Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects

A conjecture about the structure of polynomial neural networks has been disproven. Researchers identified counterexamples to the unimodal minimal filling architecture conjecture, which predicted that the hidden layers of such networks would always follow a specific width pattern [1]. The work, posted on the arXiv preprint server, focuses on polynomial neural networks (PNNs) with power activation functions. The conjecture in question held that for any minimal filling architecture—a network configuration that achieves a certain representational capacity with the fewest possible parameters—the widths of the hidden layers would be unimodal, meaning they would increase and then decrease only once [1]. The paper was submitted on 10 May 2026 and revised on 17 June 2026 [1]. Kevin Dao and colleagues found counterexamples using a combination of techniques, including a frontier search, recursive dimension bounds on neurovarieties, and symbolic computation [2]. The discovery challenges assumptions from prior literature, which had observed predominantly small-defect behavior in network architectures [2]. In contrast, several subarchitectures of the main counterexample presented in the paper exhibit a large defect, a measure of how far a network is from achieving its theoretical maximum expressivity [2]. The research was disseminated through arXiv, an open-access repository for electronic preprints in fields such as mathematics, physics, and computer science that has been operating since 1991 [6]. As of November 2024, the repository was receiving about 24,000 article submissions per month [6]. The platform also hosts experimental community tools through its arXivLabs framework, which provides features like bibliographic explorers and code finders on article pages [4][5]. These tools are developed by third-party collaborators who must adhere to arXiv's values of openness, community, excellence, and user data privacy [4]. The paper's abstract notes that the counterexamples were found by fixing the input and output widths and then searching for architectures that violate the unimodal prediction [2]. The initial version of the manuscript was 28 KB, and the revised version grew to 48 KB [1]. The findings add a new layer of complexity to the theoretical understanding of how the internal structure of polynomial neural networks relates to their function.