1D: PC1 of log population, log territory, log largest settlement, settlement hierarchy & variance share 0.821; loadings log\_pop 0.26, log\_terr 0.24, log\_cap 0.26, settl\_hier 0.24; $n=442$ \\2K: mean of four blocks (admin, rules/money, communication, records) & inter-block corr. 0.75--0.79; PC1 share 0.824; $n=344$ \\3K (2PL IRT, 30 items, MAP) & corr. with block mean 0.963; posterior SD median 0.34; $n=579$ \\4corr$(z(D), z(K))$ & 0.838 (IRT: 0.800) \\5P = $z(D)-z(K)$ & variance share of $z(D)+z(K)$ retained 0.166; corr. with D 0.39, with K -0.18, with complexity PC1 -0.04; $n=317$ polities, 127 crisis records \\6IRT invariance (Old World vs Americas/Oceania fits) & difficulty correlation 0.88 over 30 items \\7Modern K (6 V-Dem items) & PC1 share 0.605; corr$(D,K)$ 0.062; corr$(K,\text{polyarchy})$ 0.754; $n=15,697$ \\8