text\{col\}\\,\\bm\{B\}, we choose the two completions so thatcol​U=col​V\\text\{col\}\\,U=\\text\{col\}\\,V\. We claim that

Using the fact that for anyx∈ℝ3x\\in\\mathbb\{R\}^\{3\}andy∈ℝny\\in\\mathbb\{R\}^\{n\},

\(\(I−U​UT\)​y,U​x\)=\(UT​y,x\)−\(UT​U​UT​y,x\)=\(UT​y,x\)−\(UT​y,x\)=0\.\(\(I\-UU^\{T\}\)y,Ux\)=\(U^\{T\}y,x\)\-\(U^\{T\}UU^\{T\}y,x\)=\(U^\{T\}y,x\)\-\(U^\{T\}y,x\)=0\.We conclude thatU​UTUU^\{T\}is the orthogonal projection ontocol​U\\text\{col\}\\,U\. Similarly,V​VTVV^\{T\}is the orthogonal projection ontocol​V\\text\{col\}\\,V\. The two column spaces coincide by our choice, which givesU​UT=V​VTUU^\{T\}=VV^\{T\}\.

LetR=UT​VR=U^\{T\}V\. It is straightforward to verify thatR∈O⁡\(3\)R\\in O\(3\)and𝑩=𝑨​R\\bm\{B\}=\\bm\{A\}R\.

### 3\.3Existence of a complete representation

###### Lemma 12\(Gram matrix orbit characterization\)\.

On a stratumℰz0,𝐳\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}, the Gram matrix mapΨ⁡\(𝒟\)=G\\Psi\(\\mathcal\{D\}\)=GisO⁡\(3\)O\(3\)\-invariant andΓ\\Gamma\-equivariant, where the permutation groupΓ=Sn1×⋯×SnT\\Gamma=S\_\{n\_\{1\}\}\\times\\cdots\\times S\_\{n\_\{T\}\}acts on the indices via\(π⋅G\)i​k=Gπ−1​\(i\),π−1​\(k\)\(\\pi\\cdot G\)\_\{ik\}=G\_\{\\pi^\{\-1\}\(i\),\\pi^\{\-1\}\(k\)\}\.

The mapΨ\\Psidetermines the local environment up toO⁡\(3\)O\(3\)\-equivalence\. More precisely, two environments𝒟1,𝒟2\\mathcal\{D\}\_\{1\},\\mathcal\{D\}\_\{2\}in the same stratum satisfy𝒟1∼𝒟2\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}if and only if

∃π∈Γ:Gπ⁡\(i\),π⁡\(k\)\(2\)=Gi​k\(1\)∀i,k\.\\exists\\,\\pi\\in\\Gamma:\\;G^\{\(2\)\}\_\{\\pi\(i\),\\pi\(k\)\}=G^\{\(1\)\}\_\{ik\}\\qquad\\forall\\,i,k\.\(23\)

###### Proof\.

If𝒟1∼𝒟2\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}viaΔ​𝒓π⁡\(i\)\(2\)=R​Δ​𝒓i\(1\)\\Delta\\bm\{r\}\_\{\\pi\(i\)\}^\{\(2\)\}=R\\Delta\\bm\{r\}\_\{i\}^\{\(1\)\}, thenGπ⁡\(i\),π⁡\(k\)\(2\)=R​Δ​𝒓i\(1\)⋅R​Δ​𝒓k\(1\)=Gi​k\(1\)G^\{\(2\)\}\_\{\\pi\(i\),\\pi\(k\)\}=R\\Delta\\bm\{r\}\_\{i\}^\{\(1\)\}\\cdot R\\Delta\\bm\{r\}\_\{k\}^\{\(1\)\}=G^\{\(1\)\}\_\{ik\}\. Suppose a type\-preservingπ\\pigivesGπ⁡\(i\),π⁡\(k\)\(2\)=Gi​k\(1\)G^\{\(2\)\}\_\{\\pi\(i\),\\pi\(k\)\}=G^\{\(1\)\}\_\{ik\}for alli,ki,k\. Let𝑿\(1\)\\bm\{X\}^\{\(1\)\}have rowsΔ​𝒓i\(1\)\\Delta\\bm\{r\}\_\{i\}^\{\(1\)\}and𝒀\\bm\{Y\}have rowsΔ​𝒓π⁡\(i\)\(2\)\\Delta\\bm\{r\}\_\{\\pi\(i\)\}^\{\(2\)\}\. Then𝒀​𝒀T=𝑿\(1\)​\(𝑿\(1\)\)T\\bm\{Y\}\\bm\{Y\}^\{T\}=\\bm\{X\}^\{\(1\)\}\(\\bm\{X\}^\{\(1\)\}\)^\{T\}\. By[lemma11](https://arxiv.org/html/2609.00528#Thmtheorem11), there existsR∈O⁡\(3\)R\\in O\(3\)such that𝒀=𝑿\(1\)​R\\bm\{Y\}=\\bm\{X\}^\{\(1\)\}R, hence𝒟1∼𝒟2\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}\.

[lemma12](https://arxiv.org/html/2609.00528#Thmtheorem12)reduces completeness to separating theΓ\\Gamma\-orbits of Gram matrices\. SinceΓ\\Gammais finite, invariant theory supplies finitely many polynomials that do so on each stratum\.

Physical applications require continuity with respect to the atomic coordinates, in particular when an atom crosses the boundary at cutoff radiusRRand the environment passes from one stratum to another\. A representation built from the Gram matrix alone is discontinuous at such a crossing, because the entries involving a departing neighbor do not decay to zero as\|Δ​𝒓k\|→R\|\\Delta\\bm\{r\}\_\{k\}\|\\to R\.

To restore continuity, we damp each displacement with a switch function that vanishes smoothly at the cutoff radius\. Lets:\[0,R\]→\[0,1\]s\\colon\[0,R\]\\to\[0,1\]be a smooth function, positive on\[0,R\)\[0,R\), satisfyings⁡\(R\)=s′​\(R\)=0s\(R\)=s^\{\\prime\}\(R\)=0, and such thatr↦s⁡\(r\)​rr\\mapsto s\(r\)\\,ris injective on\(0,R\)\(0,R\)\. We then define the*switched Gram matrix*by replacing each displacement vector by its damping counterpartΔ​𝒓~k=s⁡\(\|Δ​𝒓k\|\)​Δ​𝒓k\\Delta\\tilde\{\\bm\{r\}\}\_\{k\}=s\(\|\\Delta\\bm\{r\}\_\{k\}\|\)\\,\\Delta\\bm\{r\}\_\{k\}:

G~i​k=Δ​𝒓~i⋅Δ​𝒓~k=s⁡\(di\)​s​\(dk\)​Gi​k\.\\tilde\{G\}\_\{ik\}=\\Delta\\tilde\{\\bm\{r\}\}\_\{i\}\\cdot\\Delta\\tilde\{\\bm\{r\}\}\_\{k\}=s\(d\_\{i\}\)\\,s\(d\_\{k\}\)\\,G\_\{ik\}\.The switched Gram matrix glues the strata together in the proof of the following proposition\.

###### Proposition 13\(Existence of a complete representation\)\.

For any cutoffR\>0R\>0there exists a complete representationΦ∗:ℰ\(R\)→ℝd\\Phi^\{\*\}:\\mathcal\{E\}^\{\(R\)\}\\to\\mathbb\{R\}^\{d\}whose components satisfy the cross\-stratum matching of condition 3 in[definition6](https://arxiv.org/html/2609.00528#Thmtheorem6), withRcR\_\{c\}replaced byRR\.

###### Proof\.

*Step 1: a single stratum\.*Work on a fixed stratumℰz0,𝒛\(R\)\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(R\)\}withnnneighbors\. The Gram matrixΨ⁡\(𝒟\)=G\\Psi\(\\mathcal\{D\}\)=Gtakes values in the space ofn×nn\\times nreal symmetric matrices𝒱≅ℝn⁡\(n\+1\)/2\\mathcal\{V\}\\cong\\mathbb\{R\}^\{n\(n\+1\)/2\}\. By[lemma12](https://arxiv.org/html/2609.00528#Thmtheorem12),𝒟1∼𝒟2\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}if and only ifΨ⁡\(𝒟1\)\\Psi\(\\mathcal\{D\}\_\{1\}\)andΨ⁡\(𝒟2\)\\Psi\(\\mathcal\{D\}\_\{2\}\)lie in the sameΓ\\Gamma\-orbit under the actionG↦Pπ​G​PπTG\\mapsto P\_\{\\pi\}GP\_\{\\pi\}^\{T\}\.

SinceΓ\\Gammais a finite group acting on𝒱\\mathcal\{V\}, by Noether’s theorem\[[26](https://arxiv.org/html/2609.00528#bib.bib26)\]the ring ofΓ\\Gamma\-invariant polynomialsℝ​\[𝒱\]Γ\\mathbb\{R\}\[\\mathcal\{V\}\]^\{\\Gamma\}is finitely generated, so there existΓ\\Gamma\-invariant polynomialsp1,…,pd:𝒱→ℝp\_\{1\},\\ldots,p\_\{d\}:\\mathcal\{V\}\\to\\mathbb\{R\}such that everyΓ\\Gamma\-invariant polynomial is a polynomial inp1,…,pdp\_\{1\},\\ldots,p\_\{d\}\. Moreover, these generators separateΓ\\Gamma\-orbits, in the sense that ifpi​\(G\(1\)\)=pi​\(G\(2\)\)p\_\{i\}\(G^\{\(1\)\}\)=p\_\{i\}\(G^\{\(2\)\}\)for alli=1,…,di=1,\\ldots,d, thenG\(1\)G^\{\(1\)\}andG\(2\)G^\{\(2\)\}lie in the sameΓ\\Gamma\-orbit\. Indeed, for anyG\(2\)∉Γ⋅G\(1\)G^\{\(2\)\}\\notin\\Gamma\\cdot G^\{\(1\)\}, the polynomialP⁡\(X\)=∏M∈Γ⋅G\(1\)‖X−M‖2P\(X\)=\\prod\_\{M\\in\\Gamma\\cdot G^\{\(1\)\}\}\\\|X\-M\\\|^\{2\}isΓ\\Gamma\-invariant, vanishes onΓ⋅G\(1\)\\Gamma\\cdot G^\{\(1\)\}, and is strictly positive onΓ⋅G\(2\)\\Gamma\\cdot G^\{\(2\)\}\. SincePPisΓ\\Gamma\-invariant, it is expressible in the generators, so at least one generator differs between the two orbits\.

Define

Φ⁡\(𝒟\)=\(p1​\(Ψ⁡\(𝒟\)\),…,pd​\(Ψ⁡\(𝒟\)\)\)∈ℝd\.\\Phi\(\\mathcal\{D\}\)=\\bigl\(p\_\{1\}\(\\Psi\(\\mathcal\{D\}\)\),\\;\\ldots,\\;p\_\{d\}\(\\Psi\(\\mathcal\{D\}\)\)\\bigr\)\\in\\mathbb\{R\}^\{d\}\.Φ\\Phiis continuous because its components are polynomials in the entries of the continuous mapΨ\\Psi,O⁡\(3\)O\(3\)\-invariant becauseΨ\\Psiis, andΓ\\Gamma\-invariant because eachpip\_\{i\}is, henceGG\-invariant\. IfΦ⁡\(𝒟1\)=Φ⁡\(𝒟2\)\\Phi\(\\mathcal\{D\}\_\{1\}\)=\\Phi\(\\mathcal\{D\}\_\{2\}\), then all generators agree onΨ⁡\(𝒟1\)\\Psi\(\\mathcal\{D\}\_\{1\}\)andΨ⁡\(𝒟2\)\\Psi\(\\mathcal\{D\}\_\{2\}\), soΨ\(𝒟1\)∼ΓΨ\(𝒟2\)\\Psi\(\\mathcal\{D\}\_\{1\}\)\\sim\_\{\\Gamma\}\\Psi\(\\mathcal\{D\}\_\{2\}\), hence𝒟1∼𝒟2\\mathcal\{D\}\_\{1\}\\sim\\mathcal\{D\}\_\{2\}by[lemma12](https://arxiv.org/html/2609.00528#Thmtheorem12)\.

*Step 2: assembly across strata\.*The per\-stratum maps are now glued through the switched Gram matrix\. PadG~\\tilde\{G\}by zeros to a fixed size, reserving for each typettas many index slots as the maximal numberMtM\_\{t\}of type\-ttneighbors, and writeG^​\(𝒟\)\\hat\{G\}\(\\mathcal\{D\}\)for the result\. When a neighbor leaves through the boundary, its switched entries vanish, soG^​\(𝒟\)→G^​\(𝒟−\)\\hat\{G\}\(\\mathcal\{D\}\)\\to\\hat\{G\}\(\\mathcal\{D\}^\{\-\}\)along the transitions of condition 3\. The pair\(z0,G^\)\(z\_\{0\},\\hat\{G\}\)still determines the environment\. The positive diagonal entries reveal the number of neighbors of each type, hence the stratum, and the injectivity ofr↦s⁡\(r\)​rr\\mapsto s\(r\)\\,rrecovers the distances, then the valuess⁡\(di\)\>0s\(d\_\{i\}\)\>0, and finallyGi​k=G^i​k/\(s⁡\(di\)​s​\(dk\)\)G\_\{ik\}=\\hat\{G\}\_\{ik\}/\(s\(d\_\{i\}\)\\,s\(d\_\{k\}\)\), so[lemmas11](https://arxiv.org/html/2609.00528#Thmtheorem11)and[12](https://arxiv.org/html/2609.00528#Thmtheorem12)apply\. Repeating Step 1 for the finite groupΓ¯=∏tSMt\\bar\{\\Gamma\}=\\prod\_\{t\}S\_\{M\_\{t\}\}, which permutes the slots within each type block ofG^\\hat\{G\}, and appending the center typez0z\_\{0\}, yieldsΦ∗\\Phi^\{\*\}\. Its components are polynomials in the entries ofG^\\hat\{G\}, so they inherit the continuity and the boundary matching, and the center type is constant along every condition\-3 transition\.

Φ∗\\Phi^\{\*\}is the representation invoked in the converse direction of[theorem9](https://arxiv.org/html/2609.00528#Thmtheorem9)\.

## 4HGNN Approximation of Complete Representations

We have thus established the existence of complete representations, yet the construction of[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13)is information\-theoretic\. It assemblesΦ∗\\Phi^\{\*\}from invariant polynomials rather than from a trainable network\. We now turn to the question of whether such a representation can be realized in practice\. Specifically, we show that the HGNN architecture is expressive enough to approximate any such complete representation to arbitrary accuracy on compact sets\.

### 4\.1DeepSets universality

We first record a form of the DeepSets universality theorem\[[40](https://arxiv.org/html/2609.00528#bib.bib40),[36](https://arxiv.org/html/2609.00528#bib.bib36)\]that covers vector\-valued elements\. It is the tool underlying every approximation result below\. The HGNN aggregation \([17](https://arxiv.org/html/2609.00528#S2.E17)\) is a sum of per\-triplet messages, and DeepSets universality is what lets such a sum represent an arbitrary symmetric function of the triplet multiset\.

For a compactX⊂ℝdX\\subset\\mathbb\{R\}^\{d\}, letℳM​\(X\)≡\(X≤M\)\\mathcal\{M\}\_\{M\}\(X\)\\equiv\\binom\{X\}\{\\leq M\}222The notation\(X≤M\)\\binom\{X\}\{\\leq M\}follows the convention in which\(Xm\)\\binom\{X\}\{m\}denotes the collection ofmm\-element sub\-multisets ofXX; the subscript≤M\\leq Madmits every size up toMM\.denote the set of all multisets of elements ofXXhaving size at mostMM\. Each fixed\-size stratumℳm​\(X\)=Xm/Sm\\mathcal\{M\}\_\{m\}\(X\)=X^\{m\}/S\_\{m\}carries the quotient metricd⁡\(S,S′\)=minπ∈Sm⁡maxi​‖xi−yπ⁡\(i\)‖d\(S,S^\{\\prime\}\)=\\min\_\{\\pi\\in S\_\{m\}\}\\max\_\{i\}\\\|x\_\{i\}\-y\_\{\\pi\(i\)\}\\\|, whereS=\{x1,…,xm\}S=\\\{x\_\{1\},\\ldots,x\_\{m\}\\\}andS′=\{y1,…,ym\}S^\{\\prime\}=\\\{y\_\{1\},\\ldots,y\_\{m\}\\\}are any ordered representatives, andℳM​\(X\)=⨆m=0Mℳm​\(X\)\\mathcal\{M\}\_\{M\}\(X\)=\\bigsqcup\_\{m=0\}^\{M\}\\mathcal\{M\}\_\{m\}\(X\)has the disjoint\-union topology\. SinceXXis compact,XmX^\{m\}is compact as a finite product of compact sets, andℳm​\(X\)\\mathcal\{M\}\_\{m\}\(X\)is compact as the continuous image ofXmX^\{m\}under the quotient map\.

###### Lemma 14\(DeepSets universality\)\.

LetX⊂ℝdX\\subset\\mathbb\{R\}^\{d\}be compact\. There is an integermmdepending only onMMandddsuch that, for any continuous functionf:ℳM​\(X\)→ℝf\\colon\\mathcal\{M\}\_\{M\}\(X\)\\to\\mathbb\{R\}, there exist continuous functionsϕ:X→ℝm\\phi\\colon X\\to\\mathbb\{R\}^\{m\}andρ:ℝm→ℝ\\rho\\colon\\mathbb\{R\}^\{m\}\\to\\mathbb\{R\}such thatf⁡\(S\)=ρ⁡\(∑x∈Sϕ⁡\(x\)\)f\(S\)=\\rho\\bigl\(\\sum\_\{x\\in S\}\\phi\(x\)\\bigr\)for everyS∈ℳM​\(X\)S\\in\\mathcal\{M\}\_\{M\}\(X\), where∑x∈S\\sum\_\{x\\in S\}denotes the sum over elements ofSSwith multiplicity\. One may takeϕ⁡\(x\)=\(xα\)0≤\|α\|≤M\\phi\(x\)=\(x^\{\\alpha\}\)\_\{0\\leq\|\\alpha\|\\leq M\}, the monomials of degree at mostMM, so thatm=\(M\+dd\)m=\\binom\{M\+d\}\{d\}\.

###### Proof\.

The coordinateα=0\\alpha=0recovers the size ofSS, and for each fixed size the sums∑x∈Sxα\\sum\_\{x\\in S\}x^\{\\alpha\}with1≤\|α\|≤M1\\leq\|\\alpha\|\\leq Mgenerate the polynomial invariants of the symmetric group acting on tuples of vectors\[[39](https://arxiv.org/html/2609.00528#bib.bib39)\], which separate its orbits as in Step 1 of the proof of[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13)\. HenceS↦∑x∈Sϕ⁡\(x\)S\\mapsto\\sum\_\{x\\in S\}\\phi\(x\)is injective, and, being a continuous bijection from a compact space onto its image, a homeomorphism\. Composingffwith the inverse and extending by the Tietze theorem yieldsρ\\rho\.

[lemma14](https://arxiv.org/html/2609.00528#Thmtheorem14)produces continuousϕ\\phiandρ\\rho, whereas the HGNN’s message functions are MLPs\. The following lemma may be viewed as an approximate version of DeepSets universality\.

###### Lemma 15\(DeepSets MLP approximation\)\.

LetX⊂ℝdX\\subset\\mathbb\{R\}^\{d\}be compact,K⊂ℳM​\(X\)K\\subset\\mathcal\{M\}\_\{M\}\(X\)compact, andf∈C0​\(K\)f\\in C^\{0\}\(K\)a continuous symmetric function\. For everyδ\>0\\delta\>0, there exist MLPsϕθ:X→ℝm\\phi\_\{\\theta\}\\colon X\\to\\mathbb\{R\}^\{m\}andρθ:ℝm→ℝ\\rho\_\{\\theta\}\\colon\\mathbb\{R\}^\{m\}\\to\\mathbb\{R\}, withmmas in[lemma14](https://arxiv.org/html/2609.00528#Thmtheorem14), such that

‖f−ρθ∘\(∑iϕθ\)‖C0​\(K\)<δ\.\\bigl\\\|f\-\\rho\_\{\\theta\}\\circ\\textstyle\\bigl\(\\sum\_\{i\}\\phi\_\{\\theta\}\\bigr\)\\bigr\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

###### Proof\.

For a multisetS∈ℳM​\(X\)S\\in\\mathcal\{M\}\_\{M\}\(X\), we writeelts⁡\(S\)⊂X\\mathrm\{elts\}\(S\)\\subset Xfor the set of elements appearing inSS\(forgetting multiplicities\) and∑x∈S\\sum\_\{x\\in S\}for the sum over elements with multiplicity\.

By[lemma14](https://arxiv.org/html/2609.00528#Thmtheorem14),f⁡\(S\)=ρ⁡\(∑x∈Sϕ⁡\(x\)\)f\(S\)=\\rho\\bigl\(\\sum\_\{x\\in S\}\\phi\(x\)\\bigr\)for continuousϕ:X→ℝm\\phi\\colon X\\to\\mathbb\{R\}^\{m\}andρ:ℝm→ℝ\\rho\\colon\\mathbb\{R\}^\{m\}\\to\\mathbb\{R\}\. DefineXK=⋃S∈Kelts⁡\(S\)⊂XX\_\{K\}=\\bigcup\_\{S\\in K\}\\mathrm\{elts\}\(S\)\\subset X, which is compact becauseKKis compact, andΣ=\{∑x∈Sϕ⁡\(x\):S∈K\}⊂ℝm\\Sigma=\\bigl\\\{\\sum\_\{x\\in S\}\\phi\(x\):S\\in K\\bigr\\\}\\subset\\mathbb\{R\}^\{m\}, which is also compact because it is a continuous image ofKK\.

By MLP universality onXKX\_\{K\}, there existsϕθ\\phi\_\{\\theta\}with‖ϕθ−ϕ‖C0​\(XK\)<ϵ\\\|\\phi\_\{\\theta\}\-\\phi\\\|\_\{C^\{0\}\(X\_\{K\}\)\}<\\epsilon\. Then for everyS∈KS\\in K,‖∑x∈Sϕθ​\(x\)−∑x∈Sϕ⁡\(x\)‖≤M​ϵ\\bigl\\\|\\sum\_\{x\\in S\}\\phi\_\{\\theta\}\(x\)\-\\sum\_\{x\\in S\}\\phi\(x\)\\bigr\\\|\\leq M\\epsilon\. LetΣϵ\\Sigma\_\{\\epsilon\}be the closedM​ϵM\\epsilon\-neighborhood ofΣ\\Sigma, which is compact\. By MLP universality onΣϵ\\Sigma\_\{\\epsilon\}, there existsρθ\\rho\_\{\\theta\}with‖ρθ−ρ‖C0​\(Σϵ\)<ϵ\\\|\\rho\_\{\\theta\}\-\\rho\\\|\_\{C^\{0\}\(\\Sigma\_\{\\epsilon\}\)\}<\\epsilon\. Sinceρ\\rhois continuous on the compact setΣϵ\\Sigma\_\{\\epsilon\}, it is uniformly continuous: for anyδ′\>0\\delta^\{\\prime\}\>0there existsη\>0\\eta\>0such that‖s−s′‖<η⇒\|ρ⁡\(s\)−ρ⁡\(s′\)\|<δ′\\\|s\-s^\{\\prime\}\\\|<\\eta\\Rightarrow\|\\rho\(s\)\-\\rho\(s^\{\\prime\}\)\|<\\delta^\{\\prime\}for alls,s′∈Σϵs,s^\{\\prime\}\\in\\Sigma\_\{\\epsilon\}\. Setδ′=δ/2\\delta^\{\\prime\}=\\delta/2, letη\\etabe the corresponding modulus, and chooseϵ\\epsilonsmall enough thatM​ϵ<ηM\\epsilon<\\etaandϵ<δ/2\\epsilon<\\delta/2\. For everyS∈KS\\in K, writings=∑x∈Sϕ⁡\(x\)∈Σs=\\sum\_\{x\\in S\}\\phi\(x\)\\in\\Sigmaands~=∑x∈Sϕθ​\(x\)∈Σϵ\\tilde\{s\}=\\sum\_\{x\\in S\}\\phi\_\{\\theta\}\(x\)\\in\\Sigma\_\{\\epsilon\},

\|ρθ​\(s~\)−ρ⁡\(s\)\|≤\|ρθ​\(s~\)−ρ⁡\(s~\)\|\+\|ρ⁡\(s~\)−ρ⁡\(s\)\|<δ2\+δ2=δ,\|\\rho\_\{\\theta\}\(\\tilde\{s\}\)\-\\rho\(s\)\|\\leq\|\\rho\_\{\\theta\}\(\\tilde\{s\}\)\-\\rho\(\\tilde\{s\}\)\|\+\|\\rho\(\\tilde\{s\}\)\-\\rho\(s\)\|<\\tfrac\{\\delta\}\{2\}\+\\tfrac\{\\delta\}\{2\}=\\delta,where the first term is bounded by the choice ofρθ\\rho\_\{\\theta\}and the second by uniform continuity, since‖s~−s‖≤M​ϵ<η\\\|\\tilde\{s\}\-s\\\|\\leq M\\epsilon<\\eta\. Since the bound holds for everyS∈KS\\in K,‖f−ρθ∘\(∑ϕθ\)‖C0​\(K\)<δ\\\|f\-\\rho\_\{\\theta\}\\circ\(\\sum\\phi\_\{\\theta\}\)\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

### 4\.2Single\-layer approximation

[lemma15](https://arxiv.org/html/2609.00528#Thmtheorem15)shows that a pair of MLPs can approximate any symmetric function of a multiset\. To apply this result to a single HGNN layer, we require that the multiset over which the layer aggregates encodes the same information as the Gram matrix\. The following identities demonstrate that this is indeed the case\. The map from the Gram matrixGGto distances and angles,

di=Gi​i,cosθi​j​k=Gi​kdi​dk\(i≠k\),d\_\{i\}=\\sqrt\{G\_\{ii\}\},\\qquad\\cos\\theta\_\{ijk\}=\\frac\{G\_\{ik\}\}\{d\_\{i\}\\,d\_\{k\}\}\\quad\(i\\neq k\),\(24\)is a bijection with inverse

Gi​k=didkcosθi​j​k,Gi​i=di2\.G\_\{ik\}=d\_\{i\}\\,d\_\{k\}\\,\\cos\\theta\_\{ijk\},\\qquad G\_\{ii\}=d\_\{i\}^\{2\}\.\(25\)
The two descriptions of the local geometry are thus interchangeable\. The computation performed by a single HGNN layer can therefore be equivalently formulated in terms of the Gram matrix\. The following lemma establishes that, under the condition that distinct neighbors carry distinct labels, this layer can approximate any continuousΓ\\Gamma\-invariant function of that Gram matrix and the incoming node features\.

###### Lemma 16\(Architectural expressiveness\)\.

On a stratumℰz0,𝐳\(rc\)\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\_\{c\}\)\}withn≥2n\\geq 2neighbors and permutation groupΓ\\Gamma, consider a single HGNN layer \(\([16](https://arxiv.org/html/2609.00528#S2.E16)\)–\([17](https://arxiv.org/html/2609.00528#S2.E17)\)\)\. Suppose the pairs\(𝐧k\(in\),dj​k\)\(\\bm\{n\}\_\{k\}^\{\(\\mathrm\{in\}\)\},d\_\{jk\}\),k∈𝒩⁡\(j\)k\\in\\mathcal\{N\}\(j\), are pairwise distinct\. Suppose further that the node features of neighbors of different types take values in disjoint sets\. Then, with sufficient MLP capacity and sufficiently large angle\-feature dimensiondad\_\{a\}, the HGNN layer can approximate any continuousΓ\\Gamma\-invariant function of the labeled dataℒ=\(G\(j\),\(𝐧1\(in\),…,𝐧n\(in\)\)\)\\mathcal\{L\}=\(G^\{\(j\)\},\(\\bm\{n\}\_\{1\}^\{\(\\mathrm\{in\}\)\},\\ldots,\\bm\{n\}\_\{n\}^\{\(\\mathrm\{in\}\)\}\)\), whereG\(j\)G^\{\(j\)\}is the Gram matrix \([12](https://arxiv.org/html/2609.00528#S2.E12)\) ofjj’s environment andΓ\\Gammaacts byπ⋅\(G,𝐧\)=\(Pπ​G​PπT,𝐧π−1\)\\pi\\cdot\(G,\\bm\{n\}\)=\(P\_\{\\pi\}GP\_\{\\pi\}^\{T\},\\bm\{n\}\_\{\\pi^\{\-1\}\}\)\.

The argument proceeds in three steps\. First, a single HGNN layer has DeepSets form over the multiset of neighbor triplets, so[lemma15](https://arxiv.org/html/2609.00528#Thmtheorem15)applies\. Second, every continuousΓ\\Gamma\-invariant function of the labeled data factors through that multiset\. Third, the two combine by composition\.

Before delving into the proof, we first fix notation\. Let𝒱=ℝsymn⁡\(n\+1\)/2\\mathcal\{V\}=\\mathbb\{R\}^\{n\(n\+1\)/2\}\_\{\\mathrm\{sym\}\}denote the space of Gram matrices and𝒲=𝒱×\(ℝd\)n\\mathcal\{W\}=\\mathcal\{V\}\\times\(\\mathbb\{R\}^\{d\}\)^\{n\}the space of labeled dataℒ=\(G,\(𝒏1,…,𝒏n\)\)\\mathcal\{L\}=\(G,\(\\bm\{n\}\_\{1\},\\ldots,\\bm\{n\}\_\{n\}\)\)\. The groupΓ\\Gammaacts on𝒲\\mathcal\{W\}byπ⋅\(G,𝒏\)=\(Pπ​G​PπT,𝒏π−1\)\\pi\\cdot\(G,\\bm\{n\}\)=\(P\_\{\\pi\}GP\_\{\\pi\}^\{T\},\\bm\{n\}\_\{\\pi^\{\-1\}\}\)\. LetX=ℝd×ℝd×\(0,rc\)×\(0,rc\)×\[−1,1\]X=\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\times\(0,r\_\{c\}\)\\times\(0,r\_\{c\}\)\\times\[\-1,1\]be the space of a single tuple\(𝒏i,𝒏k,dj​i,dj​k,cos⁡θi​j​k\)\(\\bm\{n\}\_\{i\},\\bm\{n\}\_\{k\},d\_\{ji\},d\_\{jk\},\\cos\\theta\_\{ijk\}\)\. Define the mapμ:𝒲→ℳM⁡\(M−1\)​\(X\)\\mu\\colon\\mathcal\{W\}\\to\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)by

μ⁡\(G,𝒏\)=\{\(𝒏i\(in\),𝒏k\(in\),dj​i,dj​k,cos⁡θi​j​k\)\}i≠k,\\mu\(G,\\bm\{n\}\)=\\bigl\\\{\(\\bm\{n\}\_\{i\}^\{\(\\mathrm\{in\}\)\},\\,\\bm\{n\}\_\{k\}^\{\(\\mathrm\{in\}\)\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\)\\bigr\\\}\_\{i\\neq k\},\(26\)where distances and angles are obtained fromGGvia \([24](https://arxiv.org/html/2609.00528#S4.E24)\) and the index pair\(i,k\)\(i,k\)runs over all ordered pairs of distinct neighbors\. WriteCsym0​\(ℳM⁡\(M−1\)​\(X\)\)C^\{0\}\_\{\\mathrm\{sym\}\}\(\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)\)for the continuous functions on the multiset space,CΓ​\(𝒲\)C\_\{\\Gamma\}\(\\mathcal\{W\}\)for the continuousΓ\\Gamma\-invariant functions on𝒲\\mathcal\{W\}, andμ∗:g↦g∘μ\\mu^\{\*\}\\colon g\\mapsto g\\circ\\mufor the pullback alongμ\\mu\.

###### Proof\.

*Step 1: DeepSets universality\.*The HGNN computes

𝒏j\(out\)=ψ3​\(∑i,k∈𝒩⁡\(j\)i≠kρ~3​\(𝒏j\(in\),𝒏i\(in\),𝒏k\(in\),dj​i,dj​k,cos⁡θi​j​k\)\),\\bm\{n\}\_\{j\}^\{\(\\mathrm\{out\}\)\}=\\psi\_\{3\}\\Bigl\(\\sum\_\{\\begin\{subarray\}\{c\}i,k\\in\\mathcal\{N\}\(j\)\\\\ i\\neq k\\end\{subarray\}\}\\tilde\{\\rho\}\_\{3\}\\bigl\(\\bm\{n\}\_\{j\}^\{\(\\mathrm\{in\}\)\},\\,\\bm\{n\}\_\{i\}^\{\(\\mathrm\{in\}\)\},\\,\\bm\{n\}\_\{k\}^\{\(\\mathrm\{in\}\)\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\\bigr\)\\Bigr\),whereρ~3\\tilde\{\\rho\}\_\{3\}absorbs the switch factorssa​\(dj​i\)​sa​\(dj​k\)s\_\{a\}\(d\_\{ji\}\)\\,s\_\{a\}\(d\_\{jk\}\)intoρ3\\rho\_\{3\}, both being continuous functions of the inputsdj​id\_\{ji\}anddj​kd\_\{jk\}ofρ3\\rho\_\{3\}\. Conversely, sincesas\_\{a\}is positive on\[0,rc\)\[0,r\_\{c\}\), on compact sets the desired summand is realized by lettingρ3\\rho\_\{3\}approximate the quotient by the switch factors\. This has DeepSets formψ3∘∑∘ρ~3\\psi\_\{3\}\\circ\\sum\\circ\\tilde\{\\rho\}\_\{3\}overμ⁡\(ℒ\)\\mu\(\\mathcal\{L\}\), where each summandρ~3\\tilde\{\\rho\}\_\{3\}acts on one element of the multiset\. The multiset has at mostM⁡\(M−1\)M\(M\-1\)elements\. For anyg∈Csym0​\(ℳM⁡\(M−1\)​\(X\)\)g\\in C^\{0\}\_\{\\mathrm\{sym\}\}\(\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)\), compactK′⊂ℳM⁡\(M−1\)​\(X\)K^\{\\prime\}\\subset\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\), andδ\>0\\delta\>0,[lemma15](https://arxiv.org/html/2609.00528#Thmtheorem15), applied with sufficiently largedad\_\{a\}and withXXreplaced by a compact subset containing the tuple values overK′K^\{\\prime\}, yields MLP weights forρ3\\rho\_\{3\}andψ3\\psi\_\{3\}such that‖𝒏j\(out\)−g‖C0​\(K′\)<δ\\\|\\bm\{n\}\_\{j\}^\{\(\\mathrm\{out\}\)\}\-g\\\|\_\{C^\{0\}\(K^\{\\prime\}\)\}<\\delta\.

*Step 2:μ∗\\mu^\{\*\}mapsCsym0​\(ℳM⁡\(M−1\)​\(X\)\)C^\{0\}\_\{\\mathrm\{sym\}\}\(\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)\)intoCΓ​\(𝒲\)C\_\{\\Gamma\}\(\\mathcal\{W\}\)and is surjective\.*

*μ∗\\mu^\{\*\}is well\-defined\.*The mapμ\\muis continuous andΓ\\Gamma\-invariant\. Continuity holds because each element of the multiset depends continuously onℒ\\mathcal\{L\}via \([24](https://arxiv.org/html/2609.00528#S4.E24)\)\. For invariance, anyπ∈Γ⊂Sn\\pi\\in\\Gamma\\subset S\_\{n\}permutes the ordered pairs\(i,k\)\(i,k\)withi≠ki\\neq kamong themselves, and the multiset is unordered, soμ⁡\(π⋅ℒ\)=μ⁡\(ℒ\)\\mu\(\\pi\\cdot\\mathcal\{L\}\)=\\mu\(\\mathcal\{L\}\)\. Hence forg∈Csym0​\(ℳM⁡\(M−1\)​\(X\)\)g\\in C^\{0\}\_\{\\mathrm\{sym\}\}\(\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)\), the pullbackg∘μg\\circ\\muis continuous \(composition of continuous maps\) andΓ\\Gamma\-invariant, i\.e\.,g∘μ∈CΓ​\(𝒲\)g\\circ\\mu\\in C\_\{\\Gamma\}\(\\mathcal\{W\}\)\.

*μ∗\\mu^\{\*\}is surjective\.*It suffices to show thatμ\\muis injective moduloΓ\\Gamma, i\.e\.,μ\(ℒ1\)=μ\(ℒ2\)⇒ℒ1∼Γℒ2\\mu\(\\mathcal\{L\}\_\{1\}\)=\\mu\(\\mathcal\{L\}\_\{2\}\)\\Rightarrow\\mathcal\{L\}\_\{1\}\\sim\_\{\\Gamma\}\\mathcal\{L\}\_\{2\}\. Since\(𝒏k\(in\),dj​k\)\(\\bm\{n\}\_\{k\}^\{\(\\mathrm\{in\}\)\},d\_\{jk\}\)is distinct for distinctkk\(by hypothesis\), each element\(𝒏i,𝒏k,dj​i,dj​k,cos⁡θi​j​k\)\(\\bm\{n\}\_\{i\},\\bm\{n\}\_\{k\},d\_\{ji\},d\_\{jk\},\\cos\\theta\_\{ijk\}\)of the multiset uniquely identifies the atomsiiandkk\. Equal multisets therefore imply a permutationπ∈Sn\\pi\\in S\_\{n\}with𝒏π⁡\(k\)\(2\)=𝒏k\(1\)\\bm\{n\}\_\{\\pi\(k\)\}^\{\(2\)\}=\\bm\{n\}\_\{k\}^\{\(1\)\},dj,π⁡\(k\)\(2\)=dj​k\(1\)d\_\{j,\\pi\(k\)\}^\{\(2\)\}=d\_\{jk\}^\{\(1\)\}, andcos⁡θπ⁡\(i\)​j​π​\(k\)\(2\)=cos⁡θi​j​k\(1\)\\cos\\theta\_\{\\pi\(i\)j\\pi\(k\)\}^\{\(2\)\}=\\cos\\theta\_\{ijk\}^\{\(1\)\}for alli≠ki\\neq k, where the superscripts refer toℒ1\\mathcal\{L\}\_\{1\}andℒ2\\mathcal\{L\}\_\{2\}\. Since the node features of neighbors of different types lie in disjoint sets, the matching𝒏π⁡\(k\)\(2\)=𝒏k\(1\)\\bm\{n\}\_\{\\pi\(k\)\}^\{\(2\)\}=\\bm\{n\}\_\{k\}^\{\(1\)\}forcesπ\\pito be type\-preserving, that is,π∈Γ\\pi\\in\\Gamma\. By \([25](https://arxiv.org/html/2609.00528#S4.E25)\), the matched distances and angles giveGπ⁡\(i\),π⁡\(k\)\(2\)=Gi​k\(1\)G^\{\(2\)\}\_\{\\pi\(i\),\\pi\(k\)\}=G^\{\(1\)\}\_\{ik\}, soℒ1∼Γℒ2\\mathcal\{L\}\_\{1\}\\sim\_\{\\Gamma\}\\mathcal\{L\}\_\{2\}\.

Now letf∈CΓ​\(𝒲\)f\\in C\_\{\\Gamma\}\(\\mathcal\{W\}\)andK⊂𝒲K\\subset\\mathcal\{W\}compact\. Sinceμ\\muis injective moduloΓ\\Gamma,μ⁡\(ℒ1\)=μ⁡\(ℒ2\)⇒f⁡\(ℒ1\)=f⁡\(ℒ2\)\\mu\(\\mathcal\{L\}\_\{1\}\)=\\mu\(\\mathcal\{L\}\_\{2\}\)\\Rightarrow f\(\\mathcal\{L\}\_\{1\}\)=f\(\\mathcal\{L\}\_\{2\}\), sofffactors asf=g0∘μf=g\_\{0\}\\circ\\mufor a well\-defined functiong0g\_\{0\}on the imageμ⁡\(K\)\\mu\(K\)\. The functiong0g\_\{0\}is continuous onμ⁡\(K\)\\mu\(K\)\. The restrictionμ\|K\\mu\|\_\{K\}is a continuous map from the compact setKKto the metrizable spaceℳM⁡\(M−1\)​\(X\)\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\), hence a closed map, and for any closedC⊂ℝC\\subset\\mathbb\{R\},g0−1​\(C\)=μ⁡\(f−1​\(C\)∩K\)g\_\{0\}^\{\-1\}\(C\)=\\mu\(f^\{\-1\}\(C\)\\cap K\)is closed inμ⁡\(K\)\\mu\(K\)\. Sinceμ⁡\(K\)\\mu\(K\)is a compact subset of the metrizable spaceℳM⁡\(M−1\)​\(X\)\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\), the Tietze extension theorem provides a continuous extensiong:ℳM⁡\(M−1\)​\(X\)→ℝg\\colon\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)\\to\\mathbb\{R\}withg\|μ⁡\(K\)=g0g\|\_\{\\mu\(K\)\}=g\_\{0\}\.

*Step 3: approximation\.*Letf∈CΓ​\(𝒲\)f\\in C\_\{\\Gamma\}\(\\mathcal\{W\}\),K⊂𝒲K\\subset\\mathcal\{W\}compact, andδ\>0\\delta\>0\. By Step 2,f=g∘μf=g\\circ\\muonKKwithg∈Csym0​\(ℳM⁡\(M−1\)​\(X\)\)g\\in C^\{0\}\_\{\\mathrm\{sym\}\}\(\\mathcal\{M\}\_\{M\(M\-1\)\}\(X\)\)\. By Step 1, there exist MLP weights such that the HGNN outputg~\\tilde\{g\}satisfies‖g~−g‖C0​\(μ​\(K\)\)<δ\\\|\\tilde\{g\}\-g\\\|\_\{C^\{0\}\(\\mu\(K\)\)\}<\\delta\. For everyℒ∈K\\mathcal\{L\}\\in K,

\|\(g~∘μ\)​\(ℒ\)−f⁡\(ℒ\)\|=\|g~​\(μ⁡\(ℒ\)\)−g⁡\(μ⁡\(ℒ\)\)\|<δ,\|\(\\tilde\{g\}\\circ\\mu\)\(\\mathcal\{L\}\)\-f\(\\mathcal\{L\}\)\|=\|\\tilde\{g\}\(\\mu\(\\mathcal\{L\}\)\)\-g\(\\mu\(\\mathcal\{L\}\)\)\|<\\delta,so the HGNN approximatesffuniformly onKK\.

Applying this procedure to each component of the complete representation of[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13), we prove that a single HGNN layer approximates the representation on generic environments\.

###### Proposition 17\(Single\-layer HGNN approximates a complete representation\)\.

Let the HGNN have a single layer withrc\(3\)=rc\(2\)=rcr\_\{c\}^\{\(3\)\}=r\_\{c\}^\{\(2\)\}=r\_\{c\}and sufficient feature dimensionsdad\_\{a\}anddd\. For any compactK⊂ℰ\(rc\)∩ℰ∗K\\subset\\mathcal\{E\}^\{\(r\_\{c\}\)\}\\cap\\mathcal\{E\}^\{\*\}whose environments have at least two neighbors, and anyδ\>0\\delta\>0, there exist MLP weights such that the HGNN output approximates the complete representationΦ∗\\Phi^\{\*\}of[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13)with‖Φ−Φ∗‖C0​\(K\)<δ\\\|\\Phi\-\\Phi^\{\*\}\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

###### Proof\.

Work on a fixed stratumℰz0,𝒛\(rc\)\\mathcal\{E\}\_\{z\_\{0\},\\bm\{z\}\}^\{\(r\_\{c\}\)\}\. The cross\-stratum case follows because type embeddings place each stratum’s data in a disjoint region of the MLP input space, and by MLP universality a single set of weights can approximate independent targets on each of the finitely many strata\.

By[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13), the components of the complete representationΦ∗\\Phi^\{\*\}, restricted to the stratum, are polynomials in the entries of the switched Gram matrix\. These entries are continuous functions of the distances and angles, so each component is a continuousΓ\\Gamma\-invariant function of the labeled data\.

For the single\-layer case, the input node features are type embeddings𝒏k\(0\)=type​\_​embed​\(zk\)\\bm\{n\}\_\{k\}^\{\(0\)\}=\\mathrm\{type\\\_embed\}\(z\_\{k\}\)\. For generic configurations, distinct same\-type atoms have distinct distancesdj​kd\_\{jk\}, so the pair\(𝒏k\(0\),dj​k\)\(\\bm\{n\}\_\{k\}^\{\(0\)\},d\_\{jk\}\)is distinct for distinctkk\. The type embeddings are chosen distinct, so the features of neighbors of different types lie in disjoint sets, and both hypotheses of[lemma16](https://arxiv.org/html/2609.00528#Thmtheorem16)are satisfied\.

By[lemma16](https://arxiv.org/html/2609.00528#Thmtheorem16)applied to each component, there exist MLP weights such that the HGNN outputΦ⁡\(𝒟\)\\Phi\(\\mathcal\{D\}\)approximatesΦ∗​\(𝒟\)\\Phi^\{\*\}\(\\mathcal\{D\}\)to withinδ\\deltauniformly onKK\.

### 4\.3Multi\-layer approximation

We now extend the single\-layer result to the multi\-layer HGNN\. We first introduce theLL\-hop neighborhoods and environments that anLL\-layer network sees, then the two conditions on the configuration that the main theorem requires\. These are connectivity, which makesLLlayers reach the physical rangeRcR\_\{c\}, and overlap, which lets each layer reconstruct the extended geometry from local data\. We then show that this reconstruction is possible in[lemma23](https://arxiv.org/html/2609.00528#Thmtheorem23)and prove the multi\-layer approximation theorem in[theorem24](https://arxiv.org/html/2609.00528#Thmtheorem24)\. Two collections of Gram matrices recur in what follows\. For a center atomjj,

ℐj=\(G\(j\),\{G\(k\)\}k∈𝒩⁡\(j\)\)\\mathcal\{I\}\_\{j\}=\\bigl\(G^\{\(j\)\},\\;\\\{G^\{\(k\)\}\\\}\_\{k\\in\\mathcal\{N\}\(j\)\}\\bigr\)\(27\)gathers the Gram matrices ofjjand of each of its neighbors, whileG¯\(j\)\\bar\{G\}^\{\(j\)\}denotes the Gram matrix of the extended environment obtained by merging all the neighbors’ environments\.

###### Definition 18\(LL\-hop neighborhood and environment\)\.

For cutoffrcr\_\{c\}, define the 1\-hop neighbor set𝒩\(j\)=\{k:k≠j,\|𝐫k−𝐫j\|<rc\}\\mathcal\{N\}\(j\)=\\\{k:k\\neq j,\\;\|\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\}\|<r\_\{c\}\\\}\. TheLL\-hop neighbor set is defined inductively by𝒩1​\(j\)=𝒩​\(j\)\\mathcal\{N\}^\{1\}\(j\)=\\mathcal\{N\}\(j\)and𝒩l\+1​\(j\)=\(𝒩⁡\(j\)∪⋃k∈𝒩⁡\(j\)𝒩l​\(k\)\)∖\{j\}\\mathcal\{N\}^\{l\+1\}\(j\)=\\bigl\(\\mathcal\{N\}\(j\)\\cup\\bigcup\_\{k\\in\\mathcal\{N\}\(j\)\}\\mathcal\{N\}^\{l\}\(k\)\\bigr\)\\setminus\\\{j\\\}, so that the center never counts among its own neighbors\. TheLL\-hop local environment is

𝒟j\[rc,L\]=\(zj,\{\(Δ​𝒓j​k,zk\):k∈𝒩L​\(j\)\}\),Δ​𝒓j​k=𝒓k−𝒓j\.\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\]\}=\\bigl\(z\_\{j\},\\;\\\{\(\\Delta\\bm\{r\}\_\{jk\},\\,z\_\{k\}\):k\\in\\mathcal\{N\}^\{L\}\(j\)\\\}\\bigr\),\\qquad\\Delta\\bm\{r\}\_\{jk\}=\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\}\.\(28\)

TheLL\-hop neighborhood𝒩L​\(j\)\\mathcal\{N\}^\{L\}\(j\)contains every atom reachable fromjjby a chain of at mostLLsteps, each of length less thanrcr\_\{c\}\. Note that𝒩L​\(j\)\\mathcal\{N\}^\{L\}\(j\)is in general a*subset*of the ball\{k:\|𝒓k−𝒓j\|<L​rc\}\\\{k:\|\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\}\|<Lr\_\{c\}\\\}\. An atom within distanceL​rcLr\_\{c\}may not be reachable if there are gaps in the neighbor structure\.

###### Definition 19\(LL\-hop environment space\)\.

The*LL\-hop environment space*ℰ\[rc,L\]\\mathcal\{E\}^\{\[r\_\{c\},L\]\}is the set of allLL\-hop local environments𝒟j\[rc,L\]\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\]\}, asjjranges over the atoms of all configurations\.

Since every atom reachable fromjjin at mostLLhops of length less thanrcr\_\{c\}lies within distanceL​rcLr\_\{c\}ofjj, we have the inclusionℰ\[rc,L\]⊂ℰ\(L​rc\)\\mathcal\{E\}^\{\[r\_\{c\},L\]\}\\subset\\mathcal\{E\}^\{\(Lr\_\{c\}\)\}\. The membership condition is intrinsic\. An environment𝒟∈ℰ\(L​rc\)\\mathcal\{D\}\\in\\mathcal\{E\}^\{\(Lr\_\{c\}\)\}lies inℰ\[rc,L\]\\mathcal\{E\}^\{\[r\_\{c\},L\]\}if and only if every neighbor of the central atom is connected to it by a chain of at mostLLsteps, with each step linking two atoms at distance less thanrcr\_\{c\}\. This condition can be expressed as a finite disjunction over all such chains of finitely many strict inequalities\. Henceℰ\[rc,L\]\\mathcal\{E\}^\{\[r\_\{c\},L\]\}is an open subset ofℰ\(L​rc\)\\mathcal\{E\}^\{\(Lr\_\{c\}\)\}, inheriting its stratification, metric, and differentiable structure\. We call anLL\-hop environment*generic*if the local environment of every atom in it is generic in the sense of[definition10](https://arxiv.org/html/2609.00528#Thmtheorem10)and no two of its atoms lie at distance exactlyrcr\_\{c\}\.

###### Definition 20\(Connectivity condition\)\.

A configuration satisfies the*connectivity condition*with respect to cutoffrcr\_\{c\}and physical rangeRcR\_\{c\}if, for every atomjjand every atomk≠jk\\neq jwith\|𝐫k−𝐫j\|<Rc\|\\bm\{r\}\_\{k\}\-\\bm\{r\}\_\{j\}\|<R\_\{c\}, there existsL≥1L\\geq 1such thatk∈𝒩L​\(j\)k\\in\\mathcal\{N\}^\{L\}\(j\)\.

Since a configuration has finitely many atoms, the hop depths required for the individual pairs admit a finite maximum\. Hence, a singleLLsuffices for all pairs simultaneously\. Consequently,𝒟j\[rc,L\]⊇𝒟j\(Rc\)\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\]\}\\supseteq\\mathcal\{D\}\_\{j\}^\{\(R\_\{c\}\)\}for every atomjj, meaning that anLL\-layer HGNN effectively covers the entire physical interaction rangeRcR\_\{c\}\. See[fig\.2](https://arxiv.org/html/2609.00528#S4.F2)for an illustration\. In condensed matter systems, where atoms typically form a connected network at cutoffrcr\_\{c\}, this condition is generally satisfied\.

Figure 2:Extending the receptive field via multi\-layer message passing\. The central atomjj\(star\) has a per\-layer cutoffrcr\_\{c\}\. The dashed circle marks the physical interaction rangeRcR\_\{c\}\. \(a\) After one layer, only the 1\-hop neighbors \(blue\) are reached\. \(b\) After two layers, the 2\-hop neighborhood \(green\) extends further, but atoms near the boundary ofRcR\_\{c\}remain unreached\. \(c\) After three layers, the 3\-hop neighborhood \(orange\) covers all atoms withinRcR\_\{c\}\. Note that theLL\-hop neighborhood is in general a*subset*of the ball of radiusL​rcLr\_\{c\}\(dotted circles\), since an atom within distanceL​rcLr\_\{c\}may not be reachable inLLgraph hops if intermediate neighbors are absent\. The connectivity condition guarantees that sufficiently many layers coverRcR\_\{c\}\. The overlap condition ensures that each layer faithfully reconstructs the extended geometry from local data\.###### Definition 21\(Overlap condition\)\.

Write𝒩⁡\[j\]=𝒩⁡\(j\)∪\{j\}\\mathcal\{N\}\[j\]=\\mathcal\{N\}\(j\)\\cup\\\{j\\\}for the closed neighborhood of atomjj\. A configuration satisfies the*overlap condition*with respect torcr\_\{c\}if every atom has a neighbor and, for every ordered pair of atoms\(j,k\)\(j,k\)with\|𝐫j−𝐫k\|<rc\|\\bm\{r\}\_\{j\}\-\\bm\{r\}\_\{k\}\|<r\_\{c\}, the displacements\{𝐫m−𝐫k:m∈𝒩⁡\[j\]∩𝒩⁡\[k\]\}\\\{\\bm\{r\}\_\{m\}\-\\bm\{r\}\_\{k\}:m\\in\\mathcal\{N\}\[j\]\\cap\\mathcal\{N\}\[k\]\\\}spanℝ3\\mathbb\{R\}^\{3\}\.

Bothjjandkklie in𝒩⁡\[j\]∩𝒩⁡\[k\]\\mathcal\{N\}\[j\]\\cap\\mathcal\{N\}\[k\], since\|𝒓j−𝒓k\|<rc\|\\bm\{r\}\_\{j\}\-\\bm\{r\}\_\{k\}\|<r\_\{c\}\. Spanningℝ3\\mathbb\{R\}^\{3\}is equivalent to the existence of three atomsm1,m2,m3m\_\{1\},m\_\{2\},m\_\{3\}in𝒩⁡\[j\]∩𝒩⁡\[k\]\\mathcal\{N\}\[j\]\\cap\\mathcal\{N\}\[k\]whose displacements relative tokkare linearly independent, which is the form used in[lemma23](https://arxiv.org/html/2609.00528#Thmtheorem23)\. Three is exactly the number required there, since the relative orientation of the local frames ofjjandkkis recovered as a linear map ofℝ3\\mathbb\{R\}^\{3\}, and such a map is determined by its values on a basis\.

The overlap condition requiresrcr\_\{c\}to be large enough that two neighboring atoms share enough common neighbors to spanℝ3\\mathbb\{R\}^\{3\}\. In condensed matter systems, a cutoff slightly above the nearest\-neighbor distance typically suffices\.

###### Definition 22\(Overlap condition forLL\-hop environments\)\.

The*hop depth*of an atom in anLL\-hop environment is the smallest number of hops connecting it to the center\. AnLL\-hop environment satisfies the*overlap condition*if its center has a neighbor and the spanning requirement of[definition21](https://arxiv.org/html/2609.00528#Thmtheorem21)holds for every ordered pair of its atoms\(j,k\)\(j,k\)with\|𝐫j−𝐫k\|<rc\|\\bm\{r\}\_\{j\}\-\\bm\{r\}\_\{k\}\|<r\_\{c\}in whichjjhas hop depth at mostL−1L\-1\. Here the neighborhoods are computed within the environment\.

The depth restriction exempts the outermost atoms, whose shared neighbors may lie beyond theLL\-hop horizon\. For a pair whose first atom has hop depth at mostL−1L\-1, the shared neighbors lie within one hop of that atom, hence at depth at mostLL\. Consequently, everyLL\-hop environment of a configuration satisfying the overlap condition satisfies[definition22](https://arxiv.org/html/2609.00528#Thmtheorem22)\.

The inductive step of the multi\-layer theorem needs a center atom to recover its extended environment from what its neighbors already encode\. At the level of Gram matrices, we have

###### Lemma 23\(Information sufficiency\)\.

Under the overlap condition atrcr\_\{c\}, the dataℐj\\mathcal\{I\}\_\{j\}uniquely determines the extended Gram matrixG¯\(j\)\\bar\{G\}^\{\(j\)\}\. Moreover,G¯\(j\)\\bar\{G\}^\{\(j\)\}is a continuous function ofℐj\\mathcal\{I\}\_\{j\}\.

###### Proof\.

Every atomaain the extended environment belongs to the environment of at least oneka∈𝒩⁡\(j\)k\_\{a\}\\in\\mathcal\{N\}\(j\)\. We compute each entryG¯a​b\(j\)\\bar\{G\}^\{\(j\)\}\_\{ab\}for two cases\.

*Case 1:a,ba,bbelong to a common neighbor’s environment\.*Letk∈𝒩⁡\(j\)k\\in\\mathcal\{N\}\(j\)have bothaaandbbin its environment\. Sincek∈𝒩⁡\(j\)k\\in\\mathcal\{N\}\(j\)impliesj∈𝒩⁡\(k\)j\\in\\mathcal\{N\}\(k\), atomjjappears inG\(k\)G^\{\(k\)\}\. Direct computation gives

G¯a​b\(j\)=Ga​b\(k\)−Ga​j\(k\)−Gb​j\(k\)\+dj​k2,\\bar\{G\}^\{\(j\)\}\_\{ab\}=G^\{\(k\)\}\_\{ab\}\-G^\{\(k\)\}\_\{aj\}\-G^\{\(k\)\}\_\{bj\}\+d\_\{jk\}^\{2\},\(29\)wheredj​k2=Gk​k\(j\)d\_\{jk\}^\{2\}=G^\{\(j\)\}\_\{kk\}\. All quantities on the right\-hand side are known fromℐj\\mathcal\{I\}\_\{j\}\.

*Case 2:aaandbbbelong to different neighbors’ environments with no common environment containing both\.*Letaabelong tokak\_\{a\}’s environment andbbtokbk\_\{b\}’s, withka≠kbk\_\{a\}\\neq k\_\{b\}\. This requires determining the relative orientation betweenkak\_\{a\}’s andkbk\_\{b\}’s local environments via frame alignment\.

*Gram matrix reconstruction\.*FromG\(j\)=𝑿\(j\)​\(𝑿\(j\)\)TG^\{\(j\)\}=\\bm\{X\}^\{\(j\)\}\(\\bm\{X\}^\{\(j\)\}\)^\{T\}, choose a position matrix𝑿\(j\)\\bm\{X\}^\{\(j\)\}\(determined up to right\-multiplication byR∈O⁡\(3\)R\\in O\(3\)\)\. Rowkkgives a vector𝒖k\\bm\{u\}\_\{k\}withΔ​𝒓k\(j\)=R​𝒖k\\Delta\\bm\{r\}\_\{k\}^\{\(j\)\}=R\\,\\bm\{u\}\_\{k\}\. Similarly, fromG\(ka\)G^\{\(k\_\{a\}\)\}, choose𝑿\(ka\)\\bm\{X\}^\{\(k\_\{a\}\)\}\(up toRka∈O⁡\(3\)R\_\{k\_\{a\}\}\\in O\(3\)\)\. Throughout, the center of each frame is assigned the zero vector, so that Gram entries carrying the index of the center vanish\.

*Frame alignment via shared atoms\.*By the overlap condition,𝒩⁡\[j\]∩𝒩⁡\[ka\]\\mathcal\{N\}\[j\]\\cap\\mathcal\{N\}\[k\_\{a\}\]contains atomsm1,m2,m3m\_\{1\},m\_\{2\},m\_\{3\}whose displacements relative tokak\_\{a\}are linearly independent\. Each shared atommmsatisfiesRka​𝒗m\(ka\)=R⁡\(𝒖m−𝒖ka\)R\_\{k\_\{a\}\}\\,\\bm\{v\}\_\{m\}^\{\(k\_\{a\}\)\}=R\(\\bm\{u\}\_\{m\}\-\\bm\{u\}\_\{k\_\{a\}\}\)\. DefineVkaV\_\{k\_\{a\}\}andUkaU\_\{k\_\{a\}\}as the3×33\\times 3matrices with columns𝒗mi\(ka\)\\bm\{v\}\_\{m\_\{i\}\}^\{\(k\_\{a\}\)\}and𝒖mi−𝒖ka\\bm\{u\}\_\{m\_\{i\}\}\-\\bm\{u\}\_\{k\_\{a\}\}, respectively\. Linear independence impliesVkaV\_\{k\_\{a\}\}is non\-singular\. Then

Tka≡R−1​Rka=Uka​Vka−1\.T\_\{k\_\{a\}\}\\equiv R^\{\-1\}\\,R\_\{k\_\{a\}\}=U\_\{k\_\{a\}\}\\,V\_\{k\_\{a\}\}^\{\-1\}\.\(30\)
*Expansion coefficients\.*Define𝒄a=Vka−1​𝒗a\(ka\)∈ℝ3\\bm\{c\}\_\{a\}=V\_\{k\_\{a\}\}^\{\-1\}\\,\\bm\{v\}\_\{a\}^\{\(k\_\{a\}\)\}\\in\\mathbb\{R\}^\{3\}, so that the position ofaainjj’s frame is𝒖a=𝒖ka\+Uka​𝒄a\\bm\{u\}\_\{a\}=\\bm\{u\}\_\{k\_\{a\}\}\+U\_\{k\_\{a\}\}\\,\\bm\{c\}\_\{a\}\. The coefficients𝒄a\\bm\{c\}\_\{a\}satisfy the normal equations

\(VkaT​Vka\)​𝒄a=VkaT​𝒗a\(ka\),\(V\_\{k\_\{a\}\}^\{T\}V\_\{k\_\{a\}\}\)\\,\\bm\{c\}\_\{a\}=V\_\{k\_\{a\}\}^\{T\}\\,\\bm\{v\}\_\{a\}^\{\(k\_\{a\}\)\},\(31\)where\(VkaT​Vka\)i​j=Gmi,mj\(ka\)\(V\_\{k\_\{a\}\}^\{T\}V\_\{k\_\{a\}\}\)\_\{ij\}=G^\{\(k\_\{a\}\)\}\_\{m\_\{i\},\\,m\_\{j\}\}and\(VkaT​𝒗a\(ka\)\)i=Gmi,a\(ka\)\(V\_\{k\_\{a\}\}^\{T\}\\bm\{v\}\_\{a\}^\{\(k\_\{a\}\)\}\)\_\{i\}=G^\{\(k\_\{a\}\)\}\_\{m\_\{i\},\\,a\}are entries ofG\(ka\)G^\{\(k\_\{a\}\)\}\. SinceVkaV\_\{k\_\{a\}\}is non\-singular,

𝒄a=\(Gmi,mj\(ka\)\)−1​\(Gmi,a\(ka\)\),\\bm\{c\}\_\{a\}=\\bigl\(G^\{\(k\_\{a\}\)\}\_\{m\_\{i\},\\,m\_\{j\}\}\\bigr\)^\{\-1\}\\,\\bigl\(G^\{\(k\_\{a\}\)\}\_\{m\_\{i\},\\,a\}\\bigr\),a continuous function of Gram entries ofG\(ka\)G^\{\(k\_\{a\}\)\}\. Similarly for𝒄b\\bm\{c\}\_\{b\}viaG\(kb\)G^\{\(k\_\{b\}\)\}\.

*Inner product formula\.*Expanding𝒖a⋅𝒖b=\(𝒖ka\+Uka​𝒄a\)⋅\(𝒖kb\+Ukb​𝒄b\)\\bm\{u\}\_\{a\}\\cdot\\bm\{u\}\_\{b\}=\(\\bm\{u\}\_\{k\_\{a\}\}\+U\_\{k\_\{a\}\}\\,\\bm\{c\}\_\{a\}\)\\cdot\(\\bm\{u\}\_\{k\_\{b\}\}\+U\_\{k\_\{b\}\}\\,\\bm\{c\}\_\{b\}\)and using𝒖α⋅𝒖β=Gα,β\(j\)\\bm\{u\}\_\{\\alpha\}\\cdot\\bm\{u\}\_\{\\beta\}=G^\{\(j\)\}\_\{\\alpha,\\beta\}:

G¯a​b\(j\)=Gka,kb\(j\)\+𝒄aT​𝒈ka→kb\+𝒄bT​𝒈kb→ka\+𝒄aT​Mka,kb​𝒄b,\\bar\{G\}^\{\(j\)\}\_\{ab\}=G^\{\(j\)\}\_\{k\_\{a\},\\,k\_\{b\}\}\+\\bm\{c\}\_\{a\}^\{T\}\\,\\bm\{g\}\_\{k\_\{a\}\\to k\_\{b\}\}\+\\bm\{c\}\_\{b\}^\{T\}\\,\\bm\{g\}\_\{k\_\{b\}\\to k\_\{a\}\}\+\\bm\{c\}\_\{a\}^\{T\}\\,M\_\{k\_\{a\},\\,k\_\{b\}\}\\,\\bm\{c\}\_\{b\},\(32\)where

\(𝒈ka→kb\)i\\displaystyle\(\\bm\{g\}\_\{k\_\{a\}\\to k\_\{b\}\}\)\_\{i\}=Gmi,kb\(j\)−Gka,kb\(j\),\\displaystyle=G^\{\(j\)\}\_\{m\_\{i\},\\,k\_\{b\}\}\-G^\{\(j\)\}\_\{k\_\{a\},\\,k\_\{b\}\},\(33\)\(Mka,kb\)i​j\\displaystyle\(M\_\{k\_\{a\},\\,k\_\{b\}\}\)\_\{ij\}=Gmi,mj′\(j\)−Gmi,kb\(j\)−Gka,mj′\(j\)\+Gka,kb\(j\)\.\\displaystyle=G^\{\(j\)\}\_\{m\_\{i\},\\,m\_\{j\}^\{\\prime\}\}\-G^\{\(j\)\}\_\{m\_\{i\},\\,k\_\{b\}\}\-G^\{\(j\)\}\_\{k\_\{a\},\\,m\_\{j\}^\{\\prime\}\}\+G^\{\(j\)\}\_\{k\_\{a\},\\,k\_\{b\}\}\.\(34\)Herem1′,m2′,m3′m^\{\\prime\}\_\{1\},m^\{\\prime\}\_\{2\},m^\{\\prime\}\_\{3\}denote the shared atoms chosen for the pair\(j,kb\)\(j,k\_\{b\}\), in analogy with themim\_\{i\}for\(j,ka\)\(j,k\_\{a\}\)\. Every term is a continuous function of entries ofℐj\\mathcal\{I\}\_\{j\}\.

*Independence of the choice of shared atoms\.*The coefficients𝒄a\\bm\{c\}\_\{a\},𝒈ka→kb\\bm\{g\}\_\{k\_\{a\}\\to k\_\{b\}\}, andMka,kbM\_\{k\_\{a\},k\_\{b\}\}all depend on the choice of shared atomsm1,m2,m3m\_\{1\},m\_\{2\},m\_\{3\}\. However, the final valueG¯a​b\(j\)\\bar\{G\}^\{\(j\)\}\_\{ab\}does not: since𝒖a=𝒖ka\+Uka​𝒄a\\bm\{u\}\_\{a\}=\\bm\{u\}\_\{k\_\{a\}\}\+U\_\{k\_\{a\}\}\\,\\bm\{c\}\_\{a\}andTka=Uka​Vka−1T\_\{k\_\{a\}\}=U\_\{k\_\{a\}\}V\_\{k\_\{a\}\}^\{\-1\}is the unique linear map sending𝒗m\(ka\)↦𝒖m−𝒖ka\\bm\{v\}\_\{m\}^\{\(k\_\{a\}\)\}\\mapsto\\bm\{u\}\_\{m\}\-\\bm\{u\}\_\{k\_\{a\}\}for*all*shared atomsmm, the productUka​𝒄a=Tka​𝒗a\(ka\)U\_\{k\_\{a\}\}\\,\\bm\{c\}\_\{a\}=T\_\{k\_\{a\}\}\\,\\bm\{v\}\_\{a\}^\{\(k\_\{a\}\)\}is independent of which triple definesVkaV\_\{k\_\{a\}\}andUkaU\_\{k\_\{a\}\}\. Hence𝒖a\\bm\{u\}\_\{a\}is the same for any valid triple, andG¯a​b\(j\)=𝒖a⋅𝒖b\\bar\{G\}^\{\(j\)\}\_\{ab\}=\\bm\{u\}\_\{a\}\\cdot\\bm\{u\}\_\{b\}is well\-defined\.

In both cases,G¯a​b\(j\)\\bar\{G\}^\{\(j\)\}\_\{ab\}is a well\-defined continuous function ofℐj\\mathcal\{I\}\_\{j\}\. Since this holds for every pair\(a,b\)\(a,b\), the entire extended Gram matrixG¯\(j\)\\bar\{G\}^\{\(j\)\}is uniquely and continuously determined byℐj\\mathcal\{I\}\_\{j\}\.

We are ready to prove the multi\-layer theorem\. The argument proceeds by induction onLL\. At each layer,[lemma16](https://arxiv.org/html/2609.00528#Thmtheorem16)provides the required approximation capabilities, while[lemma23](https://arxiv.org/html/2609.00528#Thmtheorem23)ensures that the information reaching that layer is sufficient to determine the extended environment it must represent\. The induction then composes these layerwise approximations to establish the fullLL\-layer completeness result\.

###### Theorem 24\(Multi\-layer HGNN approximates a complete representation\)\.

Let theLL\-layer HGNN have equal cutoffsrc\(3\)=rc\(2\)=rcr\_\{c\}^\{\(3\)\}=r\_\{c\}^\{\(2\)\}=r\_\{c\}and sufficient feature dimensionsdad\_\{a\}anddd\. Then there is a continuous complete representationΦ∗,\(L\)\\Phi^\{\*,\(L\)\}onℰ\[rc,L\]\\mathcal\{E\}^\{\[r\_\{c\},L\]\}such that for every compactK⊂ℰ\[rc,L\]K\\subset\\mathcal\{E\}^\{\[r\_\{c\},L\]\}consisting of generic environments that satisfy the overlap condition, and everyδ\>0\\delta\>0, there exist MLP weights for which theLL\-layer HGNN representationΦ\(L\):𝒟j↦𝐧j\(L\)\\Phi^\{\(L\)\}:\\mathcal\{D\}\_\{j\}\\mapsto\\bm\{n\}\_\{j\}^\{\(L\)\}satisfies‖Φ\(L\)−Φ∗,\(L\)‖C0​\(K\)<δ\\\|\\Phi^\{\(L\)\}\-\\Phi^\{\*,\(L\)\}\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

###### Proof\.

By induction onLL\. We prove the statementP⁡\(L\)P\(L\):there is a continuous complete representationΦ∗,\(L\)\\Phi^\{\*,\(L\)\}onℰ\[rc,L\]\\mathcal\{E\}^\{\[r\_\{c\},L\]\}such that for every compactKKas in the statement of the theorem and everyδ\>0\\delta\>0, there exist MLP weights for theLL\-layer HGNN with‖Φ\(L\)−Φ∗,\(L\)‖C0​\(K\)<δ\\\|\\Phi^\{\(L\)\}\-\\Phi^\{\*,\(L\)\}\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

The overlap condition gives every center in such aKKat least two neighbors, since the pair formed with one neighbor has a spanning triple containing at least two atoms of𝒩⁡\(j\)\\mathcal\{N\}\(j\)\.

Base caseP⁡\(1\)P\(1\)follows from[proposition17](https://arxiv.org/html/2609.00528#Thmtheorem17)\.

AssumeP⁡\(L\)P\(L\)holds\. Fix the weights for layers1,…,L1,\\ldots,Las guaranteed byP⁡\(L\)P\(L\), at an accuracy that may be taken as small as needed\. We find weights for layerL\+1L\+1\.

Fork∈𝒩⁡\(j\)k\\in\\mathcal\{N\}\(j\), genericity keeps every interatomic distance away fromrcr\_\{c\}, so the hop structure is locally constant and the sub\-environment𝒟k\[rc,L\]\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}depends continuously on𝒟j\[rc,L\+1\]\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\+1\]\}\. The setKLK\_\{L\}of all sub\-environments arising fromKKis therefore compact\. Genericity passes to sub\-environments\. The overlap condition passes as well\. An atom of hop depth at mostL−1L\-1in a sub\-environment𝒟k\[rc,L\]\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}has hop depth at mostLLin𝒟j\[rc,L\+1\]\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\+1\]\}, so the overlap condition of the parent applies to its pairs, and the shared neighbors thus provided lie within one hop of that atom, hence at depth at mostLLand inside the sub\-environment\. Each sub\-environment centerkkretains its neighborjj, as[definition22](https://arxiv.org/html/2609.00528#Thmtheorem22)requires\. HenceKLK\_\{L\}qualifies forP⁡\(L\)P\(L\)\.

Let𝒲\(L\)=𝒱×\(ℝd\)n\\mathcal\{W\}^\{\(L\)\}=\\mathcal\{V\}\\times\(\\mathbb\{R\}^\{d\}\)^\{n\}denote the labeled data space for layerL\+1L\+1, and define the mapν:𝒟j\[rc,L\+1\]↦𝒲j=\(G\(j\),\(Φ∗,\(L\)​\(𝒟1\[rc,L\]\),…,Φ∗,\(L\)​\(𝒟n\[rc,L\]\)\)\)∈𝒲\(L\)\\nu\\colon\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\+1\]\}\\mapsto\\mathcal\{W\}\_\{j\}=\(G^\{\(j\)\},\(\\Phi^\{\*,\(L\)\}\(\\mathcal\{D\}\_\{1\}^\{\[r\_\{c\},L\]\}\),\\ldots,\\Phi^\{\*,\(L\)\}\(\\mathcal\{D\}\_\{n\}^\{\[r\_\{c\},L\]\}\)\)\)\\in\\mathcal\{W\}^\{\(L\)\}, which labels each neighbor with the exact complete representation of itsLL\-hop environment supplied byP⁡\(L\)P\(L\)\. The node features computed by the firstLLlayers approximate these labels, and we account for the difference at the end of the proof\.

We first show thatν\\nuis injective moduloGGon the environments ofKK\. Since𝒩L\+1​\(j\)=\(⋃k∈𝒩⁡\(j\)\(\{k\}∪𝒩L​\(k\)\)\)∖\{j\}\\mathcal\{N\}^\{L\+1\}\(j\)=\\bigl\(\\bigcup\_\{k\\in\\mathcal\{N\}\(j\)\}\(\\\{k\\\}\\cup\\mathcal\{N\}^\{L\}\(k\)\)\\bigr\)\\setminus\\\{j\\\}, every atom in the\(L\+1\)\(L\+1\)\-hop environment appears in somekk’sLL\-hop environment\. Thekk\-th label is a complete invariant of𝒟k\[rc,L\]\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}, so it determines the Gram matrixG\(k\)G^\{\(k\)\}and the atom types of that environment up to a relabeling of its atoms\. Genericity fixes the relabeling where it matters\. Fork∈𝒩⁡\(j\)k\\in\\mathcal\{N\}\(j\), the distances fromkkto its neighbors are pairwise distinct, and the distance fromkkto any atom of𝒩⁡\[j\]\\mathcal\{N\}\[j\]is computable fromG\(j\)G^\{\(j\)\}\. Hence every atom of𝒩⁡\[j\]∩𝒩⁡\[k\]\\mathcal\{N\}\[j\]\\cap\\mathcal\{N\}\[k\], and in particular the centerjj, occupies a determined index inG\(k\)G^\{\(k\)\}\. The labeled data𝒲j\\mathcal\{W\}\_\{j\}therefore determinesℐj=\(G\(j\),\{G\(k\)\}k∈𝒩⁡\(j\)\)\\mathcal\{I\}\_\{j\}=\(G^\{\(j\)\},\\\{G^\{\(k\)\}\\\}\_\{k\\in\\mathcal\{N\}\(j\)\}\)together with these identifications\. Suppose two environments ofKKhave the same image underν\\nuup toΓ\\Gamma\. Then they have the same collectionℐj\\mathcal\{I\}\_\{j\}of Gram matrices, so by[lemma23](https://arxiv.org/html/2609.00528#Thmtheorem23)they have the same extended Gram matrixG¯\(j\)\\bar\{G\}^\{\(j\)\}, and the labels give their atoms the same types\. By[lemma11](https://arxiv.org/html/2609.00528#Thmtheorem11)the positions then agree up to an orthogonal transformation, so the two environments are symmetry\-equivalent\.

This lets the target representation be pulled back to the labeled data\. By[proposition13](https://arxiv.org/html/2609.00528#Thmtheorem13)applied at cutoff\(L\+1\)​rc\(L\+1\)r\_\{c\}, a continuous complete representation exists onℰ\(\(L\+1\)​rc\)\\mathcal\{E\}^\{\(\(L\+1\)r\_\{c\}\)\}\. Sinceℰ\[rc,L\+1\]\\mathcal\{E\}^\{\[r\_\{c\},L\+1\]\}is aGG\-invariant subset, its restrictionΦ∗,\(L\+1\)\\Phi^\{\*,\(L\+1\)\}is continuous and complete there\. Becauseν\\nuis injective moduloGGonKK,Φ∗,\(L\+1\)\\Phi^\{\*,\(L\+1\)\}factors onKKasΦ∗,\(L\+1\)=g∘ν\\Phi^\{\*,\(L\+1\)\}=g\\circ\\nufor a continuousΓ\\Gamma\-invariant functionggon𝒲\(L\)\\mathcal\{W\}^\{\(L\)\}, by the same factorization and Tietze extension argument used in the proof of[lemma16](https://arxiv.org/html/2609.00528#Thmtheorem16)\.

It remains to approximateggwith a single layer\. As in the proof of[proposition17](https://arxiv.org/html/2609.00528#Thmtheorem17), it suffices to work on a fixed stratum\. The cross\-stratum case follows from the label separation established below\. We verify the hypotheses of[lemma16](https://arxiv.org/html/2609.00528#Thmtheorem16)for the labels\(Φ∗,\(L\)​\(𝒟k\[rc,L\]\),dj​k\)\(\\Phi^\{\*,\(L\)\}\(\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}\),d\_\{jk\}\),k∈𝒩⁡\(j\)k\\in\\mathcal\{N\}\(j\)\. Letk≠k′k\\neq k^\{\\prime\}in𝒩⁡\(j\)\\mathcal\{N\}\(j\)\. Ifzk=zk′z\_\{k\}=z\_\{k^\{\\prime\}\}, genericity givesdj​k≠dj​k′d\_\{jk\}\\neq d\_\{jk^\{\\prime\}\}, so the pairs differ in their second component\. Ifzk≠zk′z\_\{k\}\\neq z\_\{k^\{\\prime\}\}, the environments𝒟k\[rc,L\]\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}and𝒟k′\[rc,L\]\\mathcal\{D\}\_\{k^\{\\prime\}\}^\{\[r\_\{c\},L\]\}have different center types, hence lie in different strata and are not symmetry\-equivalent, so the complete representationΦ∗,\(L\)\\Phi^\{\*,\(L\)\}separates them\. The separationσ=min⁡‖Φ∗,\(L\)​\(𝒟k\[rc,L\]\)−Φ∗,\(L\)​\(𝒟k′\[rc,L\]\)‖\\sigma=\\min\\\|\\Phi^\{\*,\(L\)\}\(\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}\)\-\\Phi^\{\*,\(L\)\}\(\\mathcal\{D\}\_\{k^\{\\prime\}\}^\{\[r\_\{c\},L\]\}\)\\\|, taken overKKand over the finitely many pairs withzk≠zk′z\_\{k\}\\neq z\_\{k^\{\\prime\}\}, is positive becauseKKis compact andΦ∗,\(L\)\\Phi^\{\*,\(L\)\}is continuous\. The labels of neighbors of different types therefore lie in disjoint sets, and in either case the pairs are distinct\. Both hypotheses of[lemma16](https://arxiv.org/html/2609.00528#Thmtheorem16)are thus satisfied, so there exist MLP weights for layerL\+1L\+1whose realized mapg~\\tilde\{g\}approximatesggto withinδ/2\\delta/2onν⁡\(K\)\\nu\(K\)\.

Finally, we account for the difference between the exact labels and the computed node features\. The realized mapg~\\tilde\{g\}is uniformly continuous on a compact neighborhood ofν⁡\(K\)\\nu\(K\), so there is anϵ\>0\\epsilon\>0such that inputs withinϵ\\epsilonof each other produce outputs withinδ/2\\delta/2\. The exact labels and the computed features differ only in the feature slots, by at mostmaxk∈𝒩⁡\(j\)⁡‖𝒏k\(L\)−Φ∗,\(L\)​\(𝒟k\[rc,L\]\)‖\\max\_\{k\\in\\mathcal\{N\}\(j\)\}\\\|\\bm\{n\}\_\{k\}^\{\(L\)\}\-\\Phi^\{\*,\(L\)\}\(\\mathcal\{D\}\_\{k\}^\{\[r\_\{c\},L\]\}\)\\\|, which is the accuracy of the firstLLlayers onKLK\_\{L\}\. InvokingP⁡\(L\)P\(L\)onKLK\_\{L\}at accuracyϵ\\epsilontherefore keeps the layer\-\(L\+1\)\(L\+1\)output on the computed features withinδ/2\\delta/2ofg~∘ν\\tilde\{g\}\\circ\\nu\. HenceΦ\(L\+1\)\\Phi^\{\(L\+1\)\}approximatesg∘ν=Φ∗,\(L\+1\)g\\circ\\nu=\\Phi^\{\*,\(L\+1\)\}to withinδ\\deltaonKK, establishingP⁡\(L\+1\)P\(L\+1\)\.

By induction,P⁡\(L\)P\(L\)holds for allL≥1L\\geq 1\.

The induction shows that suitable weights exist but does not say what the network computes at each layer\. For the atoms that share a neighbor’s environment this can be made explicit, and the resulting formula shows how Gram information propagates one hop at a time\.

## 5Universal Approximation Theorem

[theorem24](https://arxiv.org/html/2609.00528#Thmtheorem24)supplies a complete representation of theLL\-hop environment, and[theorem9](https://arxiv.org/html/2609.00528#Thmtheorem9)turns completeness into universal approximation\. A combination of two results gives the main theorem\.

###### Theorem 26\(UAT for HGNN\)\.

Let theLL\-layer HGNN have equal cutoffsrc\(3\)=rc\(2\)=rcr\_\{c\}^\{\(3\)\}=r\_\{c\}^\{\(2\)\}=r\_\{c\}and sufficient feature dimensions and MLP capacity per layer\. LetK⊂ℰ\[rc,L\]K\\subset\\mathcal\{E\}^\{\[r\_\{c\},L\]\}be a compact set of generic environments that satisfy the overlap condition, each drawn from a configuration satisfying the connectivity condition, withLLlarge enough that𝒟j\[rc,L\]⊇𝒟j\(Rc\)\\mathcal\{D\}\_\{j\}^\{\[r\_\{c\},L\]\}\\supseteq\\mathcal\{D\}\_\{j\}^\{\(R\_\{c\}\)\}\. Then for everyε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}and everyδ\>0\\delta\>0there exist parametersθ\\thetawith‖fθ∘Φ\(L\)−ε‖C0​\(K\)<δ\\\|f\_\{\\theta\}\\circ\\Phi^\{\(L\)\}\-\\varepsilon\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

###### Proof\.

LetΦ∗,\(L\)\\Phi^\{\*,\(L\)\}be the complete representation of[theorem24](https://arxiv.org/html/2609.00528#Thmtheorem24), whose hypotheses onKKare part of the present assumptions\. The targetε\\varepsilondepends only on the neighbors withinRcR\_\{c\}, so it defines aGG\-invariant function onKK, continuous by condition 3 of[definition6](https://arxiv.org/html/2609.00528#Thmtheorem6)\. By completeness ofΦ∗,\(L\)\\Phi^\{\*,\(L\)\}and the argument of the forward implication of[theorem9](https://arxiv.org/html/2609.00528#Thmtheorem9), applied onℰ\[rc,L\]\\mathcal\{E\}^\{\[r\_\{c\},L\]\}in place ofℰ\(Rc\)\\mathcal\{E\}^\{\(R\_\{c\}\)\}, there exists a readoutfθf\_\{\\theta\}with‖fθ∘Φ∗,\(L\)−ε‖C0​\(K\)<δ/2\\\|f\_\{\\theta\}\\circ\\Phi^\{\*,\(L\)\}\-\\varepsilon\\\|\_\{C^\{0\}\(K\)\}<\\delta/2\. The readoutfθf\_\{\\theta\}is uniformly continuous on a compact neighborhood ofΦ∗,\(L\)​\(K\)\\Phi^\{\*,\(L\)\}\(K\), so there is anϵ\>0\\epsilon\>0such that inputs withinϵ\\epsilonproduce outputs withinδ/2\\delta/2\. By[theorem24](https://arxiv.org/html/2609.00528#Thmtheorem24)there exist layer weights with‖Φ\(L\)−Φ∗,\(L\)‖C0​\(K\)<ϵ\\\|\\Phi^\{\(L\)\}\-\\Phi^\{\*,\(L\)\}\\\|\_\{C^\{0\}\(K\)\}<\\epsilon\. Combining the two bounds gives‖fθ∘Φ\(L\)−ε‖C0​\(K\)<δ\\\|f\_\{\\theta\}\\circ\\Phi^\{\(L\)\}\-\\varepsilon\\\|\_\{C^\{0\}\(K\)\}<\\delta\.

The theorem is stated for the HGNN, which was introduced as a reference architecture\. Any network that can reproduce its layer inherits the conclusion\. We verify this for DPA3 and CHGNet in the appendices\.

###### Corollary 27\(UAT for DPA3\)\.

Under the same conditions as[theorem26](https://arxiv.org/html/2609.00528#Thmtheorem26), theLL\-layer DPA3 architecture withrc\(3\)=rc\(2\)=rcr\_\{c\}^\{\(3\)\}=r\_\{c\}^\{\(2\)\}=r\_\{c\}, sufficient feature dimension, sufficient MLP capacity, and readout networkfθf\_\{\\theta\}approximates everyε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}to arbitrary accuracy onKK\.

The proof simulates one HGNN layer by a single DPA3 layer and is given in[section1](https://arxiv.org/html/2609.00528#S1a)of the supplementary materials\.

###### Definition 28\(Strong genericity\)\.

A local environment𝒟∈ℰ\(r\)\\mathcal\{D\}\\in\\mathcal\{E\}^\{\(r\)\}is*strongly generic*if no two neighbors are equidistant from the center, that is, ifi≠ki\\neq kimplies\|Δ​𝐫i\|≠\|Δ​𝐫k\|\|\\Delta\\bm\{r\}\_\{i\}\|\\neq\|\\Delta\\bm\{r\}\_\{k\}\|\. AnLL\-hop environment is strongly generic if every local environment within it is strongly generic\.

As with[definition10](https://arxiv.org/html/2609.00528#Thmtheorem10), the excluded set is a finite union of zero sets of nonzero polynomials, so the strongly generic environments form an open dense subset of full measure in each stratum\.

###### Corollary 29\(UAT for CHGNet\)\.

Under the same conditions as[theorem26](https://arxiv.org/html/2609.00528#Thmtheorem26), with the atom graph and the bond graph sharing the cutoffrcr\_\{c\}, and with every environment inKKstrongly generic in the sense of[definition28](https://arxiv.org/html/2609.00528#Thmtheorem28), theLL\-layer CHGNet architecture of[section2](https://arxiv.org/html/2609.00528#S2a), with sufficient feature dimension, sufficient MLP capacity, and readout networkfθf\_\{\\theta\}, approximates everyε∈ℱRc0\\varepsilon\\in\\mathcal\{F\}\_\{R\_\{c\}\}^\{0\}to arbitrary accuracy onKK\.

The proof follows the same line as that of[corollary27](https://arxiv.org/html/2609.00528#Thmtheorem27)and is given in[section2](https://arxiv.org/html/2609.00528#S2a)of the supplementary materials\. The strengthening of genericity is needed because the bond message function of CHGNet receives only the center’s node features, so the proof recovers each neighbor’s features from its distance by a lookup, which requires all neighbor distances to be distinct\. ALIGNN shares the same angle→\\toedge→\\tonode topology but differs crucially in that it constructs its messages from single gated linear layers rather than MLPs, and the previous simulation argument does not carry over\. We shall describe the architecture and explain where the argument fails in[section3](https://arxiv.org/html/2609.00528#S3a)of the supplementary materials\.

## 6Discussion and perspective

The multi\-layer theory justifies the practice of stacking layers with a per\-layer cutoff smaller than the physical interaction range\. The connectivity condition sets how many layers are needed to cover that range, and the overlap condition tells one how small the per\-layer cutoff may be, for condensed matter systems typically slightly above the nearest\-neighbor distance\.

At the architectural level, the results show that an invariant HGNN relying solely on scalar geometric inputs attains universal approximation, so raising the LiGS order beyondK=2K=2, the 3\-body angular level, is not necessary for theoretical expressiveness\. It is important to emphasize, however, that this is a statement about expressiveness alone\. The HGNN is intended as a theoretical reference architecture rather than a practical proposal; universal approximation, in itself, does not address data efficiency, computational cost, or the optimization landscape\. Its primary role is to serve as a target for simulation: once a practical architecture can reproduce an HGNN layer, universal approximation follows without the need to re\-establish the completeness analysis from scratch\.

We carry this out for DPA3 and CHGNet in[corollary27](https://arxiv.org/html/2609.00528#Thmtheorem27)and[corollary29](https://arxiv.org/html/2609.00528#Thmtheorem29), respectively\. The proofs identify a concrete architectural criterion for guaranteed universal approximation, that the message functions in the aggregation steps must be*MLPs*, which are universal approximators, not single linear layers\. Both architectures satisfy it\. ALIGNN\[[11](https://arxiv.org/html/2609.00528#bib.bib11)\], which shares the same angle→\\toedge→\\tonode topology but uses single gated linear layers in its convolutions, does not \([remark30](https://arxiv.org/html/2609.00528#Thmtheorem30)\)\. This distinction has practical implications\. Replacing the single gated linear layers in ALIGNN’s convolutions with MLPs would remove the obstruction identified in[remark30](https://arxiv.org/html/2609.00528#Thmtheorem30), although establishing universality for the modified architecture would still require a simulation argument along the lines of[section2](https://arxiv.org/html/2609.00528#S2a)\.

The theorems rely on genericity\. Completeness is valid on the generic environments of[definition10](https://arxiv.org/html/2609.00528#Thmtheorem10), which excludes highly symmetric arrangements where same\-type atoms are equidistant from the center\. The restriction is in fact sharp\. The Pozdnyakov counterexample\[[29](https://arxiv.org/html/2609.00528#bib.bib29)\]exhibits four same\-species atoms on a circle, all sharing identical multisets of distances and triplet angles, yet possessing distinct Gram matrices\. No single\-layer 3\-body invariant can separate such configurations\. In practice, thermal fluctuations break exact symmetries, so sampled configurations are generic with probability one\. Moreover, in the multi\-layer HGNN, the node features of earlier layers help break residual equidistance\. The remaining case, zero\-temperature crystals, remains a fundamental open problem\. Switching to equivariant architectures does not resolve the issue, since equivariant GNNs with a fixed tensor degreellalso degenerate on symmetric graphs for specific values ofll\[[8](https://arxiv.org/html/2609.00528#bib.bib8)\]\. Both families also face hierarchies of degeneracies\. Raising the body order or increasing the tensor degree resolves progressively more symmetric cases, yet counterexamples are known already at low levels, and no finite level has been shown to achieve completeness across all configurations\.

Several other directions remain open\. The DeepSets arguments require the angle\-feature dimensiondad\_\{a\}, which serves as the latent dimension of the aggregation, to grow at least with the number of neighbor triplets, far beyond the typical practical value of6464\. Tightening this gap remains an open problem\. The present results are also qualitative rather than quantitative\. Bounding the approximation rate, that is, how the error decays with network size, would bridge the gap between theory and practical model selection\. Finally, extending the analysis fromC0C^\{0\}toCkC^\{k\}approximation, so as to match the smoothness of the potential energy surface including forces and Hessians, requires a more detailed study of the differentiability of the DeepSets representation\. The smooth symmetrization of Pozdnyakov and Ceriotti\[[28](https://arxiv.org/html/2609.00528#bib.bib28)\]offers a promising direction in this regard\.

## Acknowledgments

The AI\-driven experiments, simulations and model training were performed on the robotic AI\-Scientist platform of Chinese Academy of Sciences\. Pingbing Ming is supported by the National Key R&D Program of China \(Grant no\. 2024YFA1012502\) and by National Natural Science Foundation of China \(Grant No\. 12371438\)\. Han Wang is supported by the National Key R&D Program of China \(Grant No\. 2022YFA1004300\) and by the National Natural Science Foundation of China \(Grants No\. 12525113 and No\. 12561160120\)\. The authors used AI tools to assist with the development of proofs, manuscript editing, and reference verification\. All mathematical proofs have been checked by the authors to the best of their ability, and the authors assume responsibility for all content\.

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Supplementary Materials

## 1Proof of the DPA3 corollary

###### Proof of[corollary27](https://arxiv.org/html/2609.00528#Thmtheorem27)\.

It suffices to show that a single DPA3 layer can simulate one HGNN layer\. AnLL\-layer HGNN is then simulated by anLL\-layer DPA3, and the result follows from[theorem26](https://arxiv.org/html/2609.00528#Thmtheorem26)\. Each simulated layer realizes a map uniformly continuous on compact sets, so an input error entering a layer propagates to a controlled output error\. Since there are finitely many layers, the per\-layer simulation accuracies can be chosen so that the final node features are within any prescribed tolerance of the HGNN output onKK\.

Setrc\(3\)=rc\(2\)=rcr\_\{c\}^\{\(3\)\}=r\_\{c\}^\{\(2\)\}=r\_\{c\}\. Writeρ~3\\tilde\{\\rho\}\_\{3\}forsa​\(dj​k\)​ρ3​\(𝒏j,𝒏i,𝒏k,dj​i,dj​k,cos⁡θi​j​k\)s\_\{a\}\(d\_\{jk\}\)\\,\\rho\_\{3\}\(\\bm\{n\}\_\{j\},\\,\\bm\{n\}\_\{i\},\\,\\bm\{n\}\_\{k\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\), and define the target intermediate function as

Fj​i=∑k∈𝒩⁡\(j\)∖\{i\}ρ~3​\(𝒏j,𝒏i,𝒏k,dj​i,dj​k,cos⁡θi​j​k\),F\_\{ji\}=\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}\\tilde\{\\rho\}\_\{3\}\(\\bm\{n\}\_\{j\},\\,\\bm\{n\}\_\{i\},\\,\\bm\{n\}\_\{k\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\),so that the HGNN output \([17](https://arxiv.org/html/2609.00528#S2.E17)\) isψ3​\(∑i∈𝒩⁡\(j\)sa​\(dj​i\)​Fj​i\)\\psi\_\{3\}\\bigl\(\\sum\_\{i\\in\\mathcal\{N\}\(j\)\}s\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}\\bigr\)\. Fix a layer indexlland write𝒏\(l\)\\bm\{n\}^\{\(l\)\}for its input node features\. We exhibit weights for which the four updates \([18](https://arxiv.org/html/2609.00528#S2.E18)\)–\([21](https://arxiv.org/html/2609.00528#S2.E21)\) carry𝒏\(l\)\\bm\{n\}^\{\(l\)\}to the output of one HGNN layer\. Throughout, we maintain the invariant that a designated coordinate of𝒆j​i\(l\)\\bm\{e\}\_\{ji\}^\{\(l\)\}carries the distancedj​id\_\{ji\}and a designated coordinate of𝒂j;i,k\(l\)\\bm\{a\}\_\{j;i,k\}^\{\(l\)\}carriescos⁡θi​j​k\\cos\\theta\_\{ijk\}\. At initialization this holds by choosing distance and angle embeddings that writedj​id\_\{ji\}andcos⁡θi​j​k\\cos\\theta\_\{ijk\}into those coordinates\.

*Step 1: the self\-message loads node features into the edges\.*At initialization, the edge features depend only on distances,𝒆j​i\(0\)=ϕd​\(dj​i\)\\bm\{e\}\_\{ji\}^\{\(0\)\}=\\phi\_\{d\}\(d\_\{ji\}\)\. The self\-message \([18](https://arxiv.org/html/2609.00528#S2.E18)\) gives

𝒆j​i\(l\+12\)=𝒆j​i\(l\)\+δs\(1\)​ϕs\(1\)​\(𝒏j\(l\),𝒏i\(l\),𝒆j​i\(l\)\)\.\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}=\\bm\{e\}\_\{ji\}^\{\(l\)\}\+\\delta\_\{s\}^\{\(1\)\}\\,\\phi\_\{s\}^\{\(1\)\}\\bigl\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\,\\bm\{n\}\_\{i\}^\{\(l\)\},\\,\\bm\{e\}\_\{ji\}^\{\(l\)\}\\bigr\)\.\(35\)The MLPϕs\(1\)\\phi\_\{s\}^\{\(1\)\}receives𝒆j​i\(l\)\\bm\{e\}\_\{ji\}^\{\(l\)\}along with both endpoint features, so by MLP universality it can approximate𝒙↦\(𝒙−𝒆j​i\(l\)\)/δs\(1\)\\bm\{x\}\\mapsto\(\\bm\{x\}\-\\bm\{e\}\_\{ji\}^\{\(l\)\}\)/\\delta\_\{s\}^\{\(1\)\}for any continuous target𝒙\\bm\{x\}\. With sufficiently large feature dimensiondd, we may choose the weights so that𝒆j​i\(l\+12\)\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}encodes\(𝒏j\(l\),𝒏i\(l\),dj​i\)\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\bm\{n\}\_\{i\}^\{\(l\)\},d\_\{ji\}\)in designated coordinates\.

*Step 2: the angle self\-message\.*The angle update \([19](https://arxiv.org/html/2609.00528#S2.E19)\) gives𝒂j;i,k\(l\+1\)=𝒂j;i,k\(l\)\+δs\(2\)​ϕs\(2\)​\(𝒆j​i\(l\+12\),𝒆j​k\(l\+12\),𝒂j;i,k\(l\)\)\\bm\{a\}\_\{j;i,k\}^\{\(l\+1\)\}=\\bm\{a\}\_\{j;i,k\}^\{\(l\)\}\+\\delta\_\{s\}^\{\(2\)\}\\,\\phi\_\{s\}^\{\(2\)\}\(\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{e\}\_\{jk\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{a\}\_\{j;i,k\}^\{\(l\)\}\)\. We takeϕs\(2\)=0\\phi\_\{s\}^\{\(2\)\}=0, so the angle features pass through unchanged and the invariant is preserved\. By Step 1 and the invariant, the three arguments ofϕc\(2\)\\phi\_\{c\}^\{\(2\)\}in the next step then carry all six arguments ofρ~3\\tilde\{\\rho\}\_\{3\}\.

*Step 3: the angle aggregation producesFj​iF\_\{ji\}\.*The edge update \([20](https://arxiv.org/html/2609.00528#S2.E20)\) reads

𝒆j​i\(l\+1\)=𝒆j​i\(l\+12\)\+δu\(2\)​ϕu\(2\)​\(∑k∈𝒩⁡\(j\)∖\{i\}wj​i​k​ϕc\(2\)​\(𝒆j​i\(l\+12\),𝒆j​k\(l\+12\),𝒂j;i,k\(l\+1\)\)\)\.\\bm\{e\}\_\{ji\}^\{\(l\+1\)\}=\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}\+\\delta\_\{u\}^\{\(2\)\}\\,\\phi\_\{u\}^\{\(2\)\}\\Bigl\(\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}w\_\{jik\}\\,\\phi\_\{c\}^\{\(2\)\}\\bigl\(\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{e\}\_\{jk\}^\{\(l\+\\frac\{1\}\{2\}\)\},\\,\\bm\{a\}\_\{j;i,k\}^\{\(l\+1\)\}\\bigr\)\\Bigr\)\.By Steps 1 and 2 each summand receives all six arguments ofρ~3\\tilde\{\\rho\}\_\{3\}through a continuous encoding, so by MLP universalityϕc\(2\)\\phi\_\{c\}^\{\(2\)\}can approximate any continuous function of them\. The prefactorwj​i​k=sa\(3\)​\(dj​i\)​sa\(3\)​\(dj​k\)w\_\{jik\}=s\_\{a\}^\{\(3\)\}\(d\_\{ji\}\)\\,s\_\{a\}^\{\(3\)\}\(d\_\{jk\}\)is positive and bounded below onKK, because genericity keeps all distances belowrcr\_\{c\}, the setKKis compact, andsa\(3\)s\_\{a\}^\{\(3\)\}is positive on\[0,rc\)\[0,r\_\{c\}\)\. Lettingϕc\(2\)\\phi\_\{c\}^\{\(2\)\}approximate the quotient of the desired summand bywj​i​kw\_\{jik\}, whose arguments are available through the edge features, the productwj​i​k​ϕc\(2\)w\_\{jik\}\\,\\phi\_\{c\}^\{\(2\)\}realizes the summand\. The sum overkkcomposed with the outer MLPϕu\(2\)\\phi\_\{u\}^\{\(2\)\}has DeepSets form, so by[lemma15](https://arxiv.org/html/2609.00528#Thmtheorem15)it approximates any continuous symmetric function of the multiset\{\(𝒏k\(l\),dj​k,cos⁡θi​j​k\)\}k≠i\\\{\(\\bm\{n\}\_\{k\}^\{\(l\)\},d\_\{jk\},\\cos\\theta\_\{ijk\}\)\\\}\_\{k\\neq i\}\. The target below also depends on the parameters\(𝒏j\(l\),𝒏i\(l\),dj​i\)\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\bm\{n\}\_\{i\}^\{\(l\)\},d\_\{ji\}\), which the outer MLP recovers from the sum as follows\. The inner MLP writes the parameters and the constant11into designated coordinates, the sum accumulates the parameters times the summand count and the count itself, and the quotient of the two returns the parameters\. The count is at least one, because the spanning triple that the overlap condition provides for the pair\(j,i\)\(j,i\)contains an atom of𝒩⁡\(j\)∖\{i\}\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\. Choose the target\(𝒙−𝒆j​i\(l\+12\)\)/δu\(2\)\(\\bm\{x\}\-\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}\)/\\delta\_\{u\}^\{\(2\)\}with𝒙\\bm\{x\}encoding\(Fj​i,dj​i\)\(F\_\{ji\},d\_\{ji\}\)in designated coordinates, the distance coordinate taken unchanged from𝒆j​i\(l\+12\)\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}\. The target is symmetric because𝒆j​i\(l\+12\)\\bm\{e\}\_\{ji\}^\{\(l\+\\frac\{1\}\{2\}\)\}is constant across the sum overkk, so the residual update gives𝒆j​i\(l\+1\)≈𝒙\\bm\{e\}\_\{ji\}^\{\(l\+1\)\}\\approx\\bm\{x\}and the invariant is preserved\.

*Step 4: the edge aggregation produces the HGNN output\.*The node update \([21](https://arxiv.org/html/2609.00528#S2.E21)\) reads

𝒏j\(l\+1\)=𝒏j\(l\)\+δu\(1\)​ϕu\(1\)​\(∑i∈𝒩⁡\(j\)wj​i​ϕc\(1\)​\(𝒏j\(l\),𝒏i\(l\),𝒆j​i\(l\+1\)\)\)\.\\bm\{n\}\_\{j\}^\{\(l\+1\)\}=\\bm\{n\}\_\{j\}^\{\(l\)\}\+\\delta\_\{u\}^\{\(1\)\}\\,\\phi\_\{u\}^\{\(1\)\}\\Bigl\(\\sum\_\{i\\in\\mathcal\{N\}\(j\)\}w\_\{ji\}\\,\\phi\_\{c\}^\{\(1\)\}\\bigl\(\\bm\{n\}\_\{j\}^\{\(l\)\},\\,\\bm\{n\}\_\{i\}^\{\(l\)\},\\,\\bm\{e\}\_\{ji\}^\{\(l\+1\)\}\\bigr\)\\Bigr\)\.Since𝒆j​i\(l\+1\)\\bm\{e\}\_\{ji\}^\{\(l\+1\)\}encodes\(Fj​i,dj​i\)\(F\_\{ji\},d\_\{ji\}\), the prefactorwj​i=sa\(2\)​\(dj​i\)w\_\{ji\}=s\_\{a\}^\{\(2\)\}\(d\_\{ji\}\)is likewise positive and bounded below onKK, andϕc\(1\)\\phi\_\{c\}^\{\(1\)\}may approximate the quotientsa​\(dj​i\)​Fj​i/wj​is\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}/w\_\{ji\}, so thatwj​i​ϕc\(1\)​\(…\)≈sa​\(dj​i\)​Fj​iw\_\{ji\}\\,\\phi\_\{c\}^\{\(1\)\}\(\\ldots\)\\approx s\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}\. The sum then gives∑iwj​i​ϕc\(1\)​\(…\)≈∑isa​\(dj​i\)​Fj​i\\sum\_\{i\}w\_\{ji\}\\,\\phi\_\{c\}^\{\(1\)\}\(\\ldots\)\\approx\\sum\_\{i\}s\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}\. The target of the outer MLPϕu\(1\)\\phi\_\{u\}^\{\(1\)\}is\(ψ3​\(∑isa​\(dj​i\)​Fj​i\)−𝒏j\(l\)\)/δu\(1\)\(\\psi\_\{3\}\(\\sum\_\{i\}s\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}\)\-\\bm\{n\}\_\{j\}^\{\(l\)\}\)/\\delta\_\{u\}^\{\(1\)\}, which depends on the parameter𝒏j\(l\)\\bm\{n\}\_\{j\}^\{\(l\)\}in addition to the aggregate\. The device of Step 3 recovers𝒏j\(l\)\\bm\{n\}\_\{j\}^\{\(l\)\}from the sum, whose summand count\|𝒩⁡\(j\)\|\|\\mathcal\{N\}\(j\)\|is positive because the overlap condition grants every center a neighbor\. Hence𝒏j\(l\+1\)≈ψ3​\(∑isa​\(dj​i\)​Fj​i\)\\bm\{n\}\_\{j\}^\{\(l\+1\)\}\\approx\\psi\_\{3\}\(\\sum\_\{i\}s\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}\), the output of one HGNN layer\.

*Multi\-layer extension\.*Steps 1–4 consume the layer’s input node features𝒏\(l\)\\bm\{n\}^\{\(l\)\}together with the geometric data maintained by the invariant, produce the output𝒏\(l\+1\)\\bm\{n\}^\{\(l\+1\)\}, and preserve the invariant, while the self\-message reloads the current node features at the start of every layer\. Stacking therefore needs no separate loading layer, and anLL\-layer HGNN is simulated by anLL\-layer DPA3\.

## 2CHGNet

CHGNet\[[13](https://arxiv.org/html/2609.00528#bib.bib13)\]uses the same atom graph and bond graph as DPA3 withK=2K=2: atoms as nodes, bonds withinrcutr\_\{\\mathrm\{cut\}\}as edges, and the line graphℒ⁡\(G\)\\mathcal\{L\}\(G\)encoding angles\. We formalize the architecture as written in Eq\. \(2\) of\[[13](https://arxiv.org/html/2609.00528#bib.bib13)\], with*directed*edges, so that each bond\{j,i\}\\\{j,i\\\}carries two features𝒆j​i\\bm\{e\}\_\{ji\}and𝒆i​j\\bm\{e\}\_\{ij\}\.333The released implementation instead shares one feature per bond \(𝒆j​i=𝒆i​j\\bm\{e\}\_\{ji\}=\\bm\{e\}\_\{ij\}, a single node of the bond graph\) and aggregates the angles centered at both endpoints into it through the same message functionϕe\\phi\_\{e\}\. Our simulation argument relies on the directed form and does not cover this bidirectional variant\.CHGNet maintains three feature types: node features𝒏jl\\bm\{n\}\_\{j\}^\{l\}, edge features𝒆j​il\\bm\{e\}\_\{ji\}^\{l\}onGG, and angle features𝒂i​j​kl\\bm\{a\}\_\{ijk\}^\{l\}onℒ⁡\(G\)\\mathcal\{L\}\(G\)\. The initial features are element embeddings \(nodes\), smooth radial Bessel function expansions \(edges\), and Fourier basis expansions of angles\.

Each layer updates all three feature types\. The node update aggregates over neighbors:

𝒏jl\+1=𝒏jl\+Lv​\[∑i∈𝒩⁡\(j\)𝒆j​i0⊙ϕv​\(𝒏il,𝒏jl,𝒆j​il\)\]\.\\bm\{n\}\_\{j\}^\{l\+1\}=\\bm\{n\}\_\{j\}^\{l\}\+L\_\{v\}\\Bigl\[\\sum\_\{i\\in\\mathcal\{N\}\(j\)\}\\bm\{e\}\_\{ji\}^\{0\}\\odot\\phi\_\{v\}\\bigl\(\\bm\{n\}\_\{i\}^\{l\},\\,\\bm\{n\}\_\{j\}^\{l\},\\,\\bm\{e\}\_\{ji\}^\{l\}\\bigr\)\\Bigr\]\.\(36\)The bond update aggregates over the angles centered at the tail atomjj:

𝒆j​il\+1=𝒆j​il\+Le​\[∑k∈𝒩⁡\(j\)∖\{i\}𝒆j​i0⊙𝒆j​k0⊙ϕe​\(𝒆j​kl,𝒆j​il,𝒂k​j​il,𝒏jl\+1\)\]\.\\bm\{e\}\_\{ji\}^\{l\+1\}=\\bm\{e\}\_\{ji\}^\{l\}\+L\_\{e\}\\Bigl\[\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}\\bm\{e\}\_\{ji\}^\{0\}\\odot\\bm\{e\}\_\{jk\}^\{0\}\\odot\\phi\_\{e\}\\bigl\(\\bm\{e\}\_\{jk\}^\{l\},\\,\\bm\{e\}\_\{ji\}^\{l\},\\,\\bm\{a\}\_\{kji\}^\{l\},\\,\\bm\{n\}\_\{j\}^\{l\+1\}\\bigr\)\\Bigr\]\.\(37\)Both edge arguments ofϕe\\phi\_\{e\}are the copies directed away from the shared centerjj\. The angle update is:

𝒂i​j​kl\+1=𝒂i​j​kl\+ϕa​\(𝒆j​il\+1,𝒆j​kl\+1,𝒂i​j​kl,𝒏jl\+1\)\.\\bm\{a\}\_\{ijk\}^\{l\+1\}=\\bm\{a\}\_\{ijk\}^\{l\}\+\\phi\_\{a\}\\bigl\(\\bm\{e\}\_\{ji\}^\{l\+1\},\\,\\bm\{e\}\_\{jk\}^\{l\+1\},\\,\\bm\{a\}\_\{ijk\}^\{l\},\\,\\bm\{n\}\_\{j\}^\{l\+1\}\\bigr\)\.\(38\)HereLvL\_\{v\},LeL\_\{e\}are linear layers,ϕv\\phi\_\{v\},ϕe\\phi\_\{e\},ϕa\\phi\_\{a\}are gated MLPs \(MLPs with sigmoid gating on intermediate activations\), and⊙\\odotdenotes element\-wise multiplication\. The gate𝒆j​i0\\bm\{e\}\_\{ji\}^\{0\}is a trainable linear image of a smooth radial basis expansion of the bond distance, multiplied by a fixed envelope that vanishes at the cutoff\. By default, CHGNet stacks three full interaction blocks followed by one block with only the node update, and uses a feature dimension of6464\. The total energy isEθ=∑jMLP⁡\(𝒏jL\)E\_\{\\theta\}=\\sum\_\{j\}\\mathrm\{MLP\}\(\\bm\{n\}\_\{j\}^\{L\}\)\.

###### Proof of[corollary29](https://arxiv.org/html/2609.00528#Thmtheorem29)\.

We show that two CHGNet layers can approximate the HGNN’s per\-layer computation up to the outer readoutψ3\\psi\_\{3\}, which is absorbed into subsequent blocks or the readout network\. As in the proof of[corollary27](https://arxiv.org/html/2609.00528#Thmtheorem27), writeρ~3\\tilde\{\\rho\}\_\{3\}forsa​\(dj​k\)​ρ3​\(𝒏j,𝒏i,𝒏k,dj​i,dj​k,cos⁡θi​j​k\)s\_\{a\}\(d\_\{jk\}\)\\,\\rho\_\{3\}\(\\bm\{n\}\_\{j\},\\bm\{n\}\_\{i\},\\bm\{n\}\_\{k\},d\_\{ji\},d\_\{jk\},\\cos\\theta\_\{ijk\}\)andFj​i=∑k∈𝒩⁡\(j\)∖\{i\}ρ~3F\_\{ji\}=\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}\\tilde\{\\rho\}\_\{3\}, so that the HGNN layer output isψ3​\(∑i∈𝒩⁡\(j\)sa​\(dj​i\)​Fj​i\)\\psi\_\{3\}\\bigl\(\\sum\_\{i\\in\\mathcal\{N\}\(j\)\}s\_\{a\}\(d\_\{ji\}\)\\,F\_\{ji\}\\bigr\)\.

The gates are handled once and for all\. SinceKKis compact and the displacement vectors range overBrc∖\{0\}B\_\{r\_\{c\}\}\\setminus\\\{0\\\}, all interatomic distances inKKlie in an interval\[δ,rc−η\]\[\\delta,r\_\{c\}\-\\eta\]withδ,η\>0\\delta,\\eta\>0\. Choose the trainable coefficients of every gate𝒆0\\bm\{e\}^\{0\}so that each component equals the first radial basis function times the envelope, which is bounded below on\[δ,rc−η\]\[\\delta,r\_\{c\}\-\\eta\]\. Absorbing a gate into a message function then means letting the MLP approximate the quotient of the target by the gate, which is continuous onKK\.

At initialization, edge features carry only distance information:𝒆j​i\(0\)=ϕd​\(dj​i\)\\bm\{e\}\_\{ji\}^\{\(0\)\}=\\phi\_\{d\}\(d\_\{ji\}\), and node features are type embeddings:𝒏k\(0\)=type​\_​embed​\(zk\)\\bm\{n\}\_\{k\}^\{\(0\)\}=\\mathrm\{type\\\_embed\}\(z\_\{k\}\)\.

*Step 1: node update\.*The node update \([36](https://arxiv.org/html/2609.00528#S2.E36)\) computes

𝒏j\(1\)=𝒏j\(0\)\+Lv​\[∑m∈𝒩⁡\(j\)𝒆j​m\(0\)⊙ϕv​\(𝒏m\(0\),𝒏j\(0\),𝒆j​m\(0\)\)\],\\bm\{n\}\_\{j\}^\{\(1\)\}=\\bm\{n\}\_\{j\}^\{\(0\)\}\+L\_\{v\}\\Bigl\[\\sum\_\{m\\in\\mathcal\{N\}\(j\)\}\\bm\{e\}\_\{jm\}^\{\(0\)\}\\odot\\phi\_\{v\}\\bigl\(\\bm\{n\}\_\{m\}^\{\(0\)\},\\,\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\bm\{e\}\_\{jm\}^\{\(0\)\}\\bigr\)\\Bigr\],whereLvL\_\{v\}is a linear layer andϕv\\phi\_\{v\}is an MLP\. SinceLvL\_\{v\}is linear,𝒏j\(1\)\\bm\{n\}\_\{j\}^\{\(1\)\}alone does not necessarily encode the full neighbor multiset\. However,𝒏j\(1\)\\bm\{n\}\_\{j\}^\{\(1\)\}will serve as input to the subsequent bond update MLPϕe\\phi\_\{e\}, and it is the*composition*that provides universality\.

*Step 2: bond update approximatesρ~3\\tilde\{\\rho\}\_\{3\}per summand\.*The bond update \([37](https://arxiv.org/html/2609.00528#S2.E37)\) for the directed edge\(j,i\)\(j,i\)is computed withLe=IL\_\{e\}=I:

𝒆j​i\(1\)=𝒆j​i\(0\)\+∑k∈𝒩⁡\(j\)∖\{i\}𝒆j​i\(0\)⊙𝒆j​k\(0\)⊙ϕe​\(𝒆j​k\(0\),𝒆j​i\(0\),𝒂k​j​i\(0\),𝒏j\(1\)\)\.\\bm\{e\}\_\{ji\}^\{\(1\)\}=\\bm\{e\}\_\{ji\}^\{\(0\)\}\+\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}\\bm\{e\}\_\{ji\}^\{\(0\)\}\\odot\\bm\{e\}\_\{jk\}^\{\(0\)\}\\odot\\phi\_\{e\}\\bigl\(\\bm\{e\}\_\{jk\}^\{\(0\)\},\\,\\bm\{e\}\_\{ji\}^\{\(0\)\},\\,\\bm\{a\}\_\{kji\}^\{\(0\)\},\\,\\bm\{n\}\_\{j\}^\{\(1\)\}\\bigr\)\.\(39\)Substituting the expression for𝒏j\(1\)\\bm\{n\}\_\{j\}^\{\(1\)\}, we obtain that each summand has the form

𝒆j​i\(0\)⊙𝒆j​k\(0\)⊙ϕe​\(ϕd​\(dj​k\),ϕd​\(dj​i\),𝒂k​j​i\(0\),𝒏j\(0\)\+Lv​\[∑m𝒆j​m\(0\)⊙ϕv​\(𝒏m\(0\),𝒏j\(0\),𝒆j​m\(0\)\)\]\)\.\\bm\{e\}\_\{ji\}^\{\(0\)\}\\odot\\bm\{e\}\_\{jk\}^\{\(0\)\}\\odot\\phi\_\{e\}\\Bigl\(\\phi\_\{d\}\(d\_\{jk\}\),\\,\\phi\_\{d\}\(d\_\{ji\}\),\\,\\bm\{a\}\_\{kji\}^\{\(0\)\},\\;\\bm\{n\}\_\{j\}^\{\(0\)\}\+L\_\{v\}\\bigl\[\\textstyle\\sum\_\{m\}\\bm\{e\}\_\{jm\}^\{\(0\)\}\\odot\\phi\_\{v\}\(\\bm\{n\}\_\{m\}^\{\(0\)\},\\,\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\bm\{e\}\_\{jm\}^\{\(0\)\}\)\\bigr\]\\Bigr\)\.This is a function of the per\-angle data\(dj​k,dj​i,cos⁡θk​j​i\)\(d\_\{jk\},d\_\{ji\},\\cos\\theta\_\{kji\}\)and the multiset𝒮j=\{\(𝒏m\(0\),dj​m\)\}m∈𝒩⁡\(j\)\\mathcal\{S\}\_\{j\}=\\\{\(\\bm\{n\}\_\{m\}^\{\(0\)\},d\_\{jm\}\)\\\}\_\{m\\in\\mathcal\{N\}\(j\)\}\. The dependence on𝒮j\\mathcal\{S\}\_\{j\}enters through𝒏j\(0\)\+Lv​\[∑mgm\]\\bm\{n\}\_\{j\}^\{\(0\)\}\+L\_\{v\}\[\\sum\_\{m\}g\_\{m\}\], wheregm=𝒆j​m\(0\)⊙ϕv​\(𝒏m\(0\),𝒏j\(0\),𝒆j​m\(0\)\)g\_\{m\}=\\bm\{e\}\_\{jm\}^\{\(0\)\}\\odot\\phi\_\{v\}\(\\bm\{n\}\_\{m\}^\{\(0\)\},\\bm\{n\}\_\{j\}^\{\(0\)\},\\bm\{e\}\_\{jm\}^\{\(0\)\}\)\. The compositionϕe​\(…,𝒏j\(0\)\+Lv​\[∑mgm\]\)\\phi\_\{e\}\(\\ldots,\\;\\bm\{n\}\_\{j\}^\{\(0\)\}\+L\_\{v\}\[\\sum\_\{m\}g\_\{m\}\]\)has the extended DeepSets form\. The inner functiongmg\_\{m\}is parametrized by an MLPϕv\\phi\_\{v\}\. With sufficiently large feature dimension, the type embeddings are chosen supported on one coordinate block and the image ofLvL\_\{v\}on the complementary block, so the fourth argument ofϕe\\phi\_\{e\}carries the center feature𝒏j\(0\)\\bm\{n\}\_\{j\}^\{\(0\)\}and the aggregateLv​\[∑mgm\]L\_\{v\}\[\\sum\_\{m\}g\_\{m\}\]in disjoint blocks\. By[lemma15](https://arxiv.org/html/2609.00528#Thmtheorem15), there exist weights forϕv\\phi\_\{v\}andϕe\\phi\_\{e\}such that each summand approximates any continuous function of\(dj​k,dj​i,cos⁡θk​j​i\)\(d\_\{jk\},d\_\{ji\},\\cos\\theta\_\{kji\}\), the center feature, and the multiset𝒮j\\mathcal\{S\}\_\{j\}, symmetric in the multiset argument\. The lemma applies in this parametrized form because the per\-angle data and the center feature enter the outer MLP as additional inputs, and the approximation is uniform on their compact range\.

The target per\-angle function isρ~3​\(𝒏j\(0\),𝒏i\(0\),𝒏k\(0\),dj​i,dj​k,cos⁡θi​j​k\)\\tilde\{\\rho\}\_\{3\}\(\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\bm\{n\}\_\{i\}^\{\(0\)\},\\,\\bm\{n\}\_\{k\}^\{\(0\)\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\), depending on specific entries𝒏i\(0\)\\bm\{n\}\_\{i\}^\{\(0\)\},𝒏k\(0\)\\bm\{n\}\_\{k\}^\{\(0\)\}of the multiset\. Under strong genericity \([definition28](https://arxiv.org/html/2609.00528#Thmtheorem28)\), all neighbors ofjjhave distinct distances\. Define the lookup functionλ:ℳ≤M​\(ℝd×ℝ\>0\)×ℝ\>0→ℝd\\lambda:\\mathcal\{M\}\_\{\\leq M\}\(\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}\_\{\>0\}\)\\times\\mathbb\{R\}\_\{\>0\}\\to\\mathbb\{R\}^\{d\}byλ⁡\(𝒮j,d\)=𝒏m\(0\)\\lambda\(\\mathcal\{S\}\_\{j\},d\)=\\bm\{n\}\_\{m\}^\{\(0\)\}, wheremmis the unique element of𝒩⁡\(j\)\\mathcal\{N\}\(j\)satisfyingdj​m=dd\_\{jm\}=d\. On strongly generic configurations this is well\-defined and symmetric in𝒮j\\mathcal\{S\}\_\{j\}\. On the compact setKKthe distances of distinct neighbors are separated by a positive gap, soλ\\lambdais continuous\. Rewriting:

ρ~3​\(𝒏j\(0\),𝒏i\(0\),𝒏k\(0\),…\)=ρ~3​\(𝒏j\(0\),λ⁡\(𝒮j,dj​i\),λ⁡\(𝒮j,dj​k\),…\),\\tilde\{\\rho\}\_\{3\}\(\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\bm\{n\}\_\{i\}^\{\(0\)\},\\,\\bm\{n\}\_\{k\}^\{\(0\)\},\\,\\ldots\)=\\tilde\{\\rho\}\_\{3\}\\bigl\(\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\lambda\(\\mathcal\{S\}\_\{j\},d\_\{ji\}\),\\,\\lambda\(\\mathcal\{S\}\_\{j\},d\_\{jk\}\),\\,\\ldots\\bigr\),which is a continuous symmetric function of𝒮j\\mathcal\{S\}\_\{j\}and the per\-angle data\. The gate prefactors𝒆j​i\(0\)⊙𝒆j​k\(0\)⊙\(⋅\)\\bm\{e\}\_\{ji\}^\{\(0\)\}\\odot\\bm\{e\}\_\{jk\}^\{\(0\)\}\\odot\(\\cdot\)are absorbed by the gate convention fixed at the beginning of the proof, andθk​j​i=θi​j​k\\theta\_\{kji\}=\\theta\_\{ijk\}\. Therefore, each summand in \([39](https://arxiv.org/html/2609.00528#S2.E39)\) approximatesρ~3\\tilde\{\\rho\}\_\{3\}\. Since the sum in \([39](https://arxiv.org/html/2609.00528#S2.E39)\) is a direct linear sum of per\-angle terms, we obtain

𝒆j​i\(1\)≈ϕd​\(dj​i\)\+∑k∈𝒩⁡\(j\)∖\{i\}ρ~3​\(𝒏j\(0\),𝒏i\(0\),𝒏k\(0\),dj​i,dj​k,cos⁡θi​j​k\)\.\\bm\{e\}\_\{ji\}^\{\(1\)\}\\approx\\phi\_\{d\}\(d\_\{ji\}\)\+\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}\\tilde\{\\rho\}\_\{3\}\(\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\bm\{n\}\_\{i\}^\{\(0\)\},\\,\\bm\{n\}\_\{k\}^\{\(0\)\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\)\.That is,𝒆j​i\(1\)≈ϕd​\(dj​i\)\+Fj​i\\bm\{e\}\_\{ji\}^\{\(1\)\}\\approx\\phi\_\{d\}\(d\_\{ji\}\)\+F\_\{ji\}, and the oppositely directed copy stores𝒆i​j\(1\)≈ϕd​\(dj​i\)\+Fi​j\\bm\{e\}\_\{ij\}^\{\(1\)\}\\approx\\phi\_\{d\}\(d\_\{ji\}\)\+F\_\{ij\}\. The summands are written into coordinates on whichϕd\\phi\_\{d\}vanishes, so𝒆j​i\(1\)\\bm\{e\}\_\{ji\}^\{\(1\)\}carriesϕd​\(dj​i\)\\phi\_\{d\}\(d\_\{ji\}\)andFj​iF\_\{ji\}in disjoint blocks\.

*Step 3: node update computes the HGNN double sum\.*The node update \([36](https://arxiv.org/html/2609.00528#S2.E36)\) withLv=IL\_\{v\}=Icomputes

𝒏j\(2\)=𝒏j\(1\)\+∑i∈𝒩⁡\(j\)𝒆j​i\(0\)⊙ϕv​\(𝒏i\(1\),𝒏j\(1\),𝒆j​i\(1\)\)\.\\bm\{n\}\_\{j\}^\{\(2\)\}=\\bm\{n\}\_\{j\}^\{\(1\)\}\+\\sum\_\{i\\in\\mathcal\{N\}\(j\)\}\\bm\{e\}\_\{ji\}^\{\(0\)\}\\odot\\phi\_\{v\}\\bigl\(\\bm\{n\}\_\{i\}^\{\(1\)\},\\,\\bm\{n\}\_\{j\}^\{\(1\)\},\\,\\bm\{e\}\_\{ji\}^\{\(1\)\}\\bigr\)\.Each summand receives𝒆j​i\(1\)\\bm\{e\}\_\{ji\}^\{\(1\)\}, the copy directed away fromjj, which stores the inner sumFj​iF\_\{ji\}by Step 2\. By MLP universality,ϕv\\phi\_\{v\}can extract this inner sum from𝒆j​i\(1\)\\bm\{e\}\_\{ji\}^\{\(1\)\}and multiply bysa​\(dj​i\)s\_\{a\}\(d\_\{ji\}\), where the distancedj​id\_\{ji\}is available through the retained blockϕd​\(dj​i\)\\phi\_\{d\}\(d\_\{ji\}\)of𝒆j​i\(1\)\\bm\{e\}\_\{ji\}^\{\(1\)\}, and the gating factor𝒆j​i\(0\)⊙\(⋅\)\\bm\{e\}\_\{ji\}^\{\(0\)\}\\odot\(\\cdot\)is absorbed by the gate convention\. The sum overiithen gives

𝒏j\(2\)≈𝒏j\(1\)\+∑i,k∈𝒩⁡\(j\)i≠ksa​\(dj​i\)​ρ~3​\(𝒏j\(0\),𝒏i\(0\),𝒏k\(0\),dj​i,dj​k,cos⁡θi​j​k\),\\bm\{n\}\_\{j\}^\{\(2\)\}\\approx\\bm\{n\}\_\{j\}^\{\(1\)\}\+\\sum\_\{\\begin\{subarray\}\{c\}i,k\\in\\mathcal\{N\}\(j\)\\\\ i\\neq k\\end\{subarray\}\}s\_\{a\}\(d\_\{ji\}\)\\,\\tilde\{\\rho\}\_\{3\}\(\\bm\{n\}\_\{j\}^\{\(0\)\},\\,\\bm\{n\}\_\{i\}^\{\(0\)\},\\,\\bm\{n\}\_\{k\}^\{\(0\)\},\\,d\_\{ji\},\\,d\_\{jk\},\\,\\cos\\theta\_\{ijk\}\),which is the HGNN double sum \([17](https://arxiv.org/html/2609.00528#S2.E17)\)*without*the outer readoutψ3\\psi\_\{3\}\.

*Step 4: absorbingψ3\\psi\_\{3\}\.*The HGNN layer output isψ3​\(∑i,ksa​\(dj​i\)​ρ~3\)\\psi\_\{3\}\(\\sum\_\{i,k\}s\_\{a\}\(d\_\{ji\}\)\\,\\tilde\{\\rho\}\_\{3\}\), but the CHGNet layer produces𝒏j\(1\)\+∑i,ksa​\(dj​i\)​ρ~3\\bm\{n\}\_\{j\}^\{\(1\)\}\+\\sum\_\{i,k\}s\_\{a\}\(d\_\{ji\}\)\\,\\tilde\{\\rho\}\_\{3\}\. With sufficiently large feature dimension, the increment of Step 3 is written into coordinates on which𝒏j\(1\)\\bm\{n\}\_\{j\}^\{\(1\)\}vanishes, so the double sum is recoverable from𝒏j\(2\)\\bm\{n\}\_\{j\}^\{\(2\)\}\. The missing nonlinearityψ3\\psi\_\{3\}is handled differently in intermediate and final layers\. In an intermediate layer, the next CHGNet block begins with a node update \([36](https://arxiv.org/html/2609.00528#S2.E36)\), whose MLPϕv​\(𝒏m\(2\),𝒏j\(2\),𝒆j​m\(2\)\)\\phi\_\{v\}\(\\bm\{n\}\_\{m\}^\{\(2\)\},\\bm\{n\}\_\{j\}^\{\(2\)\},\\bm\{e\}\_\{jm\}^\{\(2\)\}\)receives each neighbor’s double\-sum feature and the center’s own\. Being an MLP, it can applyψ3\\psi\_\{3\}to these inputs as part of its computation, so𝒏j\(3\)\\bm\{n\}\_\{j\}^\{\(3\)\}encodesψ3\\psi\_\{3\}\-transformed neighbor information, and the subsequent bond update can compute the next layer’sρ~3\\tilde\{\\rho\}\_\{3\}from it by the extended DeepSets argument of Step 2\. In the final layer, the readout network is an MLP applied after the last CHGNet layer, so it can approximate the composition of the extraction of the double sum,ψ3\\psi\_\{3\}, and the HGNN readoutfθf\_\{\\theta\}\.

*Multi\-layer extension\.*Each HGNN layer is simulated by two CHGNet layers\. The first performs the node and bond updates of Steps 1–2, computing the per\-angleρ~3\\tilde\{\\rho\}\_\{3\}and storing the inner sums in the edge features, absorbing the previous layer’sψ3\\psi\_\{3\}whenl≥2l\\geq 2as in Step 4\. The second performs the node update of Step 3, which accumulates the double sum into the node features\. The angle updates are set to approximate zero, so the angle features retain the initial encoding ofcos⁡θi​j​k\\cos\\theta\_\{ijk\}at every layer\. The bond update of the second layer of each pair is not used and is suppressed by settingLe=0L\_\{e\}=0\. Each simulated layer realizes a map uniformly continuous on compact sets, so an input error entering a layer propagates to a controlled output error\. Since there are finitely many layers, the per\-layer simulation accuracies can be chosen so that the final node features are within any prescribed tolerance of the HGNN output onKK\. Therefore, anLL\-layer HGNN is simulated by2​L2LCHGNet layers, with the finalψ3\\psi\_\{3\}absorbed into the readout, and the result follows from[theorem26](https://arxiv.org/html/2609.00528#Thmtheorem26)\.

## 3ALIGNN

ALIGNN\[[11](https://arxiv.org/html/2609.00528#bib.bib11)\]operates on two graphs: an*atom graph*GG, whose nodes are atoms and whose edges are the bonds joining pairs of atoms within the cutoffrcr\_\{c\}, and its*line graph*ℒ⁡\(G\)\\mathcal\{L\}\(G\)\.444We describe all three architectures with the distance cutoff used throughout this paper\. The reference implementation of ALIGNN instead joins each atom to its1212nearest neighbors within a cutoff of88Å\. This affects neither[remark30](https://arxiv.org/html/2609.00528#Thmtheorem30)nor any statement below, which depend only on the form of the message functions\.As in[section2](https://arxiv.org/html/2609.00528#S2a), edges are directed, and each bond\{j,i\}\\\{j,i\\\}carries two features𝒆j​i\\bm\{e\}\_\{ji\}and𝒆i​j\\bm\{e\}\_\{ij\}, matching the reference implementation\. The line graph, see[fig\.1](https://arxiv.org/html/2609.00528#S2.F1)\(c\), is constructed by making each directed edge ofGGa node inℒ⁡\(G\)\\mathcal\{L\}\(G\), and connecting two nodes inℒ⁡\(G\)\\mathcal\{L\}\(G\)if the head of the first edge is the tail of the second\. The edges ofℒ⁡\(G\)\\mathcal\{L\}\(G\)thus correspond to bond angles\. ALIGNN maintains three feature types: node features𝒏jl\\bm\{n\}\_\{j\}^\{l\}, edge features𝒆j​il\\bm\{e\}\_\{ji\}^\{l\}onGG, and angle features𝒂i​j​kl\\bm\{a\}\_\{ijk\}^\{l\}onℒ⁡\(G\)\\mathcal\{L\}\(G\)\. The initial node features are element embeddings, the initial edge features are radial basis function \(RBF\) expansions of bond distances, and the initial angle features are RBF expansions of bond\-angle cosines\.

Each ALIGNN layer performs two sequential edge\-gated graph convolutions: first onℒ⁡\(G\)\\mathcal\{L\}\(G\)to update the bond and angle features using angular information, then onGGto update the node and bond features using the enriched bonds\. We write the two convolutions out in turn, since the update order determines which features reach the node aggregation\.

*1: Line graph convolution\.*The angle features are updated using the two adjacent bond features, then the bond features are aggregated with angle\-derived gates:

𝒂i​j​kl\\displaystyle\\bm\{a\}\_\{ijk\}^\{l\}=𝒂i​j​kl−1\+SiLU⁡\(Norm⁡\(Aa​𝒆j​il−1\+Ba​𝒆j​kl−1\+Ca​𝒂i​j​kl−1\)\),\\displaystyle=\\bm\{a\}\_\{ijk\}^\{l\-1\}\+\\mathrm\{SiLU\}\\bigl\(\\mathrm\{Norm\}\(A\_\{a\}\\,\\bm\{e\}\_\{ji\}^\{l\-1\}\+B\_\{a\}\\,\\bm\{e\}\_\{jk\}^\{l\-1\}\+C\_\{a\}\\,\\bm\{a\}\_\{ijk\}^\{l\-1\}\)\\bigr\),\(40\)𝒆~j​il\\displaystyle\\tilde\{\\bm\{e\}\}\_\{ji\}^\{l\}=𝒆j​il−1\+SiLU⁡\(Norm⁡\(Ws,a​𝒆j​il−1\+∑k∈𝒩⁡\(j\)∖\{i\}σi​j​k​Wd,a​𝒆j​kl−1\)\),\\displaystyle=\\bm\{e\}\_\{ji\}^\{l\-1\}\+\\mathrm\{SiLU\}\\Bigl\(\\mathrm\{Norm\}\\bigl\(W\_\{s,a\}\\,\\bm\{e\}\_\{ji\}^\{l\-1\}\+\\sum\_\{k\\in\\mathcal\{N\}\(j\)\\setminus\\\{i\\\}\}\\sigma\_\{ijk\}\\,W\_\{d,a\}\\,\\bm\{e\}\_\{jk\}^\{l\-1\}\\bigr\)\\Bigr\),\(41\)whereAa,Ba,Ca,Ws,a,Wd,aA\_\{a\},B\_\{a\},C\_\{a\},W\_\{s,a\},W\_\{d,a\}are trainable weight matrices,Norm\\mathrm\{Norm\}is a feature normalization layer,σ\\sigmadenotes the sigmoid function, andSiLU⁡\(x\)=x​σ​\(x\)\\mathrm\{SiLU\}\(x\)=x\\,\\sigma\(x\)is the sigmoid linear unit\. All edge features entering these updates are the copies directed away from the centerjj\. Both convolutions use the same normalized gate: for features𝒙s\\bm\{x\}\_\{s\}aggregated over an index setSS,

σ⁡\[𝒙s\]=σ⁡\(𝒙s\)∑s′∈Sσ⁡\(𝒙s′\)\+ϵ,\\sigma\[\\bm\{x\}\_\{s\}\]=\\frac\{\\sigma\(\\bm\{x\}\_\{s\}\)\}\{\\sum\_\{s^\{\\prime\}\\in S\}\\sigma\(\\bm\{x\}\_\{s^\{\\prime\}\}\)\+\\epsilon\},\(42\)withϵ\>0\\epsilon\>0a small constant for numerical stability\. In \([41](https://arxiv.org/html/2609.00528#S3.E41)\) the gate isσi​j​k=σ⁡\[𝒂i​j​kl\]\\sigma\_\{ijk\}=\\sigma\[\\bm\{a\}\_\{ijk\}^\{l\}\], aggregated over the angles at the bond\(j,i\)\(j,i\)\. In \([44](https://arxiv.org/html/2609.00528#S3.E44)\) it isσj​i=σ⁡\[𝒆j​il\]\\sigma\_\{ji\}=\\sigma\[\\bm\{e\}\_\{ji\}^\{l\}\], aggregated over the neighbors ofjj\.

*2: Atom graph convolution\.*The edge update loads*both*endpoint node features into the bond, then the node update aggregates over neighbors with edge\-derived gates:

𝒆j​il\\displaystyle\\bm\{e\}\_\{ji\}^\{l\}=𝒆~j​il\+SiLU⁡\(Norm⁡\(A​𝒏jl−1\+B​𝒏il−1\+C​𝒆~j​il\)\),\\displaystyle=\\tilde\{\\bm\{e\}\}\_\{ji\}^\{l\}\+\\mathrm\{SiLU\}\\bigl\(\\mathrm\{Norm\}\(A\\,\\bm\{n\}\_\{j\}^\{l\-1\}\+B\\,\\bm\{n\}\_\{i\}^\{l\-1\}\+C\\,\\tilde\{\\bm\{e\}\}\_\{ji\}^\{l\}\)\\bigr\),\(43\)𝒏jl\\displaystyle\\bm\{n\}\_\{j\}^\{l\}=𝒏jl−1\+SiLU⁡\(Norm⁡\(Ws​𝒏jl−1\+∑i∈𝒩⁡\(j\)σj​i​Wd​𝒏il−1\)\),\\displaystyle=\\bm\{n\}\_\{j\}^\{l\-1\}\+\\mathrm\{SiLU\}\\Bigl\(\\mathrm\{Norm\}\\bigl\(W\_\{s\}\\,\\bm\{n\}\_\{j\}^\{l\-1\}\+\\sum\_\{i\\in\\mathcal\{N\}\(j\)\}\\sigma\_\{ji\}\\,W\_\{d\}\\,\\bm\{n\}\_\{i\}^\{l\-1\}\\bigr\)\\Bigr\),\(44\)whereA,B,C,Ws,WdA,B,C,W\_\{s\},W\_\{d\}are trainable weight matrices\. The total energy isEθ=∑jMLP⁡\(𝒏jL\)E\_\{\\theta\}=\\sum\_\{j\}\\mathrm\{MLP\}\(\\bm\{n\}\_\{j\}^\{L\}\)\.

A key architectural feature is that the edge update \([43](https://arxiv.org/html/2609.00528#S3.E43)\) takes both endpoint node features𝒏i\\bm\{n\}\_\{i\}and𝒏j\\bm\{n\}\_\{j\}directly\.