Sharp Hirsch-length bounds for identity supports in nilpotent groups

For a group G, let κ(G) be the supremum of the sizes of finite inclusion-minimal subsets whose positive semigroups contain the identity. For finitely generated nilpotent groups of Hirsch length h, we prove the optimal bound κ(G) ≤ F(h), where F(h) = max{h + 1, floor(4h/3)}. Equality is attained at every Hirsch length by products of integral three-dimensional Heisenberg groups and free abelian groups. If G is torsion-free of class at least three, the sharp bound improves to F(h − 1). At Hirsch lengths 3k and 3k + 1, with k ≥ 2, we classify the groups attaining F(h) by their rational Malcev Lie algebras: they are precisely the corresponding products of three-dimensional Heisenberg algebras and an abelian summand of dimension at most one. The proofs select a controlled number of original letter logarithms while preserving their entire derived Lie algebra. We also compute exact values for mixed Heisenberg, free class-two and complete-bipartite families, and prove eventual affine behavior under adjoining free central factors. For arbitrary locally nilpotent groups, classical rank theory and saturation give equivalence between finiteness of κ and a uniform bound on the Hirsch lengths of finitely generated subgroups; we obtain the same optimal upper estimate in terms of their supremum. https://github.com/eulogica/sharp-hirsch-length-bounds-nilpotent-groups AI use: ChatGPT by OpenAI, OpenAI Codex, and Claude by Anthropic were used during the preparation and revision of this manuscript for mathematical exploration, proof development, source retrieval, editorial and structural revision, and to assist in checking proofs, notation, terminology, internal consistency, cross-references, and bibliographic citations.