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

Abstract

Version 2 extends the offset-two classification to the next consecutive signed subset-sum boundary. 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. This preprint gives 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, and 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 labeled positive multiset of multiplicity at most two. For the second cutoff, a finite relation-space analysis excludes noncommensurable inputs from size six onward, exact finite classifications settle sizes six through nine, and a terminal-increment descent closes every larger size. A sharp chain lower bound gives an exact predecessor identity, overlap forces the predecessor into the same primitive lattice, and only full-interval and endpoint-one supports remain. The finite classifications are reduced to explicit bounded domains; the all-size conclusions follow from the symbolic descent. The deposit includes the revised manuscript PDF, LaTeX source, and a combined computation supplement for both boundaries. The supplement contains Python and C++20 programs, finite data tables, deterministic outputs, and reproduction scripts. The next-boundary finite seed classification contains 323 primitive profiles and 2,043 signed lifts for 6 ≤ n ≤ 9. The programs verify the finite domains identified in the manuscript; the all-size results are proved symbolically.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.