Inverse structure of short zero sums in C₃³

Which length-24 positional sequences of nonzero elements of C₃³ lie just below the known short-zero-sum threshold 25? The classification target requires every pair of nonempty zero-sum subsequences of length at most three to have zero intersection sum.

Why it matters and what counts as progress

Why it matters: The classification describes every extremal sequence immediately below the known short-zero-sum threshold, rather than only constructing examples.

What counts as progress: Classify all extremal sequences, including the proposed form of three copies of eight distinct elements with no short nonempty zero sum.

How the problem developed

2012 · Background result [1]
Yushuang Fan, Weidong Gao, Guoqing Wang, Qinghai Zhong and Jujuan Zhuang Published work on short zero-sum subsequences supplied background on the forward extremal problem; it did not state the later positional inverse classification recorded here.
9 Oct 2026 · Formal result [2]
John Fairfax-Ball Registered and Lean-verified an exact positional classification for length-24 sequences of nonzero elements of C₃³ under the stated pairwise short zero-sum intersection condition.

AI contribution

AI-assisted inverse theorem for short zero sums in C₃³

For a length-24 positional sequence of nonzero elements of C₃³, every pair of nonempty zero-sum subsequences of length at most three has zero intersection sum exactly when its positions can be permuted into three copies of eight distinct elements with no short nonempty zero sum.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4/Mathlib kernel, NanoDa and con-ron)

Human understandingFew