Quadratic maps satisfying f ∘ f = f + id: polar Lie structure and sharp dimension bounds

The equation f composed with f = f + id couples three-cycles to binary vector addition: every nonzero orbit consists of the three nonzero points of a binary plane. This preprint studies its solutions of binary algebraic degree at most two through their polar Lie algebras. The functional equation forces the Jacobi identity and an additional degree-four identity; every solvable polar algebra is nilpotent. In the metabelian case, a complete normal form gives an equivalence, including isomorphisms and automorphism groups, with metabelian F4-Lie algebras whose quotient actions on the derived ideal have square zero. The minimum binary dimensions are 6, 8 and 12 in classes 2, 3 and 4, and 2^c + 2c - 6 for every class c >= 5. The paper characterizes when the original polar bracket admits an F4-bilinear structure extending its intrinsic scalar actions. In this compatible subclass, the exact minimum is 2^c + 2c - 2 for every c >= 2, with at least c - 2 inequivalent extremizers for every c >= 4. The four additional binary dimensions for c >= 5 have a structural explanation: one additional F4-dimension in the abelianization and one in the derived ideal. Two closed classifications make the theory explicit. All eight-dimensional solutions form three binary-linear conjugacy classes. The solvable solutions with binary abelianization dimension four form an exhaustive chain of seven classes in dimensions 4, 6, 8, 10, 12, 14 and 16. The final two members are nonmetabelian; exact central-extension spaces and projective actions close the chain in every ambient dimension. The deposit supplies the 23-page manuscript, editable LaTeX, seven exact input certificates and three reproducible verification programs. The programs reconstruct all finite rank equations, check algebraic interfaces, and connect the displayed coordinate models and projective actions to the certificates. The all-class structural results and dimension bounds have written proofs in the manuscript. Fixed GitHub release: https://github.com/63331849121/yijing-mathematical-structures/releases/tag/paperG-v2026.10.30 . Manuscript and original mathematical data use CC BY 4.0; original verification programs use MIT. The source archive contains the complete project and verification materials. Earlier papers A and F remain separate records.