Stochastic partial differential equations in 4 dimensions (e.g., fluid flow) pose closely related problems, connected through analytic continuation to imaginary time. The techniques developed for an affirmative answer of the existence problem for a Yang-Mills quantum field theory (in the sense required by the Clay Millennium problem) would open a door to a constructive mathematical approach to many other 4-dimensional relativistic quantum field theories, including those describing realistic elementary particle physics. The Yang-Mills quantum field theories are believed to be the most benign ones to be constructed, but the current functional analytic tools were so far not sufficient to produce the bounds needed to show the existence of the appropriate limits. The mathematically rigorous construction of interacting 4-dimensional relativistic quantum field theories is one of the hardest problems of mathematical physics, unsolved since over 50 years, in spite of the continuous efforts of many excellent people. Physical questions are phrased as well-defined mathematical problems, to be attacked with methods from differential geometry, functional analysis, Lie groups, topology, etc. Mathematical physics is the study of physical questions from the point of view of full mathematical rigor. While there probably aren't many direct mathematical consequences to the existence of a mass gap, the techniques involved would almost assuredly be applicable towards the large number of QFT inspired results in mathematics. Solving the mass gap problem in Yang-Mills would represent the successful rigorous existence of a very non-trivial quantum field theory and the demonstration of a very nontrivial result about that field theory (that hasn't even been adequately demonstrated using physical techniques). However, while physicists may trust the manipulations they do in QFT, and the results of those manipulations have been spectacularly successful, for almost every interesting quantum field theory, there isn't even a rigorous definition or existence proof, much less a justification behind the manipulations that led to the invention of, for example, Seiberg-Witten invariants. Though the problem is officially expressed in a conventional symbolic mathematical language, the appropriate solution necessarily has an unconventional superconceptual mathematical language, just as its radical non-spatial computational physics does.There is a long, long list of mathematical subjects that were either pioneered or significantly inspired by results in quantum field theory. The new physics required to solve the problem has essentially âan idealistic frameworkâ and the new mathematics contains terms such as ânon-spatial consciousnessâ. Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.' If the property of mass gap contradicts the special relativistic law that no massive entity can travel at the speed of light, the point of this work is to understand that special relativity is not fundamental to nature. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the mass gap: the quantum particles have positive masses, even though the classical waves travel at the speed of light. As stated by the institute, 'Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. This work attempts to present the NSTP (Non - Spatial Thinking Process) theoretical (philosophy of mind) idealistic solution to the problem of Yang-Mills existence and mass gap, the millennium problem announced by the Clay Mathematics Institute.
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