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The subject of this paper is to analyse the Mathematical Principia of Economic 3D Black Holes in Roegenian economics. In detail, we study two main problems: (i) mathematical origin of economic 3D black holes; and (ii) entropy and internal political stability depending on national income and the total investment, for economic Reissner–Nordström (RN) 3D black hole. To solve these problems, it was necessary to jump from macroeconomic side to microeconomic side (a substantial approach as they are so different), to complete the thermodynamics–economics dictionary with new entities, and to introduce the flow between two macroeconomic systems. The main contribution is about introducing and studying the Schwarzschild-type metric on an economic 4D system, together with Rindler coordinates, Einstein 4D partial differential equations (PDEs), and economic RN 3D black holes. In addition, we introduce some economic Ricci type flows or waves, for further research.

The paper addresses the problem of fingerprints of fragmentation processes, showing that structuring by fragmentation can be detected and investigated fruitfully if approached with a proper methodology, of which fractal instruments represent an important part. Studied aspects include fragments size distributions, fragments size-position relations, and long-range correlations in fracture profiles and fracture patterns. The choice of experiments (comminution of flat samples and fragmentation by desiccation cracking) was directed by the aim to complement existing studies from the point of view of fragmentation energy application, being also intended to provide data from processes where different mechanisms are at work. Power-law fragments size distributions were found, but also fragments clusters on dominant size intervals that point towards a fractal character of the size distribution from the point of view of the positions of the distribution maxima. The self-affine character of fractures and of fracture patterns could be emphasised on certain scale intervals, separated by thresholds that are important for studies concerning the implied mechanisms. Fragments size-position correlations indicate a high probability of neighbouring fragments to be of comparable size. The features highlighted for structuring by fragmentation were found in all the studied experiments, their generality pointing towards useful implications for geoscientific research.

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