A Pairwise-Even Structured Heuristic for Fast Approximation of Unique Subset Sums

This paper introduces a novel heuristic-analytic method for estimating the number of unique subset sums in finite integer sets. Unlike classical approaches such as the Erdős–Turán method, which rely on probabilistic bounds but are computationally intensive, our method leverages a structured set design focused on even positive integers, optionally supplemented by a few small negative numbers. This "pairwise-even" structure reduces symmetry and overlap among subset sums, dramatically increasing the likelihood of uniqueness. The method includes algorithmic steps for constructing the set, generating subset sums, and estimating uniqueness either via direct computation or probabilistic sampling. Explicit examples demonstrate that the proposed approach consistently achieves higher accuracy than the classical Erdős–Turán estimates, particularly as the set size grows. Empirical results show that approximation errors diminish with increasing n, often reaching negligible levels. While not a rigorous theorem, the approach offers practical advantages in speed, simplicity, and accuracy for structured sets. Potential extensions include mixed parity sets, higher-dimensional vector subset sums, and connections to entropy-based analysis. This work provides a complementary tool to classical additive number theory methods, suitable for rapid estimation in combinatorial problems, simulations, and research exploration.