Extremal dimensions and critical isomorphism growth of metabelian three-Engel Lie algebras

We determine extremal dimensions and critical isomorphism growth of metabelian three-Engel Lie algebras in characteristic two and their binary quadratic orbit realizations. For class c at least five, the two binary minimum dimensions are 2^c + 2c - 6 and 2^c + 2c - 2. All equality objects and their coordinate transformations are classified. Increasing the first minimum by two binary dimensions changes a linearly growing type population into a doubly exponential one. Over finite characteristic-two fields of size at least four, a sharp square-root-of-two normalized-surplus threshold separates typical minimum-derived objects from the competing population. Macroscopic growth, typical dimension allocation and a discrete staircase refine this transition. The probability measure is uniform on unlabelled isomorphism types. This independently readable 80-page preprint develops the extremal and enumerative part of Paper G. Shared research origin: frozen Paper G, https://doi.org/10.5281/zenodo.23874173 . Independent manuscripts: Paper Q, https://doi.org/10.5281/zenodo.23874202 ; Paper E, https://doi.org/10.5281/zenodo.23874228 ; Paper C, https://doi.org/10.5281/zenodo.23874234 . Paper S continues independently, https://doi.org/10.5281/zenodo.23873752 . The deposit contains the PDF, editable LaTeX and necessary exact verification materials. Fixed GitHub release: https://github.com/63331849121/yijing-mathematical-structures/releases/tag/paperE-v2026.11.02 . Manuscript and original mathematical data: CC BY 4.0; original verification code: MIT. Bundled third-party assets retain their own licenses.