User:Virtumanity/Universal topology

In physics, Universal topology is the topological structure of Universe that, primarily in our universe, a duality of the two-sidedness of physical and virtual reality lies at the heart of all events 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, production, annihilation, or actions complementarily, reciprocally and interdependently. The natural topology consistently emerges as or entangle with a set of the fields that communicates and projects their interoperable values to its surrounding environment, alternatively arisen by or acting on its opponent through the reciprocal interactions. Therefore, it provides the context for our main philosophical interpretation to extend our fundamental physics into a duality of a oneness of nature foundation: Universal topology.

With the discovery of the dark energy , the most accepted hypothesis behind Universal topology is Many-worlds interpretation. that, at minimum, there exists a pair of the fields - one for a physical world while the other for its reciprocal virtual world. The concept of a duality of the two-sidedness of Virtual and Physical Words 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. After an observation is made, each element of the superposition becomes the combined subject–object and any object with the two "relative states" is "collapsed" at its state with the same collapsed outcome. More precisely in principle, an object possesses a pair of the fields and requires a duality of manifolds for their living entanglement. Because each object possesses a pair of the virtual and physical fields, an interruption between two objects involves two pairs of the fields, which constitute cross-entangling simultaneously and reciprocally. In mathematics, this means that, instead of a single manifold, an oneness of the real world of our universe must be modeled by a duality of the World Planes, or the Dual Manifolds of Universal topology.

The concepts of Universal Topology become important in analysing local and global aspects of topological World Planes or its simplified form: spacetime. It is especially important in physical cosmology and microcosmic or virtual ontology in terms of philosophy, physics and mathematics, which leads to groundbreaking in Unified Field Theory.

Philosophical Terminology
In physics, the objects are often virtual and physical matters, and the morphisms are dualities of the dialectical processes orchestrating a set or subsets of events, operations, and states in one regime rising, transforming, transporting, and alternating into states of the others: universal topology of the nature structure. This section abstracts some fundamental terminologies philosophically as the preliminary laws of universal topology outlined as the following:

Universe
Universe - The whole of everything in existence that operates under a topological system of natural laws for, but not limited to, physical and virtual events, states, matters, and actions. It constitutes and orchestrates various domains, called World, each of which is composed of hierarchical manifests for the events, operations, and transformations among the neighborhood zones or its subsets of areas, called Horizon.

World
World - An environment composed of events or constituted by hierarchical structures of both massless and massive objects, events, states, matters, and situations. These hierarchical structures of the global manifold are respectively defined as Virtual World, where it operates virtual event, or Physical World, where it performs physical actions. Together, the virtual and physical worlds form one integrated World as a domain of the universe and interoperates as the complementary opponents of all natural states and events. Traditionally, the virtual world is referred to as the inner world, the physical world as the outer world, and together they form holistic lives in universe. A world has a permanent form of global topology, localizes a region of the universe, and interacts with other worlds rising from one or the other with common ground in universal conservations. Furthermore, there are multiple levels of inner worlds and outer worlds. Inner worlds are instances of situations, with or without energy or mass formations, while outer worlds include physical mass of living beings and inanimate objects.

Duality
Duality - The complementary opponents of inseparable, reciprocal pairs of all natural states, energy, and events, constituted by the topological hierarchy of our world. Among them, the most fundamental duality is our domain resource of the universe, known as Yin and Yang, with neutral balance that appears as if there were nothing or dark energy. Yinyang presents the two-sidedness of any event, operations, or spaces, each dissolving into the other in an alternating stream that generates the life of situations, conceals the inanimacy of resources, operates the movement of actions through continuous helix-circulations, symmetrically and asymmetrically. Because of this yinyang nature, our world always manifests a mirrored pair in the imaginary part, a conjugate pair of a complex manifold, known as YinYang Manifolds.

Manifold
Manifold - Various states of both virtual and physical spaces are describable at global domains which emerge as object events, operate in zone transformations, and transit between state energies and matter enclaves. The universal topology consists of two manifolds: Yin Manifold, $$Y^-$$, for the events of physical supremacy and Yang Manifold, $$Y^+$$, for the events of virtual supremacy, progressively and complementarily rising through various stages of alternating streams - Entanglements.

Operation
Operation - An event is naturally initiated by and interoperated among each of horizons, worlds, and universe. Together, they form the comprehensive situations of the horizon, life steams of the world, and environments of the universe. As one of the universe domains, for example, our world is consisted by the laws of YinYang principles which represent the complementary opponents serving as the resource of the motion dynamics for all natural states and events.

Horizon
Horizon - The apparent boundary of a realm of perception or the like, where unique structures are evolved, topological functions are performed, various neighborhoods form complementary interactions, and zones of the world are composed through multi-functional transformations. Each horizon rises and contains specific fields as a construction of the symmetric and asymmetric dynamics within or beyond its own range. In other words, fields vary from one horizon to the others, each of which is part of and aligned with the universal topology of the world. In physics, for example, the microscopic and macroscopic zones are in the separate horizons, each of which emerges its own fields and aggregates or dissolves between each others.

Spacetime of Universal Topology
As the nature duality of Universal topology, our world $$ W $$ always manifests a mirrored pair in the imaginary part or a conjugate pair of the complex manifolds, such that the physical nature of $$P$$ functions is associated with its virtual or dark nature of $$ V $$ functions to constitute a duality of the real world functions. Today, the most popular topology, for example, is a duality of the virtual and physical manifolds of Entanglement Topology, which represents the foundation of Quantum entanglement as well as microcosmic ontology.

Complex manifolds
Among the real world functions, the most fundamental dynamics are our dark resources of the universal energies, known as physical Yin “−” and/or dark Yang “+” objects, with neutral balance “0” that appears as if there were nothing. Each type of the physical of virtual dark objects (−,0,+) appearing as energy fields has their own domain of the relational manifolds such that one defines a $$ Y^- $$ (Yin) manifold while the other the $$ Y^+ $$ (Yang) manifold, respectively. They jointly present the two-sidedness of any events, operations, transportations, and entanglements, each dissolving into the other in the alternating streams that generates the life of entanglements, conceals the inanimacy of resources, and operates the event actions. As a two-dimensional plane, the physical position of $$ \bf{r} $$ and virtual positions of $$ \pm i \bf{k} $$ naturally form a duality of the conjugate manifolds: $$ Y^- \{{\bf r} + i{\bf k}\} $$ and $$ Y^+ \{{\bf r} - i{\bf k}\} $$. Each of the system constitutes its world plane $$ W^\pm $$ distinctively, forms a duality of the universal topology $$ W = P \pm iV $$ cohesively, and maintains its own sub-coordinate system {$$ {\bf r} $$} or {$$ {\bf k} $$} respectively. Because of the two dimensions of the world planes {$$ {\bf r} \pm i {\bf k} $$}, each transcends its event operations further down to its sub-coordinate system with extra degrees of freedoms for either physical dimensions $$ {\bf r} = \{x_1, x_2, x_3\} $$ or virtual dimensions $$ {\bf k} = \{x^0, x^1, x^2\, \cdots\} $$. For example, in the scope of space and time duality, the compound dimensions become the tetrad coordinates, known as the following:


 * $$(2.1) \qquad x_m \in \check x \{x_0, x_1, x_2, x_3\} \subset Y^- \{{\bf r} + i{\bf k}\} \qquad : x_0=ict $$
 * $$(2.2) \qquad x^\mu \in \hat x \{x^0, x^1, x^2, x^3\} \subset Y^+ \{{\bf r} - i{\bf k}\} \qquad : x^0=-ict$$

where $$ i{\bf k} = ict = x_0 = -x^0 $$. As a consequence, a manifold appears as or is combined into the higher dimensional coordinates, which results in the spacetime manifolds in the four-dimensional spaces. Together, the two world planes $$ Y^\mp \{ {\bf r} \pm i{\bf k} \}$$ compose the two dimensional dynamics of Boost, an inertial for generators, and Spiral, a rotational contortions for stresses, which function as a reciprocal or conjugate duality transporting and transforming global events among sub-coordinates. Consequently, for any type of the events, the $$ Y^-Y^+ $$ manifolds are always connected, coupled, and conjugated between each other, a duality of which defines entanglements as the virtually inseparable and physically reciprocal pairs of all natural functions.

World planes
As a part of the Universal Topology, the communication infrastructure between the manifolds are empowered with the speed of light


 * $$(2.3) \qquad {\check \partial_t}{x_m}=\begin{pmatrix}ic, c{\bf \check b}\end{pmatrix}$$
 * $$(2.4) \qquad {\hat \partial^t}{x^\mu}=\begin{pmatrix}-ic,c{\bf \hat b}\end{pmatrix}$$

that transform and transport axiomatic commutations or entanglements of the event operations, informational transmissions or conveyable actions. Between the world planes, the two-dimensional transportations $$\{ {\bf r}\mp i{\bf k}\}$$ are naturally constructed for tunneling between the $$Y^-Y^+$$ domains as the dynamics of dark energies, which is mathematically describable by transformations among the four potential fields.



Both manifolds $${\hat x}\{{\bf r}-i{\bf k}\}$$ and $${\hat x}\{{\bf r}+i{\bf k}\}$$ simultaneously govern and alternatively perform the event operations as one integral stream of any physical and virtual dynamics. Apparently, the virtual positions $$\pm i{\bf k}$$ naturally forms a duality of the conjugate manifolds:


 * $$(2.5) \qquad {\hat x}\{ x^\nu \}\in Y^+$$
 * $$(2.6) \qquad {\check x}\{x_m\}\in Y^-$$

Each of the super two-dimensional coordinate system $$ G(\lambda)\in G\{ {\bf r}\pm i{\bf k}\}$$ constitutes its world plane $$W^\pm$$ distinctively, forms a duality of the universal topology $$W=P\pm iV$$ cohesively, and maintains its own sub-coordinate system $$\bf r$$ or $$\bf k$$ respectively. A sub-coordinate system has its own rotational freedom of either physical sub-dimensions $$\bf r$$ or virtual sub-dimensions $$\pm\bf k$$. Together, they compose two rotations as a reciprocal or conjugate duality operating and balancing the world events.

Mathematical Framework
As a part of the natural architecture of the topology, the mathematical regulation of terminology not only includes symbol notation, operators, and indices of vectors and tensors, but also classifies the mathematical tools and their interpretations under the universal topology.

Duality of Contravariance and Covariance
In order to describe the nature precisely, it is essential to define a duality of the contravariant $$Y^+=Y\{ {\bf r} - i{\bf k}\} $$ manifold and the covariant $$Y^-=Y\{ {\bf r} + i{\bf k}\} $$ manifold, respectively by the following regulations for a $$\lambda$$ event.


 * Contravariance ($$\hat \partial^\lambda$$) - One set of the symbols with the upper indices $$\{{x^\mu}, {u^\nu}, A^{\nu\sigma} \}$$, as contravariant forms, are the numbers for the $$\{{\hat x}\}$$ basis of the $$Y^+$$ manifold labelled by its identity symbols $$\{ {\hat {} }, ^+ \}$$. “Contravariance” is a formalism in which the nature laws of dynamics operates the event actions $$\hat \partial^\lambda$$, maintains its virtual supremacy of the $$Y^+$$ dynamics, and dominates the virtual characteristics under the manifold basis $$\{{\hat x}\}$$.
 * Covariance ($$\check \partial_{\lambda}$$) - Other set of the symbols with the lower indices $$\{{x_m}, {u_n}, A_{a b} \}$$, as covariance forms, are the numbers for the $$\{{\check x}\}$$ basis of the $$Y^-$$ manifold labelled by its identity symbols of $$\{ {\check {} }, ^- \}$$. “Covariance” is a formalism in which the nature laws of dynamics performs the event actions $$\check \partial_{\lambda}$$, maintains its physical supremacy of the $$Y^-$$ dynamics, and dominates the physical characteristics under the manifold basis $$\{{\check x}\}$$.

Either contravariance or covariance has the same form under a specified set of transformations to the lateral observers within the same or boost basis as a common or parallel set of references for the operational event. The communications between the manifolds are related through the tangent space of the world planes, regulated as the following operations:


 * Communications ($$\hat \partial_\lambda$$ and $$\check \partial^\lambda$$) - Lowering the operational indices $$\hat \partial_\lambda$$ is a formalism in which the quantitative effects of an event $$\lambda$$ under the contravariant $$Y^+$$ manifold are projected into, transformed to, or acted on its conjugate $$Y^-$$ manifold. Rising the operational indexes $$\check \partial^{_\lambda}$$, in parallel fashion, is a formalism in which the quantitative effects of an event $$\lambda$$ under the covariant $$Y^-$$ manifold are projected into, transformed to, or recorded at its reciprocal $$Y^+$$ manifold.

The dual variances are isomorphic to each other regardless if they are isomorphic to the underlying manifold itself, and form the norm (inner product) of the manifolds or world lines. Because of the reciprocal and contingent nature, the dual manifolds conserve their invariant quantities under a change of transform commutations and transport continuities with the expressional freedom of its underlying basis.

Event Operations
As a part of the universal topology, these mathematical regulations of the dual variances architecturally defines further framework of the event characteristics and its operational structures. In the $${Y}^\mp $$ manifolds, a potential field can be characterized by a scalar function of $${\psi} \in \{ \phi^+, \phi^-, \varphi^+, \varphi^- \} $$, named as First Horizon Fields, to serve as a state environment of entanglements. The derivative to the scalar fields are event operations of their motion dynamics, which generates a tangent space, named as Second Horizon Fields.

In order to operate the local actions, an event $$\lambda$$ exerts its effects of the virtual supremacy within its $${Y}^+$$ manifold or physical supremacy within its $${Y}^-$$ manifold, giving rise to the second horizon:


 * $$\ (3.1)\qquad{\hat \partial^\lambda} {\psi}={\dot x^\mu}{\partial^\mu} {\psi} \qquad	\qquad \qquad:

{\dot x}^{\mu}={\partial x^\mu}/{\partial \lambda}, {x^\mu} \in Y^+$$


 * $$(3.2)\qquad{\check \partial_\lambda} {\psi}={\dot x_m}{\partial_m} {\psi} \qquad \qquad \quad \ \ :

{\dot x_m}={\partial x_m}/{\partial \lambda}, {x_m} \in Y^-$$


 * $$(3.3)\qquad{\hat \partial^\lambda}=\begin{pmatrix}-ic{\partial^\kappa} & {\bf u}^+ {\nabla} \end{pmatrix} \qquad \ :

\lambda = t, {\bf u}^+=\frac {\partial x^r}{\partial t}, {\partial^\kappa}=\frac {\partial} {\partial x^0}, {\partial^r}=\nabla$$


 * $$(3.4)\qquad{\check \partial_\lambda}=\begin{pmatrix}-ic{\partial_\kappa} & {\bf u}^- {\nabla} \end{pmatrix} \qquad \ : \lambda = t,

{\bf u}^-=\frac {\partial x_r}{\partial t}, {\partial_\kappa}=\frac {\partial} {\partial x_0}, {\partial_r}=\nabla$$

The speed $${\dot x^\mu}=\{-ic, {\bf u}^+ \} $$ or $$ {\dot x_m}=\{ic, {\bf u}^- \} $$ is the contravariant or covariant velocity, observed from an inertial frame without effects of rotations and transformation. Applying to a point object, it represents a field at each point “External” to itself.

Boost Generators and Spiral Coordinators
By lowering the index, the virtual $$Y^+$$ actions manifest the first tangent potential $$\hat \partial_\lambda$$ projecting into its opponent basis of the $$Y^-$$ manifold. Because of the motion, the derivative to the vector $$\dot x^\mu$$ has the changes of both magnitude quantity $$\dot x_a \partial x^\mu/\partial x_a$$ and basis direction $$({\dot x_a} {\nabla_a {\bf b}^\mu}){x^\mu}={\dot x_a} {\Gamma}_{a \mu}^{^+\sigma} {x_\sigma}$$ transforming from one world plane $$Y^+\{{\bf r}-i{\bf k}\}$$ to the other $$Y^-\{{\bf r}+i{\bf k}\}$$. This action redefines the $$Y^+$$ event quantities of relativity and creates the Inertial Boost $${J^+_{\mu a}}$$ Generators and the Spiral Torque $${K}_{\mu a}^{+}$$ Coordinators around a central point, giving rise to the $$Y^+$$ tangent rotations of a scalar potential space:


 * $$(3.5)\qquad{\hat \partial_\lambda}{\psi}= {\dot x_a}\Big({J^+_{\mu a}}+{K}_{\mu a}^{+}\Big){\partial^\mu}{\psi} \qquad:

{J^+_{\mu a}}=\frac {\partial x^\mu}{\partial x_a}, {K}_{\mu a}^{+}={\Gamma}_{\mu a}^{+\sigma}{x_\sigma}$$
 * $$(3.6)\qquad{\dot x^\mu}\mapsto{\dot x_a}\Big ({J^+_{\mu a}}+{K}_{\mu a}^{+}\Big) \qquad \qquad:

{\Gamma}^{^+\sigma}_{\mu \nu}\equiv\frac {1}{2} g_{\sigma \epsilon}\left(\frac{\partial g^{\epsilon \mu}}{\partial x^\nu}+\frac{\partial g^{\epsilon \nu}}{\partial x^\mu}-\frac{\partial g^{\mu \nu}}{\partial x^\epsilon}\right)$$

Likewise for the $$Y^-$$ actions by raising the index, the $$Y^-$$ tangent rotations of a scalar potential space can be cloned straightforwardly.


 * $$(3.7)\qquad{\check \partial^\lambda}{\psi}={\dot x^\alpha}\left({J^-_{m \alpha}}+{K^-_{m \alpha}}\right){\partial_m} {\psi} \quad:

{J^-_{m \alpha}}=\frac {\partial x_m}{\partial x^\alpha}, {K^-_{m \alpha}}={\Gamma}_{m \alpha}^{^-s}{x_s}$$
 * $$(3.8)\qquad{\dot x_m}\mapsto{\dot x^\alpha}\left({J^-_{m \alpha}}+{K^-_{m \alpha}}\right)\qquad \quad:

{\Gamma}^{^-s}_{m n}=\frac{1}{2}{g^{s e}}\left(\frac{\partial g_{e m}}{\partial x_n}+\frac{\partial g_{e n}}{\partial x_m}-\frac{\partial g_{m n}}{\partial x_e}\right)$$

where $$g_{\sigma \epsilon}$$ or $$g^{s e}$$ is the metrics, the symbol $${\Gamma}_{\sigma \nu}^{\mp\mu}$$ is an $$Y^-$$ or $$Y^+$$ metric connection, similar but extend the meanings to Christoffel symbols, introduced in 1869.

It is worthwhile to inspire the amazing facts that it is the duality of entanglement that the Boost Generators produce Photons   and Spiral Coordinators create Gravitons.

Horizon Fields
Following the tangent curvature, the $$\lambda$$ event operates the potential vectors through the second tangent vector of the curvature, giving rise to the Third Horizon Fields, shown by the expressions:


 * $$\ \ (3.9)\qquad{\hat \partial^\lambda}{V^\mu}={\dot x}^{\nu}({\partial^\nu}{V^\mu}+{\Gamma}_{\sigma \nu}^{^+\mu} {V^\sigma}) \qquad \qquad: {V^\mu}\mapsto{\dot x^\mu}{\partial^\mu}{\psi}$$


 * $$(3.10)\quad \ \ {\check \partial_\lambda} {V_m}={\dot x}_{n}({\partial_n} {V_m}+{\Gamma}_{m n}^{^-s} {V_s})\qquad \qquad \ :{V_m}\mapsto{\dot x_m}{\partial_m}{\psi}$$


 * $$(3.11)\quad \ \ {\hat \partial^\lambda}

{\hat \partial^\lambda} {\psi}= ( {\dot x}^{\nu} {\partial^\nu} ) ( {\dot x}^{\mu} {\partial^\mu} ) {\psi} + {\dot x}^{\nu} {\Gamma}_{\mu \nu}^{^+\sigma} {\dot x}^{\sigma} {\partial^\sigma} {\psi} $$


 * $$(3.12)\quad \ \ {\check \partial_\lambda}

{\check \partial_\lambda} {\psi}= ( {\dot x}_{n} {\partial_n} ) ( {\dot x_m} {\partial_m} ) {\psi} +	{\dot x}_{n} {\Gamma}_{m n}^{^-s} {\dot x_s}{\partial_s} {\psi} $$ In the tangent space, the scalar fields are given rise to the vector fields.

Through the tangent vector of the third curvature, the events $$\hat \partial_\lambda$$ and $${\check \partial^\lambda}$$ continuously entangle the vector fields and gives rise to the forth horizon fields, shown by the formulae:


 * $$(3.13)\qquad {\hat \partial_\lambda}

{\hat \partial^\lambda} {\psi} = {\hat \partial_\lambda} {V^\mu} = {\dot x_a} \left (	{J^+_{\nu a}} + {K^+_{\nu a}} \right ) \left (	{\partial^\nu} {V^\mu} +	{\Gamma}_{\sigma \nu}^{^+\mu} {V^\sigma} \right ) $$


 * $$(3.14)\qquad{\check \partial^\lambda}

{\check \partial_\lambda} {\psi}= {\check \partial^\lambda} {V_m}= {\dot x^\alpha} \left ( {J^-_{m \alpha}} + {K^-_{m \alpha}} \right ) \left ( {\partial_n} {V_m} +{\Gamma}_{m n}^{^-s} {V_s} \right ) $$


 * $$(3.15)\qquad{\hat \partial^\lambda}

{\hat \partial^\lambda} {V^\mu}= \left ( 	{\dot x^\iota} {\partial^\iota} \right ) \left ( 	{\dot x}^{\nu} {\partial^\nu} \right ) {V^\mu} + {\dot x}^{\nu} {\Gamma}_{\mu \nu}^{^+s} {V_s} + {	{\dot x^n} {\Gamma}_{m s}^{^-n} {\dot x_\nu} {\Gamma}_{\sigma \nu}^{^+\mu} } {V^\sigma} + \left (	{\ddot x}^{\nu} 	{\Gamma}_{\mu \nu}^{^+\sigma} +	{\dot x}^{\nu} {\dot x^\iota}	{\partial^\iota \Gamma}_{\mu \nu}^{^+\sigma} +	{\dot x}^{\nu} {\dot x^\iota}	{\Gamma}_{\mu \nu}^{^+\sigma}	{\partial^\iota} \right ){V^\sigma} $$


 * $$(3.16)\qquad{\check \partial_\lambda}

{\check \partial_\lambda} {V_m} = \left ( {\dot x_e} {\partial_e} \right ) \left ( {\dot x_n} {\partial_n} \right ){V_m} + {\dot x}_{n} {\Gamma}_{m n}^{^-s} {V_s} +{	{\dot x_\nu} {\Gamma}_{\sigma \nu}^{^+\mu} {\dot x^n} {\Gamma}_{m s}^{^-n} } {V_s} + \left (	{\ddot x_n} 	{\Gamma}_{m n}^{^-s} +	{\dot x_n} {\dot x_e}	{\partial_e \Gamma}_{m n}^{^-s} +	{\dot x_n} {\dot x_e}	{\Gamma}_{m n}^{^-s}	{\partial_e} \right ){V_s} $$

As an integrity, they perform full operational commutations of inertial boosts and torque rotations operated between the $$Y^-$$ $$Y^+$$ world planes. The event processes continue to build up the further operable domain with a variety of the rank-n tensor fields. Systematically, sequentially and simultaneously, a chain of these reactions constitutes various domains, each of which gives rise to the field entanglements.

It is worthwhile to emphasize that
 * 1) the manifold operators of $$\{{\partial^\mu},{\partial_m}\}$$, including traditional “operators” of $$\{ {\partial}/{\partial t}, {\partial}/{\partial x_i}, {\nabla}\}$$ are exclusively useable as mathematical tools only, and
 * 2) the tools do not operate or perform by themselves unless they are driven or operated by an event λ, implicitly or explicitly.

Motion Operations
As a natural principle of motion dynamics, one of the flow processes dominates the intrinsic order, or development, of virtual into physical regime, while, at the same time, its opponent dominates the intrinsic annihilation or physical resources into virtual domain. Applicable to world expressions of $$W$$, the principle of least-actions derives a set of the Motion Operations:


 * $$(3.17) \qquad

{\check \partial^-} ( \frac {\partial W}    { \partial ({\hat \partial^+} {\phi^+})} ) -\frac {\partial W}{\partial {\phi^+}} =0 \qquad : {\check \partial^-} \in \{ {\check \partial}_{\lambda}, {\check \partial}^{\lambda} \}, \ {\phi^+} \in \{ 	{\phi}_{n}^+, {\varphi}_{n}^+ \} $$


 * $$(3.18) \qquad {\hat \partial^+}

( \frac {\partial W}    { \partial ({\check \partial^-} {\phi^-})} ) -\frac {\partial W}{\partial {\phi^-}} =0 \qquad : {\hat \partial^+} \in \{ {\hat \partial}^{\lambda}, {\hat \partial}_{\lambda} \}, \ {\phi^-} \in \{ 	{\phi}_{n}^-, {\varphi}_{n}^- \} $$

This set of dual formulae extends the philosophical meaning to the Euler–Lagrange equation for the actions of any dynamic system, introduced in the 1750s. The new sets of the variables of $$\phi_n^\mp$$ and the event operators of $$\check \partial^-$$ and $$\hat \partial^+$$ signify that both manifolds maintain equilibria formulations from each of the motion extrema, simultaneously driving a duality of physical and virtual dynamics.

Geodesic equation
Unlike a single manifold space, where the shortest curve connecting two points is described as a parallel line, the optimum route between two points of a curve is connected by the tangent transportations of the $$Y^-$$ and $$Y^+$$ manifolds. As an extremum of event actions on a set of curves, the rate of divergence of nearby geodesics determines curvatures that is governed by the equivalent formulation of geodesic deviation for the shortest paths on each of the world planes:


 * $$(3.19)\qquad{\ddot x}^{\mu}+{\Gamma}_{\alpha \beta}^{^+\mu}{\dot x}^{\alpha} {\dot x}^{\beta}=0 $$
 * $$(3.20)\qquad{\ddot x}_{m}+{\Gamma}^{^- m}_{a b}{\dot x}_{a} {\dot x}_{b} =0 $$

This set extends a duality to and is known as Geodesic Equation, where the motion accelerations of $${\ddot x^\mu}$$ and $${\ddot x_m}$$ are aligned in parallel to each of the world lines. It states that, during the inception of the universe, the tangent vector of the virtual $$Y^-Y^+$$ energies to the geodesic entanglements is either unchanged or parallel transport as an object moving along the world planes that creates the inertial transform generators and twist transport torsions to emerge a reality of the world.

Flux Continuity and Commutation
For the entanglement streams between the manifolds, the ensemble of an event $$\lambda$$ is in a mix of states such that each pair of the reciprocal states $$\{\phi_n^+,\varphi_n^-\}$$ or $$\{\phi_n^-,\varphi_n^+\}$$ is performed in alignment with an integrity of their probability $${p_n^\pm}={p_n}(h_n^\pm)$$, where $$h_n^\pm$$ are the $$Y^\pm$$ distributive or horizon factors, respectively. The parameter $$p^-_n$$ is a statistical function of horizon factor $${h_n^-}(T)$$ and fully characterizable by Thermodynamics. Under the event operations, the interoperation among four types of scalar fields of $${\phi_n^\pm}$$ and $${\varphi_n^\pm}$$ correlates and entangles an environment of dual densities $${\rho^+_{\phi}}={\phi_n^+}{\varphi_n^-}$$ and $${\rho^-_{\phi}}={\phi_n^-}{\varphi_n^+}$$ by means of the natural derivatives $${\dot \lambda}$$ to form a pair of fluxions $${\langle \dot \lambda \rangle}^\mp$$:


 * $$(4.1) \qquad

{\dot \lambda} {\rho^-_{\phi}}= { \langle {\check \lambda}, {\hat \lambda} \rangle }^{-} = \dot { \langle \lambda \rangle}^- = {\sum}_{n} {	{p_n^-} \Big (	{\varphi_n^+} {\check \lambda} {\phi_n^-} +	{\phi_n^-} {\hat \lambda} {\varphi_n^+} \Big ) }$$


 * $$(4.2) \qquad

{\dot \lambda} {\rho^+_{\phi}} = {\langle {\hat \lambda}, {\check \lambda} \rangle}^{+} = \dot {\langle \lambda \rangle}^+ = {\sum}_{n} p_n^+ \left (	{\varphi_n^-} {\hat \lambda} {\phi_n^+} +	{\phi_n^+} {\check \lambda} {\varphi_n^-} \right )$$

where the symbols $${\langle \ \rangle}^\mp$$ are called $$Y^-$$ or $$Y^+$$ Continuity Bracket. They represent the dual continuities of the $$Y^-Y^+$$ scalar densities, $$\rho^-_\phi$$ or $$\rho^+_\phi$$, each of which extends its meaning to an anti-commutator $$\{ \ \}$$. Considering another pair of the operational symbols $$[\dot \lambda]^\mp$$ known as commutators or Lie Bracket, introduced in 1930s [2], for respective $$Y^-$$ or $$Y^+$$ supremacy, the reciprocal entanglements of density fields are defined as Commutator Bracket $$[\cdot]^\mp$$ :


 * $$(4.3) \qquad { [

{\hat \lambda}, {\check \lambda} ] }^{+} = {\sum}_{n} {	{p_n^+} \left (	{\varphi_n^-} {\hat \lambda} {\phi_n^+} -	{\phi_n^+} {\check \lambda} {\varphi_n^-} \right ) } $$


 * $$(4.4) \qquad {[

{\check \lambda}, {\hat \lambda} ]}^{-} = {\sum}_{n} {	{p_n^-} \left (	{\varphi_n^+} {\hat \lambda} {\phi_n^-} -	{\phi_n^-} {\hat \lambda} {\varphi_n^+} \right ) } $$


 * $$(4.5) \qquad

\dot {\langle {\lambda} )}^{\pm} = {\varphi_n^\mp} {\dot \lambda} {\phi_n^\pm}, \qquad \ \dot {( {\lambda} \rangle}^{\pm} = {\phi_n^\pm} {\dot \lambda} {\varphi_n^\mp} $$

where, in addition, the symbols $$\dot {\langle {\lambda} )}^{\pm}$$ and $$\dot {( {\lambda} \rangle}^{\pm}$$ are called $$Y^-$$ or $$Y^+$$ Asymmetry Brackets. They are essential to cosmological dynamics.

Similarly, a set of the reciprocal vector fields of $${V^\pm_m}=-{\dot \partial}{\phi^\pm_m}$$ and $${\Lambda^\pm_\mu}\equiv-{\dot \partial}{\varphi^\pm_\mu}$$, has the brackets of $$Y^-$$ or $$Y^+$$ continuity and commutation:
 * $$(4.6) \qquad

{\langle {\hat \lambda},{\check \lambda} \rangle}^{\pm}_{v} \equiv {\sum}_n {p_n^\pm} \left (	{\varphi_n^\mp} {\hat \lambda} {V_n^\pm} +	{\phi_n^\pm} {\check \lambda} {\Lambda_n^\mp} \right ) $$


 * $$(4.7) \qquad

{[ {\hat \lambda},{\check \lambda}]}^{\mp}_{v} \equiv {\sum}_n {p_n^\mp} \left (	{\varphi_n^\pm} {\hat \lambda} {V_n^\mp} -	{\phi_n^\mp} {\check \lambda} {\Lambda_n^\pm} \right) $$


 * $$(4.8) \qquad

{\langle {\dot \lambda} )}^{\pm}_v = {\varphi_n^\mp} {\dot \lambda} {V_n^\pm}, \qquad \ {( {\dot \lambda} \rangle}^{\pm}_v = {\phi_n^\pm} {\dot \lambda} {\Lambda_n^\mp} $$

where the index $$n$$ is corresponds to each type of particle, and $$v$$ indicates entanglements of vector potentials, which respectively give rise to or balance each other’s horizon environment.

Entropy
A measure of the specific operations of ways is called Entropy in which states of a universe system could be arranged and balanced towards its equilibrium. As an operational duality, the entropy tends towards both extrema alternately to maintain a continuity of energy conservations, operated by each of the opponents. When a total entropy decreases, the intrinsic order, or $$Y^-$$ development, of virtual into physical regime $${\hat \partial_\lambda}{\hat \partial_\lambda} $$ is more dominant than the reverse process.

This philosophy states that for the central quantity of motion dynamics, conversely, when a total entropy increases, the extrinsic disorder, or $$Y^+$$ annihilation $${\check \partial^\lambda}{\check \partial^\lambda} $$, becomes dominant and conceals physical resources into virtual regime. There are two types of thermodynamic entropies. Each yields its own laws respectively.
 * 1) In Classic Thermodynamics, during the physical observations, the “internal” or “physical” operations result in its local effects parallel to the global domain with the entropy events of physical supremacy $$PdV$$, functions of the special transportation $$\mu_n dN_n^\pm$$, and states of symmetric property $$dE/T$$.
 * 2) In Cosmological Thermodynamics, as the virtual generators, “external” or “virtual” operations result in the projection or transform from its local effects to the neighbor domain with the events of virtual supremacy $$\Omega dJ$$, functions of general commutations $$\kappa dA/(8\pi)$$, and states of asymmetric property $$\Phi dQ$$, where $$\kappa$$ is the surface gravity, $$A$$ the horizon area, $$\Omega$$ the angular velocity, $$J$$ the angular momentum, $$\Phi$$ the electrostatic potential, and $$Q$$ the electric charge.

For an observation at long range, the commutation becomes a conservation of $$Y^-Y^+$$ thermodynamics, or is known as black hole radiations, which yields law of the area fluxions. The total entropy $$\mathcal S^+$$ represent law of conservation of area fluxions and defined by the following expression:


 * $$(4.7) \qquad

\mathcal{S}^\pm = 4 \mathcal{S}^\pm_A = {\dot \partial_\lambda} {\bf f}_s \qquad : {\dot \partial_\lambda} {\bf f}^\pm_s = {\kappa_s} \langle {\hat \partial_\lambda} {\hat \partial_\lambda} ,	{\check \partial^\lambda} {\check \partial^\lambda} \rangle^\pm $$

where $$\kappa_s$$ is factored by normalization of the potential fields. For a triplet quark system, the total entropy is at $$ 2 {\phi_a^+}({\phi_b^-}+{\phi_c^-}) \approx 4{\phi_a^+}{\phi_b^-} $$ fluxions.

Lagrangian
To seamlessly integrate with the classical dynamic equations, it is critical to interpret or promote the natural meanings of Lagrangian mechanics $$\mathcal{L}$$ in forms of the dual manifolds of Universal Topology. As a function of generalized information and formulation, Lagrangians $$\mathcal{L}$$ can be redefined as a pair of continuities, entangling between the $$Y^-Y^+$$ manifolds respectively:


 * $$(4.8) \qquad

\mathcal{L}^\pm \propto \langle {\hat \partial_\lambda} {\hat \partial_\lambda} ,	{\check \partial^\lambda} {\check \partial^\lambda} \rangle^\pm $$

The formulae generalize the Lagrangian and state that the central quantity of Lagrangian, introduced in 1788, represents the bi-directional fluxions that sustain, stream, harmonize and balance the dual continuities of entanglements of the $$Y^-Y^+$$ dynamic fields.

Artifacts
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, which are fully describable by the mathematical framework of the dual manifolds. This section provides some artifacts to demonstrate the outcomes of the Universal Topology.

Lorentz Generators
From the matrices (3.5) $${J^+_{\mu a}}={\partial x^\mu}/{\partial x_a}$$ and (3.7) $${J^-_{m \alpha}}={\partial x_m}/{\partial x^\alpha}$$, the Inertial Boosts $${J^\pm_{\mu a}}$$ of the two-dimensional world plans under the first horizon can naturally come out a pair of generators as the explicit matrix tables:


 * $$(5.1)\qquad{J^+_\mu}={L_\mu} -i{K_\mu}, \qquad \qquad \ \ {J^-_m}={L_m}+i {K_m}$$


 * $$(5.2)\qquad{K}_{x}=

\begin{pmatrix} 0 & 1 & 0 & 0 \\ 	 1 & 0 & 0 & 0 \\	 0 & 0 & 0 & 0 \\	 0 & 0 & 0 & 0 \\	\end{pmatrix}, \quad \ \ {K}_{y}= \begin{pmatrix} 0 & 0 & 1 & 0 \\ 	 0 & 0 & 0 & 0 \\	 1 & 0 & 0 & 0 \\	 0 & 0 & 0 & 0 \\	\end{pmatrix}, \quad \ \ {K}_{z}= \begin{pmatrix} 0 & 0 & 0 & 1 \\ 	 0 & 0 & 0 & 0 \\	 0 & 0 & 0 & 0 \\	 1 & 0 & 0 & 0 \\	\end{pmatrix} $$


 * $$(5.3)\qquad{L}_{x}=

\begin{pmatrix} 0 & 0 & 0 & 0 \\ 	 0 & 0 & 0 & 0 \\	 0 & 0 & 0 &-1 \\	 0 & 0 & 1 & 0 \\	\end{pmatrix}, \quad {L}_{y} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 	 0 & 0 & 0 & 1 \\	 0 & 0 & 0 & 0 \\	 0 &-1 & 0 & 0 \\	\end{pmatrix}, \quad {L}_{z}= \begin{pmatrix} 0 & 0 & 0 & 0 \\ 	 0 & 0 &-1 & 0 \\	 0 & 1 & 0 & 0 \\	 0 & 0 & 0 & 0 \\	\end{pmatrix} $$

This is similar to and known as Lorentz Generators, discovered since 1892. Conceivably, the extra $$\bf r$$-freedom is extended from the global world-planes into two of the rotations in physical space (such as $${L_x}$$ and $${L_y}$$). The Lorentz Generator is a foundation of transportation of light, also known as Photon.

Spin Generators
With one dimension r in the world planes, the manifolds are allowed to extend the extra freedom of the two dimensions to its spatial coordinates. If the $$L_\mu$$ and $$K_\mu$$ are assigned the Lorentz representation $$L_\mu \mapsto\sigma_i/2$$ and $$\pm K_\mu\mapsto\pm i\sigma_i/2$$ for a base transformation $$\hat x\mapsto\check x =\Lambda^+\hat x$$, the field $$\phi^+(\hat x)\mapsto\phi^-(\check x)=\phi^-(\Lambda^+\hat x)$$ transforms and gives rise to the spin fields $$S(\Lambda^\pm)$$ of particles:


 * $$(5.4)\qquad\phi^-(\check x)=S(\Lambda^+)\ \phi^+\Big({(\Lambda^+)}^{-1} \check x \Big)

\qquad \quad \ :\check x =\Lambda^+\hat x,\hat x=\Lambda^-\check x $$


 * $$(5.5)\qquad S(\Lambda^\pm)=

exp { \left(	{\frac 1 2} \sigma_\kappa \theta_\kappa	\mp	{\frac i 2} \sigma_\kappa \varphi_\kappa	\right) }\qquad :\Lambda^{\pm}= exp { \left(	{\frac {\omega_k} 2} J^{\pm}_\kappa \right) }$$


 * $$(5.6)\qquad{\sigma_\kappa}=

\left [ \begin{pmatrix} 1 & 0 \\ 	 0 & 1 \\ \end{pmatrix}_{0} , \ \begin{pmatrix} 0 & 1 \\ 	 1 & 0 \\ \end{pmatrix}_{1} , \ \begin{pmatrix} 0 &-i \\ i & 0 \\ \end{pmatrix}_{2} , \ \begin{pmatrix} 1 & 0 \\ 	 0 & -1 \\ \end{pmatrix}_{3} \right] $$

where $$\sigma_\kappa$$ are known as Pauli spin matrices, introduced in 1925. In chiral representation, the $$\sigma_\kappa$$ indexes of {0,3} are the diagonal elements of the matrix for a homogeneous environment. The indexes {0,1,2} define each of the rotational axes of the spatial dimensions under the tetrad-coordinates. The indexes {1,2} form a reciprocator to the {0,3} matrices. Intuitively simplified to a group of the 2x2 matrixes, the generators have the following commutation relationships:


 * $$(5.7)\qquad\sigma_a\sigma_b-\sigma_b \sigma_a=0, \quad\sigma_a\sigma_b+\sigma_b\sigma_a=2i\varepsilon_{abc}^{+}\sigma_c

\qquad : a,b, c \in (1,2,3) $$

where the Levi-Civita connection $$\varepsilon_{abc}^{+}$$ represents the right-hand chiral. In accordance with our anticipation, the zero commutator illustrates the distinct freedoms of physical supremacy that are degradable ($$\sigma_a^2=\sigma_0$$) back to the global $$\bf r$$ dimension. With the left-handed $$\varepsilon_{abc}^{+}$$ chiral, the non-zero continuity reveals the creation processes of the virtual supremacy. Therefore, defined as Spin Generator, these 2x2 tensors give rise to the quantum fields.

Chiral Entanglement
The interpretations of Figure 2.1 is that, when an axis passes through the center of an object, the object is said to rotate upon itself, or spin. Furthermore, when there are two axes passing through the center of an object, the object is said under the entanglements of the YinYang ($$Y^-Y^+$$) duality. During the first horizon, spin chirality is a type of the virtual and physical interactions that objects moving on the world lines generate the dual transformations of the $$Y^-$$ and $$Y^+$$ spinors, reciprocally, such that the nature appears the entanglement characterized by the left-handed and the right-handed chirality sourced from or driven by each of the manifolds of the virtual $$Y^+$$ and physical $$Y^-$$ dynamics. Following the trajectory, it takes in total two full rotations ($$720^o$$) from the $$W^-$$ to $$W^+$$ and then back to $$W^-$$ world plane, and vice versa, for an object to return to its original state. With its opponent companionship, the whole system yields the parity conservation by maintaining the duality reciprocally and simultaneously.

General Relativity
Generally, transportations between $$Y^-Y^+$$ manifolds are conserved dynamically. However, if the commutations between the manifolds were balanced at in a statically frozen or inanimate state $$ {[	{\hat \partial_\lambda} {\hat \partial_\lambda} ,	{\check \partial^\lambda} {\check \partial^\lambda} ]} =0 $$ without transportation contortions $$ {C}_{\nu}^{n} =0 $$, the commutation equation (4.7) and the horizon fields (3.15, 3.16) formulate General Relativity :


 * $$(5.8) \qquad G_{n \nu} = R_{n \nu} - \frac{1}{2} R g_{n \nu} $$

known as the Einstein field equations, discovered in November 1915. The theory has been one of the most profound discoveries of modern physics to account for general commutation in the context of classic forces. For a century, however, the philosophical interpretation remained a challenge until this topology was discovered in 2016.

Black Body and Black Hole
Every physical body spontaneously and continuously emits electromagnetic radiation. At near thermodynamic equilibrium, the emitted radiation is closely described by either Planck's law for black bodies or Bekenstein-Hawking radiation for black holes, or both. Because of its dependence on temperature and area, Planck and Schwarzshild radiation are said to be thermal radiation obeying area entropies. The higher the temperature or area of a body the more radiation it emits at every wavelength of light and gravitation. Since a black hole acts like an ideal black body as it reflects no light, their entropies of area law can represent radiations of light and gravitational.

Mass-energy
In mathematical formulations of entanglements, we redefine the energy-mass formations in forms of virtual complex as the following:


 * $$(6.1) \qquad {E^\mp_n}=\pm imc^2, \qquad \hbar \omega \rightleftharpoons mc^2$$

Compliant with a duality of Universal Topology $${W}=P \pm iV$$, it extends Einstein mass-energy equivalence, introduced in 1905, into the virtual energy formulae as one of the essential formulae of the topological framework.

Photon
Remarkably, an area energy fluxion of the potentials is equivalent to an entropy of the electromagnetic radiations. Applicable to the conservation of Lagrangians (4.8), it yields Planck’s law in thermal equilibrium of entropy :


 * $$(6.2) \qquad

S_A(\omega_c, T) = 4 \frac{ \hbar\omega_c^3} {4 \pi^3 c^2 k_B T} \Big( { e^{{\hbar \omega_c}/{k_BT}} - 1 } \Big)^{-1} \simeq \Big(\frac{ \omega_c^2} {\pi^3 c^2} \Big) = \eta_c \Big( \frac {\omega_c} {c} \Big)^2 \mapsto 4 \frac {E^-_c E^+_c} {( \hbar c )^2} , \qquad {\eta_c}= {\pi^{-3}} $$

where the factor 4 is compensated to account for one black body with the dual states at minimum of two physical $$Y^-$$ and one virtual $$Y^+$$ quarks. The above equivalence results in a pair of the complex formulae:


 * $$(6.3) \qquad {E^\pm_c}=\mp i \frac 1 2 {\hbar \omega_c},

\qquad {\eta_c}={\pi^{-3}}\approx 33 \% $$

The coupling constant at 33% implies that it is the triplet quarks that institute a pair of the photon energies $$\mp i {\hbar \omega_c}/2$$ for a black hole to emit lights by electromagnetic radiations.

Graviton
By associating spacetime horizon factor with Schwarzschild radius $$r_s=2GM/c^2$$ of black hole, derived in 1915, an area entropy $$S_A$$ of the quantum-gravitational radiance of a black hole is given by frequency at absolute temperature $$T$$ and constant speed $$c_g$$ as the following:


 * $$(6.4) \qquad S_A(\omega_g, T)=\frac {c^3_g} {4 \hbar G} $$

known as Bekenstein-Hawking radiation , introduced in 1974. Gravitation exhibits wave–particle duality such that its properties must acquire characteristics of both waves and particles. Integrating with the black hole thermal radiance, gravitational fluxion $$4{E^-_g E^+_g}/{(\hbar c_g)^2}$$ has the transportable commutation of area entropy $$S_A$$ and conservable radiations of a Schwarzschild black body. It is equivalent to associate it with the above Bekenstein-Hawking radiation. Therefore, it derives the graviton formula, shown as the following


 * $$(6.5) \qquad S_A(\omega_g, T)

= 4 \Big (\frac{c_g^3} {4 \hbar G}\Big ) \mapsto 4 \frac {E^-_g E^+_g} {(\hbar c_g)^2} $$

where the number 4 is factored or normalized for a dual-state system of triplet quarks similar to (6.2). Consequently, the gravitational energies $$E_g^\pm$$ contain not only a duality of the complex functions but also an irreducible unit: Graviton, introduced in 2017, as a pair of graviton units:


 * $$(6.6) \qquad {E^\pm_g}=\mp i{\frac 1 2}{E_p} \qquad : {E_p}=\sqrt {{\hbar c_g^5}/{G}} $$

where $$E_p$$ is Plank energy. It exhibits a coupling constant 1/2 to emit gravitational radiations, meaning a minimum of a pair of gravitons for a black hole emanations. Therefore, this result concludes that the entanglement duality of the Spiral Coordinators generates Gravitons.

Conclusion
Universal Topology has revealed a set of the following discoveries or groundbreakings:
 * 1) To align closely with life-streams of our natural world, the Dual complex manifolds are established that overcomes the limitations of a single spacetime manifold.
 * 2) Two pairs of the potential fields lies at the heart of the field theory for the fundamental interactions among the dark energies.
 * 3) Mathematical Framework is imperatively regulated on a new theoretical foundation by the dual variances to intimately mimic event actions of transform and transport processes.
 * 4) Motion Operations and geodesic equtions are further regulated on and performed with a new theoretical foundation of the dual events intimately mimic operational actions on the geodesic covertures, extend the meanings to the Euler-Lagrange Motion Equation.
 * 5) Seamlessly integrations with Thermal entropy and Lagrangian mechanics represents that the philosophy of Universal Topology is not only applicable to and but also extendable the meanings of our classic or mordern physics concisely aligned with the mathematical framework.
 * 6) In the center of a black hole, a system of partial differential equations forms the entanglements of gravitational and electromagnetic fields and emerges the associated forces from massive objects for internal communications.
 * 7) Essentially, on the natural two-dimensional World Planes, the inertial boost $${J^\pm_{\mu a}}$$ and spiral torque $${K^\pm_{\mu a}}$$ tensors constitute and act as the sources of “photon” and “graviton” fields being operated at the heart of energy formulations of stress strengths and twist torsions, driven by the events descending from the two-dimensional world planes of the dual manifolds and the affine connections aligning to each of the curvatures.

As a result, it has laid out a ground foundation towards a unified physics that give rise to the fields of quantum, photon, electromagnetism, graviton, gravitation, thermodynamics, cosmology, ontology and beyond.