Signed Sets with Few Subset Sums at the Offset-Two and Next Boundaries: Complete All-Size Classifications

Abstract

Revised manuscript dated 27 September 2026.For a finite set A of nonzero real numbers, let FS(A) denote its subset sums, including the empty sum, and put μ_n = floor((n + 1)²/4) + 1, K_n = μ_n + n − 2, and J_n = μ_n + n − 1. We give complete classifications at the two consecutive cutoffs K_n and J_n. At K_n, for every n ≥ 2, the classification consists of an ordinary common-lattice partition family and one balanced U–U family, together with explicit exceptional descriptions for n = 2, 3, 4. At J_n, for every n ≥ 6, the ordinary family gains one unit of partition weight, the U–U family permits the next sign imbalance, and a new oriented U–V family appears. All cardinalities are exact.The proof transfers a signed set to a labelled positive multiset of multiplicity at most two. Complete relation-space calculations settle commensurability at the two initial sizes, five and six. Cofactor bounds give finite integer seeds, and a terminal-increment descent proves the classifications for all larger sizes. Support overlap preserves the primitive lattice. A classification of the minimum terminal increment and an exact endpoint transition leave only full intervals and the two holes at distance one from their endpoints. Two exact implementations reproduce the finite lists; the unbounded conclusions use the written induction.The deposit includes the revised manuscript PDF, LaTeX source, and computation supplement for both boundaries. The supplement contains Python and C++20 implementations, finite reference data, a replay script, and a file manifest. It reconstructs both classifications through size twelve. The required finite seeds are the five-label K boundary and six-label J boundary; larger sizes follow from the manuscript’s uniform induction. The finite tables at K for sizes five through nine contain 218 primitive multisets and 1,191 signed lifts; at J for sizes six through nine they contain 323 multisets and 2,043 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.