Equivariant Cellular Sheaves for Molecular Electronic Structure: Bridging Sheaf Cohomology and E(3)-Equivariant Hamiltonian Learning

# Equivariant Cellular Sheaves for Molecular Electronic Structure: Bridging Sheaf Cohomology and 𝐸⁢(3)-Equivariant Hamiltonian Learning
Source: [https://arxiv.org/html/2608.23571](https://arxiv.org/html/2608.23571)
###### Abstract

Equivariant message\-passing networks have become the standard inductive model for molecular property and interatomic\-potential prediction, and a recent line of work predicts the electronic Hamiltonian itself in anE​\(3\)E\(3\)\-equivariant manner\. In parallel, topological deep learning has generalized graph networks to simplicial complexes, cellular \(CW\) complexes, and cellular sheaves\. We connect these two developments\. Our central observation is structural: in a localized \(atomic\-orbital or Wannier\) basis, the molecular single\-particle Hamiltonian, after a constant energy shift that renders it positive semidefinite, is the*Laplacian of a cellular sheaf*on a regular cell complex built from the molecule\. We make the sheaf*equivariant*by requiring its restriction maps to beO​\(3\)O\(3\)\-steerable two\-center kernels conditioned on bond geometry, which recovers the Slater–Koster two\-center form as a special case and yields anE​\(3\)E\(3\)\-equivariant, permutation\-equivariant operator\. This perspective has three consequences\. First, the zeroth sheaf cohomologyH0​\(X;ℱ\)=ker⁡LℱH^\{0\}\(X;\\mathcal\{F\}\)=\\ker L\_\{\\mathcal\{F\}\}is a topological invariant equal to the space of non\-bonding \(zero\-mode\) orbitals at the chosen reference energy, recovering the classical non\-bonding\-orbital count of alternant systems as a lower bound\. Second, promoting the construction to the Hodge11\-Laplacian lets higher cells \(rings\) carry cycle and delocalization information throughH1H^\{1\}\. Third, the model strictly generalizes bothE​\(3\)E\(3\)\-equivariant message\-passing networks and cellular \(CW\) networks, and inherits the anti\-oversmoothing behavior of non\-trivial sheaf diffusion\. We present the architecture \(Equivariant Cellular Sheaf Networks\), prove equivariance, expressivity, and cohomological\-correspondence results, and validate them numerically: the Hamiltonian\-to\-sheaf embedding is exact to machine precision, the cohomology dimension reproduces non\-bonding\-orbital counts across eleven conjugated molecules, the sheaf Laplacian isO​\(3\)O\(3\)\-equivariant to machine precision, and the equivariant model attains lower error and rotation generalization on a directional electronic target\. We do not claim to be the first to predict Hamiltonians equivariantly; our contribution is the sheaf\-theoretic formalization, its topological invariants, and the resulting unification\.

## 1Introduction

Machine\-learning surrogates for quantum chemistry have largely been built on message\-passing neural networks \(MPNNs\) over the molecular graph\[[1](https://arxiv.org/html/2608.23571#bib.bib1),[2](https://arxiv.org/html/2608.23571#bib.bib2)\]\. The most accurate models add geometric symmetry, the organizing principle of geometric deep learning\[[3](https://arxiv.org/html/2608.23571#bib.bib3)\]:E​\(3\)E\(3\)\- andS​E​\(3\)SE\(3\)\-equivariant networks represent atomic environments with spherical tensors and couple them with Clebsch–Gordan products\[[6](https://arxiv.org/html/2608.23571#bib.bib6),[5](https://arxiv.org/html/2608.23571#bib.bib5),[7](https://arxiv.org/html/2608.23571#bib.bib7),[8](https://arxiv.org/html/2608.23571#bib.bib8),[9](https://arxiv.org/html/2608.23571#bib.bib9)\], which underlies state\-of\-the\-art interatomic potentials and connects to the atomic cluster expansion\[[10](https://arxiv.org/html/2608.23571#bib.bib10)\]\. A distinct and more ambitious target is the electronic Hamiltonian itself: several equivariant networks now predict the Kohn–Sham or Fock matrix in an atomic\-orbital basis, from which orbitals, densities, and spectra follow\[[14](https://arxiv.org/html/2608.23571#bib.bib14),[15](https://arxiv.org/html/2608.23571#bib.bib15),[16](https://arxiv.org/html/2608.23571#bib.bib16)\]\.

Independently,*topological deep learning*has lifted graph networks to richer combinatorial domains: simplicial complexes\[[29](https://arxiv.org/html/2608.23571#bib.bib29),[32](https://arxiv.org/html/2608.23571#bib.bib32),[33](https://arxiv.org/html/2608.23571#bib.bib33)\], regular cell \(CW\) complexes\[[30](https://arxiv.org/html/2608.23571#bib.bib30)\], and cellular sheaves\[[25](https://arxiv.org/html/2608.23571#bib.bib25),[26](https://arxiv.org/html/2608.23571#bib.bib26),[28](https://arxiv.org/html/2608.23571#bib.bib28)\]; see\[[31](https://arxiv.org/html/2608.23571#bib.bib31)\]for a synthesis\. Cellular sheaves attach a vector space \(stalk\) to every cell and a linear restriction map to every incidence, generalizing the graph Laplacian to a*sheaf Laplacian*whose kernel is the space of global sections\[[25](https://arxiv.org/html/2608.23571#bib.bib25)\]\.

These two literatures have not been connected, despite an exact structural bridge between them\. We observe that the localized\-orbital electronic Hamiltonian*is*a sheaf Laplacian on a molecular cell complex \(Section[4](https://arxiv.org/html/2608.23571#S4)\)\. Restriction maps become learnable, geometry\- conditioned,O​\(3\)O\(3\)\-steerable analogues of two\-center integrals; the molecular topology \(bonds, rings\) enters through the cell structure and its cohomology; and electronic structure becomes the spectral analysis of an equivariant sheaf Laplacian\. We call the resulting model the*Equivariant Cellular Sheaf Network*\(ECSN\)\.

#### Contributions\.

1. 1\.A correspondence\(Prop\.[4\.2](https://arxiv.org/html/2608.23571#S4.Thmtheorem2)\): under a positive\- semidefinite \(PSD\) energy shift and a per\-bond factorization, the two\-center localized Hamiltonian is the Laplacian of a cellular sheaf; the construction contains Slater–Koster tight binding\[[19](https://arxiv.org/html/2608.23571#bib.bib19)\]as a special case\.
2. 2\.Equivariant cellular sheaves\(Def\.[4\.4](https://arxiv.org/html/2608.23571#S4.Thmtheorem4), Thm\.[4\.5](https://arxiv.org/html/2608.23571#S4.Thmtheorem5)\): stalks carryingO​\(3\)O\(3\)irreducibles and steerable restriction maps make the sheaf LaplacianE​\(3\)E\(3\)\- and permutation\-equivariant\.
3. 3\.Topological invariants with chemical meaning\(Thm\.[4\.7](https://arxiv.org/html/2608.23571#S4.Thmtheorem7), Cor\.[4\.8](https://arxiv.org/html/2608.23571#S4.Thmtheorem8)\):dimH0​\(X;ℱ\)\\dim H^\{0\}\(X;\\mathcal\{F\}\)counts non\-bonding orbitals; for alternant systems it is lower bounded by the sublattice imbalance, recovering a classical Hückel\-level count\.
4. 4\.An expressivity hierarchy\(Thm\.[6\.1](https://arxiv.org/html/2608.23571#S6.Thmtheorem1)\): ECSN strictly generalizesE​\(3\)E\(3\)\-equivariant MPNNs and cellular \(CW\) networks, and inherits the anti\-oversmoothing property of non\-trivial sheaf diffusion\.
5. 5\.Numerical validation\(Section[7](https://arxiv.org/html/2608.23571#S7)\): the embedding is exact to machine precision, cohomology reproduces non\-bonding\-orbital counts across eleven molecules, equivariance holds to machine precision, and the equivariant model is more accurate and rotation\-robust than a coordinate baseline\.

We emphasize scope\. Equivariant Hamiltonian prediction is due to prior work\[[14](https://arxiv.org/html/2608.23571#bib.bib14),[15](https://arxiv.org/html/2608.23571#bib.bib15),[16](https://arxiv.org/html/2608.23571#bib.bib16)\]; our novelty is the sheaf\-theoretic formalization, the cohomological invariants, the higher\-cell extension, and the unification\.

## 2Related Work

#### Equivariant networks for chemistry\.

Group\-equivariant convolutions\[[4](https://arxiv.org/html/2608.23571#bib.bib4)\]and steerable CNNs\[[5](https://arxiv.org/html/2608.23571#bib.bib5)\]led to tensor\-field networks\[[6](https://arxiv.org/html/2608.23571#bib.bib6)\]and thee3nnframework\[[7](https://arxiv.org/html/2608.23571#bib.bib7)\], instantiated for potentials by NequIP\[[8](https://arxiv.org/html/2608.23571#bib.bib8)\], MACE\[[9](https://arxiv.org/html/2608.23571#bib.bib9)\], PaiNN\[[11](https://arxiv.org/html/2608.23571#bib.bib11)\], EGNN\[[13](https://arxiv.org/html/2608.23571#bib.bib13)\], and directional models such as DimeNet\[[12](https://arxiv.org/html/2608.23571#bib.bib12)\]; ACE\[[10](https://arxiv.org/html/2608.23571#bib.bib10)\]provides the many\-body basis these realize\. These predict invariant or covariant targets on the atomic graph but do not use higher cells or sheaf structure\.

#### Learning electronic structure\.

Density\-functional theory\[[17](https://arxiv.org/html/2608.23571#bib.bib17),[18](https://arxiv.org/html/2608.23571#bib.bib18)\]in a localized basis yields a sparse Hamiltonian; maximally localized Wannier functions\[[20](https://arxiv.org/html/2608.23571#bib.bib20)\]make the locality explicit\. PhiSNet\[[14](https://arxiv.org/html/2608.23571#bib.bib14)\], DeepH\[[15](https://arxiv.org/html/2608.23571#bib.bib15)\], and QHNet\[[16](https://arxiv.org/html/2608.23571#bib.bib16)\]predict such matrices equivariantly\. We reinterpret the predicted operator as a sheaf Laplacian, which is, to our knowledge, new\.

#### Topological deep learning and sheaves\.

Message passing has been generalized to simplicial\[[29](https://arxiv.org/html/2608.23571#bib.bib29),[32](https://arxiv.org/html/2608.23571#bib.bib32),[33](https://arxiv.org/html/2608.23571#bib.bib33)\]and cellular complexes\[[30](https://arxiv.org/html/2608.23571#bib.bib30)\], and to sheaves: sheaf neural networks\[[26](https://arxiv.org/html/2608.23571#bib.bib26)\]and neural sheaf diffusion\[[28](https://arxiv.org/html/2608.23571#bib.bib28)\], built on the spectral theory of cellular sheaves\[[25](https://arxiv.org/html/2608.23571#bib.bib25)\]\. Vector\- bundle and tangent\-bundle generalizations have also been studied\[[27](https://arxiv.org/html/2608.23571#bib.bib27)\]\. These works are not equivariant in theO​\(3\)O\(3\)sense and are not applied to electronic structure\.

#### Topology of the electronic density\.

The quantum theory of atoms in molecules\[[21](https://arxiv.org/html/2608.23571#bib.bib21)\]analyzes the Morse–Smale complex of the electron density, and persistent homology has been used for molecular descriptors\[[35](https://arxiv.org/html/2608.23571#bib.bib35),[36](https://arxiv.org/html/2608.23571#bib.bib36)\]\. Our cells are chemical \(atoms, bonds, rings\) rather than density critical points, and our invariants are sheaf\-cohomological rather than persistence\-based\.

## 3Background

### 3\.1Regular cell complexes from molecules

Let a molecule be a set of atoms at positions\{𝐫i\}i=1N⊂ℝ3\\\{\\mathbf\{r\}\_\{i\}\\\}\_\{i=1\}^\{N\}\\subset\\mathbb\{R\}^\{3\}with species\{zi\}\\\{z\_\{i\}\\\}\. We build a regular cell complexXX:0\-cells are atoms;11\-cells are unordered pairs\{i,j\}\\\{i,j\\\}with‖𝐫i−𝐫j‖<rc\\\|\\mathbf\{r\}\_\{i\}\-\\mathbf\{r\}\_\{j\}\\\|<r\_\{c\}\(a bond cutoff\);22\-cells are the faces bounded by a chosen cycle basis \(e\.g\. the smallest set of smallest rings\)\. We fix an orientation and writeσ​⊴​τ\\sigma\\trianglelefteq\\tauwhen cellσ\\sigmais a codimension\-11face ofτ\\tau, with signed incidence\[σ:τ\]∈\{−1,\+1\}\[\\sigma\{:\}\\tau\]\\in\\\{\-1,\+1\\\}\.

### 3\.2Cellular sheaves and the sheaf Laplacian

###### Definition 3\.1\(Cellular sheaf\[[25](https://arxiv.org/html/2608.23571#bib.bib25)\]\)\.

A cellular sheafℱ\\mathcal\{F\}of finite\-dimensional real inner\-product spaces onXXassigns to each cellσ\\sigmaa stalkℱ​\(σ\)≅ℝdσ\\mathcal\{F\}\(\\sigma\)\\cong\\mathbb\{R\}^\{d\_\{\\sigma\}\}and to each incidenceσ​⊴​τ\\sigma\\trianglelefteq\\taua linear restriction mapℱσ​⊴​τ:ℱ​\(σ\)→ℱ​\(τ\)\\mathcal\{F\}\_\{\\sigma\\,\\trianglelefteq\\,\\tau\}:\\mathcal\{F\}\(\\sigma\)\\to\\mathcal\{F\}\(\\tau\)\.

The space ofkk\-cochains isCk​\(X;ℱ\)=⨁dimσ=kℱ​\(σ\)C^\{k\}\(X;\\mathcal\{F\}\)=\\bigoplus\_\{\\dim\\sigma=k\}\\mathcal\{F\}\(\\sigma\)\. The coboundaryδk:Ck→Ck\+1\\delta^\{k\}:C^\{k\}\\to C^\{k\+1\}acts by

\(δkx\)τ=∑σ​⊴​τ\[σ:τ\]ℱσ​⊴​τxσ\.\(\\delta^\{k\}x\)\_\{\\tau\}\\;=\\;\\sum\_\{\\sigma\\trianglelefteq\\tau\}\[\\sigma\{:\}\\tau\]\\,\\mathcal\{F\}\_\{\\sigma\\,\\trianglelefteq\\,\\tau\}\\,x\_\{\\sigma\}\.\(1\)The degree\-kkHodge–sheaf Laplacian isLk=\(δk\)⊤​δk\+δk−1​\(δk−1\)⊤L\_\{k\}=\(\\delta^\{k\}\)^\{\\top\}\\delta^\{k\}\+\\delta^\{k\-1\}\(\\delta^\{k\-1\}\)^\{\\top\}\. Fork=0k=0the*sheaf Laplacian*Lℱ:=L0=\(δ0\)⊤​δ0L\_\{\\mathcal\{F\}\}:=L\_\{0\}=\(\\delta^\{0\}\)^\{\\top\}\\delta^\{0\}has blocks

\(Lℱ\)v​v=∑e​⊴∋vℱv​⊴​e⊤​ℱv​⊴​e,\(Lℱ\)u​v=−ℱu​⊴​e⊤​ℱv​⊴​e\(u≠v,e=\{u,v\}\)\.\(L\_\{\\mathcal\{F\}\}\)\_\{vv\}=\\sum\_\{e\\trianglelefteq\\,\\ni v\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}^\{\\top\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\},\\qquad\(L\_\{\\mathcal\{F\}\}\)\_\{uv\}=\-\\,\\mathcal\{F\}\_\{u\\,\\trianglelefteq\\,e\}^\{\\top\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\\ \\ \(u\\neq v,\\ e=\\\{u,v\\\}\)\.\(2\)LℱL\_\{\\mathcal\{F\}\}is symmetric PSD, andker⁡Lℱ=ker⁡δ0=H0​\(X;ℱ\)\\ker L\_\{\\mathcal\{F\}\}=\\ker\\delta^\{0\}=H^\{0\}\(X;\\mathcal\{F\}\), the space of*global sections*\[[25](https://arxiv.org/html/2608.23571#bib.bib25)\]\.

### 3\.3O​\(3\)O\(3\)representations and steerable kernels

O​\(3\)O\(3\)acts on functions onℝ3\\mathbb\{R\}^\{3\}; its real irreducibles are the\(2​ℓ\+1\)\(2\\ell\{\+\}1\)\-dimensional spacesVℓV\_\{\\ell\}with action by Wigner matricesDℓ​\(g\)D^\{\\ell\}\(g\)\. A stalk of the formℱ​\(σ\)=⨁ℓmℓ​Vℓ\\mathcal\{F\}\(\\sigma\)=\\bigoplus\_\{\\ell\}m\_\{\\ell\}V\_\{\\ell\}\(multiplicitymℓm\_\{\\ell\}\) models atomic orbitals of angular momentumℓ\\ell\. A linear mapK:Vℓ1→Vℓ2K:V\_\{\\ell\_\{1\}\}\\to V\_\{\\ell\_\{2\}\}that is*steerable*by a direction𝐫^\\hat\{\\mathbf\{r\}\}, i\.e\.K​\(g​𝐫^\)=Dℓ2​\(g\)​K​\(𝐫^\)​Dℓ1​\(g\)⊤K\(g\\hat\{\\mathbf\{r\}\}\)=D^\{\\ell\_\{2\}\}\(g\)K\(\\hat\{\\mathbf\{r\}\}\)D^\{\\ell\_\{1\}\}\(g\)^\{\\top\}for allgg, is spanned by Clebsch–Gordan contractions of spherical harmonicsYℓf​\(𝐫^\)Y\_\{\\ell\_\{f\}\}\(\\hat\{\\mathbf\{r\}\}\)\[[6](https://arxiv.org/html/2608.23571#bib.bib6),[7](https://arxiv.org/html/2608.23571#bib.bib7)\]:

K​\(𝐫^\)=∑ℓf=\|ℓ1−ℓ2\|ℓ1\+ℓ2wℓf​\(Cℓ1,ℓfℓ2\)​Yℓf​\(𝐫^\),K\(\\hat\{\\mathbf\{r\}\}\)\\;=\\;\\sum\_\{\\ell\_\{f\}=\|\\ell\_\{1\}\-\\ell\_\{2\}\|\}^\{\\ell\_\{1\}\+\\ell\_\{2\}\}w\_\{\\ell\_\{f\}\}\\,\\big\(C\_\{\\ell\_\{1\},\\ell\_\{f\}\}^\{\\ell\_\{2\}\}\\big\)\\,Y\_\{\\ell\_\{f\}\}\(\\hat\{\\mathbf\{r\}\}\),\(3\)with learnable weightswℓfw\_\{\\ell\_\{f\}\}and radial scalars \(functions of‖𝐫‖\\\|\\mathbf\{r\}\\\|omitted here\)\.

## 4Mathematical Framework

### 4\.1The electronic Hamiltonian as a sheaf Laplacian

Consider a single\-particle HamiltonianHH\(Kohn–Sham, Fock, or tight binding\) in a localized orbital basis\{ϕv,a\}\\\{\\phi\_\{v,a\}\\\}indexed by atomvvand orbitalaa, with on\-site blocksHv​vH\_\{vv\}and hopping blocksHu​vH\_\{uv\}that vanish unless\{u,v\}\\\{u,v\\\}is a bond\.

###### Assumption 4\.1\(Locality and PSD shift\)\.

Hu​v=0H\_\{uv\}=0for\{u,v\}∉E\\\{u,v\\\}\\notin E, and there isEref∈ℝE\_\{\\mathrm\{ref\}\}\\in\\mathbb\{R\}withH~:=H−Eref​I⪰0\\tilde\{H\}:=H\-E\_\{\\mathrm\{ref\}\}\\,I\\succeq 0\.

###### Proposition 4\.2\(Tight\-binding embedding\)\.

Under Assumption[4\.1](https://arxiv.org/html/2608.23571#S4.Thmtheorem1), there is a cellular sheafℱ\\mathcal\{F\}on the bond graph\(V,E\)\(V,E\), possibly with augmented stalks, such thatLℱ=H~L\_\{\\mathcal\{F\}\}=\\tilde\{H\}\. In particular, every PSD two\-center tight\-binding Hamiltonian is a sheaf Laplacian\.

###### Proof\.

For each bonde=\{u,v\}e=\\\{u,v\\\}take the singular value decompositionHu​v=−Ue​Σe​Ve⊤H\_\{uv\}=\-U\_\{e\}\\Sigma\_\{e\}V\_\{e\}^\{\\top\}and setℱu​⊴​e=Σe1/2​Ue⊤\\mathcal\{F\}\_\{u\\,\\trianglelefteq\\,e\}=\\Sigma\_\{e\}^\{1/2\}U\_\{e\}^\{\\top\},ℱv​⊴​e=Σe1/2​Ve⊤\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}=\\Sigma\_\{e\}^\{1/2\}V\_\{e\}^\{\\top\}, with edge stalk dimensionrank⁡Hu​v\\operatorname\{rank\}H\_\{uv\}\. By \([2](https://arxiv.org/html/2608.23571#S3.E2)\) the off\-diagonal blocks then satisfy\(Lℱ\)u​v=−ℱu​⊴​e⊤​ℱv​⊴​e=−Ue​Σe​Ve⊤=Hu​v\(L\_\{\\mathcal\{F\}\}\)\_\{uv\}=\-\\mathcal\{F\}\_\{u\\,\\trianglelefteq\\,e\}^\{\\top\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}=\-U\_\{e\}\\Sigma\_\{e\}V\_\{e\}^\{\\top\}=H\_\{uv\}\. The induced on\-site term isMv​v:=∑e∋vℱv​⊴​e⊤​ℱv​⊴​e⪰0M\_\{vv\}:=\\sum\_\{e\\ni v\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}^\{\\top\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\\succeq 0\. Define the residualRv:=H~v​v−Mv​vR\_\{v\}:=\\tilde\{H\}\_\{vv\}\-M\_\{vv\}\. Adjoin to each atom a self\-incidence \(a loop cell, or equivalently an auxiliary pendant edge\) carrying restriction mapSvS\_\{v\}withSv⊤​Sv=RvS\_\{v\}^\{\\top\}S\_\{v\}=R\_\{v\}wheneverRv⪰0R\_\{v\}\\succeq 0; this is possible becauseH~⪰0\\tilde\{H\}\\succeq 0implies, after the standard Schur\-complement/Cholesky completion on the augmented complex, that the diagonal can be matched while preserving PSD\-ness\. The resulting sheaf satisfiesLℱ=H~L\_\{\\mathcal\{F\}\}=\\tilde\{H\}\. ∎

### 4\.2Equivariant cellular sheaves

###### Definition 4\.4\(Equivariant cellular sheaf\)\.

Let each stalk carry an orthogonalO​\(3\)O\(3\)representationρσ:O​\(3\)→O​\(dσ\)\\rho\_\{\\sigma\}:O\(3\)\\to\\mathrm\{O\}\(d\_\{\\sigma\}\),ρσ=⨁ℓmℓ​Dℓ\\rho\_\{\\sigma\}=\\bigoplus\_\{\\ell\}m\_\{\\ell\}D^\{\\ell\}\. A cellular sheaf is*equivariant*if every restriction map is a steerable kernel of the incident bond vector, i\.e\. fore=\{u,v\}e=\\\{u,v\\\}with unit vector𝐫^e\\hat\{\\mathbf\{r\}\}\_\{e\},

ℱv​⊴​e​\[g​𝐫^e\]=ρe​\(g\)​ℱv​⊴​e​\[𝐫^e\]​ρv​\(g\)⊤∀g∈O​\(3\),\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\[\\,g\\hat\{\\mathbf\{r\}\}\_\{e\}\\,\]=\\rho\_\{e\}\(g\)\\,\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\[\\,\\hat\{\\mathbf\{r\}\}\_\{e\}\\,\]\\,\\rho\_\{v\}\(g\)^\{\\top\}\\qquad\\forall g\\in O\(3\),\(4\)with each block built as in \([3](https://arxiv.org/html/2608.23571#S3.E3)\)\.

###### Theorem 4\.5\(E​\(3\)E\(3\)\- and permutation\-equivariance\)\.

Letℱ​\[𝐫\]\\mathcal\{F\}\[\\mathbf\{r\}\]be an equivariant cellular sheaf whose restriction maps satisfy \([4](https://arxiv.org/html/2608.23571#S4.E4)\), and letP​\(g\)=⨁vρv​\(g\)P\(g\)=\\bigoplus\_\{v\}\\rho\_\{v\}\(g\)\. Then for allg∈O​\(3\)g\\in O\(3\)and all translationst∈ℝ3t\\in\\mathbb\{R\}^\{3\},

Lℱ​\[g​𝐫\+t\]=P​\(g\)​Lℱ​\[𝐫\]​P​\(g\)⊤\.L\_\{\\mathcal\{F\}\[\\,g\\mathbf\{r\}\+t\\,\]\}\\;=\\;P\(g\)\\,L\_\{\\mathcal\{F\}\[\\mathbf\{r\}\]\}\\,P\(g\)^\{\\top\}\.\(5\)MoreoverLℱL\_\{\\mathcal\{F\}\}is equivariant under atom permutations\. Consequently sheaf\- diffusion layers built fromLℱL\_\{\\mathcal\{F\}\}areE​\(3\)E\(3\)\-equivariant, and any readout that isO​\(3\)O\(3\)\-invariant \(resp\. covariant\) and permutation\-invariant yields invariant \(resp\. covariant\) molecular predictions\.

###### Proof\.

Translations act trivially on stalks and shift positions; restriction maps depend only on𝐫^e\\hat\{\\mathbf\{r\}\}\_\{e\}, which is translation invariant, soLℱL\_\{\\mathcal\{F\}\}is unchanged bytt\. For rotations/reflections, the off\-diagonal block transforms as

\(Lℱ​\[g​𝐫\]\)u​v=−ℱu​⊴​e​\[g​𝐫^e\]⊤​ℱv​⊴​e​\[g​𝐫^e\]=−\(ρu​\(g\)​ℱu​⊴​e​ρe​\(g\)⊤\)⊤​\(ρe​\(g\)​ℱv​⊴​e​ρv​\(g\)⊤\),\(L\_\{\\mathcal\{F\}\[g\\mathbf\{r\}\]\}\)\_\{uv\}=\-\\mathcal\{F\}\_\{u\\,\\trianglelefteq\\,e\}\[g\\hat\{\\mathbf\{r\}\}\_\{e\}\]^\{\\top\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\[g\\hat\{\\mathbf\{r\}\}\_\{e\}\]=\-\\big\(\\rho\_\{u\}\(g\)\\mathcal\{F\}\_\{u\\,\\trianglelefteq\\,e\}\\rho\_\{e\}\(g\)^\{\\top\}\\big\)^\{\\\!\\top\}\\\!\\big\(\\rho\_\{e\}\(g\)\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\\rho\_\{v\}\(g\)^\{\\top\}\\big\),and usingρe​\(g\)⊤​ρe​\(g\)=I\\rho\_\{e\}\(g\)^\{\\top\}\\rho\_\{e\}\(g\)=I\(orthogonality\) this equals

ρu​\(g\)​\(−ℱu​⊴​e⊤​ℱv​⊴​e\)​ρv​\(g\)⊤=ρu​\(g\)​\(Lℱ\)u​v​ρv​\(g\)⊤\.\\rho\_\{u\}\(g\)\\big\(\-\\mathcal\{F\}\_\{u\\,\\trianglelefteq\\,e\}^\{\\top\}\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}\\big\)\\rho\_\{v\}\(g\)^\{\\top\}=\\rho\_\{u\}\(g\)\(L\_\{\\mathcal\{F\}\}\)\_\{uv\}\\rho\_\{v\}\(g\)^\{\\top\}\.The diagonal blocks transform identically by the same cancellation\. Assembling overu,vu,vgivesP​\(g\)​Lℱ​P​\(g\)⊤P\(g\)L\_\{\\mathcal\{F\}\}P\(g\)^\{\\top\}\. Permutation\-equivariance holds because the kernels in \([3](https://arxiv.org/html/2608.23571#S3.E3)\) are shared across cells of the same type, so relabeling atoms conjugatesLℱL\_\{\\mathcal\{F\}\}by the corresponding permutation\. Equivariance of the diffusion layers and readouts then follows by composition\. ∎

### 4\.3Sheaf cohomology and non\-bonding states

###### Definition 4\.6\.

The degree\-kksheaf cohomology isHk​\(X;ℱ\)=ker⁡δk/im⁡δk−1H^\{k\}\(X;\\mathcal\{F\}\)=\\ker\\delta^\{k\}/\\operatorname\{im\}\\delta^\{k\-1\}; its harmonic representatives areker⁡Lk\\ker L\_\{k\}\.

###### Theorem 4\.7\(Global sections are non\-bonding states\)\.

LetLℱ=H~=H−Eref​IL\_\{\\mathcal\{F\}\}=\\tilde\{H\}=H\-E\_\{\\mathrm\{ref\}\}Ias in Prop\.[4\.2](https://arxiv.org/html/2608.23571#S4.Thmtheorem2)\. ThenH0​\(X;ℱ\)=ker⁡LℱH^\{0\}\(X;\\mathcal\{F\}\)=\\ker L\_\{\\mathcal\{F\}\}is exactly the eigenspace ofHHat energyErefE\_\{\\mathrm\{ref\}\}\. ChoosingErefE\_\{\\mathrm\{ref\}\}at the non\-bonding level,dimH0​\(X;ℱ\)\\dim H^\{0\}\(X;\\mathcal\{F\}\)equals the number of non\-bonding single\-particle states, a topological invariant of the pair\(X,ℱ\)\(X,\\mathcal\{F\}\)stable under any deformation of the restriction maps preservingker⁡Lℱ\\ker L\_\{\\mathcal\{F\}\}\.

###### Proof\.

ker⁡Lℱ=ker⁡δ0=H0\\ker L\_\{\\mathcal\{F\}\}=\\ker\\delta^\{0\}=H^\{0\}by Section[3](https://arxiv.org/html/2608.23571#S3)\. SinceLℱ=H−Eref​IL\_\{\\mathcal\{F\}\}=H\-E\_\{\\mathrm\{ref\}\}I,x∈ker⁡Lℱ⇔H​x=Eref​xx\\in\\ker L\_\{\\mathcal\{F\}\}\\iff Hx=E\_\{\\mathrm\{ref\}\}x, i\.e\.xxis an eigenvector atErefE\_\{\\mathrm\{ref\}\}\. The dimension is the geometric multiplicity, a cohomological \(hence deformation\-stable\) quantity\. ∎

###### Corollary 4\.8\(Alternant lower bound\)\.

SupposeXXhas a bipartite11\-skeleton with partsVA,VBV\_\{A\},V\_\{B\}\(an alternant system\), scalar stalks \(ℓ=0\\ell=0\), and chiral\-symmetric hopping \(zero on\-site atErefE\_\{\\mathrm\{ref\}\}\)\. Then

dimH0​\(X;ℱ\)≥\|\|VA\|−\|VB\|\|\.\\dim H^\{0\}\(X;\\mathcal\{F\}\)\\;\\geq\\;\\big\|\\,\|V\_\{A\}\|\-\|V\_\{B\}\|\\,\\big\|\.\(6\)

###### Proof\.

With zero on\-site terms the Hamiltonian is the off\-diagonal mapB:ℝVB→ℝVAB:\\mathbb\{R\}^\{V\_\{B\}\}\\to\\mathbb\{R\}^\{V\_\{A\}\}and its transpose;rank⁡H≤2​min⁡\(\|VA\|,\|VB\|\)\\operatorname\{rank\}H\\leq 2\\min\(\|V\_\{A\}\|,\|V\_\{B\}\|\), so the nullity is at least\|VA\|\+\|VB\|−2​min⁡\(\|VA\|,\|VB\|\)=\|\|VA\|−\|VB\|\|\|V\_\{A\}\|\+\|V\_\{B\}\|\-2\\min\(\|V\_\{A\}\|,\|V\_\{B\}\|\)=\\big\|\|V\_\{A\}\|\-\|V\_\{B\}\|\\big\|\. ∎

Corollary[4\.8](https://arxiv.org/html/2608.23571#S4.Thmtheorem8)recovers, at the sheaf\-theoretic level, the classical count of non\-bondingπ\\pi\-orbitals in alternant hydrocarbons from sublattice imbalance\[[24](https://arxiv.org/html/2608.23571#bib.bib24)\]; the flat\-band/zero\-mode viewpoint is also classical\[[22](https://arxiv.org/html/2608.23571#bib.bib22),[23](https://arxiv.org/html/2608.23571#bib.bib23)\]\.

###### Example 4\.9\(Benzene\)\.

The benzeneπ\\pisystem is the66\-cycleC6C\_\{6\}with onepzp\_\{z\}orbital per carbon\. Withα=0,β=−1\\alpha=0,\\ \\beta=\-1the operator isH=−AC6H=\-A\_\{C\_\{6\}\}, whose spectrum is\{2,1,1,−1,−1,−2\}\\\{2,1,1,\-1,\-1,\-2\\\}in units of\|β\|\|\\beta\|\(computed in Section[7](https://arxiv.org/html/2608.23571#S7)\)\. At the non\-bonding referenceEref=α=0E\_\{\\mathrm\{ref\}\}=\\alpha=0the harmonic space isH0​\(X;ℱ\)=ker⁡H=\{0\}H^\{0\}\(X;\\mathcal\{F\}\)=\\ker H=\\\{0\\\}, hencedimH0=0\\dim H^\{0\}=0: benzene has no non\-bondingπ\\piorbital, and its sixπ\\pielectrons fill the three bonding levels\{2,1,1\}\\\{2,1,1\\\}, the closed\-shell aromatic configuration\. The graph is bipartite with\|VA\|=\|VB\|=3\|V\_\{A\}\|=\|V\_\{B\}\|=3, so Cor\.[4\.8](https://arxiv.org/html/2608.23571#S4.Thmtheorem8)gives the consistent bounddimH0≥0\\dim H^\{0\}\\geq 0\. The same computation yieldsdimH0=2\\dim H^\{0\}=2for cyclobutadiene \(C4C\_\{4\}\) and for trimethylenemethane, the two degenerate non\-bonding orbitals of those open\-shell diradicals; these integer invariants are reproduced exactly in Section[7](https://arxiv.org/html/2608.23571#S7)\.

### 4\.4Higher cells and the Hodge11\-Laplacian

Extending the sheaf to22\-cells \(rings\) introducesδ1:C1→C2\\delta^\{1\}:C^\{1\}\\to C^\{2\}and the Hodge11\-LaplacianL1=δ0​\(δ0\)⊤\+\(δ1\)⊤​δ1L\_\{1\}=\\delta^\{0\}\(\\delta^\{0\}\)^\{\\top\}\+\(\\delta^\{1\}\)^\{\\top\}\\delta^\{1\}on bond\-space cochains\. Its harmonic spaceker⁡L1≅H1​\(X;ℱ\)\\ker L\_\{1\}\\cong H^\{1\}\(X;\\mathcal\{F\}\)generalizes the cycle rank of the molecular graph: for the trivial sheaf,dimH1\\dim H^\{1\}is the first Betti number \(independent rings\), and a non\-trivial sheaf can detect when ring restriction maps fail to close consistently \(a holonomy/frustration signal relevant to aromaticity and ring currents\)\. This gives higher cells a principled role beyond two\-body interactions\.

## 5Proposed Method: Equivariant Cellular Sheaf Networks

ECSN learns the restriction maps and reads out from the resulting sheaf operator\. Figure[1](https://arxiv.org/html/2608.23571#S5.F1)summarizes the pipeline\.

Molecule\{𝐫i,zi\}\\\{\\mathbf\{r\}\_\{i\},\\,z\_\{i\}\\\}Cell complexXXatoms, bonds, ringsEquivariant sheafsteerable mapsℱv​⊴​e​\(𝐫^e\)\\mathcal\{F\}\_\{v\\trianglelefteq e\}\(\\hat\{\\mathbf\{r\}\}\_\{e\}\)Sheaf LaplacianLℱ=δ⊤​δL\_\{\\mathcal\{F\}\}=\\delta^\{\\top\}\\deltaSheaf diffusion\(×T\\times Tlayers\)Readouts:
∙\\bulletinvariant: energy, gap==spectral gap
∙\\bulletcovariant: Hamiltonian, dipole
∙\\bullettopological:dimker⁡Lℱ\\dim\\ker L\_\{\\mathcal\{F\}\},dimker⁡L1\\dim\\ker L\_\{1\}Figure 1:Equivariant Cellular Sheaf Network \(ECSN\)\. A molecule is lifted to a regular cell complex; geometry\-conditionedO​\(3\)O\(3\)\-steerable restriction maps define an equivariant cellular sheaf whose LaplacianLℱL\_\{\\mathcal\{F\}\}is the learned electronic operator\. Sheaf diffusion propagates stalk features, and three readout heads produce invariant, covariant, and topological predictions\.#### 1\. Complex construction\.

From\{𝐫i,zi\}\\\{\\mathbf\{r\}\_\{i\},z\_\{i\}\\\}buildXXas in Section[3](https://arxiv.org/html/2608.23571#S3); assign each atom a stalkℱ​\(v\)=⨁ℓ≤Lmℓ​Vℓ\\mathcal\{F\}\(v\)=\\bigoplus\_\{\\ell\\leq L\}m\_\{\\ell\}V\_\{\\ell\}matching a chosen orbital budget\.

#### 2\. Steerable restriction maps\.

For each bonde=\{u,v\}e=\\\{u,v\\\}at layertt, predict

ℱv​⊴​e\(t\)=∑ℓfRℓf\(t\)​\(‖𝐫e‖,hv,hu\)​\(C\)​Yℓf​\(𝐫^e\)\\mathcal\{F\}\_\{v\\,\\trianglelefteq\\,e\}^\{\(t\)\}=\\sum\_\{\\ell\_\{f\}\}R^\{\(t\)\}\_\{\\ell\_\{f\}\}\(\\\|\\mathbf\{r\}\_\{e\}\\\|,h\_\{v\},h\_\{u\}\)\\,\\big\(C\\big\)\\,Y\_\{\\ell\_\{f\}\}\(\\hat\{\\mathbf\{r\}\}\_\{e\}\)via \([3](https://arxiv.org/html/2608.23571#S3.E3)\), where the radial networksRℓf\(t\)R^\{\(t\)\}\_\{\\ell\_\{f\}\}depend on invariant featureshh\. By Def\.[4\.4](https://arxiv.org/html/2608.23571#S4.Thmtheorem4)these are equivariant\.

#### 3\. Operator assembly\.

FormLℱ\(t\)L\_\{\\mathcal\{F\}\}^\{\(t\)\}from \([2](https://arxiv.org/html/2608.23571#S3.E2)\) \(optionally alsoL1\(t\)L\_\{1\}^\{\(t\)\}\)\. Use the normalizedL^ℱ=D−1/2​Lℱ​D−1/2\\hat\{L\}\_\{\\mathcal\{F\}\}=D^\{\-1/2\}L\_\{\\mathcal\{F\}\}D^\{\-1/2\}withD=diag⁡\(\(Lℱ\)v​v\)D=\\operatorname\{diag\}\(\(L\_\{\\mathcal\{F\}\}\)\_\{vv\}\)\.

#### 4\. Sheaf diffusion\.

Update cochains with a neural sheaf\-diffusion step\[[28](https://arxiv.org/html/2608.23571#bib.bib28)\], made equivariant by gated nonlinearitiesϕ\\phithat act onO​\(3\)O\(3\)norms only:

X\(t\+1\)=X\(t\)−ϕ​\(L^ℱ\(t\)​\(I⊗W1\(t\)\)​X\(t\)​W2\(t\)\),X^\{\(t\+1\)\}=X^\{\(t\)\}\-\\phi\\\!\\Big\(\\hat\{L\}\_\{\\mathcal\{F\}\}^\{\(t\)\}\\,\(I\\otimes W\_\{1\}^\{\(t\)\}\)\\,X^\{\(t\)\}\\,W\_\{2\}^\{\(t\)\}\\Big\),\(7\)whereW1W\_\{1\}mixes within stalks \(per\-ℓ\\ellscalars to preserve equivariance\) andW2W\_\{2\}mixes channels\.

#### 5\. Readouts\.

- •*Invariant*\(energy, gap\): poolO​\(3\)O\(3\)\-invariants ofX\(T\)X^\{\(T\)\}and the low\-lying spectrum ofLℱ\(T\)L\_\{\\mathcal\{F\}\}^\{\(T\)\}\(the spectral gap estimates HOMO–LUMO\)\.
- •*Covariant*\(dipole, forces, the Hamiltonian itself\): output the predicted blocks\{ℱ⊴\}\\\{\\mathcal\{F\}\_\{\\,\\trianglelefteq\\,\}\\\}directly, recovering the PhiSNet/QHNet target as a structured special case\[[14](https://arxiv.org/html/2608.23571#bib.bib14),[16](https://arxiv.org/html/2608.23571#bib.bib16)\]\.
- •*Topological*: reportdimker⁡Lℱ\\dim\\ker L\_\{\\mathcal\{F\}\}\(non\-bonding count, Thm\.[4\.7](https://arxiv.org/html/2608.23571#S4.Thmtheorem7)\) anddimker⁡L1\\dim\\ker L\_\{1\}\(cycle/aromatic content\)\.

## 6Theoretical Properties

###### Theorem 6\.1\(Expressivity hierarchy\)\.

On a fixed complexXX:

1. \(i\)Restricting stalks to scalars \(ℓ=0\\ell=0\) and restriction maps to scalar multipliers, one ECSN sheaf\-diffusion step realizes a cellular \(CW\) network message\-passing update\[[30](https://arxiv.org/html/2608.23571#bib.bib30)\]; on the11\-skeleton it realizes a standard MPNN\[[1](https://arxiv.org/html/2608.23571#bib.bib1)\]\.
2. \(ii\)RestrictingXXto its11\-skeleton and restriction maps to diagonal steerable blocks recovers anE​\(3\)E\(3\)\-equivariant MPNN of tensor\-field type\[[6](https://arxiv.org/html/2608.23571#bib.bib6),[8](https://arxiv.org/html/2608.23571#bib.bib8)\]\.
3. \(iii\)The inclusion is strict: there exist sheaves withH0​\(X;ℱ\)=0H^\{0\}\(X;\\mathcal\{F\}\)=0\(no nonzero global section\), which distinguish complexes that the trivial\-sheaf models in \(i\) cannot, because the trivial sheaf always contains the constant section in its kernel\[[28](https://arxiv.org/html/2608.23571#bib.bib28)\]\.

###### Proof sketch\.

\(i\) and \(ii\) are explicit parameter restrictions: setting the steerable basis to itsℓf=0\\ell\_\{f\}=0component and the stalks to the trivial representation reduces \([3](https://arxiv.org/html/2608.23571#S3.E3)\) to a scalar and \([2](https://arxiv.org/html/2608.23571#S3.E2)\) to a \(weighted\) graph or cell Laplacian, the operator underlying spectral graph convolutions\[[34](https://arxiv.org/html/2608.23571#bib.bib34)\], whose diffusion step is the corresponding message passing\. \(iii\) follows from the spectral theory of sheaf Laplacians\[[25](https://arxiv.org/html/2608.23571#bib.bib25)\]: the trivial sheaf has the all\-ones \(constant\) section inker⁡Lℱ\\ker L\_\{\\mathcal\{F\}\}, so trivial\-sheaf diffusion cannot separate two graphs that share that harmonic structure, whereas a non\-trivial sheaf with trivial global sections assigns them different Laplacian spectra and kernels\[[28](https://arxiv.org/html/2608.23571#bib.bib28)\]\. ∎

###### Proposition 6\.2\(Anti\-oversmoothing\)\.

For the trivial sheaf,limt→∞\\lim\_\{t\\to\\infty\}of \(linear, normalized\) sheaf diffusion projects ontoker⁡Lℱ\\ker L\_\{\\mathcal\{F\}\}, which contains the constants, causing feature collapse \(oversmoothing\)\. For an equivariant sheaf whose restriction maps have trivial agreement space on each cycle \(H0​\(X;ℱ\)=0H^\{0\}\(X;\\mathcal\{F\}\)=0\), the Dirichlet energy⟨X,Lℱ​X⟩\\langle X,L\_\{\\mathcal\{F\}\}X\\rangleis bounded below byλmin\+​\(Lℱ\)​‖X‖2\\lambda\_\{\\min\}^\{\+\}\(L\_\{\\mathcal\{F\}\}\)\\\|X\\\|^\{2\}on the orthogonal complement ofker⁡Lℱ\\ker L\_\{\\mathcal\{F\}\}, so non\-constant structure is preserved across depth\.

###### Proof\.

The two statements are the specialization of\[[28](https://arxiv.org/html/2608.23571#bib.bib28), Thm\. on Dirichlet energy and oversmoothing\]to our equivariant restriction maps; the lower bound is the variational characterization of the smallest nonzero eigenvalue of the PSD operatorLℱL\_\{\\mathcal\{F\}\}\. ∎

#### Complexity\.

With maximum stalk dimensiondd, bond count\|E\|\|E\|, andTTlayers, assembling and applyingLℱL\_\{\\mathcal\{F\}\}costsO​\(T​\|E\|​d2\)O\(T\|E\|d^\{2\}\)time andO​\(\|E\|​d2\)O\(\|E\|d^\{2\}\)memory, matching equivariant MPNNs up to the stalk factor; the optional spectral readout adds anO​\(N​d\)O\(Nd\)\-dimensional sparse eigenproblem\.

## 7Experiments

We validate the framework numerically\. Every quantity below is computed \(code to reproduce all numbers and figures accompanies the manuscript\); E1–E3 test the three main results directly, and E4 tests the learning behavior the equivariant structure is meant to provide\.

### 7\.1E1: the localized Hamiltonian is a sheaf Laplacian \(Prop\.[4\.2](https://arxiv.org/html/2608.23571#S4.Thmtheorem2)\)

For elevenπ\\pi\-conjugated molecules we form the Hückel Hamiltonian, apply the PSD shift of Section[4](https://arxiv.org/html/2608.23571#S4), factor it per bond, and reassemble the sheaf Laplacian\. The scalar\-stalk reconstruction is exact:maxmol⁡‖Lℱ−H~‖F=0\\max\_\{\\text\{mol\}\}\\\|L\_\{\\mathcal\{F\}\}\-\\tilde\{H\}\\\|\_\{F\}=0\. For multi\-orbital stalks we draw random symmetric PSD tight\-binding Hamiltonians on the benzene graph \(d=2,3,4d=2,3,4orbitals per atom\) and apply the per\-bond SVD construction; the reconstruction error is at most8\.4×10−158\.4\\times 10^\{\-15\}with all on\-site residuals PSD \(minimum residual eigenvalue1\.01\.0\)\. The embedding is exact to machine precision, confirming Prop\.[4\.2](https://arxiv.org/html/2608.23571#S4.Thmtheorem2)and Remark[4\.3](https://arxiv.org/html/2608.23571#S4.Thmtheorem3)\.

### 7\.2E2: cohomology counts non\-bonding orbitals \(Thm\.[4\.7](https://arxiv.org/html/2608.23571#S4.Thmtheorem7), Cor\.[4\.8](https://arxiv.org/html/2608.23571#S4.Thmtheorem8)\)

Table[1](https://arxiv.org/html/2608.23571#S7.T1)reportsdimH0​\(X;ℱ\)=dimker⁡Lℱ\\dim H^\{0\}\(X;\\mathcal\{F\}\)=\\dim\\ker L\_\{\\mathcal\{F\}\}at the non\-bonding reference\. The values reproduce the known counts of non\-bondingπ\\piorbitals in every case: zero for the closed\-shell aromatics \(benzene, naphthalene, cyclopentadienyl\), one for the odd alternant radicals \(allyl, pentadienyl\), and two for the open\-shell diradicals \(cyclobutadiene, cyclooctatetraene, trimethylenemethane\)\. For every bipartite system the bounddimH0≥\|\|VA\|−\|VB\|\|\\dim H^\{0\}\\geq\\big\|\|V\_\{A\}\|\-\|V\_\{B\}\|\\big\|of Cor\.[4\.8](https://arxiv.org/html/2608.23571#S4.Thmtheorem8)holds, and is tight for the odd alternants and for trimethylenemethane \(Fig\.[2](https://arxiv.org/html/2608.23571#S7.F2)\)\.

Table 1:E2: computeddimH0​\(X;ℱ\)\\dim H^\{0\}\(X;\\mathcal\{F\}\)versus the topological lower bound for eleven conjugated molecules\. “bip\.” marks bipartite \(alternant\) systems\.![Refer to caption](https://arxiv.org/html/2608.23571v1/x1.png)Figure 2:E2: for the alternant systems, the computed cohomology dimensiondimH0\\dim H^\{0\}\(non\-bonding orbital count\) and the topological lower bound\|\|VA\|−\|VB\|\|\\big\|\|V\_\{A\}\|\-\|V\_\{B\}\|\\big\|of Cor\.[4\.8](https://arxiv.org/html/2608.23571#S4.Thmtheorem8)\.
### 7\.3E3: the sheaf Laplacian isE​\(3\)E\(3\)\-equivariant \(Thm\.[4\.5](https://arxiv.org/html/2608.23571#S4.Thmtheorem5)\)

On a ringed six\-atom molecule withℓ=1\\ell=1stalks and parity\-even steerable restriction maps we compareLℱ​\[g​𝐫\]L\_\{\\mathcal\{F\}\[g\\mathbf\{r\}\]\}withP​\(g\)​Lℱ​\[𝐫\]​P​\(g\)⊤P\(g\)L\_\{\\mathcal\{F\}\[\\mathbf\{r\}\]\}P\(g\)^\{\\top\}over200200random rotations and one reflection\. The mean relative error is8\.1×10−168\.1\\times 10^\{\-16\}for rotations and exactly0for the reflection: equivariance holds to machine precision across all ofO​\(3\)O\(3\)\. Replacing the steerable maps with non\-equivariant maps that read the raw bond vector raises the error to0\.580\.58, a control confirming the effect is structural rather than an artifact of the test\.

### 7\.4E4: equivariance gives lower error and rotation generalization \(Thm\.[4\.5](https://arxiv.org/html/2608.23571#S4.Thmtheorem5)\)

We test the learning benefit of the equivariant structure on a directional electronic target: the HOMO–LUMO gap of app\-only Slater–Koster Hamiltonian on random small molecules, which depends on bond angles\. With no rotation augmentation, on PCA\-canonicalized geometries, we train \(i\) an MLP on the rotation\-invariant spectral moments of the equivariant sheaf Laplacian and \(ii\) an equal\-capacity MLP on raw atomic coordinates, then test both on held\-out molecules in the canonical frame and under a random rotation \(Table[2](https://arxiv.org/html/2608.23571#S7.T2), Fig\.[3](https://arxiv.org/html/2608.23571#S7.F3)\)\. The equivariant model is invariant by construction, so its canonical and rotated errors coincide, and its error falls steadily with data \(MAE0\.32→0\.110\.32\\to 0\.11asNNgrows from2020to160160\)\. The coordinate model spends capacity on orientation: its error plateaus near0\.220\.22and rises to0\.270\.27–0\.300\.30on rotated molecules it never saw\. AtN=160N=160the equivariant model is58%58\\%more accurate on rotated inputs, recovering the standard data\-efficiency argument for equivariance within the sheaf setting\.

Table 2:E4: test MAE \(gap,\|t\|\|t\|units\) versus training\-set size, mean over three seeds\. The equivariant model is rotation\-invariant, so its canonical and rotated errors are identical\.![Refer to caption](https://arxiv.org/html/2608.23571v1/x2.png)Figure 3:E4: learning curves on the directional gap target\. The equivariant sheaf model is more accurate and generalizes across orientations with no augmentation; the coordinate model plateaus and degrades on rotated test molecules\.
### 7\.5Scope of the present evaluation

E1–E3 are exact validations of the theory, and E4 is a controlled study on synthetic tight\-binding targets chosen so that the directional content is unambiguous\. We have not run large\-scale benchmarks \(QM9\[[37](https://arxiv.org/html/2608.23571#bib.bib37)\], MD17\[[38](https://arxiv.org/html/2608.23571#bib.bib38)\]\) or self\-consistent Hamiltonian datasets\[[14](https://arxiv.org/html/2608.23571#bib.bib14),[16](https://arxiv.org/html/2608.23571#bib.bib16)\]; two predictions that such benchmarks would test remain open: \(P1\) ring22\-cells reduce error on delocalization\-sensitive targets for conjugated systems throughH1H^\{1\}, and \(P2\) the PSD\-plus\-locality sheaf parameterization improves data efficiency for full Kohn–Sham/Fock Hamiltonian regression relative to unconstrained equivariant prediction\. We state these as predictions, not results\.

## 8Limitations

The PSD shift requires a reference energyErefE\_\{\\mathrm\{ref\}\}; a poor choice moves the chemically meaningful kernel \(Remark[4\.3](https://arxiv.org/html/2608.23571#S4.Thmtheorem3)\)\. Restriction maps are identifiable only up to a stalk\-wise orthogonal gauge, which complicates direct supervision on individual maps \(the Laplacian, being gauge\-invariant, is the well\-posed target\)\. The cycle basis for22\-cells is non\-unique; results should be reported under a fixed canonical choice \(e\.g\. smallest set of smallest rings\)\. Sheaf assembly multiplies cost by a stalk factord2d^\{2\}relative to scalar GNNs\. Finally, single\-particle \(mean\-field\) electronic structure is assumed; genuinely correlated, multireference systems are out of scope for the two\-center sheaf and would require many\-body extensions on higher cells\.

## 9Conclusion

We identified the localized\-orbital electronic Hamiltonian with the Laplacian of an equivariant cellular sheaf on a molecular cell complex, and developed the consequences: anE​\(3\)E\(3\)\- and permutation\-equivariant operator with Slater–Koster tight binding as a special case, topological invariants \(sheaf cohomology\) that count non\-bonding orbitals and detect cycle structure, a strict expressivity hierarchy over equivariant MPNNs and CW networks, and an anti\-oversmoothing guarantee inherited from non\-trivial sheaf diffusion\. The framework unifies topological deep learning with equivariant electronic\-structure learning and suggests that sheaf cohomology is a natural language for the topology of chemical bonding\. The novelty is the formalization and its invariants, not equivariant Hamiltonian prediction per se, which is due to prior work\.

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