One of the modern areas of blockchain technology application is the Internet of Things (IoT). An important component of blockchain technology is the consensus layer. It includes consensus protocols that are used to establish and maintain consensus, as well as to ensure network security, accuracy, and protection of the registry from unauthorized access. Currently, there are a large number of different consensus protocols, including those for blockchain-based IoT networks. Therefore, choosing the most suitable consensus protocol for a specific distributed ledger system, in particular, for an IoT blockchain solution, is an important task. The problem of optimal blockchain consensus mechanism selection in IoT networks can be considered a multi-criteria decision-making problem. This paper presents the step-by-step development of a conceptual model of a system of optimal consensus protocol selection for blockchain-based IoT networks. Following this step-by-step approach, the final goal is to transform the conceptual framework into a practical, adaptive, and efficient decision-making system for blockchain-based IoT networks. The obtained results can be useful for developers and researchers working in the field of blockchain technology and the Internet of Things and contribute to improving the efficiency and security of IoT networks.

Built-in attention is often presented as a free explanation of what a forecasting transformer relies on, but the claim is rarely tested against the model's behaviour. We test it on MeteoFormer, a patch-based transformer with single-head variable and temporal attention that forecasts temperature, relative humidity, pressure and wind speed 24 h ahead from 96 h of hourly input on the Jena climate and Beijing PRSA datasets. Against ten baselines trained under one protocol with five seeds, MeteoFormer has the lowest mean absolute error on Jena (3.025 against 3.107 for the runner-up Crossformer, Diebold-Mariano p < 0.05 against every baseline after Holm correction) and the most accurate temperature forecast on both datasets, while on Beijing it is level with Crossformer in absolute error (4.151 against 4.138, p = 0.72) and Crossformer has the lower squared error, a difference that we trace to the humidity channel. The explanation claim fails. On equal budgets across all eleven models, the fidelity of the built-in variable attention does not differ from zero (0.012 on Jena, 0.046 on Beijing) and is far below post-hoc SHAP on the same model, the temporal attention is almost uniform although occlusion shows that the forecast depends mainly on the last 16 h, and the configuration selected on validation for accuracy has the least faithful attention (0.012 against 0.181 with instance normalization on Jena, p = 0.001).

Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and estimation of the number of primes on the twin primes rows are obtained. A new method of twin prime infinity is proposed.