Structural Divergence of the Inca Imperial Network, 1400--1572~CE: Persistent Homology, Substitution Topology, and the Topological Signature of Decapitation Collapse

Abstract

We apply persistent homology to a six-layer temporal network model of the Inca Empire (Tawantinsuyu), built from the Qhapaq \~{N}an road system ($6{,}546$ segments; $12{,}031$~km): 4,492 nodes (4,452 active; 40 isolated removed), $4{,}358$ edges, $57$ temporal snapshots (1400--1572~CE). The complete flag complex yields $\beta_1 = 32$ cycles ($26$ macro-continental, $6$ meso-regional) and $\beta_0 = 264$ components at the finest filtration, evaluated across six layers: infrastructure, Inca political control, Spanish alliance growth, Andean tribute commerce, and Spanish legal/effective administration. Five results. (i) Infrastructure entropy $H_{\mathrm{infra}}=1.486\pm0.0001$ stays constant throughout, indicating no detectable topological disruption of the Qhapaq \~{N}an at this resolution. (ii) Inca political entropy $H_{\mathrm{inca}}$ is stable through the Cajamarca capture (1532~CE), then collapses to zero by 1537~CE; the dominant break is at Vilcabamba ($F=120.7$), not Cajamarca ($F=11.5$). (iii) Four of six layers reach the critical regime at 1532~CE, while infrastructure stays marginally below threshold ($\mathrm{ICT}=0.495$); trade and political reach the highest values ($\mathrm{ICT}=0.976$, $0.968$), marking Cajamarca as a synchronised multi-layer phase transition. (iv) Spanish legal administration converges to the infrastructure topology by 1572~CE ($W_1=5\times10^{-6}\approx0$): Substitution Topology, infrastructure-preserving collapse via topology transfer. (v) Legal--effective divergence decays from $W_1=0.591$ (1532) to $0.002$ (1572), quantifying resolution of colonial legal fiction. Together, these results support a distinct collapse mode---Substitution Topology---differing from erosion-dominated Roman/Byzantine collapse, Han fragmentation, and Aztec military destruction.