Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks

# Spin-Weighted Spherical Harmonics Enable Complete and Scalable E(3)-Equivariant Networks
Source: [https://arxiv.org/html/2607.01408](https://arxiv.org/html/2607.01408)
Yuchao Lin1∗Andrii Kryvenko1∗Wendi Yu1Chuan Li2 Jianwen Xie2Xiaofeng Qian3,4,5Shuiwang Ji1,3,6 1Department of Computer Science and Engineering, Texas A&M University 2Lambda, Inc\. 3Department of Materials Science and Engineering, Texas A&M University 4Department of Electrical and Computer Engineering, Texas A&M University 5Department of Physics and Astronomy, Texas A&M University 6Department of Mechanical Engineering, Texas A&M University

###### Abstract

Abstract:

E​\(3\)\\mathrm\{E\}\(3\)\-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by theO​\(L6\)O\(L^\{6\}\)complexity of the Clebsch\-Gordan Tensor Product \(CGTP\)\. The recently proposed Gaunt Tensor Product \(GTP\) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity\. In this work, we presentSpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin\-Weighted Spherical Harmonics \(SWSH\)\. By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP\. It also allows for a more expressive equivariant basis that naturally accounts for the parity\-odd components of tensor products\. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE\-MACE\-OFF, and OC20\. Our results show that SpinGTP achieves accuracies comparable to full CGTP\. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non\-centrosymmetric geometries\. This work provides a complete, scalable, and mathematically rigorous path toward high\-order equivariance in large\-scale 3D atomistic system simulations\.

Keywords: AI for Science, Equivariant Networks, Spin\-Weighted Spherical Harmonics, 3D Atomistic Simulations

\* These authors contributed equally

![[Uncaptioned image]](https://arxiv.org/html/2607.01408v1/x1.png)Github Repository:[https://github\.com/divelab/SpinGTP](https://github.com/divelab/SWSH-GNN)

###### Contents

1. [1Introduction](https://arxiv.org/html/2607.01408#S1)
2. [2Related Work](https://arxiv.org/html/2607.01408#S2)
3. [3Gaunt Tensor Product of Spin\-Weighted Spherical Harmonics](https://arxiv.org/html/2607.01408#S3)1. [3\.1Gaunt Tensor Product Efficiency and the Antisymmetry Gap](https://arxiv.org/html/2607.01408#S3.SS1) 2. [3\.2Spin\-weighted Spherical Harmonics for Restoring Expressivity](https://arxiv.org/html/2607.01408#S3.SS2) 3. [3\.3Real\-Basis Spin\-Weighted Spherical Harmonics](https://arxiv.org/html/2607.01408#S3.SS3) 4. [3\.4Parity\-Equivariant Spin\-Weighted Spherical Harmonics](https://arxiv.org/html/2607.01408#S3.SS4) 5. [3\.5High\-Performance Implementation of SpinGTP](https://arxiv.org/html/2607.01408#S3.SS5) 6. [3\.6Specialized SWSH Equivariant Layers](https://arxiv.org/html/2607.01408#S3.SS6)
4. [4Experiments](https://arxiv.org/html/2607.01408#S4)1. [4\.1Chiral Tetris Classification](https://arxiv.org/html/2607.01408#S4.SS1) 2. [4\.23BPA Performance](https://arxiv.org/html/2607.01408#S4.SS2) 3. [4\.3SPICE\-MACE\-OFF Chiral Subset Performance](https://arxiv.org/html/2607.01408#S4.SS3) 4. [4\.4OC20 IS2RE Direct](https://arxiv.org/html/2607.01408#S4.SS4)
5. [5Limitations and Summary](https://arxiv.org/html/2607.01408#S5)
6. [References](https://arxiv.org/html/2607.01408#bib)
7. [ASpin\-Weighted Spherical Harmonics \(SWSH\)](https://arxiv.org/html/2607.01408#A1)1. [A\.1Explicit Formula](https://arxiv.org/html/2607.01408#A1.SS1) 2. [A\.2Equivariance and Spin\-Weighted Transformations](https://arxiv.org/html/2607.01408#A1.SS2) 3. [A\.3Nonzero Spin\-Weighted Gaunt Paths](https://arxiv.org/html/2607.01408#A1.SS3)
8. [BBest Asymptotic Runtime Cost](https://arxiv.org/html/2607.01408#A2)
9. [CReal Basis Spin\-Weighted Spherical Harmonics](https://arxiv.org/html/2607.01408#A3)1. [C\.1The Unitary𝐐\\mathbf\{Q\}Matrix](https://arxiv.org/html/2607.01408#A3.SS1) 2. [C\.2Example: The𝐐\\mathbf\{Q\}Matrix forℓ=1\\ell=1](https://arxiv.org/html/2607.01408#A3.SS2)
10. [DDerivation of the Parity\-Equivariant Basis](https://arxiv.org/html/2607.01408#A4)
11. [ESpinGTP Implementation and Time Comparison](https://arxiv.org/html/2607.01408#A5)1. [E\.1Tensor Product Implementation](https://arxiv.org/html/2607.01408#A5.SS1) 2. [E\.2Runtime Comparison](https://arxiv.org/html/2607.01408#A5.SS2)
12. [FTraining Details](https://arxiv.org/html/2607.01408#A6)1. [F\.13BPA](https://arxiv.org/html/2607.01408#A6.SS1) 2. [F\.2SPICE\-MACE\-OFF Chiral Subset](https://arxiv.org/html/2607.01408#A6.SS2)1. [F\.2\.1Chirality Classification and Parity Generalization](https://arxiv.org/html/2607.01408#A6.SS2.SSS1) 2. [F\.2\.2Energy and Force Prediction](https://arxiv.org/html/2607.01408#A6.SS2.SSS2) 3. [F\.3OC20 IS2RE](https://arxiv.org/html/2607.01408#A6.SS3)

## 1Introduction

The integration of physical symmetries into deep learning models has become a cornerstone of modern artificial intelligence for physical sciences\[zhang2025artificial,bronstein2021geometric,villar2021scalars,kondor2025principles,fei2024rotation,fu2025augmenting\]\. By enforcing equivariance, neural networks can model 3D atomic environments with superior data efficiency and generalization compared to standard architectures\. Central to these models is the Clebsch\-Gordan Tensor Product \(CGTP\), a fundamental operation that enables interactions between features of different angular frequencies, known as irreducible representations \(irreps\)\[khersonskii1988quantum,thomas2018tensor,anderson2019cormorant\]\. Despite its expressive power, the CGTP suffers from a computational complexity ofO​\(L6\)O\(L^\{6\}\)with the maximum angular degreeLL\. This complexity often forces practitioners to limitLLto small values, sacrificing the high\-order geometric information necessary for modeling complex molecular interactions and interatomic potentials\.

Recent efforts to overcome this bottleneck have focused on accelerating tensor products through alternative mathematical formulations\[passaro2023reducing,lin2025tensor\]\. A notable advance is the Gaunt Tensor Product \(GTP\)\[luo2024enabling\], which maps tensor products of irreps to pointwise multiplications of spherical functions in 2D Fourier basis\. By leveraging the convolution theorem and Fast Fourier Transforms \(FFT\), GTP reduces the asymptotic complexity fromO​\(L6\)O\(L^\{6\}\)toO​\(L3\)O\(L^\{3\}\)\. However, this efficiency comes with an expressivity loss\. GTP retains only coupling paths where the sum of irrep degreesℓ1\+ℓ2\+ℓ3\\ell\_\{1\}\+\\ell\_\{2\}\+\\ell\_\{3\}is even, and removes odd degree\-sum paths\[xie2024price\]\. These missing paths include antisymmetric interactions such as the\[ℓ1=1,ℓ2=1,ℓ3=1\]\[\\ell\_\{1\}=1,\\ell\_\{2\}=1,\\ell\_\{3\}=1\]vector cross\-product path, as well as couplings needed to form pseudoscalar and axial quantities\. This limits scalar GTP on parity\-sensitive tasks, especially those involving chiral geometries\. We refer to this limitation as the antisymmetric gap\.

To address the antisymmetric gap, recent work\[xie2026asymptotically\]shows that the antisymmetric paths in scalar GTP can be recovered by lifting scalar spherical signals to irrep\-valued signals\. Using Vector Spherical Harmonics \(VSH\), they introduce the Vector Signal Tensor Product \(VSTP\), which recovers missing antisymmetric paths through generalized Gaunt products with Wigner9​j9jsymbol\.

In this work, we pursue a more direct mathematical path to this problem by introducing the Spin\-Weighted Gaunt Tensor Product \(SpinGTP\)\. Rather than lifting scalar spherical signals to vector\-valued signals, SpinGTP works directly with Spin\-Weighted Spherical Harmonics \(SWSHs\) and their generalized Gaunt integral with Wigner3​j3jsymbols\. The spin\-weight index replaces the scalar zero\-order coupling with a signed spin selection rule, allowing odd degree\-sum paths that scalar GTP removes when the required spin sectors are present\. We implement this operator in a real, parity\-labeled SWSH basis for a rich and concrete representation\.

We evaluate SpinGTP on a suite of benchmarks, including the Tetris and large\-scale atomistic datasets such as 3BPA\[3BPA\], SPICE\-MACE\-OFF\[kovacs2025mace\], and OC20\[chanussot2021open\]\. Our results demonstrate that SpinGTP\-based networks achieves accuracy comparable to fullO​\(L6\)O\(L^\{6\}\)CGTP models while remaining as computationally efficient as the original GTP\. Furthermore, we show that our model outperforms previous tensor product methods in predicting properties of chiral materials, showing that the inclusion of antisymmetric paths through SWSH is beneficial for 3D geometric deep learning of chiral geometries\.

Our contributions are summarized as follows:

- •We propose SpinGTP, an equivariant operation based on SWSH that restores the mathematical completeness to the Gaunt Tensor Product framework\.
- •We provide mathematical derivations showing that the SpinGTP formulation efficiently recovers the previously missing parity\-odd interactions \(antisymmetric paths\) required for universalE​\(3\)\\mathrm\{E\}\(3\)equivariance\.
- •We evaluate SpinGTP across a range of tasks, including Tetris, 3BPA, SPICE\-MACE\-OFF, and OC20, demonstrating accuracy comparable to full CGTP while yielding targeted improvements in chiral geometry modeling through the recovery of antisymmetric paths\.

## 2Related Work

Equivariant Network Architectures\.The landscape ofE​\(3\)\\mathrm\{E\}\(3\)\-equivariant modeling has evolved from early invariant descriptors such as SchNet\[SchNet\]and DimeNet\[DimeNet\+\+\]toward steerable frameworks that leverage irreducible representations \(irreps\)\. Seminal architectures like NequIP\[nequip\]and Allegro\[musaelian2023learning\]demonstrated the superior sample efficiency of Clebsch\-Gordan \(CG\) tensor products, while the Equiformer series\[equiformer,equiformerv2\]extended these principles to attention\-based mechanisms\. To alleviate the computational burden of edge\-wise products, e2former\[li2026eformer\]introduced a node\-wise message\-passing scheme\. While alternative scalarization methods, such as PaiNN\[schutt2021equivariant\]and NewtonNet\[haghighatlari2022newtonnet\]offer increased throughput, they often bypass the full tensor product space, potentially sacrificing the formal universality required for complex geometric\.

Tensor Product Acceleration\.The Clebsch\-Gordan Tensor Product \(CGTP\) is a standard interaction operator inE​\(3\)\\mathrm\{E\}\(3\)\-equivariant neural networks\[khersonskii1988quantum\]\. However, direct CGTP scales asO​\(L6\)O\(L^\{6\}\)with the maximum angular degreeLL, which limits the use of high\-order features\. Several recent methods reduce this cost by exploiting structure in equivariant architectures\. eSCN\[passaro2023reducing\]aligns features with the edge direction during message passing, which sparsifies the equivariant convolution and reduces it to anSO​\(2\)\\mathrm\{SO\}\(2\)computation in the local edge frame\.\[lin2025tensor\]reduces tensor\-product cost through low\-rank tensor decomposition structure\.\[luo2024enabling\]proposed the Gaunt Tensor Product \(GTP\), which evaluates interactions through spherical harmonic transforms and pointwise products, reducing the complexity toO​\(L3\)O\(L^\{3\}\)\. GTP is efficient, but scalar GTP removes oddℓ\\ell\-sum paths, including antisymmetric interactions such as the vector cross product\[xie2024price\]\. To restore expressivity without reverting toO​\(L6\)O\(L^\{6\}\)scaling, a Vector Spherical Harmonic \(VSH\) basis and9​j9jrecoupling are introduced to recover missing antisymmetric paths\[xie2026asymptotically\]\. This framework provides complete algorithm to achieve true asymptotic speedups, reachingO​\(L4​log2⁡L\)O\(L^\{4\}\\log^\{2\}L\)complexity via fast spectral transforms\.

## 3Gaunt Tensor Product of Spin\-Weighted Spherical Harmonics

This section presents the methodology for the Spin\-Weighted Gaunt Tensor Product \(SpinGTP\)\. We first review the standard Gaunt Tensor Product and identify its main expressivity limitation, the loss of antisymmetric paths with oddℓ\\ell\-sum\. We then introduce Spin\-Weighted Spherical Harmonics \(SWSH\) and their spin\-weighted Gaunt integral\. To use this in a real\-valued neural network, we build real SWSH bases and extend them with parity labels, allowing the model to represent antisymmetric paths when the required spin channels are present\. Finally, we summarize the implementation strategy and the specialized equivariant layers in the architecture\.

### 3\.1Gaunt Tensor Product Efficiency and the Antisymmetry Gap

To reduce the𝒪​\(L6\)\\mathcal\{O\}\(L^\{6\}\)cost of standard Clebsch\-Gordan contractions, the scalar Gaunt Tensor Product \(GTP\)\[luo2024enabling\]represents equivariant interactions as pointwise products of spherical signals\. The resulting coupling is given by a Gaunt coefficient, defined as the integral of two input spherical harmonics against an output spherical harmonic overS2\\mathrm\{S\}^\{2\}such that

G\(ℓ1,m1\)​\(ℓ2,m2\)\(ℓ3,m3\)=∫S2Yℓ1​m1​Yℓ2​m2​Yℓ3​m3∗​𝑑Ω=\(2​ℓ1\+1\)​\(2​ℓ2\+1\)4​π​\(2​ℓ3\+1\)​C\(ℓ1,m1\)​\(ℓ2,m2\)\(ℓ3,m3\)​C\(ℓ1,0\)​\(ℓ2,0\)\(ℓ3,0\)\.G^\{\(\\ell\_\{3\},m\_\{3\}\)\}\_\{\(\\ell\_\{1\},m\_\{1\}\)\(\\ell\_\{2\},m\_\{2\}\)\}=\\int\_\{\\mathrm\{S\}^\{2\}\}Y\_\{\\ell\_\{1\}m\_\{1\}\}Y\_\{\\ell\_\{2\}m\_\{2\}\}Y\_\{\\ell\_\{3\}m\_\{3\}\}^\{\*\}d\\Omega=\\sqrt\{\\frac\{\(2\\ell\_\{1\}\+1\)\(2\\ell\_\{2\}\+1\)\}\{4\\pi\(2\\ell\_\{3\}\+1\)\}\}C\_\{\(\\ell\_\{1\},m\_\{1\}\)\(\\ell\_\{2\},m\_\{2\}\)\}^\{\(\\ell\_\{3\},m\_\{3\}\)\}C\_\{\(\\ell\_\{1\},0\)\(\\ell\_\{2\},0\)\}^\{\(\\ell\_\{3\},0\)\}\.\(1\)
As shown in[Equation˜1](https://arxiv.org/html/2607.01408#S3.E1), GTP retains a subset of Clebsch\-Gordan paths where the scalar couplingC\(ℓ1,0\)​\(ℓ2,0\)\(ℓ3,0\)C\_\{\(\\ell\_\{1\},0\)\(\\ell\_\{2\},0\)\}^\{\(\\ell\_\{3\},0\)\}is nonzero\. This restriction simplifies the interaction, and the spatial product formulation enables an𝒪​\(L3\)\\mathcal\{O\}\(L^\{3\}\)implementation via fast Fourier transform\. The resulting computation can scale to high spectral resolutions that are costly for direct Clebsch\-Gordan contractions\.

Despite this efficiency, scalar GTP is constrained by the factorC\(ℓ1,0\)​\(ℓ2,0\)\(ℓ3,0\)C\_\{\(\\ell\_\{1\},0\)\(\\ell\_\{2\},0\)\}^\{\(\\ell\_\{3\},0\)\}, which vanishes wheneverℓ1\+ℓ2\+ℓ3\\ell\_\{1\}\+\\ell\_\{2\}\+\\ell\_\{3\}is odd\. This removes all oddℓ\\ell\-sum coupling paths, including antisymmetric interactions such as the\[ℓ1=1,ℓ2=1,ℓ3=1\]\[\\ell\_\{1\}=1,\\ell\_\{2\}=1,\\ell\_\{3\}=1\]vector cross\-product path\. As a result, scalar GTP can fail to distinguish configurations whose signal depends on pseudoscalar or chiral interactions\. This limitation motivates the spin\-weighted extension introduced below\.

### 3\.2Spin\-weighted Spherical Harmonics for Restoring Expressivity

Spin\-Weighted Spherical Harmonics \(SWSHs\)\[goldberg1967spin\]generalize ordinary spherical harmonics by adding an integer spin weightss, alongside the degreeℓ\\elland ordermm, with\|s\|≤ℓ\|s\|\\leq\\ell\. The cases=0s=0recovers ordinary spherical harmonics\. Fors≠0s\\neq 0, SWSHs represent spin\-weighted fields onS2\\mathrm\{S\}^\{2\}and transform by a phase under rotations of the local tangent frame\. Additional details are provided in[Appendix˜A](https://arxiv.org/html/2607.01408#A1)\.

The role of SWSHs in our tensor product comes from their generalized Gaunt integral\. For three SWSHs with the spin weights satisfyings3=s1\+s2s\_\{3\}=s\_\{1\}\+s\_\{2\}, the Gaunt coefficient is

G\(ℓ1,m1,s1\)​\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)\\displaystyle G^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\)\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}=∫S2Yℓ1​m1s1​Yℓ2​m2s2​Yℓ3​m3∗s3​𝑑Ω\\displaystyle=\\int\_\{\\mathrm\{S\}^\{2\}\}\{\}\_\{s\_\{1\}\}Y\_\{\\ell\_\{1\}m\_\{1\}\}\\,\{\}\_\{s\_\{2\}\}Y\_\{\\ell\_\{2\}m\_\{2\}\}\\,\{\}\_\{s\_\{3\}\}Y\_\{\\ell\_\{3\}m\_\{3\}\}^\{\*\}\\,d\\Omega\(2\)=∏i=13\(2​ℓi\+1\)4​π​\(ℓ1ℓ2ℓ3m1m2−m3\)​\(ℓ1ℓ2ℓ3s1s2−s3\)\.\\displaystyle=\\sqrt\{\\frac\{\\prod\_\{i=1\}^\{3\}\(2\\ell\_\{i\}\+1\)\}\{4\\pi\}\}\\begin\{pmatrix\}\\ell\_\{1\}&\\ell\_\{2\}&\\ell\_\{3\}\\\\ m\_\{1\}&m\_\{2\}&\-m\_\{3\}\\end\{pmatrix\}\\begin\{pmatrix\}\\ell\_\{1\}&\\ell\_\{2\}&\\ell\_\{3\}\\\\ s\_\{1\}&s\_\{2\}&\-s\_\{3\}\\end\{pmatrix\}\.Unlike the scalar factor in[Equation˜1](https://arxiv.org/html/2607.01408#S3.E1), the spin\-weighted factor can be nonzero for oddℓ1\+ℓ2\+ℓ3\\ell\_\{1\}\+\\ell\_\{2\}\+\\ell\_\{3\}when proper nonzero spin weights are present\. SpinGTP uses this degree of freedom to recover antisymmetric paths that are absent from scalar GTP, while retaining the Gaunt product structure\. A more formal statement is presented below and the proof is provided in[appendix˜A](https://arxiv.org/html/2607.01408#A1)\.

###### Proposition 3\.1\.

Spin\-weighted path completion Let\(ℓ1,ℓ2,ℓ3\)\(\\ell\_\{1\},\\ell\_\{2\},\\ell\_\{3\}\)satisfy the Clebsch\-Gordan triangle rule\. Then there exist signed spin weights

s1∈\[−ℓ1,ℓ1\],s2∈\[−ℓ2,ℓ2\],s3∈\[−ℓ3,ℓ3\],s\_\{1\}\\in\[\-\\ell\_\{1\},\\ell\_\{1\}\],\\qquad s\_\{2\}\\in\[\-\\ell\_\{2\},\\ell\_\{2\}\],\\qquad s\_\{3\}\\in\[\-\\ell\_\{3\},\\ell\_\{3\}\],withs3=s1\+s2s\_\{3\}=s\_\{1\}\+s\_\{2\}, such that the signed\-spin path\(ℓ1,s1\)⊗\(ℓ2,s2\)→\(ℓ3,s3\)\(\\ell\_\{1\},s\_\{1\}\)\\otimes\(\\ell\_\{2\},s\_\{2\}\)\\to\(\\ell\_\{3\},s\_\{3\}\)is not identically zero\.

Local Frames and Gauge Dependency\.Spin\-weighted features depend on a local tangent\-frame gauge\. Unlike scalar spherical harmonics, an SWSH valueYℓ​ms​\(𝐱\)\{\}\_\{s\}Y\_\{\\ell m\}\(\\mathbf\{x\}\)is defined relative to a local orthonormal frame in the tangent plane at𝐱\\mathbf\{x\}\. If this frame is rotated by an angleψ\\psi, a spin\-ssfield transforms as

fs​\(𝐱\)↦ei​s​ψ​fs​\(𝐱\)\.f\_\{s\}\(\\mathbf\{x\}\)\\mapsto e^\{is\\psi\}f\_\{s\}\(\\mathbf\{x\}\)\.\(3\)This gauge dependence means that the basis functions implicitly capture the orientation of the spherical surface, yielding a richer representation than scalar spherical harmonics\. However, spin\-weighted features can be combined consistently only when their local frame conventions are compatible\. In our implementation, frame construction is treated as part of the equivariant architecture\. We use explicit geometric frames to evaluate SWSH edge features, with the details described in[Section˜A\.2](https://arxiv.org/html/2607.01408#A1.SS2)\.

Best Asymptotic Runtime Complexity\.For bounded spin weight\|s\|max=1\|s\|\_\{\\max\}=1, SpinGTP can attain the sameO​\(L4​log2⁡L\)O\(L^\{4\}\\log^\{2\}L\)asymptotic complexity as the VSTP\[xie2026asymptotically\], provided that the FFT\-based Gaunt tensor product\[luo2024enabling\]is used\. A proof is given in[Appendix˜B](https://arxiv.org/html/2607.01408#A2)\.

![Refer to caption](https://arxiv.org/html/2607.01408v1/x2.png)Figure 1:Comparison between standard and spin\-weighted spherical harmonics\. \(a\) Scalar spherical harmonics withℓm​a​x=2\\ell\_\{max\}=2\. The rows represent the basis functions forℓ=0,1,\\ell=0,1,and22, respectively, with the magnetic indexmmranging from−ℓ\-\\elltoℓ\\ell\. \(b\) Spin\-weighted spherical harmonics withℓm​a​x=2\\ell\_\{max\}=2and absolute spin\|s\|=1\|s\|=1\. The rows display the basis functions for degreesℓ=1\\ell=1andℓ=2\\ell=2\. For each degree, the basis includes components for both−s\-sand\+s\+sacross full range ofmm\.
### 3\.3Real\-Basis Spin\-Weighted Spherical Harmonics

SWSHs are naturally complex\-valued, while our network operates on real\-valued feature tensors to maintain compatibility with standard activation functions and optimize memory efficiency\. For scalar harmonics, the usual real basis can be constructed within a fixed degreeℓ\\ell\. For spin\-weighted harmonics, the construction must also account for the relation

Yℓ​m∗s=\(−1\)m\+s​Yℓ,−m−s\.\{\}\_\{s\}Y\_\{\\ell m\}^\{\*\}=\(\-1\)^\{m\+s\}\{\}\_\{\-s\}Y\_\{\\ell,\-m\}\.\(4\)
Following\[reisswig2013general\], we satisfy[Equation˜4](https://arxiv.org/html/2607.01408#S3.E4)by using areal SWSH basisthat couples the positive and negative spin weights as

Rℓ​ms=\{i2​\(Yℓ,−ms−\(−1\)−m\+s​Yℓ​m−s\)if​m<0Yℓ​0sif​m=012​\(Yℓ​ms\+\(−1\)m\+s​Yℓ,−m−s\)if​m\>0\.\{\}\_\{s\}R\_\{\\ell m\}=\\begin\{cases\}\\frac\{i\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{s\}Y\_\{\\ell,\-m\}\-\(\-1\)^\{\-m\+s\}\{\}\_\{\-s\}Y\_\{\\ell m\}\\right\)&\\text\{if \}m<0\\\\ \{\}\_\{s\}Y\_\{\\ell 0\}&\\text\{if \}m=0\\\\ \\frac\{1\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{s\}Y\_\{\\ell m\}\+\(\-1\)^\{m\+s\}\{\}\_\{\-s\}Y\_\{\\ell,\-m\}\\right\)&\\text\{if \}m\>0\.\\end\{cases\}\(5\)For any\|s\|\>0\|s\|\>0, concatenating bothYℓ​ms\{\}\_\{s\}Y\_\{\\ell m\}andYℓ​m−s\{\}\_\{\-s\}Y\_\{\\ell m\}increases the basis dimension from2​ℓ\+12\\ell\+1to2​\(2​ℓ\+1\)2\(2\\ell\+1\)because both signed spin sectors must be retained for real SWSH construction\. This doubled representation provides the real spin\-weighted features used by the network\. A visualized comparison between the real scalar spherical harmonics and the real basis SWSH is provided in[Figure˜1](https://arxiv.org/html/2607.01408#S3.F1)\.

##### Transformation via the𝐐\\mathbf\{Q\}Matrix\.

To connect real feature vectors𝐚real\\mathbf\{a\}\_\{\\text\{real\}\}with the complex Gaunt coefficients used in the tensor product, we use a unitary change of basis𝐐\\mathbf\{Q\}such that

𝐚complex=𝐐𝐚real\.\\mathbf\{a\}\_\{\\text\{complex\}\}=\\mathbf\{Q\}\\mathbf\{a\}\_\{\\text\{real\}\}\.When\|s\|=0\|s\|=0, this is the standard real\-to\-complex spherical harmonic transformation\. For a fixedℓ\\elland\|s\|\>0\|s\|\>0, the matrix satisfies𝐐∈ℂ2​\(2​ℓ\+1\)×2​\(2​ℓ\+1\)\\mathbf\{Q\}\\in\\mathbb\{C\}^\{2\(2\\ell\+1\)\\times 2\(2\\ell\+1\)\}and is organized into four blocks that couple the±s\\pm sand±m\\pm mindices\. The explicit block\-wise definitions of𝐐\\mathbf\{Q\}and the corresponding real\-form Gaunt coefficients are given in[Appendix˜C](https://arxiv.org/html/2607.01408#A3)\.

### 3\.4Parity\-Equivariant Spin\-Weighted Spherical Harmonics

AnE​\(3\)\\mathrm\{E\}\(3\)\-equivariant model must distinguish signals with different parities\. Standard scalar spherical harmonics have a fixed parityP=\(−1\)ℓP=\(\-1\)^\{\\ell\}\. For spin\-weighted harmonics, spatial inversion also flips the spin weight such that

Yℓ​ms​\(−𝐱\)=\(−1\)ℓ​Yℓ​m−s​\(𝐱\)\.\{\}\_\{s\}Y\_\{\\ell m\}\(\-\\mathbf\{x\}\)=\(\-1\)^\{\\ell\}\{\}\_\{\-s\}Y\_\{\\ell m\}\(\\mathbf\{x\}\)\.Thus inversion maps thesssector to the−s\-ssector\. As a result, the real basis functionsRℓ​ms\{\}\_\{s\}R\_\{\\ell m\}defined in[Equation˜5](https://arxiv.org/html/2607.01408#S3.E5)do not by themselves carry a fixed parity whens≠0s\\neq 0\. To obtain basis functions with an inversion parityp∈\{\+1,−1\}p\\in\\\{\+1,\-1\\\}, we combine the paired spin sectors and define aparity\-equivariant real SWSH basis, denoted byℛps,ℓ,m​\(𝐱\)\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\)\. This construction separates the parity\-even and parity\-odd components as

ℛps,ℓ,m​\(𝐱\)=12​\(Rℓ​m−s​\(𝐱\)\+p​\(−1\)ℓ​Rℓ​ms​\(𝐱\)\)\.\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\)=\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\+p\(\-1\)^\{\\ell\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\\right\)\.\(6\)By construction, this basis satisfies the required parity constraintℛps,ℓ,m​\(−𝐱\)=p​ℛps,ℓ,m​\(𝐱\)\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\-\\mathbf\{x\}\)=p\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\)\. This allows the representation to distinguish scalar from pseudoscalar channels and polar vector from axial vector channels\. A detailed derivation is provided in[Appendix˜D](https://arxiv.org/html/2607.01408#A4)\.

The parity\-labeled real basis has two roles\. First, it lets the network track the inversion parity of each irrep throughout tensor product layers\. Second, it provides the necessary geometric signals for physical properties whose sign changes under reflection, including pseudoscalar quantities associated with chiral geometry\.

### 3\.5High\-Performance Implementation of SpinGTP

The computational core of SpinGTP is an instruction\-based tensor product over SWSH irreps\. Each irrep is indexed by\(ℓ,\|s\|,p\)\(\\ell,\|s\|,p\)\. Given two input feature tensors𝐱\(1\)\\mathbf\{x\}^\{\(1\)\}and𝐱\(2\)\\mathbf\{x\}^\{\(2\)\}, the weighted tensor product maps pairs of input irreps to admissible output irreps as

\[𝐱\(1\)⊗𝐖𝐱\(2\)\]ℓ3,s3,p3\(m3,w\)=∑u,v𝐖u​v​w​∑m1,m2G\(ℓ1,m1,s1\),\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)​𝐱u,ℓ1,m1,p1\(1\)​𝐱v,ℓ2,m2,p2\(2\),p3=p1​p2,\[\\mathbf\{x\}^\{\(1\)\}\\otimes\_\{\\mathbf\{W\}\}\\mathbf\{x\}^\{\(2\)\}\]\_\{\\ell\_\{3\},s\_\{3\},p\_\{3\}\}^\{\(m\_\{3\},w\)\}=\\sum\_\{u,v\}\\mathbf\{W\}\_\{uvw\}\\sum\_\{m\_\{1\},m\_\{2\}\}G\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\),\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\\mathbf\{x\}\_\{u,\\ell\_\{1\},m\_\{1\},p\_\{1\}\}^\{\(1\)\}\\mathbf\{x\}\_\{v,\\ell\_\{2\},m\_\{2\},p\_\{2\}\}^\{\(2\)\},\\quad p\_\{3\}=p\_\{1\}p\_\{2\},\(7\)whereGGdenotes the real SWSH Gaunt contraction and𝐖\\mathbf\{W\}mixes feature multiplicities\. Internally, the contraction enumerates the signed spin sectorss=±\|s\|s=\\pm\|s\|, applies the selection rules3=s1\+s2s\_\{3\}=s\_\{1\}\+s\_\{2\}, combines the paired output spin sectors according to the output parity\. To balance expressivity with parameter efficiency, we implementfully connected,depthwise, andelement\-wisetensor products\. The implementation details are shown in[Section˜E\.1](https://arxiv.org/html/2607.01408#A5.SS1)\.

Pre\-contracted Kernel Execution\.A direct evaluation of SpinGTP involves many small sparse contractions, which is inefficient on GPUs\. Let B denote the number of edges, and let U and V denote the input multiplicities\. When computing over allm3m\_\{3\}, this contraction has time complexityO​\(B​U​V​ℓ1​ℓ2​ℓ3\)O\(BUV\\ell\_\{1\}\\ell\_\{2\}\\ell\_\{3\}\)\. In molecular message passing, the second input𝐱\(2\)\\mathbf\{x\}^\{\(2\)\}is often an edge feature with multiplicityV=1V=1\. We exploit this structure by precontracting the tensor product kernel with𝐱\(2\)\\mathbf\{x\}^\{\(2\)\}before applying it to the node channels\.

Based on[Equation˜7](https://arxiv.org/html/2607.01408#S3.E7), when the edge multiplicity is one, the sum overvvdisappears and we first form an edge\-dependent kernel

𝐊pre\(ℓ3,s3,p3\)​\(m1,m3\)=∑m2G\(ℓ1,m1,s1\),\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)​𝐱ℓ2,m2,p2\(2\)\.\\mathbf\{K\}\_\{\\mathrm\{pre\}\}^\{\(\\ell\_\{3\},s\_\{3\},p\_\{3\}\)\}\(m\_\{1\},m\_\{3\}\)=\\sum\_\{m\_\{2\}\}G\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\),\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\\mathbf\{x\}\_\{\\ell\_\{2\},m\_\{2\},p\_\{2\}\}^\{\(2\)\}\.\(8\)The remaining contraction becomes

\[𝐱\(1\)⊗𝐖𝐱\(2\)\]ℓ3,s3,p3\(m3,w\)=∑u𝐖u​1​w​∑m1𝐊pre\(ℓ3,s3,p3\)​\(m1,m3\)​𝐱u,ℓ1,m1,p1\(1\)\.\[\\mathbf\{x\}^\{\(1\)\}\\otimes\_\{\\mathbf\{W\}\}\\mathbf\{x\}^\{\(2\)\}\]\_\{\\ell\_\{3\},s\_\{3\},p\_\{3\}\}^\{\(m\_\{3\},w\)\}=\\sum\_\{u\}\\mathbf\{W\}\_\{u1w\}\\sum\_\{m\_\{1\}\}\\mathbf\{K\}\_\{\\mathrm\{pre\}\}^\{\(\\ell\_\{3\},s\_\{3\},p\_\{3\}\)\}\(m\_\{1\},m\_\{3\}\)\\mathbf\{x\}\_\{u,\\ell\_\{1\},m\_\{1\},p\_\{1\}\}^\{\(1\)\}\.\(9\)This reduces repeated geometric contractions across node channels, and gives a time complexityO​\(B​ℓ1​ℓ2​ℓ3\+B​U​ℓ1​ℓ3\)O\(B\\ell\_\{1\}\\ell\_\{2\}\\ell\_\{3\}\+BU\\ell\_\{1\}\\ell\_\{3\}\)when computing over allm3m\_\{3\}\. In implementation, the precontracted kernel is evaluated as a dense batched matrix multiplication, followed by the learned multiplicity mixing and scatter into the output irreps\. The time ablation of this implementation is provided in[Section˜E\.2](https://arxiv.org/html/2607.01408#A5.SS2)\.

### 3\.6Specialized SWSH Equivariant Layers

The SWSH representation requires standard neural network layers to respect the irrep structure indexed by\(ℓ,\|s\|,p\)\(\\ell,\|s\|,p\)\. We use SWSH linear and normalization layers that mix only compatible feature channels and preserve the type of each irrep\.

Equivariant Linear Mixing\.The linear layer performs channel mixing within each SWSH irrep\. Since features are grouped by\(ℓ,\|s\|,p\)\(\\ell,\|s\|,p\), the linear map is block diagonal across irrep types such that

\[Linear​\(𝐱\)\]ℓ,s,p=𝐖ℓ,s,p​𝐱ℓ,s,p\+𝐛​δℓ,0​δs,0​δp,1\.\[\\mathrm\{Linear\}\(\\mathbf\{x\}\)\]\_\{\\ell,s,p\}=\\mathbf\{W\}\_\{\\ell,s,p\}\\mathbf\{x\}\_\{\\ell,s,p\}\+\\mathbf\{b\}\\,\\delta\_\{\\ell,0\}\\delta\_\{s,0\}\\delta\_\{p,1\}\.\(10\)Here𝐖ℓ,s,p\\mathbf\{W\}\_\{\\ell,s,p\}mixes multiplicities within the same irrep, while the bias is applied only to invariant scalar channels\. In practice, identical irrep blocks are grouped in memory so that the blockwise mixing can be evaluated efficiently with batched matrix multiplication\.

Equivariant Layer Normalization\.Following\[equiformer\], the normalization layer treats scalar and non\-scalar irreps separately\. For scalar blocks with\(ℓ,s\)=\(0,0\)\(\\ell,s\)=\(0,0\), we subtract the mean over multiplicities before normalization\. For all other blocks, no centering is applied\. Each block is then normalized by an invariant RMS computed over multiplicities and irrep components\. Learnable gains scale each multiplicity channel and are shared across irrep components\. Learnable biases are restricted to scalar channels\.

## 4Experiments

To evaluate the expressivity of the SWSH tensor product, we benchmark our framework across four datasets covering five distinct tasks, ranging from synthetic geometric classification to large\-scale atomistic prediction\. We first use Chiral Tetris Classification to test whether spin\-weighted channels can distinguish enantiomers that scalar Gaunt\-based methods cannot separate\. We then evaluate energy and force prediction on the 3BPA dataset, which tests generalization across thermally sampled conformations\. To assess chiral molecular geometries more directly, we evaluate chirality classification and energy\-force prediction on a chiral subset of the SPICE\-MACE\-OFF dataset\. Finally, we integrate the SWSH tensor product into the Equiformer architecture and benchmark it on the Open Catalyst \(OC20\) IS2RE dataset\. Details of datasets, implementations, and comparisons are provided below\. Furthermore, to evaluate efficiency, we show the time ablation of our implementations in[Section˜E\.2](https://arxiv.org/html/2607.01408#A5.SS2)\.

### 4\.1Chiral Tetris Classification

![Refer to caption](https://arxiv.org/html/2607.01408v1/figures/tetris.png)Figure 2:3D Tetris classification\.Scalar GTP \(s=0s=0\) plateaus at 75%, consistent with its failure to distinguish the chiral mirror pair\. In contrast, the SpinGTP with nonzero spin channels reaches 100% accuracy and separates the two enantiomeric pieces, showing that spin\-weighted coupling provides the missing oddℓ\\ell\-sum interaction\.Dataset\. We first use a minimal synthetic benchmark to test whether the model can represent chiral distinctions\. Following\[xie2024price,e3nn\_software\], the task is to classify eight 3D Tetris\-like shapes built from four unit cubes, where the first two classes are non\-superimposable mirror images\. Each shape is represented as a graph whose nodes are the cube centers, with edges connecting immediately adjacent cubes, and the inputs are presented under random 3D rotations\. This makes the benchmark a direct probe of parity\-sensitive expressivity rather than a generic shape\-classification task\.

Training Details\. We train an equivariant network with four SpinGTP convolution layers, scalar node inputs, and SWSH edge features on radius\-graph edges with cutoff1\.51\.5\. The model outputs one odd scalar channel for the mirror pair and six even scalar channels for the other classes, and is trained for 500 Adam steps with fresh random rotations, learning rate10−310^\{\-3\}, and MSE loss\.

Results\. The key question is whether the model can separate the mirror pair\.\[xie2024price\]showed that CGTP\-based networks solve this task, while scalar GTP fails because it lacks the required antisymmetric interaction\. In our experiments, SpinGTP reaches 100% accuracy and correctly distinguishes the two chiral pieces\. This shows that the spin\-weighted construction provides the missing oddℓ\\ell\-sum interaction that scalar GTP cannot express\. The Tetris results are shown in Figure[2](https://arxiv.org/html/2607.01408#S4.F2)\.

### 4\.23BPA Performance

Dataset\.The 3BPA dataset\[3BPA\]consists of molecular dynamics trajectories of 3\-\(benzyloxy\)pyridin\-2\-amine, a flexible drug\-like molecule with three rotatable bonds that give rise to a complex torsional potential energy surface with many local minima\. Following\[batatia2022mace,batatia2025design,luo2024enabling\], we train on 500 geometries sampled at 300 K\. The benchmark evaluates both in\-distribution generalization at 300 K and out\-of\-distribution robustness at 600 K and 1200 K\. A dihedral test set further probes the torsional potential energy surface by scanning one dihedral angle while holding the other two fixed, directly testing whether the model correctly resolves conformer transition barriers\.

Training Details\.We adopt MACE\[batatia2022mace\]as our base architecture, extended with SWSH\-based tensor products, which we refer to as SpinGTP\. Training follows a two\-stage scheme, with the second stage fine\-tuning at an increased relative energy weight\. Full details are provided in[Section˜F\.1](https://arxiv.org/html/2607.01408#A6.SS1)\.

Results\.We compare SpinGTP against Allegro\[musaelian2023learning\], NequIP\[nequip\], BOTNet\[batatia2025design\], MACE\[batatia2022mace\], and MACE\-Gaunt\[luo2024enabling\]in Table[1](https://arxiv.org/html/2607.01408#S4.T1)\. SpinGTP consistently matches or outperforms current state\-of\-the\-art methods across all test splits\. In particular, SpinGTP maintains competitive in\-distribution performance at 300 K while showing stronger generalization on the out\-of\-distribution splits at 600 K, 1200 K, and the dihedral torsion test set\. We additionally observe that MACE\-Gaunt, despite sharing the same Gaunt\-based contraction scheme, underperforms SpinGTP across all splits, with the largest gap on the dihedral torsion test set\. This is consistent with our hypothesis that the antisymmetric spin\-paths restored by the SWSH tensor product contribute to resolving asymmetric torsional interactions, which are absent in the standard Gaunt contraction used by MACE\-Gaunt\.

Table 1:Results on the 3BPA dataset\. Root mean square error of Energy \(EE\) in meV and Forces \(FF\) in meV/Å across three temperature regimes and a dihedral scan\. Standard deviations over 3 random seeds are in parentheses\.Boldindicates best performance,underlineindicates second best\.AllegroNequIPBOTNetMACEMACE\-GauntSpinGTP300 KEE3\.84 \(0\.08\)3\.3 \(0\.1\)3\.1 \(0\.13\)3\.0 \(0\.2\)2\.9 \(0\.1\)2\.9 \(0\.1\)FF12\.98 \(0\.17\)10\.8 \(0\.2\)11\.0 \(0\.14\)8\.8 \(0\.3\)9\.2 \(0\.1\)9\.0 \(0\.3\)600 KEE12\.07 \(0\.45\)11\.2 \(0\.1\)11\.5 \(0\.6\)9\.7 \(0\.5\)10\.6 \(0\.5\)9\.6 \(0\.3\)FF29\.17 \(0\.22\)26\.4 \(0\.1\)26\.7 \(0\.29\)21\.8 \(0\.6\)22\.2 \(0\.2\)21\.0 \(0\.5\)1200 KEE42\.57 \(1\.46\)38\.5 \(1\.6\)39\.1 \(1\.1\)29\.8 \(1\.0\)30\.4 \(1\.2\)29\.0 \(1\.1\)FF82\.96 \(1\.77\)76\.2 \(1\.1\)81\.1 \(1\.5\)62\.0 \(0\.7\)63\.1 \(1\.2\)61\.4 \(3\.2\)DihedralEE––16\.3 \(1\.5\)7\.8 \(0\.6\)9\.9 \(0\.3\)8\.9 \(2\.5\)FF––20\.0 \(1\.2\)16\.5 \(1\.7\)17\.7 \(1\.1\)15\.3 \(0\.7\)
### 4\.3SPICE\-MACE\-OFF Chiral Subset Performance

Dataset\.The SPICE\-MACE\-OFF dataset\[kovacs2025mace\]is a large\-scale benchmark comprising over 950,000 unique configurations, including small molecules from PubChem, DES370K dimers, and biological systems\. To evaluate the specific impact of the SWSH tensor product on chiral expressivity, we curated a chiral subset by screening the original dataset using RDKit\[rdkit\]to identify enantiomeric pairs and chiral centers via SMILES sequences\. This allows us to probe whether the addition of antisymmetric spin\-paths translates to improved performance on geometrically sensitive tasks\.

Experimental Settings\.We evaluate our framework on two distinct categories of tasks\. First, we perform chirality classification by modifying the regression heads of\[nequip\], MACE\[batatia2022mace\]and MACE\-Gaunt\[luo2024enabling\]to distinguish between R\- and S\-chirality\. This includes both classification using mixed R/S training data and a parity generalization test, where the model is trained exclusively on R\-chirality and evaluated on unseen S\-chirality structures\. Second, we assess Machine Learning Interatomic Potentials \(MLIP\) through standard energy and force prediction across four chiral sub\-domains: PubChem, Amino Acids, DES370, and Dipeptides\.

Table 2:Chirality classification \(R vs\. S, both in training\)\. Validation accuracy \(%\) at selected epochs\. Values are reported as mean \(std\) over 3 runs\. Convergence: epochs to first reach 98% validation accuracy \(lower is better\), also mean \(std\) over 3 runs\.ModelEpoch 1↑\\uparrowEpoch 10↑\\uparrowEpoch 20↑\\uparrowBest \(%\)↑\\uparrowConvergence↓\\downarrowMACE\-Gaunt64\.9 \(0\.5\)95\.5 \(0\.3\)95\.8 \(0\.3\)98\.30 \(0\.08\)24\.3 \(1\.0\)MACE67\.8 \(0\.4\)96\.8 \(0\.3\)97\.3 \(0\.2\)98\.34 \(0\.07\)18\.2 \(0\.8\)NequIP68\.1 \(0\.4\)97\.0 \(0\.2\)97\.6 \(0\.2\)98\.43 \(0\.07\)16\.1 \(0\.6\)SpinGTP67\.4 \(0\.3\)97\.5 \(0\.2\)98\.4 \(0\.1\)98\.79 \(0\.05\)10\.9 \(0\.5\)Chirality Classification and Parity Generalization\.

Results for the chirality classification tasks are summarized in Tables[2](https://arxiv.org/html/2607.01408#S4.T2)\. In the standard classification task, SpinGTP achieves the fastest convergence, reaching 98% validation accuracy in nearly half the epochs required by MACE\-Gaunt \(10\.9±0\.510\.9\\pm 0\.5vs\.24\.3±1\.024\.3\\pm 1\.0\)\. The parity generalization test in Table[3](https://arxiv.org/html/2607.01408#S4.T3)reveals a critical failure mode in the original Gaunt tensor product \(MACE\-Gaunt\), which fails entirely \(0%0\\%accuracy\) to generalize to unseen S\-chirality\. This stems from the lack of odd\-parity paths in standard Gaunt coefficients\. In contrast, SpinGTP not only enables this generalization by encoding proper reflection symmetry but also outperforms Clebsch\-Gordan \(CG\) based models, achieving a superior94\.1%94\.1\\%accuracy on unseen mirror structures\.

Table 3:Chirality parity generalization: train on R\-only, evaluate on R\-only and S\-only\. acc\_S tests generalization to mirror structures never seen in training\. Values are mean \(std\) over 3 runs\.Modelacc\_R \(%\)acc\_S \(%\)Overall \(%\)MACE\-Gaunt100\.0 \(0\.0\)0\.0 \(0\.0\)46\.3 \(0\.0\)MACE100\.0 \(0\.0\)88\.6 \(1\.2\)93\.9 \(0\.7\)NequIP100\.0 \(0\.0\)90\.8 \(1\.0\)95\.0 \(0\.6\)SpinGTP100\.0 \(0\.0\)94\.1 \(0\.8\)97\.2 \(0\.4\)Energy and Force Prediction\.Table[4](https://arxiv.org/html/2607.01408#S4.T4)demonstrates that the completeness of the SWSH tensor product translates to higher accuracy in MLIP tasks\. Compared to MACE\-Gaunt, our model provides a reduction in both energy and force errors across most of the chiral subsets\. Notably, SWSH achieves state\-of\-the\-art performance in the Chiral DES370 and Dipeptides subsets, where sensitivity to complex torsional and chiral interactions is paramount\. Even when compared to the computationally heavier Clebsch\-Gordan implementations in NequIP and MACE, SpinGTP remains highly competitive, consistently achieving the lowest errors\.

Training Setup\.Detailed specifications regarding architecture, hyperparameters, and training protocols are provided in[Section˜F\.2](https://arxiv.org/html/2607.01408#A6.SS2)\.

Table 4:Results on SPICE\-MACE\-OFF Chiral Subset\. Test set MAE for models trained and evaluated exclusively on SPICE Chiral sub\-datasets\. Energy \(EE\) in meV/atom, Forces \(FF\) in meV/Å\. The best results are shown inboldand the second best results are shown withunderlines\.NequIPMACEMACE\-GauntSpinGTPChiral SubsetEEFFEEFFEEFFEEFFChiral PubChem6\.121\.05\.016\.04\.919\.35\.717\.6Chiral Amino Acids6\.520\.57\.519\.36\.622\.96\.619\.9Chiral DES3703\.38\.42\.66\.62\.98\.12\.46\.3Chiral Dipeptides4\.012\.04\.29\.73\.511\.83\.29\.4
### 4\.4OC20 IS2RE Direct

Dataset\. We test whether the SpinGTP implementation remains compatible with large\-scale atomistic training\. We use the Open Catalyst 2020 \(OC20\) initial\-structure\-to\-relaxed\-energy \(IS2RE\) task\[chanussot2021open\], where each system contains an adsorbate on a catalyst slab and the goal is to predict the relaxed energy from the initial structure\. These graphs are substantially larger and chemically more diverse than 3BPA and SPICE\. Following standard OC20 reporting, validation sets are divided into four sub\-splits, ID, OOD\-Ads, OOD\-Cat, and OOD\-Both\. The direct IS2RE split contains 460k training structures and 100k structures for validation\.

Training Details\. For the main comparison, we use the direct IS2RE setting without the IS2RS node\-level auxiliary task, following the comparison protocol used in Equiformer\. Our SpinGTP follows the Equiformer attention backbone and replaces its equivariant tensor\-product, linear, and normalization blocks with SWSH counterparts\. We use a six\-layer SWSH\-Equiformer with eight attention heads, cutoff radius5\.05\.0, 128 radial basis functions, on\-the\-fly periodic graphs, and a maximum of 500 neighbors\. Additional architecture and optimization details are provided in[Section˜F\.3](https://arxiv.org/html/2607.01408#A6.SS3)\.

Results\.[Table˜5](https://arxiv.org/html/2607.01408#S4.T5)compares our model against direct IS2RE baselines, including SchNet, DimeNet\+\+, GemNet\-dT, SphereNet, Equiformer, and EquiformerV2\. Our model achieves performance comparable to Equiformer, which uses Clebsch\-Gordan tensor products\. In particular, SpinGTP improves the ID split, suggesting that the spin\-weighted tensor product retains strong expressive power while remaining compatible with large\-scale OC20 training\.

Table 5:Comparison of model performance on energy predictions for OC20IS2RE\-Directvalidation set without noisy\-node auxiliary loss\. Our model is trained to compare against several baseline methods, including SchNet\[SchNet\], DimeNet\+\+\[DimeNet\+\+\], GemNet\-dT\[GemNet\-dT\], SphereNet\[SphereNet\], ComENet\[comenet\], Equiformer\[equiformer\]and EquiformerV2\[equiformerv2\]\. The best results are shown inboldand the second best results are shown withunderlines\.Energy MAE \(eV\)↓\\downarrowEwT \(%\)↑\\uparrowModelIDOOD AdsOOD CatOOD BothAverageIDOOD AdsOOD CatOOD BothAverageSchNet0\.64650\.70740\.64750\.66260\.66602\.962\.223\.032\.382\.65DimeNet\+\+0\.56360\.71270\.56120\.64920\.62174\.252\.484\.402\.563\.42GemNet\-dT0\.55610\.73420\.56590\.69640\.63824\.512\.244\.372\.383\.38SphereNet0\.56320\.66820\.55900\.61900\.60244\.562\.704\.592\.703\.64Equiformer0\.50880\.62710\.50510\.55450\.54894\.882\.934\.922\.983\.93EquiformerV20\.51610\.70410\.52450\.63650\.5953\-\-\-\-\-SpinGTP0\.50660\.65100\.51300\.58870\.56485\.072\.585\.062\.853\.89

## 5Limitations and Summary

Limitations\. One limitation of SpinGTP is that nonzero spin\-weighted features require a consistent gauge\. Without compatible frame choices, aggregating spin features can violate equivariance\. We address this through geometry\-dependent frame construction with a shared convention across nonzero spin irreps \(see[Section˜A\.2](https://arxiv.org/html/2607.01408#A1.SS2)\), though ambiguity remains for highly symmetric point clouds\. Another limitation is that we did not observe a clear acceleration from using the spherical transform, possibly because the multiplicity is large relative to the smallLL\. Resolving these limitations is a direction for future work\.

Summary\. In this work, we introduced a novel equivariant framework based on Spin\-Weighted Spherical Harmonics \(SWSH\), bridging the gap between the computational efficiency of Gaunt\-based tensor products and the mathematical completeness of the Clebsch\-Gordan Tensor Product\. By leveraging the spin\-weight degree of freedom, we recover the antisymmetric paths essential for resolving chiral geometries\. Theoretically, we demonstrate that the SWSH basis captures odd\-parity interactions through specialized spin\-selection rules\. Architecturally, we proposed equivariant SWSH linear and normalization layers that enforce strict symmetry constraints while optimizing for hardware throughput\. Empirically, our framework demonstrates superior expressivity across diverse benchmarks\. These results suggest that SWSH\-based equivariant networks offer a robust and scalable solution for high\-fidelity geometric modeling in molecular science and beyond\.

## Acknowledgments

This work was supported in part by the National Science Foundation under Grants IIS\-2243850, CNS\-2328395, and MOMS\-2331036; the National Institutes of Health under Grant U01AG070112; the Texas A&M University Division of Research Targeted Proposal Teams Funding Program; and the Texas A&M Institute of Data Science Thematic Labs Program\.

## References

## Appendix ASpin\-Weighted Spherical Harmonics \(SWSH\)

Spin\-weighted spherical harmonics \(SWSH\), denoted asYℓ​ms​\(θ,ϕ\)\{\}\_\{s\}Y\_\{\\ell m\}\(\\theta,\\phi\), are generalizations of the standard spherical harmonicsYℓ​mY\_\{\\ell m\}\. Unlike ordinary spherical harmonics, which are scalar fields, spin\-weighted harmonics are functions on the sphere that behave asU​\(1\)\\mathrm\{U\}\(1\)gauge fields, characterized by a degreeℓ\\ell, a magnetic ordermm, and a spin weightsssatisfying\|s\|≤ℓ\|s\|\\leq\\ell\.

In the standard case wheres=0s=0, these functions reduce to the conventional scalar spherical harmonics, such thatYℓ​m0=Yℓ​m\{\}\_\{0\}Y\_\{\\ell m\}=Y\_\{\\ell m\}\. Like their scalar counterparts, SWSHs form a complete orthonormal basis over the sphereS2\\mathrm\{S\}^\{2\}, satisfying the orthogonality condition:

∫S2Yℓ​ms​Yℓ′​m′∗s​𝑑Ω=δℓ​ℓ′​δm​m′\.\\int\_\{S^\{2\}\}\{\}\_\{s\}Y\_\{\\ell m\}\\,\{\}\_\{s\}\{Y\}\_\{\\ell^\{\\prime\}m^\{\\prime\}\}^\{\*\}\\,d\\Omega=\\delta\_\{\\ell\\ell^\{\\prime\}\}\\delta\_\{mm^\{\\prime\}\}\.\(11\)This orthonormality, combined with their unique transformation properties under local rotations, makes them ideal for representing high\-dimensional equivariant features that standard scalar bases cannot fully capture\.

### A\.1Explicit Formula

The spin\-weighted spherical harmonics can be calculated directly using the formula\[goldberg1967spin\]

Yℓ​ms​\(θ,ϕ\)=As​ℓ​m​sin2​ℓ⁡\(θ2\)​ei​m​ϕ×∑r=0ℓ−s\(−1\)r\(ℓ−sr\)\(ℓ\+sr\+s−m\)×cot2​r\+s−m⁡\(θ2\)\.\\begin\{split\}\{\}\_\{s\}Y\_\{\\ell m\}\(\\theta,\\phi\)&=A\_\{s\\ell m\}\\sin^\{2\\ell\}\\left\(\{\\frac\{\\theta\}\{2\}\}\\right\)e^\{im\\phi\}\\\\ &\\quad\\times\\sum\_\{r=0\}^\{\\ell\-s\}\\left\(\-1\\right\)^\{r\}\\binom\{\\ell\-s\}\{r\}\\binom\{\\ell\+s\}\{r\+s\-m\}\\\\ &\\quad\\times\\cot^\{2r\+s\-m\}\\left\(\{\\frac\{\\theta\}\{2\}\}\\right\)\.\\end\{split\}\(12\)
Specifically, the first few spin\-weighted spherical harmonics fors=1s=1andℓ=1\\ell=1are given by

Y101​\(θ,ϕ\)\\displaystyle\{\}\_\{1\}Y\_\{10\}\(\\theta,\\phi\)=38​π​sin⁡θ,Y1±11​\(θ,ϕ\)=−316​π​\(1∓cos⁡θ\)​e±i​ϕ\.\\displaystyle=\{\\sqrt\{\\frac\{3\}\{8\\pi\}\}\}\\,\\sin\\theta,\\quad\{\}\_\{1\}Y\_\{1\\pm 1\}\(\\theta,\\phi\)=\-\{\\sqrt\{\\frac\{3\}\{16\\pi\}\}\}\(1\\mp\\cos\\theta\)\\,e^\{\\pm i\\phi\}\.\(13\)
With the phase convention used in this definition, the harmonics satisfy the following conjugation and parity relations

Yℓ​m∗s\\displaystyle\{\}\_\{s\}\{Y\}\_\{\\ell m\}^\{\*\}=\(−1\)s\+m​Yℓ​\(−m\)−s,Yℓ​ms​\(π−θ,ϕ\+π\)=\(−1\)ℓ​Yℓ​m−s​\(θ,ϕ\)\.\\displaystyle=\(\-1\)^\{s\+m\}\{\}\_\{\-s\}Y\_\{\\ell\(\-m\)\},\\quad\{\}\_\{s\}Y\_\{\\ell m\}\(\\pi\-\\theta,\\phi\+\\pi\)=\(\-1\)^\{\\ell\}\{\}\_\{\-s\}Y\_\{\\ell m\}\(\\theta,\\phi\)\.\(14\)

### A\.2Equivariance and Spin\-Weighted Transformations

A defining characteristic of spin\-weighted spherical harmonics is their transformation law under the rotation groupSO​\(3\)\\mathrm\{SO\}\(3\)\. Unlike standard scalar harmonics \(s=0s=0\), which transform solely via the WignerDD\-matrices, SWSHs are sections of a line bundle and thus pick up a local phase shift corresponding to the rotation of the local tangent frame\.

##### Rotation Law\.

Given a rotationR∈SO​\(3\)R\\in\\mathrm\{SO\}\(3\), the transformation of a spin\-weighted signal at a point𝐱∈S2\\mathbf\{x\}\\in\\mathrm\{S\}^\{2\}is governed by:

Yℓ​ms​\(R​𝐱\)=ei​s​ψ​\(R,𝐱\)​∑m′=−ℓℓDm′​m\(ℓ\)​\(R\)​Yℓ​m′s​\(𝐱\)\{\}\_\{s\}Y\_\{\\ell m\}\(R\\mathbf\{x\}\)=e^\{is\\psi\(R,\\mathbf\{x\}\)\}\\sum\_\{m^\{\\prime\}=\-\\ell\}^\{\\ell\}D^\{\(\\ell\)\}\_\{m^\{\\prime\}m\}\(R\)\\,\{\}\_\{s\}Y\_\{\\ell m^\{\\prime\}\}\(\\mathbf\{x\}\)\(15\)whereDm′​m\(ℓ\)​\(R\)D^\{\(\\ell\)\}\_\{m^\{\\prime\}m\}\(R\)are the elements of the\(2​ℓ\+1\)\(2\\ell\+1\)\-dimensional irreducible representation ofSO​\(3\)\\mathrm\{SO\}\(3\)\. The termei​s​ψe^\{is\\psi\}represents aU​\(1\)\\mathrm\{U\}\(1\)gauge transformation, whereψ​\(R,𝐱\)\\psi\(R,\\mathbf\{x\}\)is the angle by which the local tangent frame at𝐱\\mathbf\{x\}rotates relative to the fixed coordinate basis after applyingRR\.

##### Geometry\-Dependent Frames\.

In our implementation, the gauge is fixed by equivariant frames constructed from the input geometry\. Before evaluating SWSHs, the input is rotated into a common frame\. We implement both the global frame and local node frame based on the geometry\.

For theglobal frame, given a graphbband point cloud\{𝐱i\}i∈b\\\{\\mathbf\{x\}\_\{i\}\\\}\_\{i\\in b\}, let

𝐜b=∑i∈bwi​𝐱i,𝐱~i=𝐱i−𝐜b\.\\mathbf\{c\}\_\{b\}=\\sum\_\{i\\in b\}w\_\{i\}\\mathbf\{x\}\_\{i\},\\qquad\\widetilde\{\\mathbf\{x\}\}\_\{i\}=\\mathbf\{x\}\_\{i\}\-\\mathbf\{c\}\_\{b\}\.We construct a graph frameFb∈SO​\(3\)F\_\{b\}\\in\\mathrm\{SO\}\(3\)from the eigensystem of the centered covariance

Cb=∑i∈bwi​𝐱~i​𝐱~i⊤\.C\_\{b\}=\\sum\_\{i\\in b\}w\_\{i\}\\widetilde\{\\mathbf\{x\}\}\_\{i\}\\widetilde\{\\mathbf\{x\}\}\_\{i\}^\{\\top\}\.\(16\)The eigenvector signs are fixed by parity\-even axial references

𝐡b\(q\)=∑i∈bwi​‖𝐫~i‖q​𝐫~i,𝐚b=𝐡b\(1\)×𝐡b\(2\),𝐛b=𝐡b\(2\)×𝐡b\(3\)\.\\mathbf\{h\}\_\{b\}^\{\(q\)\}=\\sum\_\{i\\in b\}w\_\{i\}\\\|\\widetilde\{\\mathbf\{r\}\}\_\{i\}\\\|^\{q\}\\widetilde\{\\mathbf\{r\}\}\_\{i\},\\qquad\\mathbf\{a\}\_\{b\}=\\mathbf\{h\}\_\{b\}^\{\(1\)\}\\times\\mathbf\{h\}\_\{b\}^\{\(2\)\},\\qquad\\mathbf\{b\}\_\{b\}=\\mathbf\{h\}\_\{b\}^\{\(2\)\}\\times\\mathbf\{h\}\_\{b\}^\{\(3\)\}\.If no repeated eigenvalues and vanishing axial references are present, this gives a permutation\-equivariant,SO​\(3\)\\mathrm\{SO\}\(3\)\-equivariant, and parity\-invariant graph frame\.

If the global frame encounters eigenvector ambiguity, we use thelocal node frame\. For each nodejj, we similarly construct a local node frameFj∈SO​\(3\)F\_\{j\}\\in\\mathrm\{SO\}\(3\)from the neighborhood quadrupole

Qj=∑k→jwj​k​\(‖𝐫j​k‖2​I−𝐫j​k​𝐫j​k⊤\),Q\_\{j\}=\\sum\_\{k\\to j\}w\_\{jk\}\\left\(\\\|\\mathbf\{r\}\_\{jk\}\\\|^\{2\}I\-\\mathbf\{r\}\_\{jk\}\\mathbf\{r\}\_\{jk\}^\{\\top\}\\right\),\(17\)where𝐫j​k=𝐱j−𝐱k\\mathbf\{r\}\_\{jk\}=\\mathbf\{x\}\_\{j\}\-\\mathbf\{x\}\_\{k\}\. Givne𝐠j=∑k→jwj​k​𝐫j​k\\mathbf\{g\}\_\{j\}=\\sum\_\{k\\to j\}w\_\{jk\}\\mathbf\{r\}\_\{jk\}, the eigenvector signs are fixed by axial references

𝐡j\(q\)=∑k→jwj​k​‖𝐫j​k‖q​𝐫j​k,𝐚j=𝐡j\(1\)×𝐠j,𝐛j=𝐡j\(2\)×𝐠j\.\\mathbf\{h\}\_\{j\}^\{\(q\)\}=\\sum\_\{k\\to j\}w\_\{jk\}\\\|\\mathbf\{r\}\_\{jk\}\\\|^\{q\}\\mathbf\{r\}\_\{jk\},\\qquad\\mathbf\{a\}\_\{j\}=\\mathbf\{h\}\_\{j\}^\{\(1\)\}\\times\\mathbf\{g\}\_\{j\},\\qquad\\mathbf\{b\}\_\{j\}=\\mathbf\{h\}\_\{j\}^\{\(2\)\}\\times\\mathbf\{g\}\_\{j\}\.This also gives a permutation\-equivariant,SO​\(3\)\\mathrm\{SO\}\(3\)\-equivariant, and parity\-invariant node frame whenever the local neighborhood is nondegenerate\.

### A\.3Nonzero Spin\-Weighted Gaunt Paths

The spin\-weighted Gaunt coefficient is

G\(ℓ1,m1,s1\)​\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)=∏i=13\(2​ℓi\+1\)4​π​\(ℓ1ℓ2ℓ3m1m2−m3\)​\(ℓ1ℓ2ℓ3s1s2−s3\)\.G^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\)\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}=\\sqrt\{\\frac\{\\prod\_\{i=1\}^\{3\}\(2\\ell\_\{i\}\+1\)\}\{4\\pi\}\}\\begin\{pmatrix\}\\ell\_\{1\}&\\ell\_\{2\}&\\ell\_\{3\}\\\\ m\_\{1\}&m\_\{2\}&\-m\_\{3\}\\end\{pmatrix\}\\begin\{pmatrix\}\\ell\_\{1\}&\\ell\_\{2\}&\\ell\_\{3\}\\\\ s\_\{1\}&s\_\{2\}&\-s\_\{3\}\\end\{pmatrix\}\.\(18\)The scalar GTP case fixess1=s2=s3=0s\_\{1\}=s\_\{2\}=s\_\{3\}=0, so the second Wigner symbol vanishes wheneverℓ1\+ℓ2\+ℓ3\\ell\_\{1\}\+\\ell\_\{2\}\+\\ell\_\{3\}is odd\. Allowing signed spin weights removes this limitation at the path level\.

###### Proposition A\.1\.

Spin\-weighted path completion Let\(ℓ1,ℓ2,ℓ3\)\(\\ell\_\{1\},\\ell\_\{2\},\\ell\_\{3\}\)satisfy the Clebsch\-Gordan triangle rule\. Then there exist signed spin weights

s1∈\[−ℓ1,ℓ1\],s2∈\[−ℓ2,ℓ2\],s3∈\[−ℓ3,ℓ3\],s\_\{1\}\\in\[\-\\ell\_\{1\},\\ell\_\{1\}\],\\qquad s\_\{2\}\\in\[\-\\ell\_\{2\},\\ell\_\{2\}\],\\qquad s\_\{3\}\\in\[\-\\ell\_\{3\},\\ell\_\{3\}\],withs3=s1\+s2s\_\{3\}=s\_\{1\}\+s\_\{2\}, such that the signed\-spin path\(ℓ1,s1\)⊗\(ℓ2,s2\)→\(ℓ3,s3\)\(\\ell\_\{1\},s\_\{1\}\)\\otimes\(\\ell\_\{2\},s\_\{2\}\)\\to\(\\ell\_\{3\},s\_\{3\}\)is not identically zero\.

###### Proof A\.2\.

Choose

s1=ℓ1,s2=ℓ3−ℓ1,s3=ℓ3\.s\_\{1\}=\\ell\_\{1\},\\qquad s\_\{2\}=\\ell\_\{3\}\-\\ell\_\{1\},\\qquad s\_\{3\}=\\ell\_\{3\}\.Thens3=s1\+s2s\_\{3\}=s\_\{1\}\+s\_\{2\}and−ℓ2≤ℓ3−ℓ1≤ℓ2\-\\ell\_\{2\}\\leq\\ell\_\{3\}\-\\ell\_\{1\}\\leq\\ell\_\{2\}, so all chosen spin weights are admissible\. Consider the spin Wigner factor

S=\(ℓ1ℓ2ℓ3ℓ1ℓ3−ℓ1−ℓ3\)\.S=\\begin\{pmatrix\}\\ell\_\{1\}&\\ell\_\{2\}&\\ell\_\{3\}\\\\ \\ell\_\{1\}&\\ell\_\{3\}\-\\ell\_\{1\}&\-\\ell\_\{3\}\\end\{pmatrix\}\.For this boundary case, the Wigner3​j3jsymbol has the explicit magnitude

\|S\|=\[\(2​ℓ1\)\!​\(2​ℓ3\)\!\(ℓ1−ℓ2\+ℓ3\)\!​\(ℓ1\+ℓ2\+ℓ3\+1\)\!\]1/2\.\|S\|=\\left\[\\frac\{\(2\\ell\_\{1\}\)\!\(2\\ell\_\{3\}\)\!\}\{\(\\ell\_\{1\}\-\\ell\_\{2\}\+\\ell\_\{3\}\)\!\(\\ell\_\{1\}\+\\ell\_\{2\}\+\\ell\_\{3\}\+1\)\!\}\\right\]^\{1/2\}\.All factorial arguments are nonnegative by the triangle inequalities, soS≠0S\\neq 0\. For these fixed spin weights, the spin\-weighted Gaunt coefficient factors as

G\(ℓ1,m1,s1\)​\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)=∏i=13\(2​ℓi\+1\)4​π​\(ℓ1ℓ2ℓ3m1m2−m3\)​S\.G^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\)\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}=\\sqrt\{\\frac\{\\prod\_\{i=1\}^\{3\}\(2\\ell\_\{i\}\+1\)\}\{4\\pi\}\}\\begin\{pmatrix\}\\ell\_\{1\}&\\ell\_\{2\}&\\ell\_\{3\}\\\\ m\_\{1\}&m\_\{2\}&\-m\_\{3\}\\end\{pmatrix\}S\.Taking

m1=ℓ1,m2=ℓ3−ℓ1,m3=ℓ3m\_\{1\}=\\ell\_\{1\},\\qquad m\_\{2\}=\\ell\_\{3\}\-\\ell\_\{1\},\\qquad m\_\{3\}=\\ell\_\{3\}gives the same nonzero Wigner factor in the magnetic part\. Hence

G\(ℓ1,m1,s1\)​\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)=∏i=13\(2​ℓi\+1\)4​π​S2≠0\.G^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\)\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}=\\sqrt\{\\frac\{\\prod\_\{i=1\}^\{3\}\(2\\ell\_\{i\}\+1\)\}\{4\\pi\}\}S^\{2\}\\neq 0\.Therefore the signed\-spin Gaunt tensor has a nonzero entry and the corresponding bilinear map is not identically zero\.

## Appendix BBest Asymptotic Runtime Cost

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

Runtime complexity for\|s\|max=1\|s\|\_\{\\max\}=1Given bounded multiplicities and a maximum angular degreeLL, a fully connected SpinGTP can be evaluated inO​\(L4​log2⁡L\)O\(L^\{4\}\\log^\{2\}L\)using fast spherical transforms\.

###### Proof B\.2\.

Let degrees0≤ℓ1,ℓ2≤L0\\leq\\ell\_\{1\},\\ell\_\{2\}\\leq Land spin weights−1≤s1,s2≤1\-1\\leq s\_\{1\},s\_\{2\}\\leq 1\. Define the spherical signals

f​\(Ω\)=∑m1=−ℓ1ℓ1xℓ1​m1​s1\(1\)​Yℓ1​m1s1​\(Ω\)andg​\(Ω\)=∑m2=−ℓ2ℓ2xℓ2​m2​s2\(2\)​Yℓ2​m2s2​\(Ω\)\.\\displaystyle f\(\\Omega\)=\\sum\_\{m\_\{1\}=\-\\ell\_\{1\}\}^\{\\ell\_\{1\}\}x\_\{\\ell\_\{1\}m\_\{1\}s\_\{1\}\}^\{\(1\)\}\\,\{\}\_\{s\_\{1\}\}Y\_\{\\ell\_\{1\}m\_\{1\}\}\(\\Omega\)\\quad\\text\{and\}\\quad g\(\\Omega\)=\\sum\_\{m\_\{2\}=\-\\ell\_\{2\}\}^\{\\ell\_\{2\}\}x\_\{\\ell\_\{2\}m\_\{2\}s\_\{2\}\}^\{\(2\)\}\\,\{\}\_\{s\_\{2\}\}Y\_\{\\ell\_\{2\}m\_\{2\}\}\(\\Omega\)\.\(19\)Fors3=s1\+s2s\_\{3\}=s\_\{1\}\+s\_\{2\}, the coefficient of the pointwise product off​\(Ω\)f\(\\Omega\)andg​\(Ω\)g\(\\Omega\)is

∫S2f​\(Ω\)​g​\(Ω\)​Yℓ3​m3∗s3​\(Ω\)​𝑑Ω\\displaystyle\\int\_\{\\mathrm\{S\}^\{2\}\}f\(\\Omega\)g\(\\Omega\)\\,\{\}\_\{s\_\{3\}\}Y\_\{\\ell\_\{3\}m\_\{3\}\}^\{\*\}\(\\Omega\)\\,d\\Omega=∑m1,m2G\(ℓ1,m1,s1\)​\(ℓ2,m2,s2\)\(ℓ3,m3,s3\)​xℓ1​m1​s1\(1\)​xℓ2​m2​s2\(2\)\.\\displaystyle=\\sum\_\{m\_\{1\},m\_\{2\}\}G^\{\(\\ell\_\{3\},m\_\{3\},s\_\{3\}\)\}\_\{\(\\ell\_\{1\},m\_\{1\},s\_\{1\}\)\(\\ell\_\{2\},m\_\{2\},s\_\{2\}\)\}x\_\{\\ell\_\{1\}m\_\{1\}s\_\{1\}\}^\{\(1\)\}x\_\{\\ell\_\{2\}m\_\{2\}s\_\{2\}\}^\{\(2\)\}\.\(20\)Thus, a tensor product path of\(ℓ1,s1,ℓ2,s2\)\(\\ell\_\{1\},s\_\{1\},\\ell\_\{2\},s\_\{2\}\)is evaluated by two inverse spherical transforms, one pointwise product, and one forward spherical transform\. Sinceℓ3≤ℓ1\+ℓ2≤2​L\\ell\_\{3\}\\leq\\ell\_\{1\}\+\\ell\_\{2\}\\leq 2L, the forward and backward transform costsO​\(L2​log2⁡L\)O\(L^\{2\}\\log^\{2\}L\)using fast spherical transform algorithm\[healy2003ffts\]\. The pointwise multiplication itself costsO​\(L2\)O\(L^\{2\}\)\. Therefore, a path of\(ℓ1,s1,ℓ2,s2\)\(\\ell\_\{1\},s\_\{1\},\\ell\_\{2\},s\_\{2\}\)costsO​\(L2​log2⁡L\)\+O​\(L2\)=O​\(L2​log2⁡L\)O\(L^\{2\}\\log^\{2\}L\)\+O\(L^\{2\}\)=O\(L^\{2\}\\log^\{2\}L\)\.

There are\(L\+1\)2=O​\(L2\)\(L\+1\)^\{2\}=O\(L^\{2\}\)input degree pairs of\(ℓ1,ℓ2\)\(\\ell\_\{1\},\\ell\_\{2\}\)\. These pairs are evaluated separately for the independent path weights in[Equation˜7](https://arxiv.org/html/2607.01408#S3.E7)\. Each spherical transform returns all admissible output degreesℓ3\\ell\_\{3\}simultaneously, so no additional factor ofLLis required\. Consequently, the time complexity of SpinGTP is

TSpinGTP​\(L\)\\displaystyle T\_\{\\mathrm\{SpinGTP\}\}\(L\)=O​\(L2\)×O​\(L2​log2⁡L\)\\displaystyle=O\(L^\{2\}\)\\times O\(L^\{2\}\\log^\{2\}L\)\(21\)=O​\(L4​log2⁡L\)\.\\displaystyle=O\(L^\{4\}\\log^\{2\}L\)\.This is the same asymptotic cost as the complete VSTP of\[xie2026asymptotically\]\.

## Appendix CReal Basis Spin\-Weighted Spherical Harmonics

In computational physics and deep learning, while complex\-valued representations are mathematically natural, real\-valued features are often preferred for memory efficiency and compatibility with standard nonlinearities\. We therefore store SWSH features in a real orthonormal basis and use a unitary change of basis only when evaluating complex Gaunt contractions\.

### C\.1The Unitary𝐐\\mathbf\{Q\}Matrix

Let𝐚real\\mathbf\{a\}\_\{\\text\{real\}\}denote real\-basis coefficients and𝐚complex\\mathbf\{a\}\_\{\\text\{complex\}\}denote complex SWSH coefficients\. We use a unitary matrix𝐐\\mathbf\{Q\}such that

𝐚complex=𝐐𝐚real,𝐚real=Re⁡\(𝐐†​𝐚complex\)\.\\mathbf\{a\}\_\{\\text\{complex\}\}=\\mathbf\{Q\}\\mathbf\{a\}\_\{\\text\{real\}\},\\qquad\\mathbf\{a\}\_\{\\text\{real\}\}=\\operatorname\{Re\}\\\!\\left\(\\mathbf\{Q\}^\{\\dagger\}\\mathbf\{a\}\_\{\\text\{complex\}\}\\right\)\.\(22\)For the scalar case\|s\|=0\|s\|=0,𝐐∈ℂ\(2​ℓ\+1\)×\(2​ℓ\+1\)\\mathbf\{Q\}\\in\\mathbb\{C\}^\{\(2\\ell\+1\)\\times\(2\\ell\+1\)\}\. With rows and columns indexed bym,k∈\{−ℓ,…,ℓ\}m,k\\in\\\{\-\\ell,\\ldots,\\ell\\\}, the matrix elements used in our implementation are

𝐐m,k=\{1if​m=0,k=0,12if​m\>0,k=m,i2if​m\>0,k=−m,\(−1\)\|m\|2if​m<0,k=\|m\|,−i​\(−1\)\|m\|2if​m<0,k=−\|m\|,0otherwise\.\\mathbf\{Q\}\_\{m,k\}=\\begin\{cases\}1&\\text\{if \}m=0,\\ k=0,\\\\\[2\.0pt\] \\frac\{1\}\{\\sqrt\{2\}\}&\\text\{if \}m\>0,\\ k=m,\\\\\[2\.0pt\] \\frac\{i\}\{\\sqrt\{2\}\}&\\text\{if \}m\>0,\\ k=\-m,\\\\\[2\.0pt\] \\frac\{\(\-1\)^\{\|m\|\}\}\{\\sqrt\{2\}\}&\\text\{if \}m<0,\\ k=\|m\|,\\\\\[2\.0pt\] \\frac\{\-i\(\-1\)^\{\|m\|\}\}\{\\sqrt\{2\}\}&\\text\{if \}m<0,\\ k=\-\|m\|,\\\\\[2\.0pt\] 0&\\text\{otherwise\}\.\\end\{cases\}\(23\)For\|s\|\>0\|s\|\>0, both signed spin sectorsssand−s\-sare retained\. The corresponding matrix has size

𝐐∈ℂ2​\(2​ℓ\+1\)×2​\(2​ℓ\+1\),\\mathbf\{Q\}\\in\\mathbb\{C\}^\{2\(2\\ell\+1\)\\times 2\(2\\ell\+1\)\},with the complex coefficients ordered by the−\|s\|\-\|s\|sector followed by the\|s\|\|s\|sector\. This doubled matrix couples the paired±s\\pm sand±m\\pm mcomponents using[Equation˜5](https://arxiv.org/html/2607.01408#S3.E5)\.

### C\.2Example: The𝐐\\mathbf\{Q\}Matrix forℓ=1\\ell=1

Forℓ=1\\ell=1and\|s\|=0\|s\|=0, with both rows and columns ordered as\(−1,0,\+1\)\(\-1,0,\+1\), the transformation matrix is

𝐐\(1\)=12​\(i0−1020i01\)\.\\mathbf\{Q\}^\{\(1\)\}=\\frac\{1\}\{\\sqrt\{2\}\}\\begin\{pmatrix\}i&0&\-1\\\\ 0&\\sqrt\{2\}&0\\\\ i&0&1\\end\{pmatrix\}\.\(24\)
By using this, real coefficients are mapped to the complex signed\-spin basis, the generalized Gaunt contraction is evaluated, and the result is mapped back to the real basis using𝐐†\\mathbf\{Q\}^\{\\dagger\}\.

## Appendix DDerivation of the Parity\-Equivariant Basis

A signalf​\(𝐱\)f\(\\mathbf\{x\}\)is said to possess a definite parityp∈\{\+1,−1\}p\\in\\\{\+1,\-1\\\}if, under the spatial inversion operatorP:𝐱→−𝐱P:\\mathbf\{x\}\\to\-\\mathbf\{x\}, it satisfies the equivariance relation:

f​\(P​𝐱\)=p​f​\(𝐱\)\.f\(P\\mathbf\{x\}\)=pf\(\\mathbf\{x\}\)\.\(25\)For standard scalar spherical harmonics \(s=0s=0\), parity is intrinsically tied to the degreeℓ\\ell, whereYℓ​m​\(−𝐱\)=\(−1\)ℓ​Yℓ​m​\(𝐱\)Y\_\{\\ell m\}\(\-\\mathbf\{x\}\)=\(\-1\)^\{\\ell\}Y\_\{\\ell m\}\(\\mathbf\{x\}\)\. However, spin\-weighted spherical harmonics exhibit a more complex behavior under inversion\. As shown in[Section˜A\.1](https://arxiv.org/html/2607.01408#A1.SS1), the parity transformation for SWSH involves a simultaneous inversion of the spin weight:

Yℓ​ms​\(−𝐱\)=\(−1\)ℓ​Yℓ​m−s​\(𝐱\)\.\{\}\_\{s\}Y\_\{\\ell m\}\(\-\\mathbf\{x\}\)=\(\-1\)^\{\\ell\}\{\}\_\{\-s\}Y\_\{\\ell m\}\(\\mathbf\{x\}\)\.\(26\)Because the inversion maps a function of spin weightssto a function of spin weight−s\-s, the individual basis functions do not possess a well\-defined parity unlesss=0s=0\. This same coupling persists in the real\-basis representationRℓ​ms\{\}\_\{s\}R\_\{\\ell m\}defined in Eq\.[5](https://arxiv.org/html/2607.01408#S3.E5):

Rℓ​ms​\(−𝐱\)=\(−1\)ℓ​Rℓ​m−s​\(𝐱\)\.\{\}\_\{s\}R\_\{\\ell m\}\(\-\\mathbf\{x\}\)=\(\-1\)^\{\\ell\}\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\.\(27\)To construct a basis that satisfies the parity\-equivariance constraint in Eq\.[25](https://arxiv.org/html/2607.01408#A4.E25), we seek a linear combination of the spin\-doubled basis functions\. We define the Parity\-Equivariant SWSH Basis, denotedℛps,ℓ,m​\(𝐱\)\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\), as:

ℛps,ℓ,m​\(𝐱\)=12​\(Rℓ​m−s​\(𝐱\)\+p​\(−1\)ℓ​Rℓ​ms​\(𝐱\)\)\.\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\)=\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\+p\(\-1\)^\{\\ell\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\\right\)\.\(28\)
###### Proposition D\.1\.

Parity Equivarianceℛps,ℓ,m\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}satisfies parity equivariance relation[Equation˜25](https://arxiv.org/html/2607.01408#A4.E25)\.

###### Proof D\.2\.

To verify that this construction yields a basis with paritypp, we apply the inversion operatorPPsuch that

ℛps,ℓ,m​\(−𝐱\)\\displaystyle\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\-\\mathbf\{x\}\)=12​\(Rℓ​m−s​\(−𝐱\)\+p​\(−1\)ℓ​Rℓ​ms​\(−𝐱\)\)\\displaystyle=\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{\-s\}R\_\{\\ell m\}\(\-\\mathbf\{x\}\)\+p\(\-1\)^\{\\ell\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\-\\mathbf\{x\}\)\\right\)=12​\(\(−1\)ℓ​Rℓ​ms​\(𝐱\)\+p​\(−1\)ℓ​\(−1\)ℓ​Rℓ​m−s​\(𝐱\)\)\\displaystyle=\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\(\-1\)^\{\\ell\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\+p\(\-1\)^\{\\ell\}\\,\(\-1\)^\{\\ell\}\\,\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\\right\)=12​\(\(−1\)ℓ​Rℓ​ms​\(𝐱\)\+p​Rℓ​m−s​\(𝐱\)\)\\displaystyle=\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\(\-1\)^\{\\ell\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\+p\\,\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\\right\)=p​\(12​\(Rℓ​m−s​\(𝐱\)\+\(−1\)ℓp​Rℓ​ms​\(𝐱\)\)\)\\displaystyle=p\\left\(\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\+\\frac\{\(\-1\)^\{\\ell\}\}\{p\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\\right\)\\right\)=p​\(12​\(Rℓ​m−s​\(𝐱\)\+p​\(−1\)ℓ​Rℓ​ms​\(𝐱\)\)\)\\displaystyle=p\\left\(\\frac\{1\}\{\\sqrt\{2\}\}\\left\(\{\}\_\{\-s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\+p\(\-1\)^\{\\ell\}\\,\{\}\_\{s\}R\_\{\\ell m\}\(\\mathbf\{x\}\)\\right\)\\right\)=p​ℛps,ℓ,m​\(𝐱\)\.\\displaystyle=p\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\)\.\(29\)

This derivation confirms that the basisℛps,ℓ,m​\(𝐱\)\\mathcal\{R\}\_\{p\}^\{s,\\ell,m\}\(\\mathbf\{x\}\)adheres strictly to the prescribed paritypp\. By decoupling the parity\-even \(p=\+1p=\+1\) and parity\-odd \(p=−1p=\-1\) sectors, this formulation allows the network to explicitly represent pseudoscalar and axial\-vector interactions, which are essential for resolving the antisymmetry gap in chiral geometric modeling\.

## Appendix ESpinGTP Implementation and Time Comparison

We follow a similar implementation style with respect to e3nn\[e3nn\_software\]\.

### E\.1Tensor Product Implementation

##### Multiplicity connection modes\.

All SpinGTP variants use the same real SWSH Gaunt kernel\. For one instruction involving irrepsρ1=\(ℓ1,\|s1\|,p1\)\\rho\_\{1\}=\(\\ell\_\{1\},\|s\_\{1\}\|,p\_\{1\}\),ρ2=\(ℓ2,\|s2\|,p2\)\\rho\_\{2\}=\(\\ell\_\{2\},\|s\_\{2\}\|,p\_\{2\}\), andρ3=\(ℓ3,\|s3\|,p3\)\\rho\_\{3\}=\(\\ell\_\{3\},\|s\_\{3\}\|,p\_\{3\}\)withp3=p1​p2p\_\{3\}=p\_\{1\}p\_\{2\}, let

𝐊ρ1,ρ2ρ3​\(m1,m2,m3\)\\mathbf\{K\}^\{\\rho\_\{3\}\}\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(m\_\{1\},m\_\{2\},m\_\{3\}\)\(30\)denote the real\-basis SpinGTP kernel obtained from the signed\-spin Gaunt contractionGGand parity recombination described in[Equation˜7](https://arxiv.org/html/2607.01408#S3.E7)\. Herem1,m2,m3m\_\{1\},m\_\{2\},m\_\{3\}index the real irrep components, andu,v,wu,v,windex multiplicity channels\. The different tensor\-product layers differ only in how the multiplicity indices are connected\.

Fully connected tensor product\.The fully connected mode \(modeuvw\) enumerates all admissible irrep\-block pairs\(ρ1,ρ2\)\(\\rho\_\{1\},\\rho\_\{2\}\)and all requested output irrepsρ3∈ρ1⊗ρ2\\rho\_\{3\}\\in\\rho\_\{1\}\\otimes\\rho\_\{2\}\. For each instruction, it first forms

Ru​v,m3=∑m1,m2𝐊ρ1,ρ2ρ3​\(m1,m2,m3\)​𝐱u,m1\(1\)​𝐱v,m2\(2\),R\_\{uv,m\_\{3\}\}=\\sum\_\{m\_\{1\},m\_\{2\}\}\\mathbf\{K\}^\{\\rho\_\{3\}\}\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(m\_\{1\},m\_\{2\},m\_\{3\}\)\\mathbf\{x\}^\{\(1\)\}\_\{u,m\_\{1\}\}\\mathbf\{x\}^\{\(2\)\}\_\{v,m\_\{2\}\},\(31\)and then applies a learned dense multiplicity mixing

\[𝐱\(1\)⊗𝐖𝐱\(2\)\]\(w,m3\)=∑u=1U∑v=1V𝐖u​v​w​Ru​v,m3\.\[\\mathbf\{x\}^\{\(1\)\}\\otimes\_\{\\mathbf\{W\}\}\\mathbf\{x\}^\{\(2\)\}\]^\{\(w,m\_\{3\}\)\}=\\sum\_\{u=1\}^\{U\}\\sum\_\{v=1\}^\{V\}\\mathbf\{W\}\_\{uvw\}R\_\{uv,m\_\{3\}\}\.\(32\)The parameter count per instruction isU​V​WUVW\.

Depthwise tensor product\.The depthwise mode \(modeuvu\) is fully connected over admissible irrep\-block pairs, but preserves the first input multiplicity channel\. For each instruction, the output multiplicity is tied to the first input multiplicity \(w=uw=u\), and the layer computes

\[𝐱\(1\)⊗𝐖𝐱\(2\)\]\(u,m3\)=∑v=1V𝐖u​v​∑m1,m2𝐊ρ1,ρ2ρ3​\(m1,m2,m3\)​𝐱u,m1\(1\)​𝐱v,m2\(2\)\.\[\\mathbf\{x\}^\{\(1\)\}\\otimes\_\{\\mathbf\{W\}\}\\mathbf\{x\}^\{\(2\)\}\]^\{\(u,m\_\{3\}\)\}=\\sum\_\{v=1\}^\{V\}\\mathbf\{W\}\_\{uv\}\\sum\_\{m\_\{1\},m\_\{2\}\}\\mathbf\{K\}^\{\\rho\_\{3\}\}\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(m\_\{1\},m\_\{2\},m\_\{3\}\)\\mathbf\{x\}^\{\(1\)\}\_\{u,m\_\{1\}\}\\mathbf\{x\}^\{\(2\)\}\_\{v,m\_\{2\}\}\.\(33\)In the molecular message\-passing setting, the second input is usually an edge basis withV=1V=1\. Following Equation \(8\) and \(9\) in the main text, we utilize the pre\-contracted kernel𝐊pre\\mathbf\{K\}\_\{\\mathrm\{pre\}\}to evaluate the channel\-depthwise form

\[𝐱\(1\)⊗𝐖𝐱\(2\)\]\(u,m3\)=𝐖u​∑m1𝐊preρ3​\(m1,m3\)​𝐱u,m1\(1\)\.\[\\mathbf\{x\}^\{\(1\)\}\\otimes\_\{\\mathbf\{W\}\}\\mathbf\{x\}^\{\(2\)\}\]^\{\(u,m\_\{3\}\)\}=\\mathbf\{W\}\_\{u\}\\sum\_\{m\_\{1\}\}\\mathbf\{K\}\_\{\\mathrm\{pre\}\}^\{\\rho\_\{3\}\}\(m\_\{1\},m\_\{3\}\)\\mathbf\{x\}^\{\(1\)\}\_\{u,m\_\{1\}\}\.\(34\)Thus the parameter count per instruction isU​VUV, orUUwhenV=1V=1\. The optimizedcached\_uvuandtriton\_uvubackends are specialized for this commonV=1V=1edge\-feature case\.

Elementwise tensor product\.The elementwise mode \(modeuuu\) pairs corresponding input irrep blocks and requires matching multiplicities\. For each paired block, it computes channelwise products

Ru,m3=∑m1,m2𝐊ρ1,ρ2ρ3​\(m1,m2,m3\)​𝐱u,m1\(1\)​𝐱u,m2\(2\),R\_\{u,m\_\{3\}\}=\\sum\_\{m\_\{1\},m\_\{2\}\}\\mathbf\{K\}^\{\\rho\_\{3\}\}\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(m\_\{1\},m\_\{2\},m\_\{3\}\)\\mathbf\{x\}^\{\(1\)\}\_\{u,m\_\{1\}\}\\mathbf\{x\}^\{\(2\)\}\_\{u,m\_\{2\}\},\(35\)followed by an optional per\-channel weight

\[𝐱\(1\)⊗𝐖𝐱\(2\)\]\(u,m3\)=𝐖u​Ru,m3\.\[\\mathbf\{x\}^\{\(1\)\}\\otimes\_\{\\mathbf\{W\}\}\\mathbf\{x\}^\{\(2\)\}\]^\{\(u,m\_\{3\}\)\}=\\mathbf\{W\}\_\{u\}R\_\{u,m\_\{3\}\}\.\(36\)It does not mix different multiplicity channels and only connects corresponding input block pairs\. The parameter count per instruction isUU\.

### E\.2Runtime Comparison

We benchmark SpinGTP at two levels, including the depthwise tensor\-product kernel and the full Equiformer\-style model\. All timings use a single NVIDIA H200 GPU and report time after warmup\.

##### Depthwise tensor\-product backend\.

We compare three implementations of the same depthwise tensor product, including direct Gaunt contraction, precontracted execution, and Triton\-fused precontracted execution\. The benchmark uses edge multiplicity one,smax=1s\_\{\\max\}=1, and a depthwise path\-expanded layout\. The batch dimension is the number of independent edge\-level tensor\-product samples\. We evaluate batch sizes\{1,32,128,1024\}\\\{1,32,128,1024\\\}, and selectively report over these batch sizes for each multiplicity\. ForLmax=1,2L\_\{\\max\}=1,2, both input irreps and output irreps are up toLmaxL\_\{\\max\}\. ForLmax=3L\_\{\\max\}=3, the first input is capped atLmax\(1\)=1L\_\{\\max\}^\{\(1\)\}=1, while the second input and output remain atLmax=3L\_\{\\max\}=3\.

Table 6:Depthwise SpinGTP tensor\-product runtime on H200\. Times are median milliseconds over 20 iterations\.BBis the number of independent edge\-level tensor\-product samples per call\. Speedups are relative to direct Gaunt contraction\.Lmax\(1\)L\_\{\\max\}^\{\(1\)\}Lmax\(2\)L\_\{\\max\}^\{\(2\)\}Lmax\(out\)L\_\{\\max\}^\{\(\\mathrm\{out\}\)\}MultiplicityBBDirect Gaunt\(ms\)Precontracted\(ms\)Triton Precontracted\(ms\)Gaunt / Triton111161285\.4600\.5370\.12045\.4×45\.4\\times111641285\.4930\.4660\.11946\.2×46\.2\\times1111281284\.5620\.4430\.11539\.7×39\.7\\times222163221\.5700\.9970\.25983\.15×83\.15\\times222643229\.9121\.7690\.74839\.99×39\.99\\times2221283233\.5281\.8370\.55959\.96×59\.96\\times2221612816\.1440\.9990\.25862\.5×62\.5\\times2226412829\.5611\.7710\.75239\.3×39\.3\\times22212812833\.3061\.8380\.57258\.2×58\.2\\times1331612812\.7140\.6780\.22855\.9×55\.9\\times1336412820\.6101\.0800\.60833\.9×33\.9\\times13312812822\.5461\.3440\.68133\.1×33\.1\\times22264102429\.9121\.7772\.09014\.3×14\.3\\times13364102423\.9411\.3941\.86512\.8×12\.8\\times133128102422\.6281\.3502\.24010\.1×10\.1\\timesPrecontraction gives a consistent order\-of\-magnitude improvement over direct Gaunt contraction\. Triton\-fused precontraction is fastest for most small\- and medium\-batch settings\. At largeBBand high multiplicity, the standard precontracted implementation can be faster because its batched matrix multiplications better saturate tensor cores\.

##### End\-to\-end Equiformer timing ats=0s=0\.

We also compare the standard Equiformer against an Equiformer model using the SpinGTP tensor\-product implementation\. All SWSH irreps are constrained tos=0s=0, so the comparison isolates implementation speed\. Both models use the same 6\-layer Equiformer\-style architecture,5\.0​Å5\.0\\penalty 10000\\ \\text\{\\AA \}cutoff, 500 maximum neighbors, 80 atoms per graph, no PBC, and energy\-only forward evaluation\.

The GTP\-based model is1\.21×1\.21\\times\-1\.89×1\.89\\timesfaster in forward evaluation\. At practical batch sizes of 32\-64 graphs, it sustains 65\-72k atoms/s, compared with 38\-44k atoms/s for the standard Equiformer\. Becauses=0s=0is enforced, this gain comes from the tensor\-product implementation\.

Table 7:Full\-model forward runtime for Equiformer and Equiformer with GTP ats=0s=0\. Times are median milliseconds over 10 iterations\.GraphsAtomsEquiformer\(ms\)GTP,s=0s=0\(ms\)SpeedupEquiformer\(atoms/s\)GTP,s=0s=0\(atoms/s\)18023\.43819\.3581\.21×1\.21\\times3,4134,133432027\.14419\.2831\.41×1\.41\\times11,78916,595161,28041\.72922\.0491\.89×1\.89\\times30,67458,053322,56067\.69539\.3851\.72×1\.72\\times37,81764,999645,120116\.29870\.9171\.64×1\.64\\times44,02572,197

## Appendix FTraining Details

### F\.13BPA

##### Architecture\.

For 3BPA, our model is built on top of MACE\[batatia2022mace\]with the following modifications\. First, we use SWSH withsm​a​x=1s\_\{max\}=1instead of regular spherical harmonics, so each directed edge carries both standard and spin\-weighted geometric features up toLmax=3L\_\{\\max\}=3\. Second, the equivariant tensor product in each interaction block is replaced by the fully\-connected SWSH tensor product, recovering the antisymmetric coupling paths absent from scalar GTP\. Third, all linear layers throughout the network including embedding projections, skip connections, and readout layers, are replaced with the specialized SWSH linear layers described in[section˜3\.6](https://arxiv.org/html/2607.01408#S3.SS6)\. Additionally, thes=0s=0product basis uses the same body\-ordered symmetric contraction as the MACE baseline, while\|s\|\>0\|s\|\>0channels bypass the contraction and are updated via a linear map\. Crucially, we add a graph frame as described in[section˜A\.2](https://arxiv.org/html/2607.01408#A1.SS2)that assigns all edges within a molecule a shared gauge, making the scatter\-sum over spin\-weighted messages equivariant\.

##### Training Procedure\.

We optimize a weighted energy\-and\-force loss using Adam \(AMSGrad\) at learning rate1×10−21\\times 10^\{\-2\}and weight decay5×10−75\\times 10^\{\-7\}, under a ReduceLROnPlateau schedule \(factor0\.80\.8, patience5050\)\. An exponential moving average of model weights with decay0\.990\.99is maintained throughout training\. We follow a two\-stage scheme: in Stage 1, the model is trained for up to 2000 epochs with energy and force loss weights of11and10001000respectively\. In Stage 2, we fine\-tune with an increased energy loss weight of2525to better resolve energy differences between conformers, running for up to 250 epochs\. Final results are reported from Stage 2 checkpoints selected on validation loss\. All experiments use float32 precision with RMS\-forces output scaling\. All 3BPA experiments were run on a single NVIDIA RTX A6000 \(48 GB\) GPU, with each run taking approximately 63 GPU\-hours \(56 hours for Stage 1 and 7 hours for Stage 2\)\. Full hyperparameters are summarized in Table[8](https://arxiv.org/html/2607.01408#A6.T8)\.

Table 8:Training Configuration for 3BPA\.ItemSettingRandom seeds3, 8, 9Cutoff Radius5\.0 ÅRadial basis8 Bessel, 5 polynomial cutoff basisAngular degreeLmax=3L\_\{\\max\}=3,smax=1s\_\{\\max\}=1Interactions2Correlation order3Hidden irreps256×\(0,0\)​e\+256×\(1,0\)​o\+256×\(2,0\)​e\+16×\(1,1\)​o256\{\\times\}\(0,0\)e\+256\{\\times\}\(1,0\)o\+256\{\\times\}\(2,0\)e\+16\{\\times\}\(1,1\)oReadout MLP irreps16×\(0,0\)​e16\{\\times\}\(0,0\)eOptimizerAdam \(AMSGrad\)Learning rate1×10−21\\times 10^\{\-2\}Weight decay5×10−75\\times 10^\{\-7\}LR scheduleReduceLROnPlateau \(factor 0\.8, patience 50\)Batch size5EMA decay0\.99Output scalingRMS\-forces scalingPrecisionfloat32Stage 1Stage 2Energy loss weight125Force loss weight10001000Max epochs2000250Patience256–

### F\.2SPICE\-MACE\-OFF Chiral Subset

#### F\.2\.1Chirality Classification and Parity Generalization

##### Architecture\.

For chirality tasks, we adapt the SpinGTP framework to a graph classification objective by replacing the atomic energy readout with a global MLP\-based head\. The backbone utilizes spin\-weighted spherical harmonics \(SWSH\) withLmax=2L\_\{\\max\}=2andsmax=2s\_\{\\max\}=2to encode the geometric sensitivity required for enantiomer differentiation\. The hidden representation is defined by a specific irrep set:64×\(0,0\)​e\+64×\(1,0\)​o\+8×\(1,1\)​o\+64×\(2,0\)​e64\\times\(0,0\)e\+64\\times\(1,0\)o\+8\\times\(1,1\)o\+64\\times\(2,0\)e\. To ensure a rigorous benchmark, baseline architectures including NequIP\[nequip\], MACE\[batatia2022mace\], and MACE\-Gaunt\[luo2024enabling\]are integrated into the same pipeline using identical cutoff radii \(rmax=5\.0r\_\{\\max\}=5\.0Å\) and radial basis functions \(Table[9](https://arxiv.org/html/2607.01408#A6.T9)\)\.

##### Training Procedure\.

We evaluate model performance through two distinct protocols\. First, Standard Classification involves a 2\-class task \(R vs\. S\) on a random 85/15 split, optimized via AdamW with a learning rate of1×10−31\\times 10^\{\-3\}\. Second, Parity Generalization evaluates the model’s ability to learn the underlying symmetry operation by training exclusively on R\-chiral structures with a regression target ofy=\+1y=\+1\. Generalization is measured by sign accuracy on unseen S\-chiral mirror images \(y=−1y=\-1\)\. These experiments use Adam with aStepLRschedule and early stopping based onacc​\_​S\\mathrm\{acc\}\\\_Sto capture the model’s peak generalization capability\. We train models on 144 GB NVIDIA463 H200 GPUs\. Detailed hyperparameter settings for both tasks are provided in Tables[9](https://arxiv.org/html/2607.01408#A6.T9)and[10](https://arxiv.org/html/2607.01408#A6.T10)\.

Table 9:Training Configuration for Chirality Classification\.ItemSettingTask2\-class graph classification \(R\_only vs\. S\_only\)Data splitRandom 85/15 train/validation splitBackbonesMACE, MACE\-Gaunt, NequIP, SpinGTPCutoffrmax=5\.0r\_\{\\max\}=5\.0ÅSWSH structureLmax=2L\_\{\\max\}=2, 2 interactions, 8 Bessel, 5 cutoff basisSWSH hidden irreps64×\(0,0\)​e\+64×\(1,0\)​o\+8×\(1,1\)​o\+64×\(2,0\)​e64\{\\times\}\(0,0\)e\+64\{\\times\}\(1,0\)o\+8\{\\times\}\(1,1\)o\+64\{\\times\}\(2,0\)eClassifier headMLP, hidden size 64, 2 output logitsOptimizerAdamWLearning rate1×10−31\\times 10^\{\-3\}Weight decay1×10−51\\times 10^\{\-5\}Batch size16Max epochs200LR scheduleReduceLROnPlateau \(factor 0\.5, patience 10\)Table 10:Training Configuration for Parity Generalization\.ItemSettingTaskRegression to parity target:y=\+1y=\+1\(R\),y=−1y=\-1\(S\)Train/test protocolTrain on R\-only, evaluate on R\-only and unseen S\-onlyPrimary metricSign accuracy on unseen S \(acc​\_​S\\mathrm\{acc\}\\\_S\)LossMean\-squared error \(MSE\)Batch size16Max epochs200Learning rate1×10−31\\times 10^\{\-3\}Weight decay1×10−51\\times 10^\{\-5\}Patience30 onacc​\_​S\\mathrm\{acc\}\\\_SSWSH setuprmax=5\.0r\_\{\\max\}=5\.0, mul=16, layers=2,Lmax=2,smax=2L\_\{\\max\}=2,s\_\{\\max\}=2OptimizerAdam \+ StepLR \(step=50,γ=0\.5\\gamma=0\.5\)

#### F\.2\.2Energy and Force Prediction

##### Architecture\.

We use the same SpinGTP architecture as our 3BPA experiments \([section˜F\.1](https://arxiv.org/html/2607.01408#A6.SS1), Table[8](https://arxiv.org/html/2607.01408#A6.T8)\): spin\-weighted edge spherical harmonics withs∈\{0,1\}s\\in\\\{0,1\\\}, fully\-connected SWSH tensor products in place of scalar Clebsch–Gordan contractions, SWSH linear layers throughout, body\-ordered symmetric contraction for thes=0s=0product basis with direct linear updates for\|s\|\>0\|s\|\>0channels, and a shared graph frame that fixes a consistent gauge for spin\-weighted messages \([section˜A\.2](https://arxiv.org/html/2607.01408#A1.SS2)\)\. The only substantive differences for SPICE are the hidden/readout widths \(Table[11](https://arxiv.org/html/2607.01408#A6.T11)\) and the training\-time tensor\-product backend configuration used for throughput on large\-scale dataset training\.

##### Training Procedure\.

We optimize a weighted energy\-and\-force loss with weights4040\(energy\) and10001000\(forces\), using Adam \(AMSGrad\) at learning rate1×10−21\\times 10^\{\-2\}and weight decay5×10−105\\times 10^\{\-10\}, under a ReduceLROnPlateau schedule \(factor0\.80\.8, patience5050\)\. An exponential moving average of the model weights with decay0\.990\.99is maintained throughout training\. Training runs for up to100100epochs with patience5050, using float32 precision and RMS\-forces output scaling\. We parallelize across88GPUs via PyTorch DDP with per\-GPU batch size3232\(global batch256256\)\. we train models on eight 144 GB NVIDIA463 H200 GPUs\. Final results are reported from the best validation\-loss checkpoint\. All hyperparameters are summarized in Table[11](https://arxiv.org/html/2607.01408#A6.T11)\.

Table 11:Training configuration for Energy and Force Prediction on SPICE chiral subset\.ItemSettingRandom seed9Cutoff radius5\.0 ÅRadial basis8 Bessel, 5 polynomial cutoff basisAngular degreeLmax=3L\_\{\\max\}=3,smax=1s\_\{\\max\}=1Interactions2Correlation \(body order\)3Hidden irreps192×\(0,0\)​e\+192×\(1,0\)​o\+192×\(2,0\)​e\+12×\(1,1\)​o192\{\\times\}\(0,0\)e\+192\{\\times\}\(1,0\)o\+192\{\\times\}\(2,0\)e\+12\{\\times\}\(1,1\)oReadout MLP irreps12×\(0,0\)​e12\{\\times\}\(0,0\)eProduct basis \(s=0s=0\)body\-ordered symmetric contraction \(MACE\)OptimizerAdam \(AMSGrad\)Learning rate1×10−21\\times 10^\{\-2\}Weight decay5×10−105\\times 10^\{\-10\}LR scheduleReduceLROnPlateau \(factor 0\.8, patience 50\)Energy loss weight40Force loss weight1000EMA decay0\.99Output scalingRMS\-forces scalingBatch size32 per GPU \(global 256, 8\-GPU DDP\)Max epochs190Patience50Precisionfloat32

### F\.3OC20 IS2RE

##### Architecture\.

For OC20, we use an SWSH\-Equiformer model built on the Equiformer attention backbone\. The model uses on\-the\-fly periodic graphs with cutoff radius5\.0​Å5\.0\\penalty 10000\\ \\text\{\\AA \}, at most 500 neighbors, and 128 radial basis functions\. The standard equivariant tensor products, linear layers, and normalization layers are replaced by their SWSH counterparts\. The main OC20 configuration keeps the persistent node and edge feature space bounded atLmax=1L\_\{\\max\}=1\. Spin is introduced through a local spin head constructing bounded\|s\|=1\|s\|=1SWSH templates\(1,1\)​o\(1,1\)oin the local node frame and applying a SpinGTP path

\(1,1\)​o⊗\(1,1\)​o→\(1,0\)​e\.\(1,1\)o\\otimes\(1,1\)o\\to\(1,0\)e\.This head is enabled only in the attention activation path\.

##### Training procedure\.

We train on OC20IS2RE\-Directon a single NVIDIA H200 GPU without force regression or noisy\-node auxiliary supervision\. The model is optimized with AdamW and a cosine learning\-rate schedule with warmup\. Checkpoints are selected by validation energy error\. The full configuration is summarized in[Table˜12](https://arxiv.org/html/2607.01408#A6.T12)\.

Table 12:Training configuration for OC20IS2RE\-Direct\.ItemSettingTaskOC20IS2RE\-Directrelaxed\-energy predictionNumber of layers6Attention heads8Cutoff radius5\.0​Å5\.0\\penalty 10000\\ \\text\{\\AA \}Max neighbors500Radial basis128 Gaussian radial basis functionsRadial MLP\[64, 64\]Node embedding irreps256x\(0,0\)e\+128x\(1,0\)eEdge SWSH irreps1x\(0,0\)e\+1x\(1,0\)e and 1x\(1,1\)oAttention head irreps32x\(0,0\)e\+16x\(1,0\)eMLP hidden irreps768x\(0,0\)e\+384x\(1,0\)eOutput feature irreps512x\(0,0\)eOptimizerAdamWInitial learning rate2×10−42\\times 10^\{\-4\}Weight decay1×10−31\\times 10^\{\-3\}Learning\-rate scheduleCosineLambdaLRWarmup2 epochs, warmup factor0\.20\.2Minimum LR factor10−210^\{\-2\}Batch size32Evaluation batch size32Number of workers16EMA decay0\.999Max epochs20