Have quadruple and quintuple helices emerged? : metaphors, project titles, and empirical research

In the original Triple Helix model (Etzkowitz and Leydesdorff 1995), the three "helices" of Universities, Industry and Government and the (3-way) interactions among and between them were proposed as a basis for looking at how entrepreneurship comes into being. Since the original article, other "helix-based" models have been proposed. Some vary the triple and others propose higher numbers of helix - with 4 and 5as the most typical extensions. In the article "Triple, Quadruple, and Higher-Order Helices: Historical phenomena and (neo-)evolutionary models," Leydesdorff and Lawton Smith (2022) developed an Information-Theory based approach to look more formally at the need for higher-order helices. In the case of two helices, processes of mutual shaping can generate historical trajectories that could have been otherwise: other options providing possible states, which have not yet historically been realized (Petersen et al., 2016). Adding a third helix makes a substantive change from an information point of view: a Triple Helix model is not just the sum of three sets of 2-way interactions. However, once this number has been reached further additions can be decomposed into sets of triads (Batagelj et al., 2014; Simmel, 1902). This leads to the suggestion that higher-dimensional helix structures potentially add little to discourse. Four (sets of) authors were asked by the Editors of the Triple Helix to respond to Leydesdorff and Lawton Smith (2022). Many insightful and interesting points were raised including convenience, presentation and the need to allow a more unified theory model. These are discussed in this article's "responses to just criticism" (Shostakovich, 1937). It continues the debate on triple and higher-order helices by summarising and responding to the points made by those commentators. What may be the status of triple and/or higher-order helices?