Topological Signatures of Imperial Collapse and Fragmentation: Administrative Dissolution, Territorial Reorganization and Early-Warning Observables in the Han Dynasty Network (206~BCE\,--\,220~CE)

We extend the persistent homology formalism done in references~\cite{paper1,paper2} to the Han Dynasty ($206\mathrm{B}\mathrm{C}\mathrm{E}$–$220\mathrm{C}\mathrm{E}$), testing whether the collapse threshold $\text{Hstar}=0.5241$ and the three-network decomposition methodology established for the Roman–Byzantine case generalize to a mechanistically distinct imperial system. Three complementary networks are constructed from geospatial and historical data: an administrative network ($\text{Hadm}$, 392~prefectures from CHGIS~v6), a geographic network ($\text{Hgeo}$, with node end-dates extended beyond administrative dissolution), and an 81-node Silk Road trade network ($\text{Htrd}$) as a structural control. Edge costs are derived via least-cost path analysis on a $1\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$ digital elevation model, modulated by five historical friction channels. $\text{Hadm}$ collapses to zero at 220~CE (phase~G slope $-0.0261{\mathrm{y}\mathrm{r}}^{-1}$, $R^2=0.689$) while $\text{Hgeo}$ simultaneously \emph{increases} to $3.065$, producing a divergence $\mathrm{\Delta }H=3.065$ – the topological signature of \emph{internal fragmentation}. The cross-network Wasserstein distance $W_{\times }(\mathrm{A}\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{G}\mathrm{e}\mathrm{o})$ jumps from exactly zero to $737$ at $d=190$~CE, one decade after the Yellow Turban Rebellion, and three long-lived ${\beta }_1$ cycles matching the Wei, Shu, and Wu domains emerge under the post-184~CE fragmentation model – thirty years before formal partition. Three independent early-warning indicators (Wasserstein velocity ${\dot{W}}_2$, correlation length $\xi$, and the Integrated Change Tracker) signal collapse onset 45–50~years before formal dissolution.