[Paper Review] The Universe from Nothing: A Mathematical Lattice of Empty Sets

M. Bounias, Volodymyr Krasnoholovets|ArXiv.org|Sep 25, 2003

Cold Fusion and Nuclear Reactions12 references17 citations

TL;DR

This paper proposes a mathematical framework where space-time emerges from a Boolean lattice of empty sets with fractal, non-well-founded properties. Using topological mappings of 3-D Poincaré sections in a 4D space-time, it derives quantum and gravitational phenomena from deformations of primary empty cells, showing that mass, spin, and charge arise from fractal morphologies and volume compensation in neighboring cells, with quanta of distance and fractality explicitly demonstrated.

ABSTRACT

In this work, major principles of the mathematical constitution of space and the principles of construction of the physical space are presented. 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 non-well-founded properties of the empty set is demonstrated. Space-time emerges as an ordered sequence of mappings of closed 3-D Poincare sections of a topological 4-space-time provided by the lattice of primary empty cells. The fractal kernel stands for a particle and the reduction of its volume is compensated by morphic changes of a finite number of surrounding cells. Quanta of distances and quanta of fractality are demonstrated. It is shown that the families of fractal deformations give rise to families of particle-like structures. Deformation attributes associated to mass determine the inert mass and the gravitational effects, as has previously been shown, but fractal deformations of cells are responsible for the other fundamental characteristics, namely: spin, charges, and electric and magnetic properties.

Motivation & Objective

• To establish a mathematical foundation for space and physical structure based on the empty set's non-well-founded properties.
• To derive physical space-time and particle characteristics from a topological lattice of primary empty cells without relying on conventional physical postulates.
• To unify quantum and relativistic phenomena by showing that mass, spin, and charge emerge from fractal deformations and volume compensation in a discrete-continuous lattice.
• To challenge the Big Bang cosmology by proposing an eternal, substrate-based universe arising from a self-consistent mathematical lattice.

Proposed method

• Constructing a Boolean lattice from the empty set, leveraging its non-well-founded nature to generate fractal properties.
• Defining space-time as an ordered sequence of 3-D Poincaré sections mapped from a topological 4-space-time lattice of primary empty cells.
• Using a generalized distance function and set-distance continuity to ensure topological consistency across lattice mappings.
• Modeling particles as fractal deformations of primary cells, where volume reduction is compensated by morphic changes in neighboring cells.
• Applying a combination rule for oriented sequences to generate continuous, space-time-like structures favoring aggregation into massive objects.
• Deriving quanta of distance and fractality through discrete mappings and topological invariance under homeomorphism-preserving transformations.

Experimental results

Research questions

• RQ1 Can a mathematical lattice of empty sets generate a physical space-time structure with both discrete and continuous properties?
• RQ2 How do fundamental particle properties like mass, spin, and charge emerge from topological deformations of primary cells?
• RQ3 What is the role of fractal deformations in generating quantum and gravitational effects within a topological lattice framework?
• RQ4 How does the volume reduction of a particle-like deformation get compensated, and what are the implications for inertial and gravitational mass?
• RQ5 Can the Big Bang cosmology be replaced by a model of an eternal, substrate-based universe emerging from a lattice of empty sets?

Key findings

• The empty set generates a Boolean lattice with intrinsic fractal properties due to its non-well-founded nature, forming a foundational substrate for space.
• Space-time emerges as an ordered sequence of 3-D Poincaré sections mapped from a topological 4-space-time, with continuity ensured by a set-distance function.
• Quanta of distance and quanta of fractality are explicitly derived from discrete mappings and topological invariance in the lattice.
• Mass arises from volume reduction in a central cell, compensated by morphic changes in a finite number of surrounding cells, preserving homeomorphism.
• Spin and charge are attributed to fractal deformations of cells, with spin-like behavior described by the expected moment of junction of components.
• The model predicts that a particle's influence extends to the lattice's periphery due to inerton clouds, implying non-local effects without violating locality in the lattice structure.