# WP35 — Measurement report (Phase 3)

All numbers come from results/empirical/measurement_diagnostics.json, measurement_diagnostics_modern.json, measurement_irt_item_parameters.csv, power_h2a_*.csv, produced by python -m wp35.measurement.build_indices (2026-11-01). No outcome variable was read by that module. Definitions follow research/preregistration.md §3.

# 1. Seshat (polity level)

# 1.1 Coordination demand D

Four logged scale variables (polity population, territory, largest-settlement population; settlement hierarchy levels unlogged). First principal component of their pairwise-complete correlation matrix explains 82.1% of variance with near-equal loadings (log pop 0.26, log territory 0.24, log capital 0.26, settlement hierarchy 0.24). Score = loading- weighted mean of available standardized variables, requiring ≥ 2 of 4; 442 polities have D_pc1; 299 have population alone (log_pop). Range bounds (value_from/value_to) propagate to D_pc1_lo/hi: the median half-width is 0.017 SD, the 90th percentile 0.093 SD, so range uncertainty in the scale variables is small relative to between-polity variation (the larger uncertainty is which polities have any estimate at all).

# 1.2 Coordination capacity K

Four blocks of presence indicators, each the share coded present among coded items (≥ 50% coded required): administration (5 items incl. rank-scaled administrative levels), standardized rules and money (10), communication infrastructure (7), record-keeping (9). K_mean = mean of blocks with ≥ 3 blocks: 344 polities. Inter-block correlations 0.75–0.79; first PC of the blocks explains 82.4% (loadings 0.25 each), and K_pc1 correlates 0.999 with K_mean. Cronbach α of the four blocks by macro-region ranges 0.83 (Europe) to 0.97 (South Asia); mean off-diagonal block correlation by era 0.67 (1001–1500) to 0.81 (pre-1000 BCE) — the four capacity blocks hang together in every region and era. Latent alternative: 2PL IRT (MAP, log-normal prior on discrimination, EM converged in 135 iterations) on the 30 binary items (disputed 0.5 → missing; administrative levels excluded) gives K_irt for 579 polities (≥ 3 items). It correlates 0.96 with K_mean. Item difficulties order sensibly (tokens, non-written records, roads, couriers easy; paper currency, general postal service, examination system, merit promotion hard; article is nearly universal and uninformative). Five record-keeping items (written record, script, lists/tables, calendar, practical literature) hit the discrimination ceiling (8): they are coded almost jointly in Seshat, i.e. locally dependent; the latent score is therefore dominated by the "writing cluster", which is why we keep the transparent block mean as primary. Posterior SD of K_irt: median 0.34, 90th percentile 0.61 (on a unit-variance scale) — individual-polity capacity scores carry substantial uncertainty; this is carried into the econometrics through the IRT variant and the bound runs. Cross-region invariance: refitting the IRT separately on the Old World (n = 532) and on the Americas + Southeast Asia/Oceania (n = 94) gives item difficulties correlated 0.88 across the two fits (mean absolute difference 1.16 logits, inflated by boundary estimates on the small New-World sample). Comparability is acceptable for the ordering of items but the New-World sample is too small for a strict invariance test; leave-region-out results (H5) are the operational safeguard. Leave-one-block-out: dropping any block leaves K correlated ≥ 0.98 with the full index and P correlated ≥ 0.95 with P_main.

# 1.3 Coordination pressure P and the collinearity problem

P_main = z(D_pc1) − z(K_mean) is defined for 317 polities and, in pre-crisis form, for 127 of the 166 CrisisDB records. As anticipated from Turchin et al. (2018) and Shin et al. (2020), scale and capacity are strongly collinear: corr(D_pc1, K_mean) = 0.84 (0.80 with K_irt). Consequently P_main retains only 16.6% of the variance of z(D)+z(K); it correlates 0.39 with D and −0.18 with K, and −0.04 with the Turchin-style single complexity axis (complexity_pc1, first PC of 57 social-complexity indicators, 49.7% of variance). The mismatch index is thus nearly orthogonal to overall complexity, which is exactly what makes the H2 test informative if P carries signal, and what makes it fragile: P is a residual, so measurement error in either D or K leaks into it. The pre-registered baselines (D-only, K-only, complexity-PC1-only) address this directly.

# 1.4 A-priori power (pre-registration §5.8; random outcomes, actual P distribution, n = 127)

# 2. Modern panel

K_modern = mean of six standardized V-Dem items (≥ 4 coded): rigorous/impartial administration, meritocratic appointment, transparent laws, fiscal capacity source, territorial control, EMB capacity; PC1 explains 60.5% with loadings 0.13–0.19. D_modern = log Maddison population (1820+); P_modern defined for 15,697 country-years. corr(D, K) = 0.06 (no collinearity problem in the modern panel). Caveat fixed before analysis: corr(K_modern, v2x_polyarchy) = 0.75 — the V-Dem capacity items are strongly associated with electoral democracy even though no democracy index enters K; all modern specifications therefore control for polyarchy as pre-registered, and the K-only baseline is reported.

# 3. Ethnographic Atlas

log_community_size from EA031 category midpoints (611 societies), log_population from continuous EA202 (953), jurisdictional_hierarchy_beyond = EA033 − 1 (0–4 levels; 1,155). No outcome; used for H1 only, with Glottolog family (173 families) and region controls.

# 4. What the measurement model cannot do

• Seshat K items are presence codes: they measure existence, not reach or quality, of institutions.
• Most facts are undated, so "pre-crisis" K is mostly whole-polity K (see pre-registration §3.6).
• No direct indicator of division of labour or dispersion in Polaris; D is scale only.
• IRT local dependence among writing items; invariance only coarsely checked.