User:Virtumanity/Unified field theory

In physics, a unified field theory (UFT) is a type of field theory that allows all that is usually thought of as fundamental forces and elementary particles to be written in terms of a pair of physical and virtual fields. According to the modern discoveries in physics, forces are not transmitted directly between interacting objects, but instead are described and interrupted by intermediary entities called fields. Classically, however, a duality of the fields is combined into a single physical field. For over a century, unified field theory remains an open line of research and the term was coined by Albert Einstein, who attempted to unify his general theory of relativity with electromagnetism. The "Theory of Everything" and Grand Unified Theory are closely related to unified field theory, but differ by not requiring the basis of nature to be fields, and often by attempting to explain physical constants of nature. Earlier attempts based on classical physics are described in the article on classical unified field theories.

Recently, with the discovery of the dark energy, the most accepted hypothesis to explain the entanglement is that there exists a pair of the fields - one for a physical particle while the other for its associated virtual dark energy. The concept of a duality of the two-sidedness lies at the heart of all fields as they are interrelate, opposite or contrary to one another, each dissolving into the other in alternating streams that operates a life of creation, generation, or actions complementarily, reciprocally and interdependently. More precisely, an object possesses a pair of the fields and requires a duality of manifolds for their living entanglement. Therefore, an interruption between two objects involves two pairs of the fields, which constitute cross-entangling simultaneously and reciprocally.

The goal of a unified field theory has led to a great deal of progress for future theoretical physics and continues to progress prominently. Although UFT may concern all types of the forces, it has to reveal the natural principles in connection with Quantum field theory, Quantum Chromodynamics, Gravitational Wave, General Relativity , General Symmetric Fields, and General Asymmetric Fields of Ontology and Cosmology.

Classic Theory
The first successful classical unified field theory was developed by James Clerk Maxwell. In 1820 Hans Christian Ørsted discovered that electric currents exerted forces on magnets, while in 1831, Michael Faraday made the observation that time-varying magnetic fields could induce electric currents. Until then, electricity and magnetism had been thought of as unrelated phenomena. In 1864, Maxwell published his famous paper on a dynamical theory of the electromagnetic field. This was the first example of a theory that was able to encompass previously separate field theories (namely electricity and magnetism) to provide a unifying theory of electromagnetism. By 1905, Albert Einstein had used the constancy of the speed of light in Maxwell's theory to unify our notions of space and time into an entity we now call spacetime and in 1915 he expanded this theory of special relativity to a description of gravity, general relativity, using a field to describe the curving geometry of four-dimensional spacetime.

In the years following the creation of the general theory, a large number of physicists and mathematicians enthusiastically participated in the attempt to unify the then-known fundamental interactions. In view of later developments in this domain, of particular interest are the theories of Hermann Weyl of 1919, who introduced the concept of an (electromagnetic) gauge field in a classical field theory and, two years later, that of Theodor Kaluza, who extended General Relativity to five dimensions. Continuing in this latter direction, Oscar Klein proposed in 1926 that the fourth spatial dimension be curled up into a small, unobserved circle. In Kaluza–Klein theory, the gravitational curvature of the extra spatial direction behaves as an additional force similar to electromagnetism. These and other models of electromagnetism and gravity were pursued by Albert Einstein in his attempts at a classical unified field theory. By 1930 Einstein had already considered the Einstein–Maxwell–Dirac System [Dongen]. This system is (heuristically) the super-classical [Varadarajan] limit of (the not mathematically well-defined) quantum electrodynamics. One can extend this system to include the weak and strong nuclear forces to get the Einstein–Yang–Mills–Dirac System. The French physicist Marie-Antoinette Tonnelat published a paper in the early 1940s on the standard commutation relations for the quantized spin-2 field. She continued this work in collaboration with Erwin Schrödinger after World War II. In the 1960s Mendel Sachs proposed a generally covariant field theory that did not require recourse to renormalisation or perturbation theory. In 1965, Tonnelat published a book on the state of research on unified field theories.

Modern Progress
In 1963 American physicist Sheldon Glashow proposed that the weak nuclear force, electricity and magnetism could arise from a partially unified electroweak theory. In 1967, Pakistani Abdus Salam and American Steven Weinberg independently revised Glashow's theory by having the masses for the W particle and Z particle arise through spontaneous symmetry breaking with the Higgs mechanism. This unified theory modeled the electroweak interaction as a force mediated by four particles: the photon for the electromagnetic aspect, and a neutral Z particle and two charged W particles for weak aspect. As a result of the spontaneous symmetry breaking, the weak force becomes short-range and the W and Z bosons acquire masses of 80.4 and $91.2 GeV/c^{2}$, respectively. Their theory was first given experimental support by the discovery of weak neutral currents in 1973. In 1983, the Z and W bosons were first produced at CERN by Carlo Rubbia's team. For their insights, Glashow, Salam, and Weinberg were awarded the Nobel Prize in Physics in 1979. Carlo Rubbia and Simon van der Meer received the Prize in 1984.

After Gerardus 't Hooft showed the Glashow–Weinberg–Salam electroweak interactions to be mathematically consistent, the electroweak theory became a template for further attempts at unifying forces. In 1974, Sheldon Glashow and Howard Georgi proposed unifying the strong and electroweak interactions into the Georgi–Glashow model, the first Grand Unified Theory, which would have observable effects for energies much above 100 GeV. Since then there have been several proposals for Grand Unified Theories, e.g. the Pati–Salam model, although none is currently universally accepted. A major problem for experimental tests of such theories is the energy scale involved, which is well beyond the reach of current accelerators. Grand Unified Theories make predictions for the relative strengths of the strong, weak, and electromagnetic forces, and in 1991 LEP determined that supersymmetric theories have the correct ratio of couplings for a Georgi–Glashow Grand Unified Theory. Many Grand Unified Theories (but not Pati–Salam) predict that the proton can decay, and if this were to be seen, details of the decay products could give hints at more aspects of the Grand Unified Theory. It is at present unknown if the proton can decay, although experiments have determined a lower bound of 1035 years for its lifetime.

Current Status
Theoretical physicists have not yet formulated a widely accepted, consistent theory that combines general relativity and quantum mechanics. The incompatibility of the two theories remains an outstanding problem in the field of physics.

Some theoretical physicists currently believe that a quantum theory of general relativity may require frameworks other than field theory itself, such as string theory or loop quantum gravity. Some models in string theory that are promising by way of realizing our familiar standard model are the perturbative heterotic string models, 11-dimensional M-theory, Singular geometries (e.g. orbifold and orientifold), D-branes and other branes,  flux compactification and warped geometry, and non-perturbative type IIB superstring solutions (F-theory). Theory of everything has not included gravity, simply because trying to combine the graviton with the strong and electroweak interactions runs into fundamental difficulties and the resulting theory is not renormalizable.

Recently, scientists have discovered that all physical matters are under the life entanglement between a physical matter and its associated dark energy. This behavior requires the dual potentials reciprocally, consistently, and simultaneously. In physics, this means that a duality of potentials constitutes and composites a pair of fields, one for its physical matter and the other for its associated dark energy. To translate it into a mathematical model, it must be modeled by or consisted of a duality of the manifolds. Therefore, instead of a single spacetime manifold for example, Universal Topology as one of the emerging theories has demonstrated amazingly the unification of all physical models, which lies at the heart of natural philosophy: a duality of virtual and physical worlds.

Virtual and Physical Duality
The terminology of Space and Time has been in currency since the inception of physics. Throughout the first generation of physics, space and time are individual parameters that have no interwoven relationship. From Euclidean space to Newtonian mechanics, the scientific approach known as classical physics seeks to discover a set of physical laws that mathematically describe the motion of bodies under the influence of a system of forces. In classical physics, it is reasonable to interpret a space as consisting of three dimensions, and time as a separate dimension. This regime has presented us with a basic conception for the Physical Existence of space and the Virtual Existence of time, although the virtual reality is hardly studied and their relationship remains unexpressed.

As the second generation, modern physics couples the virtual existence of time with the physical existence of space into a single interwoven continuum, known as Spacetime. By combining space and time into a manifold called Minkowski space, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supra-galactic and subatomic levels. By revealing their interwoven inferences for the events of a hierarchical universe, the manifold continuum presents us with the enhanced logic for a complex vector of the Physical dimension of space $${\bf r}=\{ x,y,z \}$$ and the Virtual dimension of time $$=ic\{t,\dots \}$$, where the constant c is the speed of light, and i marks the virtual or imaginary in mathematics. Although the virtual and physical interwoven relationship for dynamics is only limited to physical existence $$\{{\bf r}-{\bf k}\}$$ with spacetime curvature, many of the greatest minds of the twentieth century, successfully intuited the well-known theory of relativity, quantum state, fields and energy without using the interwoven continuum.

Today, with the acceptance of quantum mechanics and dark energy, contemporary physics has reached consensus on the possibility of a virtual existence beyond physical reality. When Heisenberg's uncertainty principle delimited the duality region of non-physical essence, Niels Bohr emphatically declared that “everything we call real is made of things that cannot be regarded as real.” Since 2016, the new concepts of the dual manifolds have been developed for the philosophical dialectics:
 * 1) Dark energy is entangling and associated with any physical objects, although it may or may not permeate all of space
 * 2) A matter possesses not only a physical field, but also with its companion partner - a virtual field in forms of dark energy
 * 3) An object consists of a pair of fields, each of which can be modeled by its own manifold reciprocally and simultaneously
 * 4) Because of the dual fields, the living entanglement is consistently dissolving into each other's in the alternating streams
 * 5) Interruption forces between two objects are actions of two pairs of the fields, operating on the alternating entanglement

Everywhere our world shines with a beautiful nature in duality. In every fraction of every creature, we shall find the principles and laws of physics, biology, metaphysics, information technology, and all other sciences. Nature is systematically composed of building blocks, dualities, which take on an abstract form as simple as Virtual and Physical existence.

Complex Manifolds
Worlds in universe constitutes a topological pair of manifolds, each of which represents unique dimensions, transforms objects within their neighborhood or subsets, and orchestrates events across multiple spaces. As a duality of the universal topology, these manifolds are central to many parts of worlds, allowing sophisticated structures, evolving into natural events, determining systematic solution sets, and carrying out natural laws and principles. Any objects in the universe has a sequence of events corresponding to the historical or future points of worlds appearing as a type of curve in the global manifest. Defined by global parameters of the world of universe, a universe line is curved out in a continuous and smooth coordinate system representing events as a collection of points. Each point has multidimensional surfaces, called World Plane, with analogue associations among the worlds. In our universe, scopes and boundaries among each world are composed of, but not limited to, the homeomorphic duality: virtual and physical worlds. A universe event can be graphically visualized as, for example, the lines in two-dimensions of the virtual and physical coordinates.

Because each manifold has unique representations, worlds do not exactly coincide and require transportations to pass from one to the other through commonly shared natural foundations. Therefore, our universe manifests as an associative framework of objects, crossing neighboring worlds of manifolds, illustrated as the three dimensions as the mutually orthogonal units: a coordinate manifold of physical world, $$P\left({\bf r},\lambda \right)$$, a coordinate manifold of virtual world, $$V\left({\bf k},\lambda \right)$$, and a coordinate manifold of global function, $$G({\bf r},\lambda)$$, of Word Events, $$\lambda$$, shown in figure 1.1, where $$P\left({\bf r},\lambda \right)$$ is parameterized by the coordinates of spatial vector $${\bf r}$$ and $$V\left({\bf k},\lambda \right)$$ is parameterized by the coordinates of timestate vector $${\bf k}$$, respectively


 * $$(2.1)\qquad {\bf r}={\bf r}(x_1,x_2,x_3)$$
 * $$(2.2)\qquad {\bf k}={\bf k}(y_1,y_2,\cdots)$$

The global functions in $$G({\bf r},\lambda)$$ axis is a collection of common objects and states of events $$\lambda$$, with unique functions applicable to both the virtual and physical spaces of the world W. In other words, a universe manifold is visualized as a transitional region among the associated manifolds of the worlds, which globally forms the topological hierarchy of a universe. A curve in this three-dimensional manifold $$\{ {\bf r},{\bf k}, G(\lambda)\}$$ is called a Universe Line, corresponding to intersection of world planes from virtual and physical regimes, defined as Yin ($$Y^-$$) or Yang ($$Y^+$$) manifolds. A yin $$Y^-$$ manifold is a physical supremacy of spacetime domain dependent on its virtual opponent space for resources, creations, generations, and annihilations, while a yang $$Y^+$$ manifold is a virtual supremacy of timespace domain dependent on its physical opponent space for animations, reproductions, generations, and actions. Although a yin manifold may resemble each point within its scope, it has to be globally charted by means of metric projections into its universe manifold. Likewise, a virtual world has its own yang manifold projected into the universe. Together, they form a topological infrastructure that maps events onto a global manifold of the universe.

In complex analysis, events of world planes are holomorphic functions, representing a duality of complex-conjugate functions $$W^\pm$$ of one or more complex variable set $${\check x}\{x_1,x_2,x_3\}$$ and $${\hat x}\{x^0,\cdots\}$$ in neighborhood spaces of every point in its universe regime of an open set $$\mho$$.
 * $$(2.3)\qquad

{W}^{+}({\hat x},\lambda)= P({\bf r},\lambda) {-} iV({\bf k},\lambda) \qquad : W^+ \in {Y^+}, \ {\hat x \subset \mho} $$


 * $$(2.4)\qquad

{W^-}({\hat x},\lambda)= P({\bf r},\lambda) {+} iV({\bf k},\lambda) \qquad : W^- \in {Y^-}, \ {\check x \subset \mho} $$

These two formulae are called the YinYang ($$Y^-Y^+$$) Topology of Universe. Composed into a $$Y^-$$ component, the world $$W^-$$ is in the manifold of yin supremacy which dominants the processes of reproductions or animations. Likewise, composed into a $$Y^+$$ component, the world $$W^-$$ is in the manifold of yang supremacy which dominants the processes of creations or annihilations. Therefore, both virtual and physical spaces are confined completely by the $$Y^-Y^+$$ worlds of $$W^\mp$$. A physical world functions itself under the yin manifold $$Y^-(\check x)$$ of animation or reproduction with $$Y^-$$ principles. It may convert some of the variables of virtual world as various characteristics. Vise versa, a virtual world operates as the yang manifold $$Y^+(\hat x)$$ of generation or annihilation with $$Y^+$$ principle. It may revert some of physical characteristics back to various variables. Characteristics can be expressed mathematically as special constants. Some examples are illustrated in following sections in mathematical formations of time and space fields.

A Duality of Fields
Governed by a global event $$ \lambda $$ under the universal topology, an operational environment is initiated by the scalar fields of a rank-0 tensor, a differentiable function of a complex variable in its domain at its zero derivative


 * $$(2.5)\qquad \phi(\lambda) \in \{\phi^+(\hat x,\lambda), \phi^-(\check x,\lambda)\} \qquad
 * \phi^+(\hat x,\lambda) \subset Y^+, \ \phi^-(\check x,\lambda) \subset Y^- $$

where the $$ Y^+ $$ or $$ Y^- $$ represents a Virtual or Physical manifold, respectively. A scalar function is characterized as a single magnitude with variable components of the respective coordinate sets


 * $$(2.6)\qquad \hat x\{x^0,x^1,\cdots\} \qquad \hat x \in Y^+ $$
 * $$(2.7)\qquad \check x\{x_1,x_2,x_3\} \qquad \check x \in Y^- $$

Because a field is incepted or operated under either virtual or physical primacy of an $$ Y^+ $$ or $$ Y^- $$ manifold respectively and simultaneously, each point of the fields is entangled with and appears as a conjugate function of the scalar field $$ \phi^- $$ or $$ \phi^+ $$ in its opponent manifold. A field can be classified as a scalar field, a vector field, or a tensor field according to whether the represented physical horizon is at a scope of scalar, vector, or tensor potentials, respectively.

Therefore, at the scalar potentials, the effects are stationary projected to and communicated from their reciprocal opponent, shown as the following conjugate pairs:


 * $$(2.8)\qquad \phi^+(\hat x\,,\lambda) \,, \varphi^-(\check x\,,\lambda) \qquad $$ : $$ \varphi^-(\check x\,,\lambda) \mapsto \phi^+(\hat x\,,\lambda)^* \,,\hat x \in Y^+ $$
 * $$(2.9)\qquad \phi^-(\check x\,,\lambda) \,, \varphi^+(\hat x\,,\lambda) \qquad $$ : $$ \varphi^+(\hat x\,,\lambda) \mapsto \phi^-(\check x\,,\lambda)^* \,,\check x \in Y^- $$

where * denotes a complex conjugate. A conjugate field $$ \phi^- = (\varphi^+)^* $$ of the $$ Y^+ $$ scalar potential is mapped to a field in the $$ Y^- $$ manifold, and vise versa that a conjugate field $$ \phi^+ = (\varphi^-)^* $$ of the $$ Y^- $$ scalar potential is mapped to a field in the $$ Y^+ $$ manifold. In mathematics, if f(z) is a holomorphic function restricted to the Real Numbers, it has the complex conjugate properties of f(z)=f*(z*), which leads to the above equations when $$ \hat{x}^*=\check{x}$$ is satisfied.

Fundamental Forces
All four of the known fundamental forces are mediated by fields, which in the Standard Model of particle physics result from exchange of gauge bosons. Specifically the four fundamental interactions to be unified are:
 * Strong interaction: the interaction responsible for holding quarks together to form hadrons, and holding neutrons and also protons together to form atomic nuclei. The exchange particle that mediates this force is the gluon.
 * Electromagnetic interaction: the familiar interaction that acts on electrically charged particles. The photon is the exchange particle for this force.
 * Weak interaction: a short-range interaction responsible for some forms of radioactivity, that acts on electrons, neutrinos, and quarks. It is mediated by the W and Z bosons.
 * Gravitational interaction: a long-range attractive interaction that acts on all particles. The postulated exchange particle has been named the graviton.

Modern unified field theory attempts to bring these four interactions together into a single framework.

World Equations
In mathematical analysis, a complex manifold yields a holomorphic operation and is complex differentiable in a neighborhood of every point in its domain, such that an operational process can be represented as an infinite sum of terms:


 * $$(2.10)\qquad

f(\lambda)=f(\lambda_0)+{f^'}(\lambda_0)(\lambda-\lambda_0) \cdots +f^n(\lambda_0)(\lambda-\lambda_0)^n/n! $$

known as the Taylor and Maclaurin series [2], introduced in 1715. Normally, a global event generates a series of sequential actions, each of which is associated with its opponent reactions, respectively and reciprocally. For any event operation as the functional derivatives, the sum of terms are calculated at an initial state $$\lambda_0$$ and explicitly reflected by the Event Operations $${\dot \lambda_i}\mapsto{\dot \partial}_{\lambda_i}$$ in the dual variant forms:


 * $$(2.11)\qquad

f(\lambda) = f_0 + {\kappa_1} {\dot \partial}_{\lambda_1} + {\kappa_2} {\dot \partial}_{\lambda_1} {\dot \partial}_{\lambda_2} + {\kappa_3} {\dot \partial}_{\lambda_1} {\dot \partial}_{\lambda_2} $$
 * $$\ (2.12)\qquad

\kappa_n=f^n(\lambda_0)/n!, \ \ \dot \lambda_i \in \{\dot \partial\}= \{ {\check \partial}_{\lambda}, {\check \partial}^{\lambda},

{\hat \partial}^{\lambda}, {\hat \partial}_{\lambda} \} $$

where $$\kappa_n$$ is the coefficient of each order n. The event states of world planes are open sets and can either rise as subspaces transformed from the other horizon or remain confined as independent existences within their own domain, as in the settings of $$Y^-Y^+$$ manifolds of the world planes. The operational function $$f(\lambda)$$ for an event $$\lambda$$ involves the state densities $$f(\rho_n)$$ and spacetime exposition $$\Gamma$$ of a system with N objects or particles. Assuming each of the particles is in one of three possible states: $$|-\rangle$$, $$|+\rangle$$, and $$|o\rangle$$, the system has $$N^+_n$$ and $$N^-_n$$ particles at non-zero charges with their state functions of $$\phi^+_n$$ or $$\phi^-_n$$ confineable to the respective manifold $$Y^\pm$$. Therefore, the horizon functions of the system can be expressed by:


 * $$(2.13)\qquad

{W_c} = k_w \int {W_b} { d \Gamma } \qquad {W_b} = {\sum}_n {h_n W_a} \qquad {W_a} = f(\lambda) {\rho_n} $$
 * $$(2.14)\qquad

{\rho_n}= {\phi_n^+}(\hat x, \lambda) {\phi_n^-}({\check x, \lambda}) ,\quad h_n={N_n^\pm}/{N} \qquad : {\hat x}, {\check x} \subset {Y^\pm}\{{ \bf r } \mp i{ \bf k }\} $$

where $$h_n$$ is a horizon factor, $${N_n^\pm}/{N}$$ are percentages of the $$Y^-Y^+$$ particles, and $$k_w$$ is defined as a world constant. During space and time dynamics, the density $$\phi_n^-\phi_n^+$$ is incepted at $$\lambda=\lambda_0$$ and followed by a sequence of the evolutions $$\lambda_n=\dot \partial_{\lambda_n}$$. This process engages and applies a series of the event operations of equations (2.11) to the equations of (2.13) in the forms of the following expressions, named as World Equations:


 * $$(2.15)\qquad

{W} = k_w \int { d \Gamma } \sum_n h_n \left[ {W^\pm_n} + {\kappa_1} {\dot \partial_{\lambda_1}} + {\kappa_2} {\dot \partial_{\lambda_2}} {\dot \partial_{\lambda_1}} \cdots \right] {\phi}_{n}^{+} {\phi}_{n}^{-} $$

where $${W^\pm_n} \equiv W({\hat x|\check x},\lambda_0)$$ is the $$Y^-$$ or $$Y^+$$ ground environment or an initial potential of a system, respectively. Because an event process $$\lambda_n$$ is operated in complex composition of the virtual and physical coordinates, it yields a linear function in a form of operational addition: $$f(\partial_\kappa+\partial_r)=f(\partial_\kappa)+f(\partial_r)$$, where the $$\{{\bf r},{\bf k}\}$$ vectors of each manifold $$Y^\mp \{ {\bf r} \pm i{\bf k} \}$$ constitute their orthogonal coordinate system $${\bf r}\cdot{\bf k}=0$$. As the topological framework, various horizons are defined as, but not limited to, timestate, microscopic and macroscopic regimes, each of which is in a separate zone, emerges with its own fields, and aggregates or dissolves into each other as the interoperable neighborhoods, systematically and simultaneously. Through the $$Y^-Y^+$$ communications, the expression of the tangent vectors defines and gives rise to each of the horizons.

Horizon Fields
At the first horizon, the field behaviors of individual objects or particles have their potentials of the timestate functions in the form of, but not limited to, the dual densities:


 * $$(2.16) \qquad

{\rho^+_\phi}={\phi}{({\hat x},\lambda)}{\varphi}{({\check x},\lambda)} \qquad : {\phi}^{+}\equiv{\phi}{({\hat x},\lambda)}, {\varphi}^{-}\equiv{\varphi}{({\check x},\lambda)} $$
 * $$(2.17) \qquad

{\rho^-_\phi}={\phi}{({\check x},\lambda)}{\varphi}{({\hat x},\lambda)} \qquad : {\phi}^{-}\equiv{\phi}{({\check x},\lambda)}, {\varphi}^{-}\equiv{\varphi}{({\hat x},\lambda)} $$

This horizon is confined by its neighborhoods of the ground fields and second horizons, which is characterizable by the scalar objects of $${\phi}^{\pm}$$ and $${\varphi}^{\pm}$$ fields of the ground horizon, individually, and reciprocally.

At the second Horizon, the effects of aggregated objects has their commutative entanglements of the microscopic functions in forms of


 * $$(2.18) \qquad

{\bf f}{^+_n} = {{\kappa}^{+}_{f}} {\dot \partial {\rho^+_\phi}} = \frac {\hbar c}{2} \Big (	{\varphi^-_n} 		\frac {\dot x^\nu}{E^{^+}_n}	{\partial^\nu}{\phi^+_n}	+	{\phi^+_n} 		\frac {\dot x_\nu}{E^{^-}_n}	{\partial_\nu}{\varphi^-_n} \Big ) $$
 * $$(2.19) \qquad

{\bf f}{^-_n} = {{\kappa}^{-}_{\bf f}} {\dot \partial {\rho^-_\phi}} = \frac {\hbar c}{2} \Big (	{\varphi_n^+} 	\frac {\dot x_m}{E^{^-}_n} 	{\partial_m} {\phi_n^-} +	{\phi_n^-} 	\frac {\dot x^m}{E^{^+}_n} 	{\partial^m} {\varphi_n^+} \Big )$$

defined as Fluxion Fields. This horizon summarizes the timestate functions $${\bf f}{^\pm}=\sum {\bf f}{^\pm_n}$$, confined between the first and third horizons.

At the third horizon, the integrity of massive objects characterizes their global motion dynamics of the macroscopic matrices and tensors through an integration of, but not limited to, the derivative to microscopic fields of densities and fluxions, defined as Force Fields:


 * $$(2.20) \qquad

{\bf F}^{\pm} = {\kappa}^{\pm}_{\bf F} \int {\rho_a} {\dot \partial}{\bf f}^{\pm} d{\Gamma} \qquad : {\dot \partial} \in \{ {\check \partial}_{\lambda}, {\hat \partial}^{\lambda} \} $$

where $${\kappa}^+_{\bf F}$$ or $${\kappa}^-_{\bf F}$$ is a coefficient. This horizon is confined by its neighborhoods of the second and fourth horizons and characterizable by the tensor fields of $${\dot \partial}{\bf f}_{m}$$ and $${\dot \partial}{\bf f}^{\mu}$$.

The horizon ladder continuously accumulates and gives a rise to the next objects in form of a ladder hierarchy:
 * $$(2.21) \qquad

\iiiint \cdots {\rho_c} {\dot \partial} \int {\rho_b} {\dot \partial} {\bf F}{^\pm} {d \Gamma} \mapsto {\bf W}^{\pm}_{x} $$ They are orchestrated into groups, organs, globes or galaxies.

Universal Field Equations
Following Universal Topology, world events, illustrated in the [~][~] flow diagram of Figure 4.1, operate the potential entanglements that consist of the [~] supremacy (white background) at a top-half of the cycle and the [~] supremacy (black background) at a bottom-half of the cycle. Each part is dissolving into the other to form an alternating stream of dynamic flows. Their transformations in between are bi-directional antisymmetric and transported crossing the dark tunnel through a pair of the end-to-end circlets on the center line. Both of the top-half and bottom-half share the common global environment of the state density [~] that mathematically represents the [~] for the [~] manifold and its equivalent [~] for the [~] manifold, respectively.

Law of Event Evolutions
Besides, the left-side diagram presents the event flow acted from the inception of [~] through [~] [~] [~] to intact a cycle process for the [~] supremacy. In parallel, the right-side diagram depicts the event flow initiated from the event [~] through [~] [~] [~] to complete a cycle process for the [~] supremacy. The details are described as the following: ￼ Figure 4.1: Event Flows of [~][~] Evolutional Processes

Visualized in the left-side of Figure 4.1, the transitional event process between virtual and physical manifolds involves a cyclic sequence throughout the dual manifolds of the environment: incepted at [~], the event actor produces the virtual operation [~] in [~] manifold (the left-hand blue curvature) projecting [~] to and transforming into its physical opponent [~] (the tin curvature transforming from the left-hand into right-hand), traveling through [~] manifold (the right-hand green curvature), and reacting the event [~] back to the actor.

As a duality in the parallel reaction, exhibited in the right-side of Figure 4.1, initiated at [~], the event actor generates the physical operation [~] in [~] manifold (the right-hand green curvature) projecting [~] to and transforming into its virtual opponent [~] (the tin curvature transforming from right-hand into left-hand), traveling through [~] manifold (the left-hand blue curvature), and reacting the event [~] back to the actor. With respect to one another, the two sets of the Universal Event processes, cycling at the opposite direction simultaneously, formulate the flow charts in the following mathematical expressions:

[~]		(4.1)	[~]		(4.2)

This pair of the interweaving system pictures an outline of the internal commutation of dark energy and continuum density of the entanglements. It demonstrates that the two-sidedness of any event flows, each dissolving into the other in alternating streams, operate a life of situations, movements, or actions through continuous helix-circulations aligned with the universe topology, which lay behind the context of the main philosophical interpretation of World Equations.

Universal Field Equations
The potential entanglements is a fundamental principle of the real-life streaming such that one constituent cannot be fully described without considering the other. As a consequence, the state of a composite system is always expressible as a sum of products of states of each constituents. Under the law of event operations, they are fully describable by the mathematical framework of the dual manifolds.

During the events of the virtual supremacy, a chain of the event actors in the flows of Figure 4.1 and equations (4.1)-(4.2) can be shown by and underlined in the sequence of the following processes:

[~]; [~]	(6.1)

From the event actors [~] and [~], the World Equations (5.4) becomes:

[~]	(6.2)

Meanwhile the event actors [~] and [~] turn World Equations into:

[~]	(6.3)

where [~] is the time invariant [~]-energy fluxion. Rising from the opponent fields of [~] or [~], the dynamic reactions under the [~] manifold continuum give rise to the Motion Operations of the [~] fields [~] or [~] approximated at the first and second orders of perturbations in term of the above World Equation, as an example:

[~] 	(6.4)	[~]		(6.5)	[~]		(6.6)

where the potentials of [~] and [~] give rise simultaneously to their opponent’s reactors of the physical to virtual transformation [~] and the physical reaction [~], respectively. From these interwoven relationships, the motion operations (4.3)-(4.4) determine a pair of partial differential equations of the [~][~] state fields [~] and [~] under the supremacy of virtual dynamics at the [~] manifold:

[~]	(6.7) [~]	(6.8)

giving rise to the [~] General Fields from each respective opponent during their physical interactions. In the events of the physical supremacy in parallel fashion, a chain of the event actors in the flows of Figure 4.1 can be shown by the similar sequence of the following processes:

[~] ; [~]	(6.9) [~]	(6.10) [~]	(6.11)

where [~] is the time invariant [~]-energy fluxion. Rising from its opponent fields of [~] or [~] in parallel fashion, the dynamic reactions under the [~] manifold continuum give rise to the Motion Operations of the [~] state fields [~] or [~], which determines a linear partial differential equation of the state function [~] or [~] under the supremacy of physical dynamics at the [~] manifold:

[~]	(6.12) [~]	(6.13)

giving rise to the [~] General Fields from each of the respective opponents during their virtual interactions. A homogeneous system is a trace of diagonal elements where an observer is positioned external to or outside of the objects. The source of the fields appears as a point object and has the uniform Conservations at every point without irregularities in field strength and direction, regardless of how the source itself is constituted with or without its internal or surface twisting torsions.

Whereas, a heterogeneous system is the off-diagonal elements of the symmetric tensors where an observer is positioned internal to or inside of the objects, and the duality of virtual annihilation and physical reproduction are balanced to form the local Continuity or Invariance.

The two pairs of the dynamic fields (6.7)-(6.8) and (6.12)-(6.13) are operated generically under first horizon of the World Events. Together, the four formulae are named as First Universal Field Equations, which are fundamental and general to all fields of natural evolutions.