The abstract ways you can add up to a number actually form a 'landscape' that acts like a physical object melting or freezing.
March 25, 2026
Original Paper
Simplex Stratification and Phase Boundaries in the Partition Graph
arXiv · 2603.23228
The Takeaway
Integer partitions (like how 4 can be 2+2 or 3+1) are usually seen as simple lists of combinations. However, when these combinations are mapped as a network, they reveal "phase boundaries" and distinct layers, suggesting that the world of pure numbers behaves like physical matter changing states from liquid to solid.
From the abstract
We study the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. We introduce the simplex stratification of $G_n$: for each vertex $\lambda$, let $\dim_{\mathrm{loc}}(\lambda)$ denote the largest dimension of a simplex of the clique complex $K_n = \mathrm{Cl}(G_n)$ containing $\lambda$. This defines a decomposition of $V(G_n)$ into layers $L_r(n)=\{\lambda\in V(G_n): \dim_{\mathrm{loc}}(\lambda)=r\}