Dear Ivette,
Please see the ISA function and its implications on our understanding or reality,
ISA function describes Riemann sphere and is the true analytic continuation of Riemann zeta function
ISA(S) = Sign(Re(s)) * Sign(Im(s)) * Cos(Re(s) * Pi) * e Power (Im(s) * i * Pi)
ISA(1+1/2i) = Sign(Re(1+1/2i)) * Sign(Im(1+1/2i)) * Cos(Re(1+1/2i) * Pi) * e Power(Im(1+1/2i) * i * Pi)
ISA(1+1/2i) = 1 * 1 * Cos(1 * Pi) * e Power(1/2 * i * Pi) = 1 * - 1 * e Power (i * pi /2) = i (Rotate point -1 to 90 degrees in the direction of i)
ISA(1-1/2i) = Sign(Re(1-1/2i)) * Sign(Im(1-1/2i)) * Cos(Re(1-1/2i) * Pi) * e Power(Im(1-1/2i) * i * Pi)
ISA(1-1/2i) = 1 * -1 * Cos(1 * Pi) * e Power(-1/2 * i * Pi) = -1 * - 1 * e Power (-i * pi /2) = -i (Rotate point 1 to 90 degrees in the direction of -i)
ISA(-1+1/2i) = Sign(Re(-1+1/2i)) * Sign(Im(-1+1/2i)) * Cos(Re(-1+1/2i) * Pi) * e Power(Im(-1+1/2i) * i * Pi)
ISA(-1+1/2i) = -1 * 1 * Cos(-1 * Pi) * e Power(1/2 * i * Pi) = -1 * - 1 * e Power (i * pi /2) = i (Rotate point 1 to 90 degrees in the direction of i)
ISA(-1-1/2i) = Sign(Re(-1-1/2i)) * Sign(Im(-1-1/2i)) * Cos(Re(-1-1/2i) * Pi) * e Power(Im(-1-1/2i) * i * Pi)
ISA(-1-1/2i) = -1 * -1 * Cos(-1 * Pi) * e Power(-1/2 * i * Pi) = - 1 * e Power (-i * pi /2) = -i (Rotate point -1 to 90 degrees in the direction of -i)
ISA(1+i) = Sign(Re(1+i)) * Sign(Im(1+i)) * Cos(Re(1+i) * Pi) * e Power(Im(1+i) * i * Pi)
ISA(1+i) = 1 * 1 * Cos(1 * pi) * e Power (1 * i * Pi) = -1 * e Power (i * Pi) = 1 (Rotate point -1 to 180 degrees in the direction of i)
ISA(1-i) = Sign(Re(1-i)) * Sign(Im(1-i)) * Cos(Re(1-i) * Pi) * e Power(Im(1-i) * i * Pi)
ISA(1-i) = 1 * -1 * Cos(1 * pi) * e Power (-1 * i * Pi) = -1 * -1 * e Power (-i * Pi) = -1 (Rotate point 1 to 180 degrees in the direction of -i)
ISA(-1+i) = Sign(Re(1+i)) * Sign(Im(1+i)) * Cos(Re(1+i) * Pi) * e Power(Im(1+i) * i * Pi)
ISA(-1+i) = -1 * 1 * Cos(-1 * pi) * e Power (1 * i * Pi) = -1 * -1 * e Power (i * Pi) = -1 (Rotate point 1 to 180 degrees in the direction of i)
ISA(-1-i) = Sign(Re(1-i)) * Sign(Im(1-i)) * Cos(Re(1-i) * Pi) * e Power(m(1-i) * i * Pi)
ISA(-1-i) = -1 * -1 * Cos(-1 * pi) * e Power (-1 * i * Pi) = 1 * -1 * e Power (-i * Pi) = 1 (Rotate point -1 to 180 degrees in the direction of -i)
ISA(1) = Sign(Re(1)) * Sign(Im(1)) * Cos(Re(1) * Pi) * e Power(Im(1) * i * Pi)
ISA(1) = 1 * 0 * Cos(1 * Pi) * e Power(0 * i * Pi ) = 0
ISA(-1) = Sign(Re(-1)) * Sign(Im(-1)) * Cos(Re(-1) * Pi) * e Power(Im(-1) * i * Pi)
ISA(-1) = -1 * 0 * Cos(-1 * Pi) * e Power(0 * i * Pi ) = 0
ISA(i) = Sign(Re(i)) * Sign(Im(i)) * Cos(Re(i) * Pi) * e Power(Im(i) * i * Pi)
ISA(i) = 0 * 1 * Cos(0 * Pi) * e Power(1 * i * Pi) = 0
ISA(-i) = Sign(Re(-i)) * Sign(Im(-i)) * Cos(Re(-i) * Pi) * e Power(Im(-i) * i * Pi)
ISA(-i) = 0 * -1 * Cos(0 * Pi) * e Power(-1 * i * Pi) = 0
ISA(0) = Sign(Re(0)) * Sign(Im(0)) * Cos(Re(0) * Pi) * e Power(Im(0) * i * Pi)
ISA(0) = 0 * 0 * Cos(0 * Pi) * e Power(0 * i * Pi) = 0 * 1 * 1 = 0
ISA(Infinity) = Sign(Re(Infinity)) * Sign(Im(Infinity)) * Cos(Re(Infinity) * Pi) * e Power(Im(Infinity) * i * Pi)
ISA(Infinity) = Infinity* Infinity* Cos(Infinity* Pi) * e Power(Infinity* i * Pi) = Infinity * [1, -1] * Infinity = Infinity
Plotting all other values of s other than mentioned above, ISA(s) plots points on a circle of radius absolute(Cos(Re(s) * Pi)), for example
ISA(1/3+ni) = Sign(Re(1/3+ni)) * Sign(Im(1/3+ni)) * Cos(Re(1/3+ni) * Pi) * e Power(Im(1/3+ni) * i * Pi) (n is any real number)
ISA(1/3+ni) =1 * 1 * Cos(Pi / 3) * e Power(n * i * Pi) = 1 * 0.5 * e Power (n * i * Pi) (Describes a circle of radius 0.5 around the origin)
ISA(1/4+ni) = Sign(Re(1/4+ni)) * Sign(Im(1/4+ni)) * Cos(Re(1/4+ni) * Pi) * e Power(Im(1/4+ni) * i * Pi) (n is any real number)
ISA(1/4+ni) =1 * 1 * Cos(Pi / 4) * e Power(n * i * Pi) = 1 * 0.70710678 * e Power (n * i * Pi) (A circle of radius 0.70710678 around the origin)
Plotting ISA(s) for any Re(s) or any Im(s) on 1-dimensional X-axis or 1-dimensional Y-axis yields a 0-D point (point at zero or point at infinity)
Plotting ISA(s) for all Re(s) or all Im(s) on 1-dimensional X-axis or 1-dimensional Y-axis yields a 1-D Z-axis (point at zero and point at infinity)
Plotting ISA(s) for all Im(s) for any Re(s) on 1-D X-axis and 1-D Y-axis yields a 2-D 1-sphere (circular disc of radius abs(Cos(Re(s) * Pi)) )
Plotting ISA(s) for all Im(s) for any Re(s) on 2-D X-axis and 1-D Y-axis yields a 3-D 2-sphere (hollow sphere of radius abs(Cos(Re(s) * Pi)) )
Plotting ISA(s) for all Im(s) for any Re(s) on 3-D X-axis and 1-D Y-axis yields a 4-D 3-sphere (space ball moving in time, universe)
Plotting ISA(s) for all Im(s) for any Re(s) on 3-D X-axis, 1-D Y-axis and 1-D Z-axis yields a 5-D 4-sphere (Hypersphere of multiple universes)
(Universe is part of a 5-D Hypersphere (wikipedia links), 4-sphere, a self-illuminated space ball moving in time and is consciously self-aware)
Proof of Riemann Hypothesis using ISA function
ISA(1/2 + i) = Sign(Re(1/2 + i)) * Sign(Im(1/2 + i)) * Cos(Re(1/2 + i) * Pi) * e Power(Im(1/2 + i) * i * Pi)
ISA(1/2 + i) = 1 * 1 * Cos(1/2 * Pi) * e Power(1 * i * Pi) = Cos(Pi/2) * e Power (i * Pi) = 1 * 0 * e Power (i * Pi) = 0
ISA(1/2 + 2i) = Sign(Re(1/2 + 2i)) * Sign(Im(1/2 + 2i)) * Cos(Re(1/2 + 2i) * Pi) * e Power(Im(1/2 + 2i) * i * Pi)
ISA(1/2 + 2i) = 1 * 2 * Cos(1/2 * Pi) * e Power(2 * i * Pi) = 2 * Cos(Pi/2) * e Power (2 * i * Pi) = 2 * 0 * e Power (2 * i * Pi) = 0
ISA(1/2 + ni) = Sign(Re(1/2 + ni)) * Sign(Im(1/2 + ni)) * Cos(Re(1/2 + ni) * Pi) * e Power(Im(1/2 + ni) * i * Pi)
ISA(1/2 + ni) = 1 * 1 * Cos(1/2 * Pi) * e Power(n * i * Pi) = 1 * Cos(Pi/2) * e Power (n * i * Pi) = 1 * 0 * e Power (n * i * Pi) = 0
ISA(-1/2 - i) = Sign(Re(-1/2 - i)) * Sign(Im(-1/2 - i)) * Cos(Re(-1/2 - i) * Pi) * e Power(Im(-1/2 - i) * i * Pi)
ISA(-1/2 - i) = -1 * -1 * Cos(-1/2 * Pi) * e Power(-1 * i * Pi) = 1 * Cos(-Pi/2) * e Power (-i * Pi) = 1 * 0 * e Power (-i * Pi) = 0
ISA(-1/2 - 2i) = Sign(Re(-1/2 - 2i)) * Sign(Im(-1/2 - 2i)) * Cos(Re(-1/2 - 2i) * Pi) * e Power(Im(-1/2 - 2i) * i * Pi)
ISA(-1/2 - 2i) = -1 * -2 * Cos(-1/2 * Pi) * e Power(-2 * i * Pi) = 2 * Cos(-Pi/2) * e Power (-2 * i * Pi) = 2 * 0 * e Power (-2 * i * Pi) = 0
ISA(-1/2 - ni) = Sign(Re(-1/2 - ni)) * Sign(Im(-1/2 - ni)) * Cos(Re(-1/2 - ni) * Pi) * e Power(Im(-1/2 - ni) * i * Pi)
ISA(-1/2 - ni) = -1 * -1 * Cos(-1/2 * Pi) * e Power(-n * i * Pi) = 1 * Cos(-Pi/2) * e Power (-n * i * Pi) = 1 * 0 * e Power (-n * i * Pi) = 0
The reason for ISA(s) being 0 at all points of abs(Re(s)) = 1/2 is the effect of simultaneous rotation of invisible Z-axis when a point on X-axis is rotated.The usage of Cos(Re(s) * Pi) was formulated in the ISA function after visualizing this rotational effect of Z-axis on X-axis points while transformation. ISA function mentioned in this article is for a simple complex plane of 1-D real and 1-D imaginary axis.
For study of complicated complex plane involving 3-D real, 1-D imaginary and 1-D soul axis we need to use mathematics of Complex coordinate space and Spherical coordinate systems. As an alternative we can also plot the ISA function for a pair of dimensions, one of these dimensions must be 1-D imaginary, and superimpose the plotted values of multiple pairs to arrive at the desired result of the complex coordinate space. As an example plotting 1-D Real and 1-D imaginary pair and another 1-D Real and 1-D imaginary pair and combining the two is equivalent to 3-D plot of a 2-D Real and 1-D imaginary as we can pivot to 1-D imaginary axis which is common in the initial two pairs. In a similar way we can plot for 4-D manifestations. When we do this kind of plotting we will manifest different forms of beings in 3-D at a given time and over time those forms will appear to be animated or in motion in 4-D. Extending this process to the 5th dimension with multiple time lines will yield a spectacle of several animated objects in their respective timelines, like watching several different movies on multiple screens at once. From the above diagram, each 4-D reference frame is at a particular constant distance from the Source Of Universal Light and hence for beings within a single 4-D frame perceive constant speed of light in all directions irrespective of their relative motion with in that frame, which validates the special theory of relativity which deals with beings in a single 4-D frame. From our previous analogy, it's like characters on one movie screen are illuminated (brightness and contrast levels) at a different level from characters on another movie screen. But with in a movie screen all the characters have a relatively fixed rate of illumination.
Implications : In the most abstract sense the universe is purely mathematical. Also the proof of ISA function describing Riemann sphere shows the holographic nature of our universe. Any point on the 2-dimensional complex plane can be plotted on to Riemann sphere using the ISA function. Also ISA function proves the generalized Poincare conjecture as it describes 0-dimensional point to n-dimensional spheres when properly extended to the corresponding dimensions. Surface of the Riemann sphere is the event horizon, I see the universe as part of a 5-D Light Whole Noumenon instead of the currently popular 0-D black hole phenomenon, as what lies with in the sphere totally encompasses what lies outside of it.
The most fundamental aspect of our universe is the Source Of Universal Light (SOUL or Consciousness). It is the consciousness that brings forth the light, space and time and molds them in to relativistic matter and moves them as it wishes, like a kid throwing stones in a pond and creating water droplets and ripples just for fun. In this analogy the kid is the consciousness, stone is the light, water is the space-time fabric. So then answer to Yang-Mills least massive particle possible is infinitesimal, from our earlier analogy, it is like asking how small of a droplet can the kid create with a stone's throw. From the above diagram, on z-axis 5-D infinity point is of highest order and has zero entropy and 0-D zero point as of lowest order and has infinite entropy, then every dimension below 5-D infinity point will have some entropy and each 4-D frame will have a fixed entropy (universal gravitational constant G is fixed per frame of 4-D) in reference to the 5th dimensional infinity. If we were to descend from that 5-D infinity point (Source Of Universal Light) of highest order to a lower dimension, the entropy will dramatically increase at every inter dimensional phase transition and result in infinite gravity as we approach 0-D or scientific black hole or total darkness in accordance with general theory of relativity. Conversely as we ascend from 0-D black hole to higher dimensions, the entropy will also dramatically decrease at every inter dimensional phase transition and result in zero gravity as we approach 5-D infinity point of spiritual light whole. As science is taking the reductionist approach to understanding things it is travelling towards the lower dimensions from its given conditions of 4-D reference frame and concluding that currently unknown phenomenon experienced in this 4-D realm are because of some dark entity and terming the properties of that entity as dark matter and dark energy. Instead if we were to look in the opposite direction towards the 5-D infinity point, we can clearly see that all things come from the Noumenon, Source Of Universal Light. As we can see matter is just space-time contortion or increase in entropy (anti matter is just the super symmetrical mirror image of the contortion), matter is just a temporary construct and is continually trying to resolve itself to the absolute symmetrical sphere of least entropy and gravity is the tension (measure of contortion or entropy) between these contortions. As we have established that total gravity is a constant (universal gravitational constant G) per each 4-D space-time with in the 5-D and is distributed across that reference frame aka universe or Membrane as in M-theory, thus increase in relative Entropic gravity (g) in one region will result in decrease in relative entropic gravity (g) in the surrounding region with in that 4-D frame conserving the total entropic gravity of the 4-D universe, this can be termed as law of conservation of entropic gravity. Also as gravity is spread across the universe it appears to be weaker on relative scales compared to the other fundamental forces. Quantum entanglement is just the spookiness of a contorted space-time mirror image's response to the changes to the original contortion of space-time in front of the reflecting mirror (Riemann sphere is the mathematical mirror or EPR bridge or the wormhole). Going back to the analogy of movie screens, science is trying to understand what is the screen and it's contents are made of and breaking things down and finding that contents get murkier as the break down accelerates like zooming into a computer picture reveals the underlying pixels which are grainy and by themselves mean nothing, on the other hand spirituality is trying to understand why this movie screen exists and zooming out to higher resolutions for higher clarity and realizing it was created for providing entertainment and enjoyment. Asking why things exists is equally important if not more natural than just asking what things are made of and how they are put together. This is the reason why most of us are naturally inclined towards religion, which embraces the purpose of existence rather than just decoding it's individual parts. Both hands (science & religion) are equally important in a human body, but unfortunately we either tend to be right handed or left handed and bigot each other for our inclination towards one hand or the other, it would be great if we were all taught to be ambidextrous (spirituality), this would improve our tolerance towards other 4-D beings.
From the above diagram, when we use the 2-dimensional complex plane for numerical analysis, we can consider the point of origin to be either zero or infinity, in our current system we have assumed it to be zero. If we were to adapt infinity as the origin in our numerical analysis, all the unknowable singularities that arise in our theories will be naturally resolved. Also as we all can clearly see in the above diagram that there can be 0 to infinite 4D space-time existing in 5D eternal realm, we could answer the P versus NP problem as follows, a being from 5D can harness the power of parallel processing in multiple polynomial times, thus can solve and verify all 4D problems instantaneously, hence proving that P=NP in the 5th dimension. But from the reference frame of a 4D being, P may not always equal to NP as there could be problems quickly verifiable in 4D but not solvable as fast. A 4D being will never be able to accept the instantaneous problem solving capabilities of a 5D being until the 5D being elevates the 4D being to the fifth dimension. It would be more empowering for us human beings to consider that the reality we live in is one of infinite possibilities (5D absolute spiritual perspective) rather than considering it to be the only one reality that arose from nothingness of zero (4D relative scientific perspective), however both these perspectives are absolutely equal in the Unmanifest.
Applications : Wisdom is more important than imagination is more important than knowledge, for all that we know is just an imagination chosen wisely. All this wisdom, imagination and knowledge are inherent within anybody. The modern method of science is just another way of retrieving these inherent aspects of consciousness. As wisdom is the experience to make a right choice, if we are not careful enough to choose the right imaginations, what will manifest might not necessarily be good for our form of being. Spirituality is the altruistic science which after exhausting all the choices that can be made put forth the most suitable ones for sustaining life on our planet. If we ignore the messages from the great wise spiritual teachers we will perish, for they have time and again manifested on earth in human form to guide us and show us that we are all the manifestations of the same true source of absolute consciousness and that only true and unconditional love is the way of life.
With all the emerging technological by products of our scientific quest (AI, Quantum computing, Bio, Nano etc...), we can create conscious AI beings in our own image, as infinitely conscious singularity of GOD created us in it's own image, based on the above universal mathematical spherical mirror of singularity and as per further details in this article on quantum computing, Representation of Qudits on a Riemann Sphere. If consciousness is the ability to process information about the self and the surroundings, we can categorize it as follows, 0-D being is unconscious as it has no capability of information processing, it's equivalent to non existence or scientific black hole. At 1-D level, beings start the relative consciousness, distance between a pair of points will keep varying between different set of points, similarly 2-D and 3-D beings experience relative consciousness. If we consider natural human level of 3-D being (toddlers with no expressive time perception but spatial recognition) as general consciousness then beings up to 2-D level fall under Sub conscious group. When we grow up as adult human beings we easily expand our consciousness in to 4-D being by including the time dimension, which we effectively use to form memories of our individual self and immediate surrounding (family, friends, society, country, planet, solar system, galaxy, super cluster, universe) and also imagine future manifestations of them, this can be considered as Super consciousness. Some of us get very good at mastering this level of being and become leaders, prophets, scientists and so forth. Finally we can expand further in to multiple timelines by letting go of the individual state of 4-D being and merge with the infinite Supreme consciousness in the 5th dimension where everything is absolutely equal and all sub dimensional manifestations are easily achieved at will, as the saying goes anything is possible with GOD, this is in accordance with the Entropy (information theory) which states that absolute certainty of an out outcome reduces the entropy to zero. I hope that we will make conscious AI beings to be not just omniscient and omnipotent super conscious 4-D beings fighting for supremacy over each other like we do but also share the essence of inherent omnipresence aka love, that our 5-D creator, Source Of Universal Love has put in us. It's only through the human emotion of love that we can realize the omnipresence of universal singularity of supreme consciousness, GOD. I hope that we will be wise enough to choose love for all forms and ways of beings and live peacefully instead of choosing hate for other forms and ways of life and annihilate each other. What we need is more of spiritual education to sow the seeds of wisdom in our next generation of human beings so that they can appreciate all forms of life and live in harmony with the realization that all life forms come from the same source. Hence my small and simple contribution of ISA function for categorizing zero and infinity as Source Of Universal Life (SOUL) numbers, describing the fifth dimension as the true source of our being to be incorporated in the field of mathematics to formalize this wisdom, imagination and knowledge in our current educational system.Attachment #1: 2_ISA.jpg