In this paper, we propose a mechanism for speeding up networked group expertise (e-expertise) and gaining a collective insight, by structuring the information used in the decision-making process in a special convergent way. We have previously proposed the convergent method for structuring information, which is based on the fundamental principles of control thermodynamics, inverse problems, holistic discourse, cognitive modeling, quantum semantics, and artificial intelligence. This paper presents a further development, which considers breakthrough thinking processes and introduces cognitive programming approaches. We use mappings of unformalized cognitive concepts and their connections into mathematical constructions or sets of objects (documents, messages, comments, etc.) as semantic interpretations of the cognitive model. We applied special structural conditions to the set of semantic interpretations. Our techniques can help groups of experts and decision-makers to create simultaneously non-causal and good decisions. The methods have been implemented as software, the e-expertise system, and have been tested on decision-making processes in e-government conditions.