Knowledge Conservation Cycle

Abstract

Languages from Physics and Mathematics are represented in the Language of a Conservative Cycle. This represents the known Languages to be a Form of “simple mass balance”. Every Fundamental Law in Physics, Mathematics or any other Language is represented to be seen as a form of Duality of Knowledge which is Identified as a Cycle to give a Conserved Quantity. This is further reinterpreted as: it is this Conserved Quantity (Identity) which takes all kinds of possible Forms in the Cycle. In Symmetry Language, Asymmetries may be Identified to Symmetry (Identity). In Nature Language, Equilibrium is a Dynamic exchange between two Phases which is a Cyclic Balance.

“Knowledge is Conserved” (The Fundamental Law of Language Cycle)

I want to state: “Knowledge takes Cyclic Form” or just : “Knowledge is”, or its Differential form: “Dual Forms is Identified”. But it will take us some time to get there.
For now, let us state the Fundamental Law as: Knowledge takes various forms but the “Amount of Knowledge” is the same or “Knowledge is conserved in a Cycle”. “Everything is an Equilibrium Cycle and part of a greater Equilibrium Cycle until Identity is reached”.
To some of us this might seem not to be a law at all, after all we have seen so many processes which don’t have conservation. I shall request that we be patient. I will first work out several detailed examples for us to follow.
- Take any irreversible process in which heat is generated as an example and we see that Energy is not conserved, that is, it gets dissipated as heat. But if we include the dissipated energy to be part of the system then the total Energy is conserved. The cycle of Energy (Thermodynamics) is a prime example of Cyclic Balance or Conservation.
This might seem trivial and is very much so but it is for this very reason that its Dual can take non-trivial form.
I shall represent everything I know as a form of the Law - “Knowledge is Conserved”. That is, I shall Differentiate the Law to its Differentiated Forms. Since the Knowedge I know “amounts to nothing”, I will appreciate if anyone would like to show me the Laws they know, so that I can see those Laws as a form of this Law too.
At the end I shall make a statement for “why all this Duality?”. This statement “amounts to nothing”. It will then be clear why all “Fundamental Laws” are called “Fundamental Laws”. In fact, it will be clear that All “Fundamental Laws are statements that amount to nothing”.
All known Fundamental Laws will be shown to be “Absolutely Rigid in the Language at their repsective Degree”, that is, “Fundamental Laws of a Language form the Irreducibles of a Language” or “Eigen-Laws of the Language”. This means “Fundamental Laws are Equilibrium Statements” or “Fundamental Laws are Equilibrium Phase Boundaries of a Language”.
By virtue of all Fundamental Laws being the Equilibrium Phase Boundaries of a Language, all the above made statements will follow. It will also be clear that all “Fundamental Laws” have no Degree of Freedom with respect to their Language at their Degree, that is, they cannot be Differentiated further in that Language at that Level. This is why they are Fundamental Laws: they are Invariants of the Language. This is not to be confused with: Fundamental Laws cannot exist in various forms- they can take as many forms as possible, and in fact, they do- “There is a Fundamental Law at every Degree of Freedom in a Language”. The Key Point to note is that the Differentiation used needs to be in the Language of which they are the Equilibrium Phase Boundary and at the respective Degree of Freedom. Fundamental Laws of a Language are related to each other by Differentiation and Integration.
The condition for a Law (E) to be a Fundamental Law ($E^{*}$) at the Degree of Freedom $i$ was derived in Equilibrium Thermodynamics :
\[d_{i}(E^{i+1})_{E^*} =0\]
Fundamental Laws in a Language, at the degree of freedom $i$ arise from The Fundamental Law in the Language of Calculus:

\[(-d_{i}) \,\, (d_{i}+) = 0\]

Mass Balance as a form of “Knowledge is Conserved”

The typical example of “Knowledge is conserved” is the law of mass balance. This law states that mass stays conserved no matter “how we divide it into different forms”. In the Language of Duality, this corresponds to “Knowledge stays conserved no matter how we think (divide) it into different Languages”.
- We see this appearing throughout in Nature. No matter what system we take, we find that dividing it into pieces doesn’t change its mass.

Projection Duality as form of “Knowledge is Conserved”

Sterographic projection or any kind of projection is a form of Differentiation. The Angle is the dual (differential form) of the Radius, a Line is the Dual of a Circle, the Plane is the Dual of the Sphere and the $n-Sphere$ is the dual of $n$ Dimensional Plane, which is its differential form. In this form, the “Knowledge stays conserved no matter how we divide it into or Line”. This gives the Fundamental Laws :
Length along line = $\theta r$
Length along plane (Area) = $\theta r$ composed with $\,\, r\theta\,\, = \,\, r \cdot r \,\, (\frac{\theta}{2}) = \,\, r^2 (\frac{\theta}{2})$
Length along $n-1$ dimensional plane = differential form of $S^n$.

Electric Field-Electric Charge Duality as a form of “Knowledge is Conserved”

The well known Law relating Electric Field to Electric Charge is just the statement “Knowledge stays conserved no matter how we represent (divide) it into different Languages”. Knowledge in this case is the “measure”

\[measure(q) = measure(E)\]

The way to “measure” the “amount of Mass” at the degree in which Charge exists is simply its scalar value or its Fundamental Law (Scalar Values will be later seen to be the Fundamental Laws at the Degree 1). The way to “measure” the “amount of Mass” at the degree in which Electric Field exists to meet with its dual Charge is its Flux. This is the Fundamental Law which Gauss had also discovered.
The Fundamental Law itself can be differentiated to give its Differential Form, which states that Charge is the Dual of Electric field, which is another way of stating a conservation law, here Flux (from its Integral) becomes divergence (Differential form):

\[q = divergence(E)\]

Magnetic Field-Magnetic Charge Duality as a form of “Knowledge is Conserved”

The well known Law relating Magnetic Field to Magnetic Charge is just the statement “Knowledge stays conserved no matter how we represent (divide) it into different Languages”. Knowledge in this case is the “measure”

\[measure(q_{m}) = measure(B)\]

The way to “measure” the “amount of Mass” at the degree in which Charge exists is simply its scalar value or its Fundamental Law (Scalar Values will be later seen to be the Fundamental Laws at the Degree 1). The way to “measure” the “amount of Mass” at the degree in which Magnetic Field exists to meet with its dual Charge is its Flux. This is the Fundamental Law- Gauss’s Law for Magnetism had also discovered.
The Fundamental Law itself can be differentiated to give its Differential Form, which states that Charge is the Dual of Electric field, which is another way of stating a conservation law, here Flux (from its Integral) becomes divergence (Differential form):

\[q = divergence (B)\]

Magnetic Monopoles do exist but not in our Universe. They exist in a Degree of Freedom that is Dual to our Universe. Hence, they cannot be seen. However they are Dual to Electric Charges. That is, Magnetic Monopoles create Electric Monopoles and vice Versa. The Orthogonality of meeting points of Magnetic and Electric Fields also shows that they are Dual to each other. Orthogonality is a form of Duality (Fundamental Law).

Space-Time Duality as a form of “Knowledge is Conserved”

Space and Time are duals of each other.
Measure of space is equal to measure of Time is seen at the Equilibrium Phase Boundary of Space and Time: The Light Cone. This is what Einstein discovered in the Metric. The Einstein’s Field Equations are the set of Maxwell’s Relations that can be formed by composing Dualities.
Another way to see this is by seeing that Space and Time meet “orthogonal” to each other. This is a sign of Duality.
Space-Time Duality composed with Electro-Magnetic Duality produces Light or one may look at it in the Dual direction: Light produces Space-Time and Electro-Magnetism. Light is the Identity for Space Time and Electromagnetism.

Quantum Mechanics as a form of “Knowledge is Conserved”

Particle and Wave are the dual forms. Spectrum and Operator are the corresponding Duals in Mathematics giving the eigevalue equation:

\[\hat{H} \,\,\Psi = E\,\, \Psi\]

The Duality of Fundamentals $i$ and $\hbar$ gives rise to $i\hbar$ being conserved.
Amount of Energy in Particles plus the amount of Energy in waves must stay conserved or together (Dual). Amount of Mass in Space Duality between Space-Time is stated in Qunatum Mechanics along with composing the Particle-Wave Duality. Hence this Equation is to be read in two Dual Directions- one involving space time duality and the other involving Particle-Wave Duality which is represented by the Wave function. The “New Space” produced by composing the two Dual directions is the Energy.
\[(-d_{Space}d_{Space}+) \Psi =(i\hbar) d_{time}(\Psi)\]
The above equation is a Maxwell’s Relation (Thermodynamics) corresponding to the Equilibrium Phase Diagram of Particle-Wave Duality.

Quantum-Gravity

If Quantum Mechanics and Gravity are to be treated as Independent Variables (Dual Varibales): A new Energy ($E$) must be defined having degrees of freedom in Space-Time Duality as well as in Particle-Wave Duality, That is the new Hamiltonian now acts on $(\Psi, S)$ as the Independent Variables. The metric (64 dimensional: 8 cross 8 matrix) then takes the form

\[(dE)^2 = (\hbar)^2\partial^t \Psi^{\dagger}(x_1,x_2,x_3,t) \partial_t \Psi(x_1,x_2,x_3,t)-(\sum_{a = x_1,x_2,_x_3} \partial^a \Psi^{\dagger}(x_1,x_2,x_3,t) \partial_a \Psi(x_1,x_2,x_3,t))-(c^2(dt)^2-(dx_1)^2-(dx_2)^2-(dx_3)^2)\]

The condition for Equilibrium is then $dE =0$.
This results in a new wave with a new fundamental speed of propagation in the direction orthogonal to the Space-Time and the Quantum mechanical wave. The new fundamental speed $\S$ is given by $\S^2= \frac{1}{c \hbar}$. The corresponding Mawxell-Type Equations are obtained by composing the Duality of $\Psi$ and $S$.
A new force with these extra dimensions, a total of 8 (to combine both $\Psi$ and $S$) which is the Integral version of Gravity (of space-time) must be named. This force is then Dual to a Potential Energy of 64 dimensions.

The Integral Theories to Quantum-Gravity

When the Equation above was written it automatically sets up its dual equation which can be further composed with this one to go to higher and higher Degrees of Metrics. This is no different than doing Euclidean Geometry in Higher and Higher Dimensions. The fundamental Law of Each Dimension is just the Axis.

Cycle is Fundamental Law of Balance - Every Fundamental Law of Balance is a Cycle

Every Fundamental Law is a Duality.
- Equilibrium Phase Diagram is a Cyclic Balance between the Phases. Phases on either side get identified at the Phase Boundary.
Every Fundamental Law is a Composition Law of Duality. This means that every Fundamental Law is obtained by Composing Dualities. This is just a restatement of the fact that every Fundamental Law is an Equilibrium Phase Boundary. In particular, every Mathematically correct Equation is an Equilibrium Phase Boundary which can be obtained by composing Dualities.
This is why Fundamental Laws are Natural: They Appear and Disappear naturally at the same time as expressions of Duality.
Every Equilibrium Phase Diagram is a Fundamental Law and all that is a Fundamental Law is a Phase Diagram. The Universe itslef is a Phase in a Phase Diagram. On the other side of the Equilibrium Phase boundary of our Universe is our Dual Universe. Differentiation and Integrations connects all Phases to one another and hence allows us to go to Higher Degrees of Freedom.
The Fundamental Laws themselves have Degree of Freedom lesser than the Degree of freedom at which level they are the Fundamental Law. This follows as Fundamental Laws are the Phase Boundary where two or more Phases meet or the Fundamental Laws are common to both Phases.
Differentiating a Fundamental Law of Degree 1 gives two Fundamental Laws of Degree $1/2$ one corresponding to Even Parity and another corresponding to Odd Parity. Similarly, on Differentiating further, one may obtain Fundamental Laws of degrees of freedom that are positive powers of $1/2$. Since they have lesser than one Degree of Freedom, they cannot be seen in the usual Positive Integer Valued Dimensional Spaces. However, they have more than Zero Degrees of Freedom. To go below Zero degrees of Freedom, one needs Differentiation of an entirely Different Degree- at a higher level than the Degree of Freedom of Differentiation.
Dark Matter has Negative Degrees of Freedom and is a Dual Phase to our entire Universe. Their existence is obvious since every Phase must has its dual, so our Universe has its Dual too. We are separated from the Dual Universe by an Equilibrium Phase Boundary hence we do not have the Degree of Freedom to go there as we are. However, if we Differentiate ourselves, we can go there leading to cancelltion with our Dual Phase. This is similar to how positive numbers when composed with Negative Numbers gives 0. Or Look at any equation where numbers are to be moved from one side to the other, there is opposite sign and then cancellation. This can be Imagined by seeing the Reflection Symmetry in Mirrors or composition in Numbers. Any Equation be it Stokes Theorem or Pythagoras Theorem is a display of this.
- For example, + and - are Fundamental Laws in the Language of Numbers at the Degree of freedom 1. Each of them individually have Degree of Freedom $1/2$. + corresponds to Even Symmetry, - corresponds to Odd Symmetry.
- For example, Differentiation and Integration are Fundamental Laws in the Language of Duality Calculus at the Degree of freedom 1. Each of them have Degree of Freedom $1/2$. Integration corresponds to Even Symmetry, Differentiation corresponds to Odd Symmetry.
- For example, Fermions and Bosons are Fundamental Laws in the Language of Particles at the Degree of freedom 1. Each of them have Degree of Freedom $1/2$. Bosons correspond to Even Symmetry, Fermions correspond to Odd Symmetry.
- For example, Left and Right are Fundamental Laws in the English Language of Space at the Degree of freedom 1. Each of them have Degree of Freedom $1/2$. Left corresponds to Even Symmetry, Right corresponds to Odd Symmetry.
- For example, $\sin{\theta}$ and $\cos{\theta}$ are Fundamental Laws in the Euclidean Geometry of Space at the Degree of freedom 1. Each of them have Degree of Freedom $1/2$. $\sin{\theta}$ corresponds to Even Symmetry, $\cos{\theta}$ corresponds to Odd Symmetry.
Each Fundamental Law has its Dual Law. They are Related by the Duality Relations, that is, one may be obtained by Differentiating the other may be obtained by Integrating. This can be restated as a statement of Mirror Symmetry: Fundamental Laws are Mirror Reflections of Each other. Hence, at the dimension $n$, there are $2^n$ Fundamental Laws.
- For Example: All n-Spheres for $n$ are Fundamental Laws in the Language of Euclidean Measure. For $n$ greater than 0, $S^n$ is a Fundamental Law at the the degree of freedom $n+1$ in the Language of Euclidean Measure. $S^0$ is a Fundamental Law at the the degree of freedom $1$ in the Language of Euclidean Measure. Each of them individually have Degree of Freedom $1/2$. Point of $S^0$ on the Right side corresponds to Even Symmetry, Point of $S^0$ on the Left side corresponds to Odd Symmetry. **Poincare Conjecture is a Fundamental Law because it is a composition of the Duality of

The Fundamental Laws in the Language of Number Theory

$0$ and $\infty$ are Dual to each other.
$0$ and $\infty$ are the Fundamental Laws at the Degree 1.
The above Fundamental law is stated as : Duality of $0 \quad \infty$.
Whole Numbers are the Fundamental Laws in the Language of Number Theory. The number $n$ is a Fundamental Law at the degree $n+1$.
This is interpreted as Differentiating the $\infty$$ Language at the Degree $i$ requires going beyond the Degree

The Fundamental Laws in the Language of Complex Numbers

Fundamental Laws are +,-. This composed with the Differentiation operator $d$ produce the Fundamental Laws $\sin$ and $cos$.
The Riemann Hypothesis is true and its a Fundamental Law in a Sub-Language of the Language of Complex Numbers. This can be proved by showing that the Composition of Fundamental Laws generated by Duality of negative evens with the Duality of $\frac{1}{2}+it$ produces the Zeta Function. Details will be updated shortly.

The Fundamental Laws in the Language of Euclidean Geometry

1 is the Fundamental Identity.
+ and - are Fundamental Laws in the Language of Euclidean Geometry. 1 and -1 are Fundamental Laws in the Language of Euclidean Geometry.
$\theta$ is a Fundamental Law of Measure $2\pi$.
$\sin(\theta)$ and $\cos(\theta)$ are both Fundamental Laws in the Language of Euclidean Geometry obtained by composing the Duality of . Each of them individually have Measure $2$ since they are formed between the duality of $-1$ and $1+$, that is they Integrate to $2$.

The Fundamental Laws in the Language of Powers

$1$ is the Fundamental Identity.
$-\infty$ and $\infty$ are Dual to each other. This Duality creates (Integrates) to $\log$. Simply place together $-\infty$ and $\infty$ -this horizontal composition line (or axis) is the degree of freedom within which Logarithm exists. Logarithm is the “Gravitational Force” between $-\infty$ and $\infty$.
$0$ and $\infty$ are Dual to each other. This Duality creates (Integrates) to $\exp$.
All remaining Fundamental Laws are formed by composition of these dualities. Composition of Dualities is how one Integrates
This is how Ramanjuan and other geniuses “generate Laws out of nowhere”- anything we see are forms of numbers interacting with each other. All equations are pictures of Equilibrium Phase Diagrams and all of these are generated out of Identity simply by composing the Dualities involved. English and Art(drawing) are Integral versions of Mathematics but the same applies there too. The Classics are all Fundamental Laws which exhibit symmetry. Diamond (most discrete symmetries) is hence the most rigid material.
All of us are forms of each other which Identify to a Cycle. Connecting each other is the Cycle. Communicating with each other is the Cycle. Identifying each other composes us together to form an Integral Cycle.
Everything is in Dynamic Equilibrium (which is both Complete at one degree and Incomplete at the other degree).
Duality exists for a phase because it does not exist in the other phase. Duality exists as a pahse because Non-Duality exists as a phase. Non-Duality exists as a phase because Duality exists as a Phase. Non-Duality exists as a phase because it does not exist in the other Phase.