Scanning the structure of ill‐known spaces: Part 1. Founding principles about mathematical constitution of space

Abstract

Abstract. Some necessary and sufficient conditions allowing a previously unknown space to be explored through scanning operators are reexamined with respect to measure theory. Some generalized conceptions of distances and dimensionality evaluation are proposed, together with their conditions of validity and range of application to topological spaces. The existence of a Boolean lattice with fractal properties originating from nonwellfounded properties of the empty set is demonstrated. This lattice provides a substrate with both discrete and continuous properties, from which existence of physical universes can be proved, up to the function of conscious perception. Spacetime emerges as an ordered sequence of mappings of closed 3-D Ponicaré sections of a topological 4-space provided by the lattice, and the function of conscious perception is founded on the same properties. Self-evaluation of a system is possible against indecidability barriers through anticipatory mental imaging occurring in biological brain systems; then our embedding universe should be in principle accessible to knowledge. The possibility of existence of spaces with fuzzy dimension or with adjoined parts with decreasing dimensions is raised, together with possible tools for their study.The work presented here provides the introductory foundations supporting a new theory of space whose physical predictions (suppressing the opposition of quantum and relativistic approaches) and experimental proofs are presented in details in Parts 2 and 3 of the study.

The Light-Dark Dual Universe for the Big Bang and Dark Energy

“…The second step introduces kinetic energy (detachment space) in the slicing of particles for the reduction of the mass dimension number. Bounias and Krasnoholovets [16] propose another explanation of the reduction of >4 D space-time into 4D space-time by slicing >4D space-time into infinitely many 4D quantized units surrounding the 4D core particle. Such slicing of >4D space-time is like slicing 3-space D object into 2-space D object in the way stated by Michel Bounias as follows: "You cannot put a pot into a sheet without changing the shape of the 2-D sheet into a 3-D dimensional packet.…”

Variation in Mass of Entities in Condensed Media

“…its fractal deformed cell, is treated as a canonical particle. Subatomic mechanics, or submicroscopic mechanics of the motion of a particle in the tessel-lattice developed in a series of works (see the review paper by Krasnoholovets, 2003a) has a direct comparison with conventional quantum mechanics, which, as is well known, operates in an abstract phase space. Submicroscopic mechanics is developed in real space and represents an inner sub structure of the ψ -wave function giving expressions that interlink the particle's de Broglie wavelength λ with the range of space Λ covered by the spatial excitations that accompany the moving particle; in particular, the amplitude of inerton cloud is tied with the de Broglie wavelength by means of relationship…”

An extended interpretation of the thermodynamic theory, including an additional energy associated with a decrease in mass

“…Very recently a rigorous mathematical theory of the real physical space has been developed by Bounias and Krasnoholovets [12][13][14]. The theory shows that the real space represents a mathematical lattice, called the tessellattice, packing with elementary cells whose size can be estimated as the Planck one, ∼ 10 −35 m. The tessellattice represents a degenerate space-time, i.e.…”