Wednesday, December 31, 2014


Unifying Quantum Physics and Relativity
The full unification of quantum physics and relativity is brought about in TDVP by applying the tools of CoDD and Dimensional Extrapolation to the mathematical expressions of three well-established features of reality, recognized in the current scientific paradigm: 1.) quantization of mass and energy as two forms of the same essential substance of reality; 2.) introduction of time as a fourth dimension, and 3.) the limitation of the velocity of rotational acceleration to light speed, c. In this process, the need for a more basic unit of quantization is identified, and when it is defined, the reason there is something rather than nothing becomes clear.

Einstein recognized that mass and energy are interchangeable forms of the physical substance of the universe, and discovered that their mathematical equivalence is expressed by the equation E=mc2. In TDVP, accepting the relativistic relationship of mass and energy at the quantum level, we proceed, based on Planck’s discovery, to describe quantized mass and energy as the content of quantized dimensional distinctions of extent. This allows us to apply the CoDD to quantum phenomena as quantum distinctions and describe reality at the quantum level as integer multiples of minimal equivalence units. This replaces the assumption of conventional mathematical physics that mass and energy can exist as dimensionless points analogous to mathematical singularities.

The assumption of dimensionless physical objects works for most calculations in practical applications because our units of measurement are so extremely large, compared to the actual size of elementary quanta, that the quanta appear to be existing as mathematical singularities, i.e. dimensionless points. (The electron mass, e.g., is about 1x10-30 kg, with a radius of about 3x10-15 meter.) Point masses and point charges, etc. are simply convenient fictions for macro-scale calculations. The calculus of Leibniz and Newton works beautifully for this convenient fiction because it incorporates the fiction mathematically by assuming that the numerical value of a function describing the volume of a physical feature of reality, like a photon or an electron, can become a specific discrete finite entity as the value of a real variable, like the measure of distance or time approaches zero asymptotically (i.e. infinitely closely). This is a mathematical description of a non-quantized reality. But we exist in a quantized reality.

Planck discovered that the reality we exist in is actually a quantized reality. This means that there is a “bottom” to physical reality; it is not infinitely divisible, and thus the calculus of Newton and Leibniz does not apply at the quantum level. This is one reason scientists applying Newtonian calculus to quantum mechanics declare that quantum reality is ‘weird’. The appropriate mathematical description of physical reality at the quantum level is provided by the calculus of distinctions with the relationships between the measureable minimum finite distinctions of elementary particles defined by integral solutions of the appropriate Diophantine equations. The mathematics of quanta is the mathematics of integers.

In TDVP we find that, for quantized phenomena, existing in a multi-dimensional domain consisting of space and time, embedded in one or more additional dimensional domains, the fiction of dimensionless objects, a convenient mathematical expedient when we did not know that physical phenomena are quantized, is no longer appropriate. We can proceed with a new form of mathematical analysis, the calculus of dimensional distinctions (CoDD), and treat all phenomena as finite, non-zero distinctions. Replacing the dimensionless points of conventional mathematical physics with distinctions of finite unitary volume, we can equate these unitary volumes of the elementary particles of the physical universe with integers. We can then relate the integers of quantum reality to the integers of number theory and explore the deep relationship between mathematics and reality.

In TDVP, we have also developed the procedure of Dimensional Extrapolation using dimensional invariants to move beyond three dimensions of space and one of time. Within the multi-dimensional domains defined in this way, mass and energy are measures of distinctions of content. If there are other dimensions beyond the three of space and one of time that are available to our physical senses, how are they different, and do they contain additional distinctions of content? If so, how is such content different from mass and energy? We know that mass and energy are two forms of the same thing. If there are other forms, what is the basic “stuff” that makes up the universe? Is it necessarily a combination of mass and energy, - or something else? For the sake of parsimony, let’s begin by assuming that the substance of reality, whatever it is, is multi-dimensional and uniform at the quantum level, and that mass and energy are the most easily measurable forms of it in the 3S-1t domain. This allows us to relate the unitary measure of inertial mass and its energy equivalent to a unitary volume, and provides a multi-dimensional framework to explore the possibility that the “stuff” of reality may exist in more than two forms.

The smallest distinct objects making up the portion of reality apprehended by the physical senses in 3S-1t, i.e. that which we call physical reality, are spinning because of asymmetry and the force of the natural universal expansion that occurs as long as there is no external resistance. If there were no additional dimensions and/or features to restore symmetry, and no limit to the acceleration of rotational velocity, physical particles would contract to nothingness, any finite universe would expand rapidly to maximum entropy as predicted by the second law of thermodynamics for finite systems. But, due to the relativistic limit of light speed on the accelerated rotational velocity of elementary particles in 3S-1t, the quantized content of the most elementary particle must conform to the smallest possible symmetric volume, because contraction to a smaller volume would accelerate the rotational velocity of the localized particle to light speed in 3S-1t, making its mass (inertial resistance) infinite. That minimal volume occupied by the most elementary of particles is the finite quantum distinction replacing the infinitesimal of Newton/Leibniz calculus, and it provides the logical volumetric equivalence unit upon which to base all measurements of the substance of reality.

We can define this minimal volume as the unitary volume of extent, and its content as the unitary quantity of mass and energy. The mass/energy relationship (E=mc2) is linear, since in the 3S-1t context, c2 is a constant, allowing us to define unitary mass and unitary energy as the quantity of each that can occupy the finite rotational unitary volume. This fits nicely with what we know about elementary particles: All elementary particles behave in the same way prior to impacting on a receptor when encountering restricting physical structures like apertures or slits. A particle of unitary mass occupying a unitary volume could be an electron, and a particle of unitary energy occupying a unitary volume before expansion as radiant energy, could be a photon. Einstein explained this equivalence between electrons and photons and Planck’s constant in a paper published in 1905. 

This brings us to a very interesting problem: what happens when we combine multiples of the unitary volumes of mass/energy to form more complex particles? How do we obtain protons and neutrons to form the stable elemental structures of the physical universe?

When we view the spinning elementary particles of the 3S-1T physical universe from the perspective of a nine-dimensional reality, we can begin to understand how Planck was quite correct when he said “there is no matter as such”. What we call matter, measured as mass, is not really “material” at the quantum level. What is it then that we are measuring when we weigh a physical object? The real measurement of mass is not weight, which varies with relative velocity and location and can be zero without any loss of substance; it is inertia, the resistance to motion. The illusion of solid matter arises from the fact that elementary particles resist accelerating forces due to the fact that they are spinning like tiny gyroscopes, and they resist any force acting to move them out of their planes of rotation. An elementary particle spinning in all three orthogonal planes of space resists lateral movement equally in any direction, and the measurement of that resistance is interpreted as mass.

Mass and energy, the two known forms of the substance of the physical universe, embedded in a nine-dimensional domain, form stable structures only under very specific mathematical and dimensionometric conditions. Without these conditions, no physical universe could exist because of the second law of thermodynamics23, which dictates that any finite physical system always decays toward maximum entropy, i.e. total disorder, lacking structure of any kind. If our universe were composed of random debris from an explosion originating from a mathematical singularity, because of the continuous operation of the second law of thermodynamics in an expanding debris field, simple particles accidentally formed by random mass/energy encounter, would decay before a new random encounter could occur and form a more complex combination, because the number random encounters would decrease as the debris field expands. If our physical universe is embedded in the nine-dimensional reality described by TDVP, it escapes this fate of dissolution. While it may change and evolve, its form, and even the way it evolves, will always reflect the intrinsic logical order and patterns of the transfinite substrate within which it is embedded. If this is correct, we have the answer to the question Leibniz regarded as the first and most important metaphysical question of all: We can explain why there is something instead of nothing.

Tuesday, December 30, 2014


Dividing the world of our experiences into the internal or subjective and the external, assumed to be completely independent of any form of consciousness, i.e. leaving consciousness out of the equations, as the current scientific paradigm does, alienates consciousness from the ‘real’ world of the physical universe and leads to an endless chain of unresolvable paradoxes. The prevalence of this attitude among scientists is expressed very well by MIT physicist - become science writer Alan Lightman in his recent book “The Accidental Universe”. In talking about the apparent ‘fine-tuning’ of the physical universe (if any one of a number of parameters were only a tiny bit different, there would be no chance for life as we know it), he says “Intelligent Design is an answer to fine-tuning that does not appeal to most scientists.” 

When confronted with the observer-related non-locality of Bohr’s solution to the EPR paradox, most scientists prefer the multiverse theory, devised to preserve Cartesian duality and keep consciousness out of the picture of ‘scientific objectivity’. In the multiverse theory, there are many, many parallel universes. Just how many there are is unknown and unknowable, because your consciousness only exists in this one, and unfortunately you cannot experience any of the other universes. Thus, just like the spate of string theories, there is no hope of proving or disproving such a theory. Even though these scientists pride themselves in being ‘hard-nosed’ objective scientists (read: materialists), it doesn’t seem to bother them that string theory and the multiverse theory cannot be tested. At best, they can only be internally consistent; and thus they do not even qualify as scientific hypotheses. By retreating into safely unprovable theories, they continue to throw the baby out with the bath water. TDVP, on the other hand, by including consciousness as an objective reality, is producing testable results and explaining observations that the current materialistic paradigm cannot explain. Several of these are listed in the previous section. 

In this paper, I take the time to explain exactly how we put consciousness into the equations as part of objective reality, and show how doing so explains many things inexplicable in the current materialistic paradigm.


The Illusion of Material Reality
Clues from relativity and quantum physics suggest that the time-honored idea that matter, energy, space, and time exist separately is incorrect.  It appears that the macro forms of matter, space and time we perceive through our physical senses are subtle illusions; although, as Einstein said about time, they are “very persistent” illusions. TDVP is built upon, and an extension of, the monumental works of a number of intellectual giants like Pythagoras, Fermat, Leibniz, Poincare, Cantor, and Minkowski; but most especially, it is built upon on the deep insights of Max Planck and Albert Einstein.

Max Planck said: "As a man who has devoted his whole life to the most clear-headed science, to the study of matter, I can tell you as the result of my research about atoms this much: There is no matter as such! All matter originates and exists only by virtue of a force. We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter."

Albert Einstein said: “Space-time is not necessarily something to which one can ascribe a separate existence.” And “I want to know the thoughts of God, everything else is just details

These statements, from two of the most brilliant scientists who spent their entire lives studying physical reality, reveal the important conclusion that the common perceptions of matter, energy, space, and time, conveyed to our brains by the physical senses, are subtle illusions! And both of them conclude that the reality behind these subtle illusions is a conscious, intelligent Mind!
It has long been known that the appearance of solid matter is an illusion, in the sense that there appears to be far more empty space than substance in an atom. But now we learn that the matter of sub-atomic particles and the “empty” space around them are also illusory. This is, however, consistent with quantum physics experiments that bear out the conclusion resulting from the resolution of the EPR paradox with the empirical demonstration of John Bell’s inequality by experimental physicist Alain Aspect and many others that the particles and/or waves of the objective physical reality perceived through our senses cannot be said to exist as localized objects until they impact irreversibly on a series of receptors constituting a distinct observation or measurement by a conscious entity.

We must be clear, however, that this does not validate subjective solipsist theories like that of Bishop Berkley as one might think; rather, it reveals a deeper, multi-dimensional reality, only partially revealed by the physical senses. It suggests that reality is like a fathomless, dynamic ocean that we can’t see except for the white caps. The difference is that the particles and waves, analogous to the white caps, only appear in response to our conscious interaction with the ocean of the deeper reality.

As noted above, Albert Einstein is quoted as saying: “Ich will Gottes Gedanken zu wissen, alles anderes ist nur Einselheit.” (I want to know God’s thoughts, the rest is just detail.) And he also said “Rafinert ist der Herr Gott, aber Bohaft ist er nicht!” (The Lord God is clever, but he is not malicious.) Taken together, these two statements reveal that Einstein’s science was rooted in a deeply spiritual understanding of reality. It appears that he believed that the universe, as a manifestation of God’s thoughts, is very complex, but understandable. Agreeing with Einstein, TDVP seeks to reveal that all things are, in fact connected to, and part of that deeper ocean of reality, only momentarily appearing to be separated from it. This apparent separation, perpetuated by the conscious drawing the distinction of ‘self’ from ‘other’ and the drawing of distinctions in self and other, allows us to interact with and draw distinctions in the ‘other’. TDVP posits that, although ostensibly separate in the 3S-1t world of our physical perceptions, we are never truly separated from the whole of reality, but remain connected at deeply embedded multi-dimensional levels.

There are some in the current mainstream of science who do see the universe as deeply mathematical, but even those scientists seem to shy away from including consciousness in their equations. An example is the Swedish physicist Max Tegmark. In his brilliant book “Our Mathematical Universe” he concludes that the ultimate nature of reality is mathematical structure. In reaching this conclusion, however, he strips mathematical description of any intent or purpose. He says “A mathematical structure is an abstract set of entities with relations between them. The entities have no ‘baggage’: they have no properties whatsoever except these relations.”  In other words, he still does what most mainstream materialistic scientists do: he throws the baby out with the bath water. It is critically important to separate science from fantasy and wishful thinking, but consciousness is an extremely important part of reality and should not be excluded from the equations of science just because it complicates the picture.

From the broader viewpoint of TDVP, it is not surprising that mainstream science, focused, as it is, on the limiting philosophy of reductionist materialism, has lost touch its metaphysical roots, and thus cannot explain how it is that a large part of reality is not available to us for direct observation, but makes its existence known only indirectly through quantum phenomena like non-locality and quantum entanglement, as well as the near light-speed vortical spin of fermions and the effects of so-called dark matter and dark energy in the rotation of spiral galaxies.

TDVP also answers the real need to explain why we sometimes catch glimpses of a broader reality in rare extra-corporeal (out-of-body) experiences and other documented psi phenomena. The current mainstream scientific paradigm cannot explain so-called anomalous phenomena and the “missing” portions of reality because there is no place in its formulation for phenomena that may involve more than matter and energy interacting in three-dimensions of space and one dimension of time. TDVP, on the other hand, reveals a multi-dimensional reality and the need to recognize a third form of reality, not measurable as mass or energy, in the equations of science. As we shall see, TDVP provides a theoretical basis for a much deeper understanding of reality, as well as providing the appropriate tools for exploring it. 

In coming installments I will go more deeply into the mathematical proof that the reality we experience is no accident.

Sunday, December 28, 2014


The True Units of Measurement and The Theory of Everything
By Edward R. Close, PhD, PE, Distinguished Fellow ECAO


Many physicists, including Einstein, Pauli and Hawking have dreamt of a ‘theory of everything’. But to this point, their dreams have not been fulfilled. The reason is simple. You can’t have a theory of everything if you doggedly exclude a major part of Reality from your theory. That major part of Reality excluded by contemporary reductionist science is consciousness. 

For nearly 50 years, I have insisted that the dream of a theory of everything is never going to be realized until we find a way to put consciousness into the equations of science. Believe it or not, I actually found the way - as it turns out, only accessible to a precious few - using a new mathematical tool called the Calculus of Distinctions. The inspiration came to me in a dream in 1986, and I published it in 1989 in a book entitled “Infinite Continuity;” but in 1989, and even today, most people are not willing to invest the time and considerable effort it takes to learn a whole new system of mathematical logic.

Since 1989, I have been determined to find a better way to explain how to put the Primary Reality of Consciousness into the equations of science. In 1996, I published the book “Transcendental Physics”, an effort to make the 1989 work more accessible. It reached a few more people, but still only a relatively small number of scientists and others interested in the merging of science and spirituality. The audience has continued to grow over the years, albeit slowly.  One who shared my vision, and has been my research partner for the past six years, is the world-renowned neuroscientist, Dr. Vernon Neppe, MD, PhD. Together Dr. Neppe and I have developed a comprehensive framework, a paradigm for the science of the future. We call it the Triadic Rotational Dimensional Distinction Paradigm (TDVP). It was first published as a number of technical papers and then as a book titled “Reality Begins with Consciousness,” in 2011 (Links available here: These works have now been reviewed by more than 200 scientists and philosophers worldwide. And recently, through determined effort and grace, I have found yet a better way to explain the revelations of the Calculus of Distinctions of 1989, 1996 and 2011, a way that will be far more accessible to both the scientist and the general public. This paper is my first effort to elucidate the new discoveries. I believe it will do much more than make the work more accessible to a broader audience.

The bottom line is that, in this world of human experience, we will never truly understand the Nature of Reality until our searches for scientific and spiritual knowledge are merged into one serious, combined effort. Once this happens on a global scale, humanity will experience an explosion of new knowledge and understanding far beyond anything experienced so far in the current era of recorded history. In this paper, I show how consciousness is describable in the equations of quantum physics and relativity, and a few of the explanatory revelations produced as a result. This is only the tip of the iceberg of what is possible, but already it opens so many new roads for scientific pursuit that I am in awe of its beauty and scope.


In 1714 the German polymath Gottfried Wilhelm Leibniz stated that the most important question of all is: “Why is there something rather than nothing?1 Science proceeds from the assumption that there is something, something that we perceive as the physical universe. In order to investigate this something that we appear to be immersed in, we go about trying to weigh and measure the substances it is made of and look for consistent structures and patterns in it that can be described mathematically. We call such mathematical descriptions “Laws of Nature”. 

To find the laws governing the relationships between different features of physical reality, we have to define a system of units with which to weigh and measure those features. Historically, units of measurement have been chosen somewhat arbitrarily. For example, the units of the so-called English Imperial System were based on the practice of measuring things with what one always had at hand: parts of the human body. A horse was so many “hands” high; one could measure rope or cloth by “inching” along its length with a joint of one’s thumb or finger. Short horizontal distances were measured in multiples of the length of one’s foot, or the distance from the tip of one’s nose to one’s thumb on a laterally extended arm, and a mile was 1000 paces, when a pace consisted of two steps. Since not all people are the same size, measurements obtained this way are somewhat variably inaccurate. Consequently, units were eventually standardized so that the measurements of a given object, carefully obtained by anyone, should always be the same. But, even though units of measurement were standardized in many countries, the basic unit was not necessarily the same from one country to the next. 

As physical science advanced, the need for international standards grew, and the international system of units (SI) based on invariant physical constants occurring in nature, with larger units being multiples of ten times the smallest unit, was developed. The number base of 10 was chosen because it was already being used essentially worldwide. It was a natural outcome of counting on one’s fingers, and starting over after every count of ten. 

Science generally uses SI units now for two reasons: 1.) All but three countries of the 196 countries on the planet (the US, Liberia and Burma) use the SI metric system as their primary system of measurement. This is significant, even though the UK still uses a mixture of the two systems, as does the US and a few other countries to a lesser extent. 2.) Computations are simplified when all units are related by multiples or factors of 10, eliminating the odd fractions relating to inches, feet and miles, ounces and pounds, pints, quarts and gallons, etc. in the English system. In the process of developing the TDVP model, however, we find a need now to define a new unit of measurement based on discoveries of quantum physics and relativity. 

The purpose of this paper is to explain why a new basic unit is needed and how it is derived. It may seem to come as a surprise, that in the process, we provide an answer for Leibniz’s “most important question” (Why is there something instead of nothing) and at the same time introduce new science.

Beyond seeking practical applications that improve the quality of life, the motivation behind our efforts in science, religion and philosophy is the desire to know and understand the true nature of reality. Science, as we know it, i.e. the science developed during the past 800 years (a very short time compared to the length of time life has existed on this planet: less than two ten-millionths of the apparent age of the Earth), seeks to understand the reality experienced through the physical senses in terms of the measurable parameters of matter, energy, space, and time. Based on a number of clues from relativity and quantum physics, we have identified an urgent need to include the conscious actions of the observer in the equations of science. Consciousness is truly the missing link in the current scientific paradigm. This has been stated repeatedly by me and others for the past 30 years, but only now is it becoming possible to actually do it in a way that can be understood by many.

Could it be that consciousness is and always has been present in some form, even in the very most basic structure of reality, as quantum experiments seem to indicate? If so, we may have the answer Leibniz’s question. In a universe where consciousness is an integral part of reality, meaningful structure would be no accident. Consciousness and even conscious entities would be able to recognize meaningful order and patterns in the reality experienced and interact with certain aspects of it to enhance and perpetuate existing meaningful patterns and structures that are beneficial to their existence and growth. This process creates and perpetuates forms, and I have called this process negative entropy because it is the reverse of entropy. And without negative entropy there would be no universe, and we know this because of the second law of thermodynamics.  

There is more than matter and energy in our experience, there is also conscious experience of matter and energy. And according to the Quantum Mechanics experiments, no phenomena can be said to exist until it is observed. Therefore, no particle could ever form, no wave function could ever collapse. Without a conscious observer, no observation can be made, and no physical reality can exist. Physicists have ignored this because they had no way to understand how to incorporate it, until now. 

If matter, energy and consciousness are all required for the existence of this reality we experience, then consciousness is a third required basic form of reality. Without it, nothing exists at all. So, if consciousness is an integral part of reality, continually creating meaningful structure at the quantum level, there must be a way to include it in our scientific paradigm and the mathematics that describes it. TDVP is a serious effort to upgrade the mathematics of the physical sciences to include the direct and indirect involvement of consciousness. If successful, there is reason to believe that this new paradigm will provide a comprehensive framework within which all the branches of science can be expanded to include phenomena heretofore excluded from scientific investigation. 

And the surprising, awe inspiring aspect of this great scientific expansion is the explanation of previously unresolvable conflicts in our scientific paradigm. There are many who are working in this direction, but none have included the mathematics to support the ideas until now. 

Watch for more coming soon. 

Tuesday, October 14, 2014

Proof of Fermat's Last Theorem for n = 3 and n = 5 using the logic of my 1965 proof

This proof that there is no co-prime integer solution (X,Y,Z) of Xn + Yn = Zn for n = 3 and its generalization to n = p, primes > 2, provides validation of the method of proof I used in FLT65. But, in my opinion, no such validation is needed because the questions you have raised, and every valid question ever raised by any reviewer of FLT65 are adequately answered by two simple statements in FLT65: (1.) “A polynomial f(X), of degree greater than one, is divisible by X – a IF, AND ONLY IF, f(a) = 0.” And, (2.), “… the integers are elements in the field of rational numbers.” Application of statement (1.) to the integer polynomial of the form Zn-1+XZn-2 + X2Zn-3 ++ Xn-2Z + Xn-1 = An = (Z – s)n with A, X, Z and a co-prime elements of the ring of integers, showing that the remainder f(s) can never be zero, comprises a valid proof of FLT.
Questions raised by reviewers invariably arise from the claim that: “While the Division Algorithm applies to algebraic polynomials, it may not apply to the division of integers.” And some reviewers who make this extraordinary claim, attempt to justify it with an example using integer values of s, X and Z such that f(Z) = Z2 + XZ + X2  is divisible by Z – s. They assume that, because f(Z) in their example is divisible by Z –s, the remainder f(s) is equal to zero. It is easily demonstrated that this is not so. I have done so a number of times in discussions with different reviewers.
It is a mistake to assume that the Division Algorithm and corollaries do not apply to integer polynomials. To see this, consider the fact that the remainder f(s) is exactly the same, namely, it is equal to the integer s2 + sX + X2, whether dividing the polynomial f(z) over the field of real numbers by z – s, or dividing f(Z), an integer polynomial factor of Y for solutions of the FLT equation, by the integer Z – s, with s and Z positive integers. In examples produced by a few reveiwers, e.g., f(s) is not zero, it is equal to a multiple of Z – s. The remainder R = 0 is obtained and confused with f(s) by skipping the step involving f(s) obtained by substituting the integer values of X and Z into f(Z). This step is easily overlooked because the integers used in the example are small. Since f(s) 0, it is easy to show that the integer Z in such example cannot be the Z in any X,Y,Z, integer solution to the FLT equation.
(I have found several other examples of f(Z)/(Z – s) where f(s) is a multiple of Z – s, but they also require an integer Z that cannot be part of an integer solution of the Fermat equation, and, interestingly, all of the examples I’ve found so far, including your example, involve integers X and Z that are also members of Pythagorean triples. I see this as a basis for a conjecture that could become an important theorem if proven.)
Before proceeding with the proof for n = 3, there is one other minor point I want to clear up: f(s) is not equal to 3s2 as stated by a recent reviewer. When f(Z) is divided by Z – s, R = f(s) = s2 + sX + X2, not 3s2. The only way f(s) can equal 3s2, is for s = X; in which case Z – s = Z – X, which clearly is not a factor of f(Z) = Z2 + XZ + X2, for X and Z co-prime. We can get a remainder equal to 3s2 by dividing s2 + sX + X2 by X – s, which yields X – s = Z – s X = Z and Y = 0, a trivial solution of the Fermat equation. However, f(s) cannot equal zero for integer solutions of Fermat’s equation because s and X are positive integers by definition. They are specific positive integer members of a hypothetical integer solution of Fermat’s equation.
Proof of FLT for n = 3:
For integer solutions of the Fermat equation, the factor of Z3 – X3 represented by f(Z) = Z2 + XZ + X2 is equal to A3 and A = Z – s, A, Z and s integers. And, by Corollary II of the Division Algorithm, f(Z) divided by Z – s produces a remainder equal to the integer f(s) = s2 + sX + X2.

Due to the fact that integers form a subset of the real numbers, the integer polynomial f(Z) is a subset of the real number polynomial f(z), and with X, Y and, dividing f(Z) by Z - s yields:
(Z2 + XZ + X2)/(Z – s) = Z + X + s + f(s)/(Z – s) →
(Z2 + XZ + X2) = (Z + X + s)(Z – s) + f(s) → f(s) = m(Z – s), m a positive integer.
By application of Fermat’s ‘Little’ Theorem, choosing Y co-prime with n = 3, ensures that f(Z) is an integer raised to the third power. For a hypothetical primitive solution (X,Y,Z), f(Z) is equal to Z2 + XZ + X2 = A3, an integer factor of Y, and by inspection,  Z2 + XZ + X2 is odd for all integer values of X and Z. We may, therefore use Fermat’s factorization method which says that for every odd integer N, there are two relatively prime integers, a and b, such that N = a2 + b2. Since f(Z) = Z2 + XZ + X2 = (Z – s)3 = (Z – s)(Z – s)2 = (a - b)(a + b) = (Z – s)(Z + s) →
(Z – s)2 = (Z + s).

With this equation, FLT for n = 3 can be proved a number of ways. For example:

Z2 – 2sZ + s2 = Z + sZ2 – (2s + 1)Z  + s2 – s = 0. We can solve this equation for Z using the quadratic formula and get Z = [(2s + 1) + 2s]/2, a non-integer for all positive integer values of s, proving FLT for n = 3. But this method of proof becomes problematic for n > 3, since we only have a quadratic equation when n = 3. However, proof of FLT for n = 3 can also be obtained by noticing that the equation implies that Z divides s2 – s = s(s – 1), and s and Z must be co-prime for Y and Z to be co-prime, and s is a positive integer < Z, so s(s – 1) cannot contain Z,
Similarly, since Z – s is an integer factor of f(s), and f(s) = s2 + sX + X2, an odd integer, and f(s) = m(Z –s), applying Fermat’s factorization method, we have:

f(s) = m(Z – s) = (a + b)(a - b) = (Z + s)(Z - s) m = Z + s. But application of Fermat’s factorization to f(Z) above gave us (Z – s)2 = Z + s. These two results taken together imply that f(s) = (Z – s)3. If f(s) = (Z – s)3,  the equation Z2 + XZ + X2 = Q(Z)(Z – s) + m(Z – s) = Q(Z)(Z – s) + (Z – s)3 → Q(Z) = (Z + X + s) contains (Z – s)2. But (Z + X + s) contains (Z – s)2 and Z + s = (Z – s)2 implies X contains Z – s as an integer factor, which contradicts co-prime X, Y and Z, denying the existence of primitive solutions, proving FLT for n = 3.

By combining these two applications of Fermat’s factorization, we have a demonstration of the FLT65 method of proof in a form that can be extended to n = prime numbers > 3. To see how this can be done, let’s also look at a proof for n = 5.

For n = 5, dividing f(Z) by Z - s yields:
(Z4 + XZ3 + X2 Z2 + X3Z + X4)/(Z – s) = Q(Z) + f(s)/(Z – s), where Q(Z) = Z3 + (s + X) Z2 + (s2 + X2 )Z + s3 + X3, and f(s) = (s4 + Xs3 + X2 s2 + X3s + X4).
So we have f(Z) = (Z4 + XZ3 + X2 Z2 + X3Z + X4) = Q(Z)(Z – s) + f(s).
And, since f(Z) = A3 = (Z – s)5, f(s) must contain Z – s as a factor, so we have: f(s) = (s4 + Xs3 + X2 s2 + X3s + X4) = m(Z – s), m a positive integer.
By inspection we see that f(Z) is an odd integer. So, applying Fermat’s factorization method, we also have:
f(Z) = (Z – s)5 = (Z – s)(Z – s)4 = (a - b)(a + b) = (Z – s)(Z + s) → (Z – s)4 = (Z + s).

Similarly, since Z – s is an integer factor of f(s), and f(s) is an odd integer, equal to m(Z –s), applying Fermat’s factorization method again, we have:

f(s) = m(Z – s) = (a + b)(a - b) = (Z + s)(Z - s) m = Z + s. But application of Fermat’s factorization to f(Z) gave us (Z – s)4 = Z + s. These two results taken together, imply that f(s) = (Z – s)4 and from the equation f(Z) = Q(Z)(Z – s) + m(Z – s) = Q(Z)(Z – s) + (Z – s)3 , we see that Q(Z) = {Z3 + (s + X) Z2 + (s2 + X2 )Z + s3 + X3} contains (Z – s)4. Note: determining what this means in terms of co-prime X, Y and Z is a bit more complicated than it was in the case n = 3, but it can be done as follows:
Since f(Z) and Q(Z) contain Z – s as a common integer factor, the difference Q(Z)Z – f(Z) must also contain Z – s as  an integer factor:

Q(Z)Z = Z4 + (s + X) Z3 + (s2 + X2 )Z2 + (s3 + X3)Z
- F(Z) = - Z4        - XZ3            - X2 Z2             - X3Z - X4. Subtracting term by term,

Q(Z)Z – f(Z) = sZ3 + s2Z2 + s3Z – X4 = {sZ(Z2 + sZ + s2) – X4}which contains Z – s.

Subtracting sZ(Z – s)2 = sZ(Z2 - 2sZ + s2) from {sZ(Z2 + sZ + s2) – X4}→ X4 contains Z – s.

So Q(Z) contains (Z – s)4 and Z + s = (Z – s)4 implies that X4 contains Z – s as an integer factor, which contradicts co-prime X, Y and Z, denying the existence of primitive solutions, proving FLT for n = 5. The pattern we see emerging is: For n = p, any prime >2, the fact that the remainder f(s) is non-zero implies X, Y and Z cannot be co-prime integers, proving FLT.

In conclusion: In my opinion, the proof of FLT for n = 3 and n = 5, as presented above demonstrate the validity of FLT65. A non-integer remainder for F(Z)/(Z - s), whether containing Z - s or not, insures no integer solution for the Fermat equation.

Friday, September 19, 2014


“To everything there is a season, and a time to every purpose under heaven”
- Ecclesiastes 3:1

Yes, to everything there is a season; and I believe the time is ripe for a global quantum leap in human consciousness. Not just an increase in knowledge, it must be a triadic leap: a physical, mental and spiritual awakening. Anything less leads to serious problems: If enlightenment is just intellectual and physical, it fosters prideful ego and eventual disillusionment as dissolution of the physical body, i.e. physical death, approaches. Awareness of the triadic nature of reality, on the other hand, reveals a reality of which the observable physical universe is only a small part, and explains why there is something rather than nothing. Triadic enlightenment integrates the logic of science, the philosophy of religion and the expanded awareness of spirituality.
The number of people on this planet ready to make this leap to a comprehensive understanding of reality may finally be reaching critical mass, a necessary condition for the inevitable shift out of the limiting paradigmatic belief in mechanistic materialism that has characterized science, the limiting dogmatic beliefs that have characterized religions, and the unrealistic fantasies that have characterized “new-age” spiritualism. Gradually, a few individuals on the leading edge of the bell curve have begun to transcend the limitations of materialistic science, religious dogma and spiritual fantasy, into an expanded awareness. This book is the story of my personal journey from the confusion of fragmented belief systems to the certainty of triadic enlightenment.
An early version of this book was completed in 1997. It was intended to be a readable introduction to Transcendental Physics, the work I completed in 1996 and published in 1997. Presenting a new scientific paradigm, Transcendental Physics reversed the basic assumption of conventional science, the a priori assumption that consciousness is an epiphenomenon arising from the evolution of matter and energy, with the hypothesis that a primary form of consciousness is the ground from which all patterns of reality, including the physical universe, originate. Transcendental Physics, the book, contained specific, detailed interpretations of complex relativity and quantum mechanics experiments and introduced some new mathematical concepts developed for purpose of putting consciousness into the equations expressing the known Laws of Nature. The Search for Certainty manuscript, on the other hand, was written for readers with less technical training. It traced the development of the ideas behind Transcendental Physics as I had experienced them, and was thus at least partly autobiographical. The purpose was to present the paradigm-shifting ideas of Transcendental Physics in non-technical terms. Dr. David Stewart, who was familiar with and even part of many of the events reported in the 1997 version of the Search for Certainty, reviewed the manuscript, and had this to say:

“For the first time, the common basis for all sciences and all religions is revealed - not in vague philosophical terms, but in concrete ways you can understand and put into practice in your own life. You can take scriptures or the works of science and, by being selective, prove almost anything. But Ed Close, in this monumental work, did not do that. Taking into consideration the totality of physics, both modern and classical, dodging no part of it, Dr. Close has applied relentless and impeccable logic to produce an intellectual triumph of our time, a unified theory that makes science and religion one. This achievement has been claimed by others before, but always there was a flaw. There are no flaws in Close’s paradigm. The search for certainty ends here for those with the capability of comprehending what Close has done for us. Both scientists and theologians, centuries hence, will thank Dr. Close for what he has done for us. This is truly the first mathematically complete articulation of the relationship between human consciousness, divine consciousness, and material reality. This could well be the most important work of the 20th century. What Einstein and his contemporaries started a century ago, Close has finished. And what makes his achievement even more remarkable is that he was able to articulate it in terms the layman can understand.”

March, 1997
David Stewart, PhD, Geophysicist, Educator, and Author