Book 5: The Developmental Metric and the Cone

Book 5 develops the continuous core of developmental geometry from first principles. Starting from three axioms—stability, coherence curvature, and the minimal path principle—the book derives the balance law, developmental density, geometric time, finite propagation, mass divergence, the developmental metric, curvature, geodesics, the developmental cone, and the axis field. The theory is presented as a closed variational system: the balance law arises as an Euler–Lagrange equation, geometric time normalizes developmental motion, finite propagation imposes a universal speed bound, and mass divergence forces maximal‑velocity paths to converge to geodesics of the developmental metric. The axis field is characterized as a fixed point of the axis‑extraction operator, establishing global self‑consistency of the developmental geometry. The final chapter outlines structural correspondences with the discrete developmental program developed in companion works. This volume provides the complete continuous foundation for the developmental geometry series.