Central-depth stability and optimal central realizations of three-Engel Lie algebras

We study how the central depth of the second derived ideal controls nilpotency class in ordinary three-Engel Lie algebras over characteristic-two fields with more than two elements. We determine the exact centre-by-metabelian class law max(5, d + 2), complete top central covers and rank-uniform stable ranges through sixth central depth. Fifth-depth stability starts optimally at rank nine; sixth-depth stability holds from rank ten. The rank-seven and rank-eight boundaries carry distinct cubic representations, including a nonsplit Frobenius structure. The least center dimension realizing the prescribed free metabelian quotient is ceil((d + 1)/2), attained by a three-Engel construction. Symbolic propagation is separated from exact finite starting certificates. This independently readable 28-page preprint develops the ordinary central-depth 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/paperC-v2026.11.02 . Manuscript and original mathematical data: CC BY 4.0; original verification code: MIT. Bundled third-party assets retain their own licenses.