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Gary,

Your equation 4 is very close to the universal wave function I use. P=exp(iEt/H)*exp(-iEt/H). It is what MIT calls unitary evolution based on the Schrodinger equation. You can look it up by searching MIT22 Evolution of Function Chapter 6. It is in Heading 6.1.2, Unitary Evolution. The Hamiltonian can be simply Energy since it is time dependent.

The interesting parallel...

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Gary,

Your equation 4 is very close to the universal wave function I use. P=exp(iEt/H)*exp(-iEt/H). It is what MIT calls unitary evolution based on the Schrodinger equation. You can look it up by searching MIT22 Evolution of Function Chapter 6. It is in Heading 6.1.2, Unitary Evolution. The Hamiltonian can be simply Energy since it is time dependent.

The interesting parallel between our work is that we are both evaluating exponents.

P=1 and iEt/H=1 and we have to look inside the 1’s. The equation E=e0*exp(N) that gives the E’s is easily derived from the Schrodinger equation but I have never found any use of the equation in physics. [Barbee, Gene H., Schrodinger Fundamentals for Mesons and Baryons, October 2017, vixra:1710.0306v1].

Below is an excerpt from the proton model described in the reference (quad 2 out of 5 quads). The values of E that satisfy P=1 are 13.797, 5.076, 101.947 and 0.687 MeV. For example 5.076 MeV comes from the equation E-2.02e-5*exp(12.432).



With these 4 E’s, P=1=psi*psi*psi*psi=exp(13.797it/H)*exp(5.076it/H)*exp(-101.
947it/H)*exp(-0.687it/H). The imaginary numbers divide out and each Et/H=1. I labelled the E’s mass, kinetic energy, strong field, and grav field and they describe what I call a quantum circle. There is an equal amount of positive energy and negative energy in the proton model, for example 13.8+88.15=101.947+ 0.687 MeV.

I tried to relate this to your work but you are looking for dimensions and I am looking for information that describes a quantum circle. The circle is also a wave and can be described by your sine and cosine functions. The circle also represents simple properties, spin, parity, charge and fields as indicated in my essay. In my work, the exponents are also probabilities and the dimensions are formed when the surface of a three dimensional sphere is divided into exp(180) individual surfaces with a neutron with kinetic energy on each surface. There is a good discussion of geometry in Principles of Cosmology by P.J.E. Pebbles. I think the use of a surface is justified by dx^2+dy^2+dz^2-(cdt)^2=0. Again, there is a parallel with your four square usage and I like your two time dimensions. Time repeats and counts forward but the time ratio we called gamma gives nature freedom to move (ke=m/g-m).

Thanks for introducing me to quaternions. My question to you is “are we looking at the same thing from a different perspective?” Engineers unite!

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Hi Gary D. Simpson…..

“The importance of Lagrange’s Four Squares Theorem, and the arguments that the vacuum is a 5-D Quantum-Space-Time and that the vacuum is fundamental ”are nicely argued and derived nicely dear Simpson…. Best wishes for your essay….

…..….. very nice idea…. I highly appreciate your essay and hope for reciprocity.

I request you please...

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Hi Gary D. Simpson…..

“The importance of Lagrange’s Four Squares Theorem, and the arguments that the vacuum is a 5-D Quantum-Space-Time and that the vacuum is fundamental ”are nicely argued and derived nicely dear Simpson…. Best wishes for your essay….

…..….. very nice idea…. I highly appreciate your essay and hope for reciprocity.

I request you please spend some of the valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance

Some of the Main foundational points of Dynamic Universe Model :

-No Isotropy

-No Homogeneity

-No Space-time continuum

-Non-uniform density of matter, universe is lumpy

-No singularities

-No collisions between bodies

-No blackholes

-No warm holes

-No Bigbang

-No repulsion between distant Galaxies

-Non-empty Universe

-No imaginary or negative time axis

-No imaginary X, Y, Z axes

-No differential and Integral Equations mathematically

-No General Relativity and Model does not reduce to GR on any condition

-No Creation of matter like Bigbang or steady-state models

-No many mini Bigbangs

-No Missing Mass / Dark matter

-No Dark energy

-No Bigbang generated CMB detected

-No Multi-verses

Here:

-Accelerating Expanding universe with 33% Blue shifted Galaxies

-Newton’s Gravitation law works everywhere in the same way

-All bodies dynamically moving

-All bodies move in dynamic Equilibrium

-Closed universe model no light or bodies will go away from universe

-Single Universe no baby universes

-Time is linear as observed on earth, moving forward only

-Independent x,y,z coordinate axes and Time axis no interdependencies between axes..

-UGF (Universal Gravitational Force) calculated on every point-mass

-Tensors (Linear) used for giving UNIQUE solutions for each time step

-Uses everyday physics as achievable by engineering

-21000 linear equations are used in an Excel sheet

-Computerized calculations uses 16 decimal digit accuracy

-Data mining and data warehousing techniques are used for data extraction from large amounts of data.

- Many predictions of Dynamic Universe Model came true….Have a look at

http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.h
tml

I request you to please have a look at my essay also, and give some of your esteemed criticism for your information……..

Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.

In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from “http://vaksdynamicuniversemodel.blogspot.in/ ”

I request you to please post your reply in my essay also, so that I can get an intimation that you replied

Best

=snp

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Hello Gary,

Have your essay open now, commenting stream-of-consciousness while browsing...

hmmmm. pondering this

"...in a 4-D geometry, if a right triangle is constructed from an integer

number of basis lengths in each of the four dimensions (a, b, c, and d), then the hypotenuse (f) that traverses through the 4-D space will also have an integer number of the basis...

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Hello Gary,

Have your essay open now, commenting stream-of-consciousness while browsing...

hmmmm. pondering this

"...in a 4-D geometry, if a right triangle is constructed from an integer

number of basis lengths in each of the four dimensions (a, b, c, and d), then the hypotenuse (f) that traverses through the 4-D space will also have an integer number of the basis lengths."

this seems to me to be quite extraordinary, foundational so to speak. Something very simple and basic about 'quantization' in 4D spacetime that i've not seen before. thank you.

then you add a scaling factor that permits you to look at the Planck length. Nicely done. You've brought a form of 4D spacetime quantization to gravity. Starting with the assumption that your present 7.0 is fair, this is a good nudge upwards imo.

excellent beginning of the next section, most appropriate here to ask “What is waving?”. Change of dimension that you mention follows from outer product, likely what you had in mind in taking us by the hand and leading us along. Good to keep in mind tho that in the algebra it comes not from bending things but rather from interactions of the fundamental geometric objects of the algebra.

and then you drift off into curved space. Big mistake imo. The Lasenby Cambridge group showed back in the 90s that gauge theory gravity of geometric Clifford algeba is flat space equivalent of curved space GR (and the only route to quantum gravity imo).

you recover nicely by going to vacuum and electron wavefunction, then spin out by playing the numerology game. Need to stay conscious of the fact that you're playing in an elegant web, where possibilities such as you suggest in attempting to jump to the proton mass have what are 'hidden' connections in the things we do not yet understand. Near as i can tell the goal is to get things in some sort of matrix format such as you show for the octonion, then then try to drop all the coefficents and see what numerical values remain. That's the raw web of interconnections, without pi, spin-related factors of 1/2 and 2, sqrt of 2 and 3 in projection factors,... That's the most basic part. I think if you try to do that in your model and are successful you will find you can take all of them to unity except one of them has to be the number 4. That comes from the fact that dimensionality of the 4D pseudoscalar of the Dirac algebra generated by the product of two 3D Pauli algebra wavefunction.

point is forget the numerology, that's what earned us both a 1 last year and i hope the point is well taken. We both paid the price for it.

i like what you're doing, seems like you have good intuition, am going to push your rating up a bit. Suggest you look at adding the appropriate geometric properties - scalar, vector, bivector, trivector - to the generators of your 'octonion' matrix. It may be that you will then start to see the full power of what you're doing, and hopefully carry it deeper into the math. As you point out, my math sucks. No argument there.

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I'm not bound to integers. However, I found it to be very coincidental that Lagrange's Theorem was composed of four squares and the length of a quaternion is also four squares. Such coincidences mEssentially, things can appear to be quantized (i.e., integer valued) without actually being quantized because both methods reduce to four squares.ake me suspect that something deeper is underfoot....

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I'm not bound to integers. However, I found it to be very coincidental that Lagrange's Theorem was composed of four squares and the length of a quaternion is also four squares. Such coincidences mEssentially, things can appear to be quantized (i.e., integer valued) without actually being quantized because both methods reduce to four squares.ake me suspect that something deeper is underfoot.

I have to agree totally with this statement since that is what I found as well. My essay What is fundamental is the area of the imaginary unit" is also based on the four square theorem, though that comes out as a result not an assumption. "The basic idea in the essay is that a number is a square area (not lengths squared), and from this square area we can form four other areas off each side of the number's area. The basic idea is that any number can be viewed as the sum of four squares." And that the imaginary unit can be thought as +i and -i obtained from i² the area of the imaginary unit. And yes it uses the fact that "Lagrange's Theorem was composed of four squares and the length of a quaternion is also four squares." In the first post is an attachment FAQ which has all the details. Very similar ideas. The essay goes into how to form a dual of the purely imaginary unit. Then using the folk-wisdom which in the essay is called "the basic multiplication table for the definitional imaginary units"

1. -ve times -ve gives +ve

2. +ve times +ve gives +ve

3. -ve times +ve gives -ve

4. +ve times -ve gives -ve,

We can show how the quantised (i.e. "the basic multiplication table for definitional units") can produce a continuum as in space-time. Or as you say "Such coincidences mEssentially, things can appear to be quantized (i.e., integer valued) without actually being quantized because both methods reduce to four squares."

Reading your insightful essay has clarified many questions that I had about quaternions and octonians. Your matrix is most ingenious. I think you should read another essay as well Unified Field which has equations of the form +i⁺¹ and -i^-1 and goes through a lot of concise ideas towards a "math" unification of reality.

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Every time I read your essay I seem to understand, it more and more.

I have a couple of questions about Equation 1

(a² + b² + c² + d²)u² = f²u²

A quote page 3

"The meaning of Equation 1 is that in a 4-D geometry, if a right triangle is constructed from an integer number of basis lengths in each of the four dimensions (a, b, c, and d), then the...

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Every time I read your essay I seem to understand, it more and more.

I have a couple of questions about Equation 1

(a² + b² + c² + d²)u² = f²u²

A quote page 3

"The meaning of Equation 1 is that in a 4-D geometry, if a right triangle is constructed from an integer number of basis lengths in each of the four dimensions (a, b, c, and d), then the hypotenuse (f) that traverses through the 4-D space will also have an integer number of the basis lengths."

In Equation 1

Clearly it is the area u² that is common to both sides. Since its area's four squares when summed gives a transcendent "number" to both (a² + b² + c² +d²) and the area f². So if we have a 5-d hypotenuse cut from area f² within our 4-d space-time based on a well understood four squares geometry with an invariant length "the square root of s²". How do you avoid this "cut" being s and not the area s²=(a² + b² + c² +d²) which what equation 1 is saying. That the total area of (a² + b² + c² +d²) times the common area u² equals the common of area of u² times the area f². And ever body knows that (the sign of s²) times (the sign of area u²) equals (the sign of area u²) times (the sign of the area f²).

"Yes, I am treating an octonion as a bi-quaternion. That is what makes the multiplication table work.

The matrix multiplication is interesting. If the complex i commutes normally with the unit vectors, the coefficient matrix uses B. But if the complex i anti-commutes with the unit vectors, the coefficient matrix uses B*."

Bi-quaternions are just directed areas, that is, an area with a + and - sign. Clearly the matrix works because we have the invariant area ijk which then allows us to use octonian logic "based on + and - signs" which are attached to the bi-quaternions' areas. Hence in equation 1 the need of the 5-d hypotenuse cut from the area f² in our 4-d world which is based on an invariant four squares space-time summation.

Your 5-d area's four squares summation gives us the length of 4-d hypotenuse "the invariant length of the square root s²" not the total invariant area summation. You have 4-d areas with a 5-d hypotenuse length of the four squares for the area f². We have literally have a 5-d hypotenuse length within our 4-d space-time that any four square summation must obey. Since the area of u² is the one common transcendental number that bridges both sides of Equation 1, while the 5-d hypotenuse is an invariant 4-d length that any summation must have available to have closure for the geometry of the area of f².

A number (which is a perfect square) is the summation of four squares. If the area of f² is n square metres d²ct, then the physical manifestation of that area is a n invariant unit lengths of dct in our 4-d space-time. Not an area. We have an area f² on the right RHS, then on the LHS, equation 1 has a 5-d hypotenuse cut -- length c(metre) -- an invariant length that, by the 4-S theorem and equation 1 - each and every, any and, all - four square invariant summations must obey within our space-time.

Of course your multiplication matrices Eq 5.4 and Eq 5.5, clearly ties "i" with c(metre), clearly via the common area u² which is on both sides, where we have units of the summation of transcendental i if we use the 4-S on both sides at once but using your multiplication rules A,B*,A,B* for – and + sign matrix Eq 5.3, which is, after all, a + and - sign summation using "octonian" logic directed bi-quaternion areas i.e. the column [C,D], using Eq 4.1 about a stationary "ijk" invariant the area f², using f a length “the square root of the area of f²” to transverse the equal sign, Equation 1 uses a 5-d length, so cannot be associated 1-1 with a summation of four square labelled A,B,C,D thought of as a “a perfect number as an area” . It is the area u² that is, the common “four square summation” i.e. the perfect square, that spans the equal sign using the 4-S theorem on both sides of Equation 1. A number (which is a perfect square) is the summation of four squares). Your Eq 5.3 is a dance using A,B,C,D where A,B,C,D do integral steps on directed areas ALL on the geometry of the area of ijk. More simply the dance is with the directed areas which have a + or – sign, that is, i and * are not moving, i.e. they don't lead! It is --- i and * --- that are stationary and it is Eq 5.3 that moves areas that equal + or – throughout a basic multiplication table page 6, clearly Eq 5.3 only gives the square root of s², a length not an area for how the multiplication table works in your matrices Eq 5.4 and Eq 5.5.

The full 4-S multiplication "of the areas on both sides of Equation 1" is:-

(the sign of the area (a² + b² + c²+d²)) times (the sign of the area u² on the LHS) = (the sign of the area u² on the RHS) times (the sign of the area f²).

You will find Eq 5.3 octonian area + and – logic uses only the “square roots for the area u²” on the LHS for the bi-quaternions areas plus and minus signs attachment. That is, it is the common area of the transcendent “number” (a summation of four squares) which transverses the equal sign in Eq 1. as perfect numbers). Not your A,B*,A,B*,-,+ matrix dance Eq 5.3. which is after all + and - sign summation using "octonian" logic directed bi-quaternion areas i.e. the column [C,D]; clearly uses Eq 4.1 a stationary "ijk" invariant the area f².

More simply the area of f² is ijk equals -1 and then we take the square root of the area of ijk that is √-1 the imaginary unit. Clearly the full 4-S multiplication table for the "equal sign" invariant + and - unit count across the equal sign for Equation 1 is a transcendent dimensional process with "a unit of the square root of the area u² (see below)”; we will call the invariant unit of the times table a "sec"" for the area of the total summation of the area of the four squares of space-time. Then the 5-d hypotenuse cut would have a pure number a "transcendental" 5-d number c=i and it's "4-d length" of i(sec). The full 4-S sign multiplication times table used for how the LHS and RHS signs of the area u² common area behave across the equal sign, are; same signs on the LHS and RHS give +ve while different signs on the RHS and LHS give -ve. Or the appearance of the bridge (common area) across the equal sign is in units of -- +i and -i -- that is, how we cross the equal sign using the area of u² on the LHS and using the area of u² on the RHS.

You said in my comments

You have some interesting ideas but they are very speculative. Essay contests such as this are a good place to present such ideas:-) yes you are right about that

I don't think you can set i=c or i=h but I do think you can construct something similar to the following:

PSI = exp(omega) = sqrt[1 - (v/c)^2] + (v/c)i

Then for v=c, PSI=i. I looked at your work instead, to see how you bridged with a common 5-d length (of the square root of f²) the areas on both sides of of the equal sign. Your method mixes lengths with areas across the equal sign. While in the full 4-S, it is the four sums of +i and -i that are the "invariant count" lengths of the area u². The hypotenuse of the area geometry of f² is an invariant 5-d length "f" which isn't an area on the LHS.

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