General Formulation of Topos Many-Node Theory

TL;DR

This paper develops a comprehensive mathematical framework for modeling the early universe's created entities as a topos many-node system, introducing Hamiltonians and linking the theory to cosmic background data.

Contribution

It presents a novel general formulation of many-node theory with structural and nonstructural Hamiltonians, and proposes a method to connect the model with cosmic data.

Findings

Identifies an optimized critical dimension with characteristic spectra.

Defines a procedure for state-space and truth value determination.

Proposes a dynamic map between successive graphs.

Abstract

We consider the created entities (events) in the first moments of universe creation. It is assumed that there exists a causal energetic relationship between all events (nodes) such that all nodes are placed on a world line and each node occupies a region (instead of a point) in space-time, called locale, in mathematical terms. The set of locale nodes form a topos many-node system. Using some basic assumptions, we introduce two kinds of Hamiltonians. By attributing a general structural Hamiltonian to the system, it is shown that the system has an optimized critical dimension with a probable Raman and infrared spectrums. Also, we consider a general nonstructural Hamiltonian which includes a set of commutative self-adjoint operators and an interaction terms due to the spin, charge, or other kinds of probable degrees of freedoms for each $n^{th}$-optimized graph. For finding the state-space, truth values and quantity valued objects of the many-node system, a general procedure is introduced. The set of these values is a classical snapshot of the $n^{th}$-optimized graph which forms its kinematic. We show that the dynamic of the system can be explained by defining a combined map between the $n^{th}$- state-space belongs to the $n^{th}$-graph and the $({n+1)}^{th}$-state-space belong to $({n+1)}^{th}$-graph. Finally, by providing an interpretation of the general formulation of many-node theory, we discuss and explain how one can use the data of the cosmic background radiations and cosmic rays for finding a detailed model of both general structural and nonstructural introduced Hamiltonian. Here, time is no more than the change in truth value during comparison between $n^{th}$ and $({n+1)}^{th}$-graph.