Because the universal cover of a connected 5-regular graph is the infinite regular tree $T_5$, one mathematical completion is the boundary at infinity $\partial T_5$ equipped with a visual metric.

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MIC-T02 — Boundary Visual Metric Theorem

Let $T_5$ be the infinite 5-regular tree rooted at $o$. The boundary $\partial T_5$ is the space of infinite self-avoiding rays starting at $o$. For visual parameter $\alpha >0$, the visual metric is

$$d_{\alpha }(\xi ,\eta )=e^{-\alpha |\xi \wedge \eta |},$$

where $|\xi \wedge \eta |$ is the length of the common initial segment of rays $\xi$ and $\eta$.

Properties of $(\partial T_5,d_{\alpha })$

1. Compactness: $(\partial T_5,d_{\alpha })$ is a compact, totally disconnected, perfect metric space (a Cantor set).
2. Hausdorff Dimension:

$${\dim }_H(\partial T_5,d_{\alpha })=\frac{\ln 4}{\alpha }.$$

3. Self-Similarity Tuning: Any finite dimension $D>0$ can be realized by selecting visual scaling parameter $\alpha$:

$$\alpha =\frac{\ln 4}{D}.$$

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The Physical Obstruction for Pure Tree Boundaries

$$\boxed{{\displaystyle \text{The boundary }\partial T_5\text{ is not the vertex set of any graph in the refinement sequence.}}}$$

1. The vertices of $T_5$ (the "bulk") are entirely absent from $\partial T_5$.
2. The nearest-neighbor graph Laplacian on $T_5$ does not induce the standard Dirichlet Laplacian on $\partial T_5$.
3. Pure boundary points represent asymptotic directions at infinity, not local spatial coordinates.

Therefore, boundary completion is mathematically sound as an abstract metric space, but physically distinct from a refinement that retains bulk local interactions.