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arXiv · 2608.12310

A Multiplicative Fourier Proof of the Length-Four Index Conjecture

Hongjian Li·Pingzhi Yuan·Shijie Yuan·Weilin Zhang
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Abstract

Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157\varphi(n)/1440<\varphi(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.

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Hongjian Li, Pingzhi Yuan, Shijie Yuan, Weilin Zhang. 2026-08-12. A Multiplicative Fourier Proof of the Length-Four Index Conjecture. https://arxiv.org/abs/2608.12310

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