Subset-Sum Growth under Linear Concentration Constraints

This preprint studies the minimum number of distinct subset sums of finite subsets of real vector spaces subject to bounds on the number of points in proper linear subspaces through the origin. A support-sensitive progression reduction gives sharp asymptotic formulas throughout the admissible concentration ranges in dimensions two, three and four, including the three-dimensional general-position question of Carpenter, Defant and Kravitz. In every fixed dimension, the paper determines the concentration boundary near the full set size. Above a dimension-dependent fixed cap threshold, it obtains matching bounds through a finite envelope of progression ranks and describes the transition points. It also identifies limitations of this envelope at small fixed caps in higher dimensions. The methods combine reserved subsets, lattice-zonotope rounding, total sensitivity, progression comparison and finite-field constructions. The deposit includes the manuscript PDF, LaTeX source and a computational supplement with a standard-library Python verification program and reference JSON. Exact finite calculations cover modular curves, boundary constructions, independent subfamilies, rank envelopes, mixed-radix injectivity, sign folding, progression contraction and reserve correction. These finite checks complement the general proofs in the manuscript.