Sharp ideal growth of metabelian restricted Lie algebras

We determine sharp ideal-growth terms and typical quotient structures for metabelian restricted Lie algebras over finite fields. For fixed generator rank d at least three, the base-q logarithm of the number of codimension-n restricted ideals is n^2/4 + gamma_d n^(2-1/d) + O(n^(2-2/d)), with an explicit coefficient. Rank two has the sharper expansion n^2/4 + n^(3/2)/2 - n log_p(n)/2 + O(n). These coarse formulas also hold for p-nilpotent quotients and restricted isomorphism types. An exact outer-compatibility mixture yields typical quotient layers and classical Hall partition laws under uniform ideal sampling. We prove a canonical central torus convolution and, for the two-generator free source, the whole-quotient nilpotence proportion q^(-n/2 + O(sqrt(n))). Homogeneous metabelian sources have corrections determined by Krull dimension and Hilbert multiplicity; graph commutation relations encode vertex connectivity and weighted minimum cuts. This independent 34-page preprint includes complete editable LaTeX and exact verification programs. It is a companion to Paper G, which develops the quadratic orbit equation and its Lie structure; the present work independently develops unrestricted-height ideal growth and probability proofs. Fixed GitHub release: https://github.com/63331849121/yijing-mathematical-structures/releases/tag/paperS-v2026.11.01 . Manuscript and original mathematical data: CC BY 4.0; original verification code: MIT.