Open collaboration is increasingly used to advance scientific discovery, yet sustaining participation remains difficult when leadership is informal and problems are open-ended. We study projects from Polymath—a large-scale open collaborative initiative in which mathematicians jointly work on unsolved mathematical problems under conditions of extreme epistemic and social uncertainty. Drawing on public online discussion data from four Polymath projects and using computerized text analysis, we examine how leaders’ emotional expressions facilitate continued participation. We find that leaders’ emotional expressions are associated with higher crowd participation. This effect does not happen through emotional contagion but through cognitive inference. Participants interpret leaders’ emotions as informational cues that help them contribute despite radical uncertainty. The relationship weakens when leaders’ emotions are framed as questions rather than affirmative statements, suggesting that clarity in emotional signaling matters for sustaining participation. These findings show how emergent epistemic leadership operates as leaders shape collective inquiry not through formal coordination but by signaling meaning and direction through emotions.

While much innovation work is focused on pragmatic product concepts that solve a problem, at times the goal is instead to set out a vision for possible products of the future, such as with “concept cars” in automotive firms or the work of corporate innovation labs. Visionary product concepts—conveyed by verbal descriptions and visual prototypes—provide ideas from which subsequent innovations will be developed. The paradoxical tensions for these concepts are that they need to be open to incomplete convergence to a large set of ideas while still being coherent, and they must integrate both novelty and feasibility. Our exploratory research questions were: What practices support the generation of visionary product concepts? What are the relationships among the ideas that result? We studied ten teams developing visionary product concepts and examined the links they made between ideas, discovering differences in network structure among them. Effective visionary concepts relied on three practices that together shifted from initial ideas, tightly linked novel ideas, and selectively integrated ideas for coherence. We contribute an understanding of visionary product concepts generation practices that help to integrate paradoxical tensions between openness and coherence, and novelty and feasibility.