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

Abstract

We extend the persistent homology formalism done in references~\cite{paper1,paper2} to the Han Dynasty ($206~\mathrm{BCE}$--$220~\mathrm{CE}$), testing whether the collapse threshold $\Hstar = 0.5241$ 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 ($\Hadm$, 392~prefectures from CHGIS~v6), a geographic network ($\Hgeo$, with node end-dates extended beyond administrative dissolution), and an 81-node Silk Road trade network ($\Htrd$) as a structural control. Edge costs are derived via least-cost path analysis on a $1~\mathrm{km/pixel}$ digital elevation model, modulated by five historical friction channels. $\Hadm$ collapses to zero at 220~CE (phase~G slope $-0.0261~\mathrm{yr}^{-1}$, $R^2 = 0.689$) while $\Hgeo$ simultaneously \emph{increases} to $3.065$, producing a divergence $ΔH = 3.065$ -- the topological signature of \emph{internal fragmentation}. The cross-network Wasserstein distance $W_\times(\mathrm{Admin},\mathrm{Geo})$ jumps from exactly zero to $737$ at $d=190$~CE, one decade after the Yellow Turban Rebellion, and three long-lived $β_1$ 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 $ξ$, and the Integrated Change Tracker) signal collapse onset 45--50~years before formal dissolution.