SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Once defined a tensor on  $\mathbf{R}^3$ , it's easy to extend it to a continuous manifold or a discrete graph. A tensor field of order  $s$  on a 3D graph  $G = (V, E)$  is a tensor-valued function  $f$  which assigns to each 3D node  $\mathbf{x}_i$  an order  $s$  tensor, denoted by  $f(\mathbf{x}_i)$ .
SE(3) Tensor Representations. Let  $V$  be a vector space, then the group  $\mathrm{SE}(3)$  is said to act on  $V$  if there is a mapping  $\phi : SE(3) \times V \to V$  satisfying the following two conditions:
1. if  $e \in SE(3)$  is the identity element, then
$$
\phi (e, x) = x \quad \text {f o r} \forall x \in V.
$$
2. if  $g_{1}, g_{2} \in SE(3)$ , then
$$
\phi \left(g _ {1}, \phi \left(g _ {2}, x\right)\right) = \phi \left(g _ {1} g _ {2}, x\right) \quad \text {f o r} \forall x \in V.
$$
If we further require  $\phi(g, \cdot)$  is a linear map for all  $g \in SE(3)$ , then  $\phi$  becomes a group representation of  $SE(3)$ . From now on, we only consider the rotation subgroup SO(3) and its group representations. When  $V = \mathbf{R}^3$ , there is a natural representation of  $SO(3)$  by rotating vectors in  $\mathbf{R}^3$ . In this way, an element  $g \in SO(3)$  is identified with a Rotation matrix, denoted by  $\{g_i^j\}_{1 \leq i,j \leq 3}$ .
From the tensor definition (10), this natural representation on  $\mathbf{R}^3$  induces a tensor representation on  $T$ . Still take  $\mathbf{T} = \{T_{ij}\}_{1\leq i,j\leq 3}$  as an example, we have
$$
T _ {k l} = \sum_ {i} \sum_ {j} g _ {k} ^ {i} g _ {l} ^ {j} T _ {i j}, \quad 1 \leq k, l \leq 3, \tag {12}
$$
for  $\forall g\in SO(3)$ . It's easy to check that (12) is indeed a SO(3) representation on the vector space spanned by second-order tensors.
Relation with Spherical Harmonics. For the  $SO(3)$  group, all representations (including the tensor representations) can be decomposed as a direct sum of irreducible representations. For each type of irreducible representations, there is a subset of spherical harmonics formulating a basis for this specific representation. However, in terms of representing equivariant geometric quantities, the theorem in [55] claims that tensor representations and irreducible representations are equally powerful: They all form a complete basis in the space of continuous  $E(3)$  equivariant functions.
B Equivariant frame construction
Equivariant Frame Definition. An equivariant frame on a 3D graph  $G$  consists of three orthonormal vectors that transform equivariantly under  $SE(3)$ . In the following, we present specific constructions of equivariant frames. However, it is important to note that the proof of Theorem 4.1 does not depend on our particular choice of local equivariant frames.
It is worth noting that the number of vectors required in the equivariant frame is equal to the dimension of the space. For example, in the case of the four-dimensional Lorentz space, where the symmetry group is replaced by the Lorentz group, the equivariant frame consists of four independent vectors.
Edge-wise frame. From this point onward, we assume that the 3D graph's mass has been translated to zero. This translation ensures that the center of mass of the graph is located at the origin, ensuring the translation invariance of the system. Consider node  $i$  and one of its neighbors  $j$  with positions  $\mathbf{x}_i$  and  $\mathbf{x}_j$ , respectively. The orthonormal equivariant frame  $\mathcal{F}_{ij} \coloneqq (\mathbf{e}_1^{ij}, \mathbf{e}_2^{ij}, \mathbf{e}_3^{ij})$  is defined with respect to  $\mathbf{x}_i$  and  $\mathbf{x}_j$  as follows:
$$
\left(\frac {\mathbf {x} _ {i} - \mathbf {x} _ {j}}{\| \mathbf {x} _ {i} - \mathbf {x} _ {j} \|}, \frac {\mathbf {x} _ {i} \times \mathbf {x} _ {j}}{\| \mathbf {x} _ {i} \times \mathbf {x} _ {j} \|}, \frac {\mathbf {x} _ {i} - \bar {\mathbf {x}} _ {j}}{\| \mathbf {x} _ {i} - \bar {\mathbf {x}} _ {j} \|} \times \frac {\mathbf {x} _ {i} \times \mathbf {x} _ {j}}{\| \mathbf {x} _ {i} \times \mathbf {x} _ {j} \|}\right). \tag {13}
$$
Node-wise frame. Consider node  $i$  with 3D position  $\mathbf{x}_i$ , and let  $\bar{\mathbf{x}}_i \coloneqq \frac{1}{N} \sum_{\mathbf{x}_j \in \mathcal{N}(\mathbf{x}_i)} \mathbf{x}_j$  be the center of mass around the 1-hop neighborhood of  $\mathbf{x}_i$ . The orthonormal equivariant frame  $\mathcal{F}_i \coloneqq (\mathbf{e}_1^i, \mathbf{e}_2^i, \mathbf{e}_3^i)$  is defined with respect to  $\mathbf{x}_i$  as follows:
$$
\left(\frac {\mathbf {X} _ {i} - \bar {\mathbf {X}} _ {i}}{\| \mathbf {X} _ {i} - \bar {\mathbf {X}} _ {i} \|}, \frac {\bar {\mathbf {X}} _ {i} \times \mathbf {X} _ {i}}{\| \bar {\mathbf {X}} _ {i} \times \mathbf {X} _ {i} \|}, \frac {\mathbf {X} _ {i} - \bar {\mathbf {X}} _ {i}}{\| \mathbf {X} _ {i} - \bar {\mathbf {X}} _ {i} \|} \times \frac {\bar {\mathbf {X}} _ {i} \times \mathbf {X} _ {i}}{\| \bar {\mathbf {X}} _ {i} \times \mathbf {X} _ {i} \|}\right). \tag {14}
$$
Note that node frames can also be obtained by averaging the edge-wise frames and applying the Gram-Schmidt orthogonalization process. This approach provides an alternative method for constructing equivariant frames at the node level.
C Algorithm for the proposed LEFTNet
The detailed LEFTNet is shown in Algorithm 1. Note that for simplicity, the message  $\mathbf{m}$  in this algorithm could include both invariant and equivariant terms in Fig. 3.
The design of LEFTNet (LSE only) used in the ablation study is shown in Algorithm 2.
Algorithm 1 LEFTNet
1: Input: 3D graph with equivariant positions  $\mathbf{X} = (\mathbf{x}_1, \ldots, \mathbf{x}_n) \in \mathbb{R}^{n \times 3}$ , invariant node features  $h_i \in \mathbb{R}^d$ , invariant relative distances  $d_{ij} \in \mathbb{R}^1$ , equivariant edge features  $\mathbf{e}_{ij} \in \mathbb{R}^c$ .
2: Centralize the positions:  $\mathbf{X}\gets \mathbf{X} - \mathrm{CoM}(\mathbf{X})$
3: for  $(i = 1; i <= n; i++)$  do
4: Compute node-wise equivariant frames  $\mathcal{F}_i$  via Eq. 14.
5: for  $j\in \mathcal{N}(i)$  do
6: Compute edge-wise equivariant frames  $\mathcal{F}_{ij}$  via Eq. 13
7: Get the mutual 3D structure  $\mathbf{S}_{i - j}$ , perform local scalarization through  $\mathcal{F}_{ij}$ :
$$
t _ {i j} = \left\{\mathbf {S c a l a r i z e} \left(\mathbf {S} _ {i - j}, \mathcal {F} _ {i j}\right) \right\}
$$
8: Calculate the SE(3)-invariant structural coefficients:  $A_{ij} = g(t_{ij},d_{ij})$
9: Perform equivariant message passing as in Eq. 4:
$$
\mathbf {m} _ {i j} = \phi_ {m} ^ {1} \left(h _ {i}, A _ {i j} \odot h _ {j}, d _ {i j}\right) + \phi_ {m} ^ {2} \left(h _ {i}, A _ {i j} \odot h _ {j}, d _ {i j}\right) \cdot \mathbf {e} _ {i j} + \phi_ {m} ^ {3} \left(h _ {i}, A _ {i j} \odot h _ {j}, d _ {i j}\right) \cdot \mathcal {F} _ {i j}
$$
11: Equivariant message aggregation:  $\mathbf{m}_i = \sum_{j\in \mathcal{N}(i)}\mathbf{m}_{ij}$
12: Transform equivariant node features through  $\mathcal{F}_i$ :
$$
t _ {i} = \mathbf {S c a l a r i z e} (\mathbf {m} _ {i}, \mathcal {F} _ {i})
$$
13: Update invariant node features:
$$
h _ {i} = \phi_ {h} (h _ {i}, t _ {i})
$$
14: Equivariant Output: Perform tensorization through  $\mathcal{F}_i$ :
$$
\mathbf {h} _ {i} = \operatorname {T e n s o r i z e} \left(h _ {i}, \mathcal {F} _ {i}\right).
$$
15: end for
16: Output: for invariant properties (e.g. energy): AvgPooling  $(h_1, \ldots, h_n)$ , for equivariant properties (e.g. force):  $(\mathbf{h}_1, \ldots, \mathbf{h}_n)$ .
Algorithm 2 LEFTNet (LSE only)
1: Input: 3D graph with equivariant positions  $\mathbf{X} = (\mathbf{x}_1, \dots, \mathbf{x}_n) \in \mathbb{R}^{n \times 3}$ , invariant node features  $h_i \in \mathbb{R}^d$ , invariant relative distances  $d_{ij} \in \mathbb{R}^1$ .  
2: Centralize the positions:  $\mathbf{X} \gets \mathbf{X} - \mathrm{CoM}(\mathbf{X})$ .  
3: for  $(i = 1; i <= n; i++)$  do  
4: for  $j \in \mathcal{N}(i)$  do  
5: Compute edge-wise equivariant frames  $\mathcal{F}_{ij}$  via Eq. 13:  
[ \mathcal{F}_{ij} = \text{EquiFrame}(\mathbf{x}_i, \mathbf{x}_j). ]  
6: Get the mutual 3D structure  $\mathbf{S}_{i-j}$ , perform local scalarization:  
[ t_{ij} = \{\text{Scalarize}(\mathbf{S}_{i-j}, \mathcal{F}_{ij})\}. ]  
7: Calculate the SE(3)-invariant structural coefficients:  $A_{ij} = g(t_{ij}, d_{ij})$   
8: Perform invariant message passing:  
[ m_{ij} = \phi_m(h_i, A_{ij} \odot h_j, d_{ij}) ]  
9: end for  
10: Update invariant node features:  
[ h_i = \phi_h(h_i, \sum_{j \in \mathcal{N}(i)} m_{ij}) ]  
11: end for  
12: Output: AvgPooling  $(h_1, \dots, h_n)$ .
D Related proofs and discussions of Section 3
In section 3, we proposed a novel hierarchy of local geometric isomorphisms that further motivates the design of incorporating the mutual 3D substructure's information into equivariant GNNs. Different from our fine-grained local characterization, a cocurrent work GWL [22]) proposes to measure geometric isomorphism from local to global by the k-hop partition.
From another point of view, we essentially demonstrated that encoding mutual 3D substructures expands the capacity of the transformation function class with respect to an equivariant GNN. [23] put forward the Completeness concept for characterizing these transformation functions. However, it mainly concentrates on testing whether a function can discriminate global geometric isomorphism (in the sense of Eq. 3).
Discussion on the Completeness Concept. Following our terminology in the preliminary section, completeness of a transformation  $f$  can be translated into claiming that  $f$  is invariant among 3D graphs if and only if they are globally isomorphic (see definition (3)). Therefore, it's easy to refine the notion of completeness that adapts to our local version by replacing the global isomorphism to local isomorphism:
$$
f (\mathbf {X}) = f (\mathbf {Y}),
$$
if and only if  $\mathbf{X}$  and  $\mathbf{Y}$  are local {tree-, triangular-, subgraph-} isomorphic. Then, in terms of function class capacities, the following relation holds:
Global complete  $\subset$  Subgraph complete  $\subset$  Triangular complete  $\subset$  Tree complete.
Note that our equivalent description of complete transformation reveals the fact that the completeness concept in [23] is defined from the global 3D isomorphism point of view. Therefore, we shall claim that the above series of completeness notions belong to the structure completeness. Indeed, the theory developed in section 3 indicates that a GNN which can express structure complete functions may not be sufficient in expressing general tensor potential functions on a 3D graph.
On the other hand, a non-negligible proportion of 3D graph tasks may not be sensitive to the global 3D non-isomorphism. For example, some chemical properties (formulated as a function defined on molecular graphs) are characterized by local substructures [56]. In these scenarios, we are looking for a geometric transformation  $f$  that is global non-complete, but (tree-, triangular-, subgraph-) local complete.
1. When compared to the construction of topological structural coefficients that take the discrete adjacency matrix of the graph as input, as proposed in [27], the input 3D structural coefficients we seek are of a continuous nature. For instance, as illustrated in Fig. 1, the key features that differentiate these 3D local shapes are the dihedral angle and relative distance, both of which are continuous functions of the 3D positions. Therefore, we require the existence of universal approximators for continuous functions to ensure that the parametrization of  $A_{ij}$  has sufficient expressiveness.
2. The reason we desire the LSE module to be invariant is because it ensures the local injectivity of message passing, following the standard argument proposed in Xu et al. [25]. This local injectivity is crucial for proving the second part of Theorem 3.1.
3. From a mathematical perspective, the common approach to proving the universal approximator property is by demonstrating that a neural network can express a complete basis in the function space, typically leveraging the Stone-Weierstrass theorem. Two successful examples are: a. Piecewise linear function bases achieved by wide Multi-layer perceptrons (MLP); b. Polynomial bases. In the 3D GNN setting, equivariant polynomials that are also permutation-invariant can be rigorously constructed using ACE [29]. Additionally, through local scalarization and the Kolmogorov representation theorem [57], the universal approximator property of 3D GNNs reduces to the universal approximator property of MLP, which has been proven for a broad range of non-linear activation functions.
Proof. The first part of the theorem is proved by providing an explicit example. Consider the first pair of 3D shapes in Figure 1, where the mutual 3D substructures  $\mathbf{S}_{i - q}$  and  $\mathbf{S}_{j - q^{\prime}}$  both contain two triangles:  $(x_{i},x_{p},x_{q})$ ,  $(x_{i},x_{m},x_{q})$  and  $(x_{j},x_{p^{\prime}},x_{q^{\prime}})$ ,  $(x_{j},x_{m^{\prime}},x_{q^{\prime}})$  respectively.
The difference between these two triangular non-isomorphic shapes can be captured by the distance function:
$$
d (\mathbf {x} _ {p}, \mathbf {x} _ {m}) = \| \mathbf {x} _ {p} - \mathbf {x} _ {m} \| ^ {2},
$$
and similarly,  $d(\mathbf{x}p', \mathbf{x}m')$  for the second shape. Note that this function cannot be expressed solely by tree-level features since there is no edge connecting  $\mathbf{x}_p(\mathbf{x}_{p'})$  and  $\mathbf{x}_m(\mathbf{x}_{m'})$ . The fact that this function produces different output values for the two tree isometric but triangular non-isometric 3D shapes implies that  $\phi$  is capable of distinguishing 3D shapes beyond tree isomorphism.
Therefore, if the structural coefficients  $A_{ip} \left( A_{jp'} \right)$  take the value of  $d(\mathbf{x}_p, \mathbf{x}_m) \left( d(\mathbf{x}_{p'}, \mathbf{x}_{m'}) \right)$ , then we know that it can differentiate these two triangular non-isomorphic shapes. Thus, it suffices to prove that the encoder  $\phi$  is a universal approximator of continuous functions (which obviously contain the distance functions) that takes the mutual 3D substructures  $\mathbf{S}_{i-q} \left( \mathbf{S}_{j-q'} \right)$  as input.
According to the Stone-Weierstrass theorem, it is sufficient to demonstrate that  $\phi$  can express a complete basis within the continuous function space. In the 3D GNN setting, this basis should satisfy two additional requirements: 1) permutation equivariance, and 2)  $SE(3)$  equivariance. An example of such a basis is the set of  $E(3)$  equivariant polynomials that are also permutation equivariant, as demonstrated in Drautz [29]. To obtain invariant coefficients from the combination of equivariant basis functions, we can employ a projection layer, similar to the one used in Dym and Maron [55].
Alternatively, a simpler approach to achieve the universal approximator property is by leveraging local scalarization. Specifically, we use an equivariant frame on the edge  $e_{iq}$  (as described in the previous section) to scalarize the mutual 3D structure  $\mathbf{S}_{i - q}(\mathbf{S}_{j - q^{\prime}})$ , which includes  $\mathbf{x}_p(\mathbf{x}_{p'})$  and  $\mathbf{x}_m(\mathbf{x}_{m'})$ . Let  $\tilde{S}_{i - q}$  denote the scalarized 3D positions of  $\mathbf{S}_{i - q}$ . By quoting the information lossless theorem and the Kolmogorov representation theorem for permutation-invariant functions from Zaheer et al. [57], Du et al. [30], we know that any  $\varphi$  that is a  $SE(3)$  and permutation invariant function of  $\mathbf{S}_{i - q}$  can be expressed as:
$$
\varphi \left(\mathbf {S} _ {i - q}\right) = f \left(\sum_ {x \in \tilde {S} _ {i - q}} g (x)\right). \tag {15}
$$
where  $f$  and  $g$  are certain continuous functions that can be approximated by wide MLP, following the universal approximator property of MLP [25].
The second part of the proof demonstrates that there exists an enhanced message passing framework (under injectivity assumptions, which can be realized similarly to [25, 27]) that incorporates the structural coefficients  $\{\{A_{ij} := \phi(\mathbf{S}_{i-j})\}_{e_{ij} \in E}\}$  to map two distinct local 3D subgraphs with isometric local tree structures in Figure 1 to different embeddings. Since we specifically construct  $\phi$  as an invariant encoder, the collection  $\{\{A_{ij} := \phi(\mathbf{S}_{i-j})\}_{e_{ij} \in E}\}$  consists of invariant numbers, and the construction of the message passing framework follows a similar approach to [27]. For completeness, we provide a brief review of the injectivity condition and the message passing construction.
To establish the injectivity condition and prove the second part, we introduce the multi-set notation  $\{\cdot \cdot \}$ , following Xu et al. [25]. A basic equivariant GNN within our enhanced framework consists of at least two steps: 1. Message passing, defined by (4); 2. Node-wise update:
$$
h _ {i} ^ {t + 1} = \mathbf {M L P} \left(m _ {i} ^ {t}, h _ {i} ^ {t}\right).
$$
For simplicity, we denote the composition of these two steps as  $\Psi$ . Then, the additional injectivity condition requires that  $\Psi$  satisfies the following:
$$
\Psi \left(\left\{\left\{h _ {i} ^ {t}, A _ {j i} h _ {i} ^ {t},, h _ {j} ^ {t} \mid j \in \mathcal {N} _ {i} \right\} \right\}, \left\{\left\{A _ {i j} h _ {j} ^ {t} \mid j \in \mathcal {N} _ {i} \right\} \right\}\right) \tag {16}
$$
is injective for each layer  $t$  and each node  $i$ . It is worth noting that this condition can be trivially achieved by incorporating weighted residue terms, similar to formula (5) in Wijesinghe and Wang [27]. Consequently, based on the injectivity condition, it is evident that two non-identical collections of  $\{\{A_{ij}\}_{e_{ij} \in E}\}$  would yield two different embedded feature vectors.
Furthermore, in the first part, we have proven the existence of at least two distinct local 3D subgraphs with isometric local tree structures such that the corresponding geometric weights  $\{\{A_{ij}\}_{e_{ij} \in E}\}$  generated by the encoder  $\phi$  are different. Combining both parts, we have successfully demonstrated the theorem.
E Related proofs and discussions of Section 4
Torsion Angle is Secretly Hidden in FT
Recall the edge-wise (signed) torsion angle  $\tau_{ij}$  [58] involves the 1-hop atom pairs  $i$  and  $j$  and two 2-hop atoms  $k$  and  $l$ , then  $\tau_{ij}$  is defined to be the dihedral angle between plane  $k - i - j$  and plane  $l - j - i$ . Although exhausting all torsion angles requires  $O(k^2)$  complexity, Wang et al. [23] reduces the computation to  $O(k)$  order by selecting a canonical 2-hop atom  $k$  and  $l$ , which is enough for detecting the relative orientations between atoms (insufficient for general tasks like many body interactions).
Now we show how  $\tau_{ij}$  naturally appears as one of the derivatives from frame transition functions. For node  $i$ , define the equivariant frame  $\mathcal{F}_i$  by
Figure 5:  $\tau_{ij}$  indicates the relative rotation of two frames along the z-axis.
$$
\left(\mathbf {e} _ {1} ^ {i}, \mathbf {e} _ {2} ^ {i}, \mathbf {e} _ {3} ^ {i}\right) = \left(\mathbf {x} _ {i} - \mathbf {x} _ {j}, \mathbf {x} _ {i} - \mathbf {x} _ {k}, \mathbf {e} _ {1} ^ {i} \times \mathbf {e} _ {2} ^ {i}\right).
$$
$\mathcal{F}_i$  is normalized through the Gram-Schmidt algorithm. For node  $\mathrm{j}$ ,  $\mathcal{F}_j$  is defined similarly by
$$
(\mathbf {e} _ {1} ^ {j}, \mathbf {e} _ {2} ^ {j}, \mathbf {e} _ {3} ^ {j}) = (\mathbf {x} _ {j} - \mathbf {x} _ {i}, \mathbf {x} _ {j} - \mathbf {x} _ {l}, \mathbf {e} _ {1} ^ {j} \times \mathbf {e} _ {2} ^ {j}).
$$
Then following the transition formula (5),
$$
R _ {i j} = \left(\mathbf {e} _ {1} ^ {i}, \mathbf {e} _ {2} ^ {i}, \mathbf {e} _ {3} ^ {i}\right) \cdot \left(\mathbf {e} _ {1} ^ {j}, \mathbf {e} _ {2} ^ {j}, \mathbf {e} _ {3} ^ {j}\right) ^ {T}.
$$
Note that for an orthonormal matrix, its inverse is equal to its transpose. Therefore, based on the standard definition of a dihedral angle, we have:
$$
\tau_ {i j} = \mathbf {e} _ {3} ^ {i} \cdot \mathbf {e} _ {3} ^ {j} \equiv R _ {i j} (3, 3).
$$
In conclusion,  $\tau_{ij}$  represents only one component of the transition matrix  $R_{ij}$ . To fully determine  $R_{ij}$ , we still require two additional angles, as a transition matrix is uniquely determined by three Euler angles.
Proof of Theorem 4.1.
1. The crucial property we leverage to prove this theorem is that all local scalars can be expressed through local scalarization using one local frame and a nonlinear transformation. However, it is important to note that the transition between local frames reveals the relative change of local geometries, which is inherently non-local.
2. Interestingly, stemming from the fact that invariant scalars can be expressed using scalarization and a universal approximator, we can deduce that by incorporating the frame transition matrix into the union set of  $S_{\mathbf{B}}$  and  $S_{\mathbf{C}}$ , all (continuous) invariant interactions  $f_{a}(\mathbf{B},\mathbf{C})$  can be expressed.
Proof. This theorem is proved in two steps:
1. The first step characterizes all scalars  $S_{\mathbf{B}}(S_{\mathbf{C}})$  determined by isolated local clusters  $\mathbf{B}$  and  $\mathbf{C}$  through equivariant frames and local scalarization;
2. The second step constructs a specific invariant function  $f_{a}(\mathbf{B},\mathbf{C})$  that cannot be expressed by taking the union of  $S_{\mathbf{B}}$  and  $S_{\mathbf{C}}$ :  $\{S_{\mathbf{B}}\parallel S_{\mathbf{C}}\}$ .
Let  $G$  denote a 3D point cloud. As proven in Du et al. [30], when equipped with an equivariant frame  $\mathbf{F}_G$ , all equivariant features of  $G$  can be transformed into scalar features through scalarization without information loss. More precisely, follow the convention in the main text, let  $\tilde{G}$  be the output of performing scalarization (using any equivariant frames purely determined by  $G$ , see Section B for examples) on the 3D point cloud  $G$ . For any invariant function  $f(G)$ , there exists a corresponding function  $\tilde{f}$  such that:
$$
f (G) = \tilde {f} (\tilde {G}). \tag {17}
$$
Since  $S_{G}$  represents the collection of all invariant scalars produced by  $G$ , which is equivalent to the collection of all invariant functions that depend on the 3D point cloud  $G$ , Eq. 17 implies that  $S_{G}$  can be generated by a finite set of invariant scalars  $\tilde{G}$ . To apply this insight to our current theorem, we have two different 3D clouds  $\mathbf{B}$  and  $\mathbf{C}$ . Therefore, we need to build two local equivariant frames  $\mathcal{F}_{\mathbf{B}} = (e_1^{\mathbf{B}}, e_2^{\mathbf{B}}, e_3^{\mathbf{B}})$  and  $\mathcal{F}_{\mathbf{C}} = (e_1^{\mathbf{C}}, e_2^{\mathbf{C}}, e_3^{\mathbf{C}})$ . Importantly, the frame  $\mathcal{F}_{\mathbf{B}}$  itself doesn't depend on  $\mathbf{C}$ , and scalarization through  $\mathcal{F}_{\mathbf{B}}$  is only performed on the local 3D point cloud  $\mathbf{B}$ . Therefore, performing operations like scalarizing equivariant information of  $\mathbf{C}$  through  $\mathcal{F}_{\mathbf{B}}$  would violate the assumptions of the theorem.
We are now ready to construct an explicit counterexample  $f_{a}(\mathbf{B},\mathbf{C})$  using  $\mathcal{F}_{\mathbf{B}}$  and  $\mathcal{F}_{\mathbf{C}}$ :
$$
f _ {a} (\mathbf {B}, \mathbf {C}) := e _ {1} ^ {\mathbf {B}} \cdot e _ {1} ^ {\mathbf {C}}.
$$
Since  $f_{a}$  is the inner product of two equivariant vectors, it automatically becomes an invariant function. Next, we check whether  $f_{a}(\mathbf{B},\mathbf{C})$  can be expressed as a function of  $f_{a}(\{S_{\mathbf{B}}\parallel S_{\mathbf{C}}\})$ . The equivariant component of  $f_{a}$  related to  $\mathbf{B}$  is precisely  $e_1^{\mathbf{B}}$ . Applying the above local scalarization principle, we scalarize  $e_1^{\mathbf{B}}$  through  $\mathcal{F}_{\mathbf{B}}$  and obtain:
$$
e _ {1} ^ {\mathbf {B}} \rightarrow \tilde {e} _ {1} ^ {B} = (1, 0, 0).
$$