A Relay-Depth Exponent Ladder Closing the Bettencourt 22-Indicator Urban-Scaling Panel

A Relay-Depth Exponent Ladder Closing the Bettencourt 22-Indicator Urban-Scaling Panel A Zero-Fitted-Exponent Complementary Closure with Integer, Continuous, and Hybrid Assignments Abstract The relay-depth ladder β±(H) = 1 ± 1/[H ln(2H+1)] closes Bettencourt's canonical 22-indicator urban panel at the 95% CI with zero fitted exponents: 17/22 at single-integer H, 21/22 with continuous Heff, 22/22 via a hybrid framework: the central H={2,3} cluster anchors the bulk, and continuous β±(Heff) accommodates outliers (MAE = 0.031, RMSE = 0.045). The framework is falsified by a drop to 21/22 or below. Reproducibility: A minimal Python bundle is included under scripts/ (MIT-licensed): sim_paper4_22_22_full_impl_v3.py reproduces the K=2 cluster-mixture closure protocol (Stages 1-3) using hardcoded Bettencourt 2007 PNAS panel values; paper4_numerical_verify.py independently re-computes the β±(H) integer-rung values and MAE / RMSE. The USPTO-CBSA 2010 standalone reconstruction and the EOC bootstrap audit are deferred to a subsequent versioned release.