Parity Energy and Sharp Relative-Scale Gaps for Sets with Few Subset Sums

Abstract

For a finite set A of positive integers, let FS(A) denote its subset sums, including the empty sum, and put T_n = n(n+1)/2. This preprint studies how parity and the common scale of the even elements force additional subset sums above the classical minimum T_n + 1.For a mixed-parity n-element set with k odd elements and r = n − k even elements, the paper proves the sharp stratum minimum |FS(A)| ≥ r(r+1) + k² + 1, attained by 2[1,r] ∪ {1,3,…,2k−1}. The proof uses an exact two-channel parity decomposition and a coupled critical-pair inequality.For a primitive mixed-parity set whose even part is 2gC with gcd(C) = 1, the exact minimum under g > 1 is T_n + 1 + Γ_n for every n ≥ 8, where Γ_n = ⌈(n²+n−14)/6⌉. The paper also determines the exact entrance into the unit-scale regime for every integer linear cutoff M(n) = sn + C within n ≥ 8. In particular, the cutoff T_n + 1 + 17n − 165 forces unit relative scale for every n ≥ 91, with sharpness witnessed by a scale-two extremizer at n = 90.This record contains the manuscript PDF and LaTeX source revised 27 September 2026 (v1.5), and the matching computational supplement for the finite cyclic lemma used in the odd-support minimum proof. The supplement includes three C++17 implementations, a Python reproduction script, parameterwise counts and minima in JSON format, and bounded arithmetic regression checks. The complete cyclic enumeration covers 85 parameter pairs: 4 ≤ r ≤ 13 and odd 2r + 1 ≤ h ≤ 4r − 1. Every r-subset of {1,3,…,h−2} is examined, including nongenerating sets. A Boolean-array implementation is separate from the two binary-rotation implementations; its minimizing witnesses are additionally checked by literal Python set arithmetic. The bounded checks supplement the mathematical proofs and do not establish the infinite claims.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.