First-Gap Rigidity for Multiplicity-Two Subset Sums

Abstract

Revised manuscript dated 27 September 2026.For a nondecreasing labelled multiset B = (b_1, …, b_n) of positive integers with multiplicity at most two, write L = sum B and let FS(B) include the empty sum. Assume that FS(B) has an internal first gap g = min({1, …, L} minus FS(B)) ≥ 2. This preprint proves the sharp envelope L ≥ E_n = floor(n²/4) + 2n − 1 for n ≥ 6, with a unique equality profile for n ≥ 7. For every n ≥ 10, E_n ≤ L ≤ E_n + 8 implies FS(B) = ([0,L] intersect Z) minus {g,L−g}; the two displayed holes coalesce when 2g = L. The constant eight is best possible: at excess nine, (1,4,4,5,5,6,6,7,7,8) has four holes.Put μ_n = floor((n + 1)²/4) + 1 and J_n = μ_n + n − 1. For n ≥ 10, if |FS(B)| = J_n, then L ≥ E_n + 9, and the ambient integer interval [0,L] has at least floor(n/2) + 9 missing points. Its hole count is odd exactly when L is even and L/2 is missing, by complement symmetry. The proof combines a first-gap defect identity, an exact classification of the low split defects, a bounded-suffix lemma, an exhaustive finite core, and a symbolic endpoint transition.The first-gap envelope and stability theorem concern total sums. The consequence at the next signed cardinality is conditional and does not assert that such internal-first-gap profiles exist.Files include the revised manuscript PDF, LaTeX source, and a computation supplement with Python and C++20 source programs, saved verification outputs, a canonical profile stream, and a reproducibility script. The finite core covers 10 ≤ n ≤ 23 and excesses zero through eight: 176 cells and 2,337 profiles. An additional direct C++20 enumeration generates the bounded integer-multiset domain without using the split, low-defect gate, or saved catalog and reproduces the same 2,337 profile rows. Auxiliary checks are bounded tests; the unbounded conclusions rely on the written symbolic proofs.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.