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The Fundamental Existent

by Jack Martinelli


This paper presents the Lorentz/Einstein coordinate space as a quantum sub-space and the universe as the intersection of these sub-spaces. The geometry of these fundamental existents, presents novel concepts for invariant measurements, space, time, mass, energy, causality, field, quanta, position and momentum . These concepts are used to develop a descriptive theory explaining how mass "acts" on space to cause it to "bend" and how the curvature of space yields a causal action, i.e., a generic force field. The theory is supported by derivations of Hubble’s constant and correspondence to Bohr’s equations for atomic spectral emissions. Also given are derivations for Einstein’s and Planck’s quantum energy equations as well as a deeper understanding of their equivalence.

Equalizing the frames
Definition of a measurement
Definition of a Clock

Definition of scale
Definition of Invariant Measurement

Derivation of Snell’s law

The three most fundamental equations
‘Length is Arbitrary’ or ‘The Bootstrap Space’
Newton’s laws
Definition of a stable distance
Relativistic expansion
The Stable intercoordinate distance
Bohr’s Atom
The Fine structure constant

Element Skateboards, Completes, Decks, Wheels and more 4-less!



A Deeply Unified Field Theory

Rather than try to explain the fundamental forces in terms of known physics, this paper tries to explain existence and causality by unifying the three basic units of measurement -- length mass and time. The attempt is to relate mathematical abstractions to substantial existents and their dynamics. The result is a fundamental understanding of mass, causality, force, energy, their interactions and how they are physically and mathematically related.


Einstein’s equations for length contraction, time dilation and mass expansion are:

(1)     (2)    (3)

By setting the square root to the ratio of the primed to un-primed masses you get:

(4)    (5)

Note that the product of mass times length remains constant independently of velocity. It is invariant. You can show that the product of scale and length has an identical invariant expression. The balance of this paper is an exploration of this idea.

Equalizing the frames

Einstein said that "there are no special frames" that "the laws of physics are the same for every frame". If, however, you look at space and matter you can recognize two special frames. The space frame, which is the container frame, and the matter frame, which is the contained frame. I call this the container/frame problem.

My approach to solve this problem was to remove the space-time continuum - the special container frame -- and propose the frame as the quantum existent -- the fundamental existent. The universe then is the intersection of   frames. Nothing really new here except that I propose that these frames are substantial and are the fundamental structure of the universe.  I call these frames sub-space particles and the aggregate (the universe) the sub-space plasma or the ambient sub-space.

There are three qualities that are implicit in this proposition. The first is that action at a distance simply becomes "action" or "interaction." There is no concept of distance separating objects. All objects intersect each other. This is identical to the concept of a coordinate system extending infinitely and I assert is the basis for non-local interaction.

The second quality to recognize is the quantization of a real space. My quantization differs significantly from that used in the standard quantum model. The standard model defines an abstract particle – the virtual particle – to explain the mechanics of interaction. This particle assumes the existence of the container space as the medium for all "interactions" and that an interaction involves a transaction across some quantity of this space. Many of the interactions of the Standard Model seem to violate the principle taught to us by Einstein, Maxwell, Lorentz, Michelson, Morley and many others. This is the principle: "Nothing can travel faster than the speed of light." Quantum theory and many experiments have shown that many interactions can and do occur at exchange velocities exceeding the speed of light. My model supports this kind of interaction. Because every quantum intersects every other quantum, there is no distance to traverse. Consequently, my Sub-Space Model does not break the universe’s speed limit. In fact, one of the most exciting results of my model is that the speed of light must be constant as a result of these relative sub-space particles.

The third observation is that because there is only one kind of existent – the sub-space particle – there is no question about what contains the universe. There is no hierarchy of containers. The universe simply is.

The volume of work to perfect an argument of this magnitude goes beyond the scope of this paper (and probably my lifetime). To demonstrate that this theory is plausible, I have presented an argument for Planck’s constant and Hubble’s constant. Furthermore, my model demonstrates correspondence with Newton’s conservation laws, Einstein’s SR & GR laws, and shows that the speed of light must be constant. It also shows that force, fields and energy are geometric properties directly related to a quantum geometry. The sub-space particle demonstrates the necessary measures, constraints and conditions to explain wave-like as well as particle-like behaviors. Fundamental to this argument, as Heisenberg demonstrated, is a deep understanding of the measurement.

Definition of a measurement

A measurement is the quotient an object divided by a reference object of exactly the same kind.


Notice that the definition does not refer to a number but does refer to the "quotient of objects". In physics, a measurement maps a property into an abstract space -- a mathematical space. The measuring process is identical to finding the quotient of two numbers. Additionally, it is important to realize that the magnitude of a measurement is relative to the reference object. Its value and more importantly, its meaning is not intrinsic to the measured object. That is, a measurement is a model -- a representation of a real property.

Throughout this paper I refer to a reference quantity (like a ruler) with a zero subscript and non-reference quantities with a non-zero subscript.

Definition 1 applies to a delta measurement as well. That is, if we are to measure a change, say Dr1 it must be measured (divided) by a well defined reference object of exactly the same kind -- say Dr0.

One of the primary goals of my work is to find one or more universal referents that are related to the structure and dynamics of existence. With this reference set you should be able to construct a reference dependant coordinate system – as an abstraction – which demonstrates a high degree of semantic alignment with reality.

Definition of a Clock

We do know that a clock changes. Its hands will move, its digits will change, electrons will move  -- something must change for a clock to be a clock. So we define a clock as an object that gives us a reference change. Different kinds of clocks can be made to measure different kinds of relative changes. My clock is a linear clock. That is, the amount of space traversed by an object divided by the amount of space traversed by my clock (Imagine a slider control – like on the front of a stereo.) gives a measure of a relative change in position. Then the clock is like a ruler & a measurement by this ruler is a measure of a change in position – velocity. And a difference between two velocities – another positional difference – also measured by this "ruler" is acceleration.

You should really have a Java capable browser these days.

The idea behind a measurement is to find the number of reference objects that are "contained" in the object being measured.  Thus, to measure a change in position we use a reference change in position.


An important property of a change in position is that it is linearly independent of position. This property gives it the illusion of a spatial axis -- not something we can easily say. As a convenience, it can however be incorporated into basis functions, manifolds, tensors or parametric functions. Also notice that a linearly independent parameter can behave like a flat dimension as time seems in relativity. Furthermore, a change in position can never be negative, (a coordinate change can be). Time, as a reference object also can never be negative. This is important to understand, since, when we start to deal with coordinates we do need a negative sign.

Definition of scale

The scale of a frame is the relative size of a reference length with respect to a reference length residing in another frame - E.g., the primed frame.


It is equivalent to measuring a reference object in our frame with an identical reference object in a primed frame. There are no length restrictions on the reference objects (they could be shovels or boots), only that if they are in the same frame that their measurement be unity and that they be identical in composition. Thus if we choose an arbitrary length rx for our reference object for measuring scale, and duplicate it and put one copy in frame one (primed) and the other in frame two (double primed) we get for scales

or (8) and or (9)

Combining (1), (2) and relabeling… and , we have:


That is, if we place a test length in some frame and measure its length and scale with respect to another frame, the product of this scale and length will be the same for all frames. Which motivates the following definition.

Definition of Invariant Measurement

(Invarient Measurement)=(Scale)(Measurement)

The invariant measurement expanded is:


The invariant measurement is a measurement with respect to the other frame. In other words, it is a "frame native" measurement. The scale of the ‘foreign’ frame acts as to transform the reference object of your frame to an exact equivalent reference object of another frame. In yet another respect it acts to normalize a measurement. That is, it quantifies a measurement as an ‘amount of an existent’, similar to the product of a mass density and volume, only here we get the "amount of measurement" rather than a conventional mass measurement. The salient point however is that, with each measurement expressed in common units we can perform standard arithmetic and geometric operations in a common space.

Take the following example:

Derivation of Snell’s law

A straight line is the least amount of space between two points.

The line from point A to point B is a "straight" line (geodesic) through the two spaces 1 & 2. The total invariant length or amount of space between points A & B is

: (12)


: (13)         (14)

and so...    ( 15)

finding the minimum of by taking the derivative of with respect to x...


canceling & rearranging we have

Snell’s law: (17)

Note that this can also be writen as:


Note that l is exactly analogous to Snell's wavelength.   Note also that length is a simpler concept than wavelength.  (see "What is Occams' razor?")

Line is what Einstein called a geodesic - a line representing the least distance between two points. In undergraduate physics, Snell’s law is presented as something of an enigma - that is - light traveling through a transparent medium somehow seems to find a path that is shortest with respect to time. Here, we express Snell’s law as a function of the least amount of space between two points in an aggregate space - i.e. a straight line. Then, it becomes quite clear how and why a particle seems to find the path that takes the least time. Furthermore, if we view matter - like glass - as space, its transparent quality becomes fairly intuitive..

The three most fundamental equations

Throughout this document I refer to the three following definitions:




Or, each is constant in any frame:




Invariant velocity is identical to momentum and invariant acceleration is identical to force. Note, however, that the terms sc and sa, also express relative change and change in relative change.  (see also "How a change in scale affects length, velocity and acceleration")


The units that energy is expressed in are . Interpreting this in terms of my model, you would say that this is an invariant area. The reasoning for this is that you have square meters divided by a reference area – square seconds. And this is multiplied by kilograms to make it invariant. But what makes this such an interesting measurement is that it represents a measurement of a dynamic area. In other words it is like momentum only in two dimensions rather than one.

So, consider energy as a change with respect to position as a corroborating measure.

Let measurement A represent a change and be measured with respect time - a reference change. And measurement B also represent a change and is measured with respect to a change in B’s position. A’s body is moving away from B’s (or visa versa).

The measure of A’s change is just

  and B’s is


The amount of change in A’s position with respect to B is identical to the physical amount of change in B’s position with respect to A independently of the measuring technique used. That is:


And solving for dB we get:


Integrating both sides gives:


Then if A=sv then B is energy – a measure of a change with respect to second kind of reference change – dr. (Also note that there are other possibilities for B when A is sr or sa.)

‘Length is Arbitrary’ or ‘The Bootstrap Space’

In geometry we have an abstract notion of a point in a space. In a real space – our real space – we cannot find these so called points. We can however measure distances between objects (physical points or test points) & so we might infer the existence of a  space that we can represent geometrically. However to measure a distance from something to something there must be two physical existents present. One at each end of a ‘length’. To construct a coordinates  that are physically significant, what we would like is a coordinate space based on the presence of these point ‘test’ objects. Furthermore, to separate the notion of an abstract point and a real point. We define a real location as a place where we can place a real and unique test object.

Visualize a point in a 0 dimensional space. Then visualize n intersecting points in the same space. You should see that the two images are indistinguishable. That is, there is nothing special about any of these geometric points. The single point and n-intersecting points are visually and conceptually identical. If you imagine the same image for one line in a one dimensional space then n parallel lines in this space you get the same observation, i.e., since the intersection of a point is a point each of the individual line’s points looses their individuality. You can continue to apply this process to two and three (or more if you want) dimensions with the same results. Each individual object vanishes. However, in two or more dimensions, if one of these objects is spinning you have one object with an axis which defines one unique point (but no boundaries!). Call this a physical point. This point is shared by all objects in the collection. (notice that with respect to this spinning object the rest of the objects are rotating in the opposite direction and it is not rotating with respect to it self). Then if you have two objects in the collection that are spinning whose centers are not concentric you end up with two uniquely identifiable points. Note that if you are a member of this collection, in this state, you cannot perform a distance measurement. You have no ruler. Consequently you cannot assign a position to these points. You need at least two more points for a ruler. Say you have four points. You need to choose two of these points to use for a reference length. Naturally you would want to choose two which are not moving with respect to each other - especially not moving away from or towards each other. But to make that determination you need to perform at least two distance measurements. And you’re stuck. Adding more points just makes more points. You are still stuck with choosing which are not moving with respect to each other. It turns out that any two points will do. You can use these two points as your ruler to find any number of points that are as stationary with respect to each other as are your ruler points. However, because the choice of your ruler points is arbitrary, any other pair could have equally been used. That is in all cases any and all of these possible reference distances represent one unit of reference length. To some readers this may seem absurd – it did to me. But it is unavoidable. Each of these possible references are mathematically equivalent. In other words "length" depends upon an arbitrary choice or some reference length. It is not an intrinsic property of space. Then if every one of these lengths is equivalent how do you measure anything? Its simple of course.

In the development of invariant length we used a frame as a container space. But to actually have a real container you need real boundaries. A spinning space can only give a point but does not define  containing walls. But then if we have two unique points we do have the required boundaries for a one dimensional space. That is our reference length defines a frame which represents a length. A length which is equivalent to all other lengths in a space of two or more dimensions. Furthermore, this one dimensional frame gives us an object of exactly the same kind as all other one dimensional objects contained in a space of 2 or more dimensions. I.e., it satisfies the requirements of an object used to measure other one dimensional objects.

Also when you are looking at a one dimensional frame you don’t know if the distance you are measuring is of a larger or smaller scale than that of your ruler.

Thus we can use equation (7) to calculate the scale of an abstract line in our real space. Then use the invariant length to represent this quantity for all lines in real space. I.e.,


Newton’s laws

Notice that a spinning space within multiple spaces creates only points (and axis'), not boundaries. Furthermore existence itself cannot be contained, since the only logical choice for such a container would define a transition from space (something that possesses the property of location and distance) to that which cannot be space (remember intersecting spatial objects lose their identity). Since this skin covering the universe cannot posses the property of location, it cannot have a size; or its size=0. In other words the universe cannot be bounded since the size of the universe is apparently >0.

Without a boundary, there cannot be absolute position velocity or acceleration. A relative change in the position of a point with respect to another point however, is all that is possible. I.e.,:


the v’s represent a change in position as measured by a reference change in position. The scale parameter makes the measure invariant. The sv’s quantify the amount of "object" that the other space traverses or the amount of space that the other object traverses. These must be the same. I.e., If frame A moves away from frame B it is exactly the same as frame B moving away from frame A and SV is the amount of change in position that occurs. This can be generalized for n frames as:


Frame i moving in an arbitrary direction with respect to frame j is identical to frame j moving with respect to frame i in an equal and opposite direction. Because of this equality the sum of the two quantities is zero.

Eq. (31) also implies a general principle (and Newton already kinda said this) That "For every change there must be an equal and opposite change." Which would also include the following.


Definition of a stable distance

A stable distance is one that does not change with respect a well defined reference distance.

That is, it does not matter if the endpoints of the measured object are moving away from each other if the measuring object is behaving in exactly the same way. Furthermore, if two frames do not seem to be expanding, we can only conclude that their expansion velocities are the same. (see "Is Matter Shrinking")

For example, imagine looking at someone in deep space and observing that he is growing. To him you are shrinking (all of your atoms are moving inward). From either frame, either observation is true. You can arbitrate who is growing and who is shrinking only by choosing one frame as the reference frame.

We can express this expansion/contraction model analytically using our set of invariant measurement equations.

Choosing frame 0 as our reference frame and combining equations (21) & (22) we have:


From this we can calculate the amount of acceleration of one frame with respect to another arbitrary frame of another scale.

Because is constant the derivative of Eq 33 with respect to t is 0.


Multiplying through by and rearranging…


Substituting and we have:




Notice that this looks exactly like the acceleration of uniform circular motion.

Substituting this back into equation (23) we get


or simply


In a flat but "expanding" space, the expansion velocity is given by the product of an expansion constant and the distance from some point. In our universe it is Hubble’s constant, H0 , times r, the distance from us. When we make this substitution into Eq. 39, we get:


This says that in a single flat space expanding with respect to one other frame, the force is constant everywhere. That is, there is no potential difference between locations. Now we need to look at how scale changes with respect to an expansion velocity.

Relativistic expansion

If mass is scale in disguise, then the equation for relativistic mass expansion should apply to equation 2 as well. Thus, for a space expanding with a velocity v with respect to another space we can write:


Applying this to an expanding frame with some expansion constant H, we have:


After a little algebra we get:


Our expansion velocity then becomes:


Notice that, in Eq. 42, as l approaches infinity that l’ approaches c/H. Consequently the expansion velocity, Eq. 43 approaches the constant velocity c and space becomes spherical for large l. Or in general, for any expansion constant H, when:


Note that with a very large radius () our generic force equation then becomes:


The completely general equation for all r is:


Where H represents the expansion constant (/s ala Hubble’s constant) of a sub-space with respect to a reference frame. Each field equation is given more detail in the following sections.

Note that a graph of this function resembles a soletonic wave..  In the following graph, the purple line represents the clasical electrostatic force and the black line represents its solitonic form as predicted when the electron is viewed as a SubSpace Field.

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The Stable intercoordinate distance

To be able to measure the scale of any SSP we need a universal reference length, which we can identify in any other SSP. This length must represent the same physical amount of space in any SSP of any scale. Although the simple measured length may vary from one particle to the another, we can use this variance to calculate the relative scale of that particle.

Note that Eq 5. Is similar to Eq. 10. Because of the similarity, we shall explore the hypothesis, that mass is the scale of a frame. For breviety let .   One constant equal to another.

we can re-write Eq. (45) as


Note that Eq (37) looks exactly like the inward acceleration of uniform circular motion. (when the sign is negative it is a contracting space. A positive sign means the space is expanding.) We can balance the expanding or contracting force by rotation. (If one frame is spinning the other is "orbiting". E.g., the sun orbiting around the earth. I call this spin-orbit symmetry see also "Mach's Principle")

We can express the orbital velocity as:




combining 47 & 49 we have ( inward = outward force when v = c and acceleration is inward )


The field strength of a sub-space particle is proportional to its rotation frequency.

The energy of a stable orbit is:


The antiderivitive for a constant frequency f is:


Replacing r with we have:


Which corresponds to Planck’s energy equation for a single quantum of energy.

Calculating the same from Eq. (39):


The antiderivitive is:

Expansion Energy:


Thus, as Einstein had asserted in 1915:


More generally, if the two stable objects are comprised of i, and z SSP’s in each group we would write:


The Planck/Einstein equation implicitly defines a stable & universal distance. That is, this distance represents our ruler distance and it is embedded within our SSP. Thus we can use this distance to measure other distances and more importantly the scale of other particles.

Bohr’s Atom

This theory, like Neils Bohr’s relies on the idea of stable orbits for predictions with respect to spectral emissions. Take the ratio of the electrostatic force to my force equation:


We can use this result to express the electrostatic force of a single SSP as:


and since force vectors add then for a group of i SSP’s:


The energy for a group of j electrons can be expressed as:


The energy of i sub-space particles in stable orbits around this j electrons is:


And under our stability constraint these must be equal:




Which is the result that Bohr couldn’t explain.

Then solving for c…

And since:


after canceling and rearranging:


Substituting this back into Eq (65) and reducing terms:


This result essentially reproduces Bohr’s famous "energy of an atomic orbit" equation. Except that here we have spinning Sub-space particles and electrons in stable positions with respect to each other.

To express the change in energy going from i to j sub-space particles:


This equation is exactly like Bohr’s equation for predicting the spectral lines for atoms.

We can calculate this from another perspective. Take Eq. (64) and expand .



Notice the similarity to Eq 21. Now, solving for s we have:


Writing Eq. 22 as


combining these we get the equilibrium velocity for our set of particles:


Putting this into Einstein’s energy equation:


In the following section we show that the ratio of radii is the fine structure constant. Thus giving us exactly the same result.

The Fine structure constant

We can write equation (21) as:


The radius of the classical electron is:


The radius of a sub-space particle with mass m equal to an electron’s is:


The ratio of these two expresses the relative scale of a single electron to our SSP:


An expectation you might have (as I did) is that this ratio should be 1. That is the radius of the electron no matter how it is expressed, when divided by an exactly equivalent expression should always equal 1. The same is true for the ratio for the mass and scale. Unless the expressions have different reference constants that are assumed to be equal. This mistake could easily occur if there are two different spaces of two different scales. Where the measurements are only consistent within the respective frames.

Another possibility is that we are calculating the ratio of the radii of one electron (my electron) to two. The reasoning for this is that the classical electrostatic model depends on two electrons. That is, the electrostatic force can only be measured by a reference electron and a test electron. Calculations using the force of one electron implicitly assumes the presence of another.

Take two SSP’s of scales and . It is easy to show that the scale resulting from the intersection of these two SSP’s is the product of these two scales.

Interactive scale


Eq. 76 suggests that a single electron changes the scale in its vicinity by the factor , or the scale of a single electron with respect to the ambient sub space is


and for two electrons in each other’s presence


And since scale is scale, whether it is squared or not and by invariant (21)


Measuring both side by the scale of ambient space gives:


In this equation, the scale on the right hand side is the scale relative to a scale of 1. This scale represents the scale of ambiend space.  But is not in the units that we typically work with -- if mass is equivalent to scale. To express this as mass then me=s0/a2, and then we can express s0 in mass units.

The net scale of the electron is the scale of ambient space divide by alpha squared.

The scale of the universe with respect to the mass of the electron then is:


Take Hubble expansion. is the reference length at the scale at which the expansion velocity of a galaxy equals the speed of light.




represents the relative scale of two galaxies (one interacting and receding from another) with respect to the reference space. The scale of an individual galaxy is the square root of Eq. (84) or,


From this result we can predict a value for Hubble’s constant that is quite precise.


My value for Hubble’s constant is 2.005418631e-18 /s. CODATA gives this as about 2.1e-18 /s.

Notice that the rest energy for s0 is about -(2)13.6 eV.

Then, expressing as a function of scales:


We can then write our special force equation as:


If this field equation is relatively general the electron mass could be replaced with an arbitrary mass. Then, if we have n sub-space particles that are stable with respect to z arbitrary massive particles, the magnitude of the forces between them must be equal. Therefore:


canceling and rearranging we have:


Which is interesting enough but the point is that the "rest" energy of s is:


And Once again we have correspondence with the energy of Bohr’s atomic orbitals.