Hierarchical clustering of integers under the p-adic norm |x - y|_p = p^-v_p(x-y)
The $p$-adic numbers $\mathbb{Q}_p$ redefine arithmetic distance using the $p$-adic norm $|x|_p = p^{-v_p(x)}$, where $v_p(x)$ is the highest power of $p$ dividing $x$. Under this valuation, two integers are topologically close if their difference is divisible by a high power of $p$, yielding a non-Archimedean ultrametric space.
Ultrametric spaces satisfy the strong triangle inequality $|x - z|_p \le \max(|x - y|_p, |y - z|_p)$, forcing every triangle to be isosceles with a short base. This property organizes integers into a fractal hierarchy of disjoint, compact valuation balls $\mathbb{Z}_p$, where every point inside a ball functions as its center.
This visualization renders the radial branching of $\mathbb{Z}_p$ rings up to depth $k$. In the Trinary Framework, composite spokes map directly to closed sub-balls within $\mathbb{Z}_3$ and $\mathbb{Z}_5$, leaving coprime channels as open boundary paths across the $p$-adic boundary.