A set of 12 numbers is not determined by its set of 4-sums

TL;DR

This paper shows that a set of 12 integers is not uniquely determined by its 4-sums, providing counterexamples and correcting a long-standing mathematical misconception from 50 years ago.

Contribution

It presents explicit counterexamples of two different 12-integer sets sharing the same 4-sums, refuting a previous proof claiming uniqueness.

Findings

Two distinct 12-integer sets with identical 4-sums

Refutation of a 50-year-old proof of uniqueness

Correction of an incorrect calculation in prior work

Abstract

We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect calculation in it.