Signed Sets with Few Subset Sums at the Offset-Two Boundary: A Complete All-Size Classification

Abstract

This preprint classifies finite sets A of nonzero real numbers with n elements whose set of subset sums FS(A), including the empty sum, satisfies |FS(A)| ≤ μ_n + n − 2, where μ_n = floor((n + 1)²/4) + 1, for every n ≥ 2. For n ≥ 5, the classification consists of ordinary signed integer templates on a common positive scale, governed by a defect combining sign imbalance and partition weights, together with a balanced exceptional family formed from the intervals U_s = {2, 3, …, s + 1}. The cases n = 2, 3, and 4 are described by explicit continuous and finite exceptional families. The proof combines a stable theorem valid for n ≥ max{M + 2, 10} with an exact signed-to-positive-multiset transfer and finite classifications using exact integer and rational arithmetic. The deposit includes the manuscript PDF, its LaTeX source, and a computational supplement containing C++20 and Python programs, finite data tables, a README, and a Bash reproduction script. The supplement covers stable boundary calculations, noncommensurable relation spaces, six- and seven-magnitude classifications, and the primitive multiplicity-two classification and signed lifts. The reported finite classification contains 218 primitive profiles and 1,191 signed lifts. Research methodology and AI assistance: This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology combining persistent research archives, literature and prior-art investigation, iterative proof exploration, and exact verification procedures. Generative AI was used for mathematical exploration, computational reasoning, literature research, proof development, and manuscript preparation.