SlowGuess/NeurIPS_2023_OCR huggingface dataset:
An polymer graph  $P_{k}$  with  $k \geq 2$  and finite  $n$ -vertex monomer  $M = (V_{M}, E_{M})$  is the graph on  $kn$  vertices that contains  $k$  copies  $M_{1}, \ldots, M_{k}$  of  $M$ . Since  $M$  is finite, we can choose a start  $s_{1}$  and end  $e_{1}$  vertices in  $V_{M_{1}}$ .  $M_{i}$ 's being copies of  $M$  also means that there is an isomorphism  $\phi_{i}$  from  $V_{M_{i-1}}$  to  $V_{M_{i}}$  for each  $i \in [k-1]$ . Let  $s_{i}$  and  $e_{i}$  be inductively defined as  $\phi_{i}(s_{i-1})$  and  $\phi_{i}(e_{i-1})$  for  $i$  from 2 to  $k$ . Beside edges in the copies of  $M$ , we also add all edges of the form  $(e_{i-1}, s_{i})$  for each  $i$  from 2 to  $k$ . See Figure 2 for an illustration.
(a) Cayley graph of  $\mathbb{Z}$
(c) Polymer graph whose unit is a graph with 5 vertices. The pattern extends ad infinitum on both sides.
(b) Cayley graph of  $\mathbb{Z}^2$ . (()) is computation mod 1
(d) Graphing of (c).  $((\cdot))$  is computation mod 0.2.  
Edges drawn are from a single unit in the polymer.
Figure 2: Examples of limit objects. The vertex set is the interval [0,1]. Example edges are the arcs connecting points on the intervals.  $a$  and  $b$  are distinct irrational numbers. In each graph, edges that miss an endpoint are identified as a single edge connecting the two existing endpoints.
A.3 Illustrations
Figure 2 shows examples of graphings for a Cayley graph of  $\mathbb{Z},\mathbb{Z}^2$  and a polynomial graph.
A.4 Experimental details
To empirically study the rate of convergence of the  $d_{M}$  metric, we study an approximation of Hausdorff metric between 1-profiles of GNNs build on small finite polymers to that built on a large one. The result is in Figure 1 and we now detail the set up.
Architecture The GNN has 2 hidden layers, uses ReLU activation at each layer, including the output layer. We fix a particular polynomial  $\mathrm{GSO} A^2 + A$  where  $A$  is the normalized (by edge vertex degree) adjacency matrix of the input graph of the GNN. Monomer for each polymer has size 5 and is the same monomer seen in Figure 2. The small polymers consist of 2, 4, 8, 16, 32 monomers. While the large monomers that take the place of the limit object has 128 monomers.
1-profile estimation Recall that the 1-profile of an operator  $P$  consists of entry distributions of  $(f, Pf)$  where  $f$  runs through all measurable functions. It is therefore, not tractable to construct the 1-profile exactly. Instead, we draw random  $F = \{f_1, f_2, \ldots, f_{10}\}$ 's to construct a set of 10 probability distributions. Repetitions of this random drawing make up the different solid lines in Figure 1. We construct each  $f_i : [0,1] \to [-1,1]$  as piecewise linear function with pieces  $((u - 1)/1000, u/1000]$  for  $u$  from 1 to 1000. We set  $f_i(0) = 0$  for each  $i$  and recursively set  $f_i(u/1000) = f_i((u - 1)/1000) + q/100$  where  $q$  is a random draw from  $\{-1,1\}$  uniformly. Finally, we linearly interpolate between two consecutive endpoints. Numpy random seed is set at 1234567.
After getting this set  $F$ , for each operator  $P$  that are either GNN built on small polymers or GNN built on the large one, we compute the set  $\{\mathcal{D}(f_i, P f_i) : f_i \in F\}$  to get the corresponding estimation of the 1-profile.
Hausdorff metric computation Given a pair of 1-profile estimations to compute  $d_{H}$  over, we use optimal transport code from (Flamary et al., 2021) to compute the earth mover distance between each pairs of distributions. This gives an approximation of  $d_{LP}$  between elements of the two profiles. Using the definition of Hausdorff distance, we obtain the result in Figure 1.
A.5 Milder assumptions
As mentioned in the main text, we will work with the following slightly less restrictive set of Assumptions.
Assumption 4.A (Maps constant pieces to constant pieces with high probability). We say that an operator  $A: \mathcal{F} \to \mathcal{F}$  maps constant pieces to constant pieces whp at resolutions in  $\mathcal{N} \subset \mathbb{N}$  if there exists a set  $E \subset [0,1]$  with Lebesgue measure  $\lambda(E) < \inf_{n \in \mathcal{N}} 1/n$ , such that for any  $n \in \mathcal{N}$ , and for any  $f \in \mathcal{F}_{[-1,1]}$  that is a.e. constant on each interval  $(u - 1/n, u]$  for  $u \in [n]/n$ ,  $Af$  is constant on  $(u - 1/n, u] \backslash E$  and  $\| Af\mathbb{1}_E\|_1 < \inf_{n \in \mathcal{N}} \frac{1}{n}$ .
Assumption 4.B (Maps Lipschitz functions to Lipschitz functions with high probability). We say that an operator  $A: \mathcal{F} \to \mathcal{F}$  maps Lipschitz functions to Lipschitz functions whp at resolutions in  $\mathcal{N} \subset \mathbb{N}$  if there exists a set  $E \subset [0,1]$  with  $\lambda(E) < \inf_{n \in \mathcal{N}} 1/n$ , such that for any  $n \in \mathcal{N}$ , and for any  $f \in \mathcal{F}_{[-1,1]}$ ,  $Af$  is  $C_v$  Lipschitz on  $[0,1] \backslash E$  and  $\| Af\mathbb{1}_E \|_1 < \inf_{n \in \mathcal{N}} \frac{1}{n}$ .
Assumption 5.A (Maps constant pieces to Lipschitz pieces). We say that an operator  $A: \mathcal{F} \to \mathcal{F}$  maps constant pieces to Lipschitz pieces at resolutions in  $\mathcal{N} \subset \mathbb{N}$  and constant  $C$  if for any  $n \in \mathcal{N}$ , and for any  $f \in \mathcal{F}_{[-1,1]}$  that is a.e. constant on each interval  $(u - 1/n, u]$  for  $u \in [n]/n$ , we have that  $Af$  is  $C$ -Lipschitz on each  $(u - 1/n, u]$ , for all  $u \in [n]/n$ .
Assumption 5.B (Maps constant pieces to Lipschitz pieces with high probability). We say that an operator  $A: \mathcal{F} \to \mathcal{F}$  maps constant pieces to Lipschitz pieces whp at resolutions in  $\mathcal{N} \subset \mathbb{N}$  and constant  $C$  if for any  $n \in \mathcal{N}$ , there exists a set  $E \subset [0,1]$  with  $\lambda(E) < \frac{1}{n}$  such that for any  $f \in \mathcal{F}_{[-1,1]}$  that is a.e. constant on each interval  $(u - 1/n, u]$  for  $u \in [n]/n$ , it holds that  $Af$  is  $C$ -Lipschitz on each  $(u - 1/n, u] \backslash E$ , for all  $u \in [n]/n$  and  $\| Af\mathbb{1}_E\|_1 < \inf_{n \in \mathcal{N}} \frac{1}{n}$ .
B Omitted proofs from Section 3
Proof of Lemma 1. Fix  $m \in \mathbb{N}$  and  $f, g \in \mathcal{F}_m$ . Since  $P$ -operators are bounded, to show that they are self-adjoint, it suffices to show that  $\langle A_m f, g \rangle = \langle f, A_m g \rangle$  where  $\langle \cdot, \cdot \rangle$  is the usual inner product in the Hilbert space  $\mathcal{F}_m$ . We have:
$$
\begin{array}{l} \langle A _ {m} f, g \rangle = \sum_ {u \in 1 / m [ m ]} (A _ {m} f) (u) g (u) (19) \\ = \sum_ {u \in 1 / m [ m ]} \int_ {u - \frac {1}{m}} ^ {u} A f ^ {\prime} \mathrm {d} \lambda \cdot g (u) (20) \\ = \sum_ {u \in 1 / m [ m ]} \int_ {u - \frac {1}{m}} ^ {u} \left(A f ^ {\prime}\right) g ^ {\prime} \mathrm {d} \lambda = \int_ {0} ^ {1} \left(A f ^ {\prime}\right) g ^ {\prime} \mathrm {d} \lambda (21) \\ = \int_ {0} ^ {1} f ^ {\prime} \left(A g ^ {\prime}\right) \mathrm {d} \lambda (22) \\ = \sum_ {u \in 1 / m [ m ]} \int_ {u - \frac {1}{m}} ^ {u} f ^ {\prime} \left(A g ^ {\prime}\right) \mathrm {d} \lambda = \langle f, A g \rangle , (23) \\ \end{array}
$$
where the first line is the definition of the inner product in  $\mathcal{F}_m$ , the second line is the definition of the discretization  $A_{m}$  (recall that for  $f\in \mathcal{F}_m$ ,  $f^{\prime}\in \mathcal{F}$  is the extension of  $f$  defined as  $f^{\prime}(x) = f(\lceil xm\rceil /m)$ ), the third line is because  $g^{\prime}$  is constant on fixed  $[u - 1 / m,u]$  intervals for each  $u$  and the fourth line is because  $A$  is self-adjoint.
C Theory of graphings
In this subsection, we highlight definitions and key characteristics of graphings so that the paper is self-contained. A much more in-depth discussion can be found in Lovász (2012).
Definition 1 (Borel graphs). Let  $\Omega$  be a topological space and  $(\Omega, \mathcal{F})$  be the corresponding Borel space. A Borel graph is a graph  $(\Omega, E)$  such that  $E \in \mathcal{F} \times \mathcal{F}$ .
The following proposition asserts that bounded-degree graphs without automorphisms are always Borel:
Lemma 3 (Proposition 18.6 from Lovász (2012)). If  $G$  is a bounded-degree graph without automorphisms then there exists a topology  $\tau$  on  $V(G)$  (called the local topology) such that  $G$  is Borel with respect to the Borel space built from  $\tau$ .
When the graph does have automorphisms, one can break the symmetries by coloring the nodes with some set of colors (the fact that the graph has bounded degree means that one only needs finitely many colors).
We next introduce the main object of interest:
Definition 2 (Graphings). A graphing is a quadruple  $G = (\Omega, \mathcal{F}, \lambda, E)$  such that  $\mathcal{F}$  is a Borel  $\sigma$ -algebra that makes  $(\Omega, E)$  a Borel graph, and  $\lambda$  is a probability measure on  $(\Omega, \mathcal{F})$  satisfying: for any  $A, B \in \mathcal{F}$ ,
$$
\int_ {A} d e g _ {B} (x) \mathrm {d} \lambda (x) = \int_ {B} d e g _ {A} (x) \mathrm {d} \lambda (x), \tag {24}
$$
where  $\deg_A(x)$  counts the number of neighbors in  $A$  of  $x$ .
As an example, we now describe paths and cycles in terms of graphings: let  $V$  be  $[0,1]$ ,  $\mathcal{F}$  the Borel  $\sigma$ -algebra generated by open intervals with rational endpoints and for each  $x \in [0,1]$ , put  $(x,x\pm a)$  in  $E$  if  $x\pm a\in [0,1]$  for some real number  $a < 1$ . For  $a < \frac{1}{2}$ , each connected component of the graphing is a finite path. If we consider the edge set  $(x,x\pm a \mod 1)$  for rational  $a$ , then it is not hard to see that connected components of the graphing are finite cycles. If  $a$  is irrational, then the resulting graphing is a two-way infinite path, i.e., a path with no "beginning" and no "end". A formal argument, with an appropriate metric on the space of graphings, can be made to show that the limit of cycles and paths coincides to be the two-way infinite path.
Graphings are not unique in representing certain graphs. There are weak equivalences between pairs of graphings that are formalized through the notion of local isomorphisms.
Definition 3 (Local isomorphisms of graphings). Let  $G_{1}$ ,  $G_{2}$  be two graphings. A measure-preserving map  $\varphi : V(G_{1}) \to V(G_{2})$  is a local isomorphism if its restriction to almost every connected component of  $G_{1}$  (outside of a set of connected components of measure 0) is a graph isomorphism with one of the connect components of  $G_{2}$ . More formally,
$$
\Pr_ {\lambda \left(G _ {1}\right)} \left(\left(G _ {1}\right) _ {x} \equiv \left(G _ {2}\right) _ {\varphi (x)}\right) = 1, \tag {25}
$$
where  $\equiv$  is rooted graph isomorphism and  $(G_1)_x$  is the connected component of  $G_{1}$  rooted at  $x$ .
Note that local isomorphism of graphings are not symmetric and the map  $\varphi$  needs not be invertible. A stronger notion of equivalence, which is symmetric and transitive is local equivalence:
Definition 4 (Local equivalence (informal)).  $G_{1}$  and  $G_{2}$  are locally equivalent if they have the same subgraph densities  $t^{*}(F,G_{1}) = t^{*}(F,G_{2})$  for every connected simple graph  $F$ .
The above definition is informal since we have not defined subgraph densities (which is done via the Benjamini-Schramm interpretation of graphings). We state Definition 4 for readers who are familiar with dense graph convergence of graphons since this is how such convergences are defined. In fact, we can conveniently bypass formally defining local equivalence by the following characterization:
Lemma 4 (Bi-local isomorphism (Theorem 18.59 in Lovász (2012))). Two graphings are locally equivalent iff there is a third graphing with a local isomorphism to each of them - a property called bi-local isomorphism.
C.1 Degeneracy of eigenvectors
We give an heuristic argument why directed graphing limits can fail to have eigenvalues and eigenvectors/eigenfunctions. Consider the sequence of one-sided directed path:  $G_{n} = (V = [n], E = \{(i, i + 1) : i \in [n - 1]\})$ , for  $n \geq 1$ . Note that this may not be a Cayley graph in certain authors' definition, which requires Caley graphs to be undirected. However, it is a realistic example in machine learning since convolutional layers in CNNs are exactly the adjacency operators of these directed graphs. The limiting adjacency operator for this sequence, for all intents and purposes, is intuitively the shift operator (say, on  $\ell^2$ )  $A : (x_1, x_2, x_3, \ldots) \mapsto (0, x_1, x_2, \ldots)$ . An easy calculation then shows that this shift operator does not have eigenvalues.
For a more formal argument and construction on failure cases of existence of eigenvectors in undirected graphings (which is what we defined in this section), refer to Remark 1.6 of Backhausz and Szegedy (2022).
D Omitted proofs from Section 4
D.1 Proof of Lemma 2
We prove each bullet point in Lemma 2 separately, in the next three lemmas.
In the following lemma, we show that graphings corresponding to infinite paths and high dimensional grids satisfy the assumptions in the main results. Similar to how isomorphic graphs represent the same graph, we only need to specify a locally isomorphic graphing that represent the equivalence class of graphings containing the infinite path and high dimensional grids.
Lemma 5 (Well-behaved GSOs - Graphings). Let  $G$  be a graphing corresponding to the Cayley graph of  $\mathbb{Z}$  (two-way infinite paths) or high-dimensional generalizations (infinite 2D and 3D grids). For a graphing  $H$ , let  $A(H)$  be its adjacency operator. If  $H$  is regular, let  $\deg(H)$  be the degree of any of its vertices. For each  $N \in \mathbb{N}$ , there exists locally equivalent graphings:
1.  $G_N'$  such that  $A(G_N')$  satisfies Assumption 3.A with resolution set  $d(N) = \{x \in \mathbb{N} : x\alpha = N \text{ for some } \alpha \in \mathbb{N}\}$ .
2.  $G_N''$  such that  $A(G_N'')$  satisfies Assumption 4.A with resolution set  $[N]$ .
3.  $G_N^{\prime \prime \prime}$  such that  $A(G_N^{\prime \prime \prime}) / \deg (G)$  satisfies Assumption 4.B with resolution set  $[N]$ .
Proof of Lemma 5. We will first show the results for two-way infinite paths. Higher dimensional versions follow almost verbatim. Fix  $a \in \mathbb{R} \backslash \mathbb{Q}$  irrational. Recall from Lovász (2012) that the graphing  $G = ([0,1], \mathcal{F}, \lambda, E)$  where  $\mathcal{F}$  is the Borel  $\sigma$ -algebra generated by open intervals with rational endpoints,  $\lambda$  is some probability measure on  $([0,1], \mathcal{F})$  and  $E \in \mathcal{F} \times \mathcal{F}$  is defined as:
$$
E := \{(x, x \pm a \mod 1) \mid x \in [ 0, 1 ] \}. \tag {26}
$$
That  $G$  is a graphing and each of its connected components is a copy of the Cayley graph of  $\mathbb{Z}$  generated by  $\{-1, 1\}$  is asserted in Lovász (2012).
1. Fix  $N \in \mathbb{N}$ , the goal is to define a graphing  $G_N'$  that is locally equivalent to  $G$  such that  $A(G_N')$  satisfies Assumption 3.A with resolution set  $d(N)$  - the divisor set of  $N$ .
Defining  $G_N'$ . Let  $G_N' \coloneqq ([0,1], \mathcal{F}, \lambda, E_N')$  where,
$$
E _ {N} ^ {\prime} := \left\{\left(x _ {j}, j - \frac {1}{N} + \left(x _ {j} \pm \frac {a}{N} \mod \frac {1}{N}\right)\right): j \in [ N ] / N, x _ {j} \in \left[ j - \frac {1}{N}, j\right) \right\}. \tag {27}
$$
Intuitively,  $G_N'$  consists of  $N$  disjoint copies of  $G$  shrunk to the space  $[0,1/N)$ . Since  $E \subset [0,1] \times [0,1]$ ,  $G_N'$  is a graphing in the same Borel space as  $G$ .
$G_N'$  is locally equivalent to  $G$ . From Lemma 4 it suffices to display a local isomorphism from a third graphing to each of them. Let  $G_{2N}'$ , defined similarly as  $G_N'$ , be the third graphing. We claim that  $\varphi_1: x \mapsto 2x \mod 1$  is a local isomorphism from  $G_{2N}'$  to  $G_N'$  and  $\varphi_2: x \mapsto 2Nx \mod 1$  is a local isomorphism from  $G_{2N}'$  to  $G$ . Intuitively,  $G_{2N}'$  contains  $2N$  copies of  $G$  while  $G_N'$  contains  $N$  copies of  $G$ . Thus, our local isomorphisms only need to make sure that a connected component in one copy of  $G$  in  $G_N'$  is mapped bijectively to a connected component in another copy of  $G$  in  $G_{2N}'$ . We give a rigorous argument below.
For  $x$  picked randomly according to  $\lambda$ , let  $j \in [2N] / (2N)$  be such that  $x \in [j - 1 / (2N), j)$ . By definition of  $G_{2N}'$ , the connected component  $(G_{2N}')_x$  consists of vertices of the form
$$
v (k) = j - \frac {1}{2 N} + (x \pm a k / (2 N) \mod 1 / (2 N)),
$$
for some  $k \in \mathbb{Z}$  and there is an edge from  $v(k)$  to  $v(k \pm 1)$ . Consider  $\varphi_1(x) = 2x \mod 1$  that sends  $v(k)$  to
$$
\varphi_ {1} (v (k)) := \left(2 j - \frac {1}{N} + \left(2 x \pm \frac {a k}{N} \mod \frac {1}{N}\right)\right) \mod 1.
$$
Now consider the connected component of  $\varphi_1(x) = 2x$  mod 1 in  $G_N'$ . Since  $x \in [j - \frac{1}{2N}, j)$ ,  $\varphi_1(x) \in [j' - 1/N, j')$  where  $[N]/N \ni j' := 2j \mod \frac{1}{N}$ . Thus, the connected component of  $(G_N')_{\varphi_1(x)}$  consists of vertices of the form:
$$
\begin{array}{l} v ^ {\prime} (k) := j ^ {\prime} - \frac {1}{N} + \left(\varphi_ {1} (x) \pm a k / N \mod 1 / N\right) (28) \\ = \left(2 j \mod \frac {1}{N}\right) - \frac {1}{N} + \left((2 x \mod 1) \pm \frac {a k}{N} \mod \frac {1}{N}\right). (29) \\ \end{array}
$$
With some modulo arithmetic manipulation, it is not hard to see that  $\varphi_1(v(k)) = v'(k)$  for all  $k \in \mathbb{Z}$ . By definition of  $G_N'$ , there is an edge  $(v'(k), v'(k \pm 1))$ . Therefore, there is an edge  $(\varphi_1(v(k)), \varphi_1(v(k \pm 1)))$ . If  $\lambda$  is the uniform measure then we can ignore vertices at the endpoints of our intervals (points in  $[N]/N$ ) and conclude that  $(G_{2N}')_x \equiv (G_N')_{\varphi_1(x)}$  with probability 1 when  $x \sim \lambda$ .
Now consider  $\varphi_2(x) = 2Nx$  mod 1 that sends  $v(k)$  to:
$$
\varphi_ {2} (v (k)) := (2 N j - 1 + (2 N x \pm a k \mod 1)) \mod 1. \tag {30}
$$
The connected component of  $\phi_2(x)$  in  $G$  is in  $[0,1]$  and consists of vertices of the form:
$$
v ^ {\prime \prime} (k) := (2 N j - 1 \mod 1) + (2 N x \pm a k \mod 1). \tag {31}
$$
Thus  $v''(k) = \varphi_2(v(k))$  for each  $k \in \mathbb{Z}$  and the definition of  $G$  implies that  $(G_{2N}')_x \equiv G_{\varphi_2(x)}$  since both are isomorphic to the two-way infinite path. If  $\lambda$  is the uniform measure then the above hold for a.e.  $x \in [0,1]$ .
$G_N'$  satisfies Assumption 3.A with resolution set  $d(N)$ . Let  $D$  be a divisor of  $N$ , then  $D\alpha = N$  for some  $\alpha \in \mathbb{N}$ . The intuition is rather straightforward, since  $D$  is a divisor of  $N$ , the partition into  $N$  equal intervals of  $[0,1]$  is simply a finer partition into  $D$  equal intervals. By construction, a vertex  $x$  of  $G_N'$  only has neighbors in the  $1/N$  interval containing it. Therefore, if  $f$  is constant on each of the  $D$  pieces, it is also constant on each of the  $N$  pieces and  $f(y) = f(x) = f(x') = f(y')$  for any  $(x,y),(x',y')$  neighbors such that  $x$  and  $x'$  come from the same  $1/D$  pieces.
Here is a more formal argument. Fix  $f \in \mathcal{F}_{[-1,1]}$  that is a.e. constant on each interval  $(u - 1 / d, u]$  for  $u \in [D] / D$ . We need to show that  $A(G_N')f$  is a.e. constant on the same pieces where  $A(H)$  is the adjacency operator of a regular graphing  $H$ . Fix  $d \in [D]$ , fix  $x, x' \in [d - 1 / D, d)$ . Let  $j$  be such that  $x \in [j - 1 / N, j)$  and  $j'$  be
such that  $x' \in [j' - 1/N, j')$ . Since  $D\alpha = N$ , we have  $[j - 1/N, j), [j' - 1/N, j') \subseteq [d - 1/D, d)$ . Furthermore, by definition of  $G_N'$ , all neighbors of  $x$  and  $x'$  are in  $[j - 1/N, j)$  and  $[j' - 1/N, j')$  respectively, and thus in  $[d - 1/D, d)$ . Therefore,  $A(G_N')f(x) = 2f(x) = 2f(x') = A(G_N')f(x')$  for all  $x, x' \in [d - 1/D, d)$ , which means that  $A(G_N')f$  is constant on each of the pieces  $[d - 1/D, d)$  for each  $d \in [D]$ .
2. Fix  $N \in \mathcal{N}$ , the goal is to define a graphing  $G_N''$  that is locally equivalent to  $G$  such that  $A(G_N'')$  satisfies Assumption 4.A with resolution set  $[N]$ .
Since  $\mathbb{Q}$  is dense in  $\mathbb{R}$ , there exists a number  $\delta(N) < \frac{1}{4N^2}$  such that  $\frac{1}{4N^2} + \delta(N)$  is an irrational number. Let  $G_N'' = ([0,1], \mathcal{F}, \lambda, E_N'')$  where,
$$
E _ {N} ^ {\prime \prime} := \left\{\left(x, x \pm \left(\frac {1}{4 N ^ {2}} + \delta (N)\right) \mod 1\right): x \in [ 0, 1 ] \right\}. \tag {32}
$$
That  $G_N''$  is locally equivalence to  $G$  is easily seen via the ambiguity of selecting  $a$  when defining  $G$ . Now we show that  $A(G_N'')$  satisfies Assumption 4.A with resolution set  $[N]$ . Pick  $M \leq N$  and  $f \in \mathcal{F}_{[-1,1]}$  that is a.e. constant on each piece  $(m - 1 / M, m]$  for each  $m \in [M] / M$ . Let
$$
E = \bigcup_ {m \in [ M ] / M} \left(m - \frac {1}{4 N ^ {2}} - \delta (N), m + \frac {1}{4 N ^ {2}} + \delta (N) \right]
$$
$$
\lambda (E) = \sum_ {m ^ {\prime} = 1} ^ {M} 2 (1 / (4 N ^ {2}) + \delta (N)) \leq \frac {M}{N ^ {2}} <   \frac {1}{N}.
$$
Pick  $x, x' \in (m - 1 / M, m] \backslash E$  then  $x, x' \in (m - 1 / M + \epsilon, m - \epsilon]$  where  $\epsilon = 1 / (4N^2) + \delta(N)$ . Thus we have both  $x \pm \epsilon$  and  $x' \pm \epsilon$  are in  $(m - 1 / M, m]$ . By definition of  $G_N''$ , all neighbors of  $x$  and  $x'$  are in the same  $1 / M$  piece as  $x$  and  $x'$ . Therefore, if  $f \in \mathcal{F}_{[-1,1]}$  are constant on these pieces, so is  $Af$ .
3. Fix  $N \in \mathcal{N}$ , the goal is to define a graphing  $G_N'''$  that is locally equivalent to  $G$  such that  $A(G_N'')$  satisfies Assumption 4.B with resolution set  $[N]$ .
Since  $\mathbb{Q}$  is dense in  $\mathbb{R}$ , there exists a number  $\delta(N) < 1/(4N)$  such that  $1/(4N) + \delta(N)$  is an irrational number. Let  $G_N''' = ([0,1], \mathcal{F}, \lambda, E_N'')$  where
$$
E _ {N} ^ {\prime \prime \prime} := \left\{\left(x, x \pm \left(\frac {1}{4 N} + \delta (N)\right) \mod 1\right): x \in [ 0, 1 ] \right\} \tag {33}
$$
That  $G_N'''$  is locally equivalent to  $G$  is easily seen via the ambiguity of selecting  $a$  when defining  $G$ . Now we show that  $A(G_N'')$  satisfies Assumption 4.B with resolution set  $[N]$ . Pick  $M \leq N$  and  $f \in \mathcal{F}_{\mathrm{reg}(C_v)}$  that is a.e.  $C_v$ -Lipschitz on  $[0,1]$ . Set  $\epsilon = \frac{1}{4N} + \delta(N)$  and let:
$$
E = [ 0, \epsilon) \cup [ 1 - \epsilon , 1). \tag {34}
$$
Then  $\lambda(E) = 2\epsilon < \frac{1}{N}$ . Pick  $x, x' \in [0,1] \backslash E$ , then  $x \pm \epsilon$  and  $x' \pm \epsilon$  do not 'loop over' in the interval  $[0,1]$  ( $y \mod 1 = y$ , for  $y \in \{x, x'\} + \{\pm \epsilon\}$ ). Thus we have:
$$
\begin{array}{l} \left| A f (x) - A f \left(x ^ {\prime}\right) \right| = \frac {\left| f (x + \epsilon) + f (x - \epsilon) - f \left(x ^ {\prime} + \epsilon\right) - f \left(x ^ {\prime} - \epsilon\right) \right|}{2} (35) \\ \leq \frac {\left| f (x + \epsilon) - f \left(x ^ {\prime} + \epsilon\right) \right| + \left| f (x - \epsilon) - f \left(x ^ {\prime} - \epsilon\right) \right|}{2} (36) \\ \leq C _ {v} | x - x ^ {\prime} |, (37) \\ \end{array}
$$
where in the last line we use Lipschitz property of  $f$ . This finishes the proof.
Lemma 6 (Well-behaved operators - General graphs). Let  $G$  be a (potentially countably infinite) graph with a coloring  $C: V(G) \to [N]$  for some  $N$  such that for each vertex  $u, v$  with the same color, the multiset of their neighbors' colors  $\{C(u') : (u', u) \in E\}$  are the same. Additionally, assume that the cardinality of vertices of each color is the same: that there is a bijection from  $\{v : C(v) = c\}$  to  $\{v : C(v) = c; \}$  for any colors  $c, c'$ . Then its adjacency operator satisfies Assumption 3.A with resolution  $N$ .