5th generation computing systems require a fundamentally new approach to the problems of both their micro and macroarchitecture, and operating systems and flexible automatic production (GAP) of their . In the preface to the Russian edition of the book: "The computer of the 3rd generation" Moscow, 1984 . , Academician K. P. Velikhov notes that the main distinguishing feature of the computer-5 is usually considered: "specific, extremely dense packaging of SBIS elements". This, in particular, requires "the creation of a fundamentally new architecture of the computers themselves".. The same applies, of course, to computers built on the basis of integral optics.. In this book (the fourth in the series [73, 74, 19-19]) the mathematical questions of the architecture of such systems are investigated. At the atom, the extreme transitions along the "dense packaging" required the introduction of new concepts of generalized convergence and generalized function. a new concept of integration and introduction of unusual scalar works depending on the nature of the optimization task and semi-modules which are 'linear/ equations of Hamilton-Jacobi, Bellman and other optimization equations. It turned out that the most important limiting tasks are reduced to the Pontryagin maximum principle .. On a number of real problems of architecture, parallel programs and GAO EMI-5, we tried to show for practitioners the effectiveness of this new approach.