Quadratic orbit maps and three-Engel Lie algebras in characteristic two

This 73-page version studies finite binary quadratic maps satisfying f composed with f = f + id. Explicit changes of operations identify all quadratic solutions with ordinary F4 three-Engel Lie algebras satisfying R = 0, retaining every linear intertwiner. The coordinate conjugacy is quadratic in both directions; the full ordinary three-Engel class has cubic encoding with third difference R. New results give the exact class- and head-rank-preserving central repair cost dim N(H), uniqueness of the minimum extension isomorphism class over a fixed bottom, half-weight constraints, and the sharp first-ascent cost over specified compact bottoms S_r: D2 = 5 and D_r = binom(r,2)+3 for r >= 3, allowing arbitrary-length central towers. These are relative costs, not global m(5). Full F4 certificates cover the rank-four exception, and an exterior-module fiber-product construction attains the bound at all larger ranks. Authors of version 2026-11-02.2: Wencong Huang, Baohua Li. Earlier versions retain their original authorship. This deposit contains the PDF, editable LaTeX and exact verification materials. Third-round class-six results remain separate local research and are not part of the article or this deposit. C and E are background comparisons; no new companion theorem is a proof input. Frozen source collection: https://doi.org/10.5281/zenodo.23874173. Earlier 65-page Q: https://doi.org/10.5281/zenodo.23874202. Fixed GitHub release: https://github.com/63331849121/yijing-mathematical-structures/releases/tag/paperQ-v2026.11.02.2. Manuscript and mathematical data: CC BY 4.0; original verification code: MIT. Third-party assets retain their own licenses.