The twin prime conjecture
Central Idea:
One can regard the overlapping shadow diagram below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
Now as seen in the overlap, two different shadows combine to form a darker shadow.
Now think about points, mathematical objects which have no extent.
If we had two objects, one in front of the other illuminated by a single light, the object closer to the light would cast part of it’s shadow on the object further away, but it could extend past the closer object in both directions. The resulting shadow cast on a table below would have one “point” everywhere, it would be similar to the situation of coincidence in mathematics.
The two overlapping shadows show a new situation of two “points” being placed together as the shadows have no height so can be thought of as points which have no extent.
This can be thought of as ‘sharing’. The two “points” are sharing one location.
There can be hidden items of no extent due to the nature of the notion of no extent.
Something of no extent can be multiple, for example doubled, there could be two items of no extent there. They would just appear to be one item there as both items have no extent. This is certain. So it is possible there could be more structure.
So we need another item to no extent. But the only items we know about are points. We know that they have no extent and also that if I place two together the result is a single point.
So what if there is another entity of no extent but if we place two of these types of items together they do not merge into a single point but share position as in the overlapping shadows.
We can call points p’s and the other new entities e’s. If I place two points together in overlap they coincide and we say we have one point. Also if two points are placed next to each other they have to touch in their entirety so we also have one point.
But now we have e’s as well which are zero-dimensional but not points. So I can place a point together with an e as poe and this is a point and an e overlapping.
This has a form of zero distance like pop, but e and p do not merge, being different entities. Yet there is another form of zero distance as we have an e and a p and not two p’s. The only other way two entities of no extent can come together is contiguously, so we write e*p as e and p together with the new zero distance in mind. E is contiguous with p and also overlapping with p. This is how e is different from p.
We can think of the analogy given near the top of one object in front of and extending past another object, both objects being lit by a single light. Then there is that shadow cast on a table. But there is some of the shadow cast on the further object from the light from the nearer object to the light. We can think of the shadow being lifted off of the combined shadow onto the further object. This is an analogy of e being both overlapping and contiguous with p. As it shows e lifted off of p.
Two items of no extent placed together can be thought of as a single position as in points, p or two e’s next to each other or overlapping like in e’s. This is how p’s and e’s are different.
Two e’s cannot be in coincidence. If we imagine a space of only e’s, eoe does not equal e otherwise e=p. Also e*e does not equal e otherwise *=o and e=p. So e*e=e*e, that is to say e*e does not resolve to a single e.
The two e’s “touch in their entirety” (since they are both at p but they are together in a serial sense or an overlapping sense).
When we tried to put two points together there was no choice but to resolve it to a single p, since p’s were all we thought of that had no extent. Since we open up the door for another possibility, already having p’s, we can have e’s here now.
At p we can have two of the same e. Or just one e, to begin with. When one e combines with a p it does so either serially or as in coincidence (overlapping) as e and p are different entities.
Two items of no extent could be coincident, as in points, or also share as in e’s. The e’s are in the same position so that they are both there. Yet they share as in the overlapping shadows, since something of no size added serially or in overlapping to something else of no size still has no size. Yet there are two types of zero size so these can all fit together.
So the idea is that points, p and items e are sharing the concept to no extent but are different in another way. The other way is that e’s share while p’s coincide.
To sum up, two p’s can be in coincidence forming a single p. E can share with p either serially or in the sense of coincidence, this can work since e’s and p’s are different entities. E’s can exist sharing with each other in a serial way (contiguous) or overlapping. E’s can also share with other e’s, serially or overlapping.
At the beginning we can have a plane of these new entities “e”, coexisting with points. Identify which e’s are in the set as is done with points. Let us start with the whole plane then if I move an e I can put it in sharing with another e, leaving a hole.
But let’s examine e, further. Suppose I think of a single e. It can’t exist singularly like p exists singularly. We can think of p being in coincidence with any number of copies of itself, leading to just a singular p. But e doesn’t coincide with itself. So it can share with itself any number of times, sharing being associated with e’s. This is not the same as coincidence as I can separate e’s away from copies of themselves but I cannot do this with p’s.
Then instead of leaving a hole, let e share with itself once and then have a copy of e, leave to share with other e’s. We can have a subset of another coexisting plane of e’s. This way we can have a fixed plane of e’s and a moving set of e’s.
So how does e move away from p? First put in a plane of e’s then add to it a plane of p’s. Then we can have a subset of another sharing plane of e’s. Then e(1) at p(1) can move off into the fixed plane so the fixed plane becomes a place of places, a next dimension of place. So we have a dimension of place co-existing with a dimension of places of places. e(1) is sharing with itself so it both moves in the e plane and stays still in the p plane. Its place stays the same but the place’s place of places changes.
So this fixed plane may be regarded as a place of places and the set of moving e’s as moving places. The fixed plane is a new dimension or level of places, places of places.
We only need the space of a point to build this and we have to have a point to start with. Then also the still e and the moving e can share position. So if I have the concept of a point, I can further build this and I have my new item to no extent. I am able to remove the moving e from the combination.
Then e and e’ are different. E’ can be fixed e’s. (e’[(e)]]. E is contained in e’.
We can then form new structures with e, which have new properties. The space is re-engineered.
Foundational Preface: Universal Concept Theory (UCT) framework:
Abstract:
A framework for the structural completion of mathematics
Objective: to propose a unified foundation framework-Universal Concept Theory-(UCT)-that resolves long-standing mathematical conjectures (e.g.., the Collatz Conjecture and Fermat’s Last Theorem) by redefining the nature of mathematical identity and coincidence.
Methodology: UCT departs from standard axiomatic set theory by introducing “Conceptual Engineering”. This process involves three primary stages.
- Scaffolding: The construction of higher level “Places of places” and “Number of numbers” that exist as containers for lower level concepts.
- Concept Removal: The systematic removal of the single occupant rule, allowing a single placement to support multiple entities.
- Concept Sharing and Separation: The introduction of a variable “Coincidence Switch”. In the 1-sharing state, the distance between distinct concepts( such as the steps in the Collatz sequence) is reduced to zero, creating a unified identity. In the 0-sharing state, concepts are “separated” into the discrete non-overlapping values found in standard arithmetic.
- Structural Capacity: UCT demonstrates that the transition from sharing to separation is governed by the “structured capacity” of the engineered space.
- The Fermat Limit: The theory explains Fermat’s Last Theorem as a geometric mismatch: while 2D squares possess the directional capacity to support 1-sharing, higher dimensional cubes (n>2) do not, forcing the coincidence switch to 0 and precluding integer solutions.
- Collatz Conjecture: By applying 1-sharing, the entire Collatz tree is revealed as a single, folded singularity where all integers are conceptually equal to 1.
Conclusion:
Universal Concept Theory provides the “missing layer” of mathematics, transitioning the field from a collection of isolated rules to a complete, structural hierarchy. By understanding the “backstage” of concept sharing, the paradoxes of standard math are revealed as simple logical certainties.
The Foundations of Universal Concept Theory: The Host and the Guest
In standard mathematics, a “point” or a “number” is an isolated entity. It is a lonely occupant of a single location, and standard rules dictate that no two distinct entities can occupy the same spot simultaneously. Universal Concept Theory (UCT) engineered a more sophisticated foundation by introducing the Host.
1. The Host (The Higher-Level Scaffolding)
Before we can understand how concepts interact, we must first build the environment. We define a Host (represented as; r in geometry or A’ in arithmetic).
The Host is not a “container” that is larger than its contents. Instead, the Host is the fundamental environment that shares the exact same space as the concepts themselves. It is the “scaffolding” that grants permission for multiple concepts to coexist. Without a Host, there is no room for sharing; with a Host, the capacity of a single location can expand.
2. The Guests (Fixed and Mobile Entities)
Once the Host environment is established, we perform Concept Removal—removing the old rule that a location must have only one occupant. This allows us to introduce our “Guests”:
- The Fixed Guest ( p or A): This is the original concept. It remains anchored to its identity, providing the base reference for the location.
- The Mobile Guest (e or B) This is the new entity (like the e iin our geometric work). Because the Host provides the room, the Mobile Guest can move or shift within the extended space while still “sharing” the same fundamental location as the Fixed Guest.
3. The 1-Sharing State (The Social Connection)
When the Host is active, we enter the 1-Sharing state. In this state, the distance between the Fixed Guest and the Mobile Guest is defined as zero. They are distinct characters, but they “coincide” perfectly.
This is the “Natural State” of mathematics. It explains why a Collatz sequence is actually a single, unified chain: every step is a different Guest sharing a seat at the same Host’s table. The sequence only looks like 111 steps long because we have “separated” the Guests.
4. The 0-Sharing State (The Standard Restriction)
What we call “Standard Math” is simply the state where the Host has restricted access. When we set the coincidence switch to 0, the Guests are no longer allowed to share the same seat. They are forced to separate into the discrete, isolated points and numbers we use for everyday arithmetic.
Mathematical concept removal, subsequent sharing and separation
Introduction:
The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?
They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”
But what if something like the contiguousness of two points could be possible. Since something of no extent combined with something of no extent would still have no extent- but there could be two items of no extent there! The two objects could be hidden as one! The two items of no extent would have to be different from points in some way so that they would not combine to be one singular item.
One can regard the overlapping shadow diagrams below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
The shadows can be regarded as two points overlapping which could be thought of as at a common point, but also as two items which are both not points being in a contiguous state. Now as seen in the overlap, two different shadows can take up the space of one location, we can regard these as e and p. e is another possible entity of no extent, p is a point. If I take away one light, one shadow still remains. The table can be like an underlying, hosting space.
If e and p are sharing a position, they do not combine because they are different. It must be that they are either next to each other, or that they overlap and do not merge because they are different entities, as they are both not p’s. Or overlap, creating one point if they were both points or creating one point if they were side by side as was Aristotle’s way of thinking of things. An item of no extent put with a different item of no extent would still have no extent but there would be two items there.
We can consider 2 e’s at a point. The two e’s would be in a sense the same, since they are at the same point, yet they are different from points, in that they can appear multiply in a contiguous state..
Think of a mathematical point not as a single solid object but as a Russian Nesting Doll. Even when the dolls are tucked inside one another and appear to occupy a single spot, their individual identities remain perfectly intact and shared. My new language allows us to unfold these nested layers, revealing the hidden structures and connections that standard math accidentally flattens out. This analogy was developed in collaboration with an AI assistant.
Start with a plane of places and a horizontal line of points labelled 1,2,3.. So that we have a unit length.
We wish to show that there is a next level to this plane and set of numbers on a line.
Let’s build the next level.
What is scaffolding?
It is the creation of a new concept based on a pre-existing concept. The new concept is said to host the pre-existing concept. It has the same basic notion but exists at the next level of conceptual space.
Since the concepts have the same basic notion there is an exact fit of the next level concept and the lower level concept sharing space together. So there is “room” for this. But then also since the two concepts share the same basic notion there is a ‘building” possible. One can build upwards.(Since the host is the next level of the original concept in the concept space as I can remove the guest. Also the host itself might also have a host). Since we only have e’s and p’s. The host level is made of e’s and can host p’s or other e’s.
Also,since there is more than one concept, there is more than one next level of concepts, so we can build outwards as well.
In points, call this new item “r” This is a set of e’s. Then this together with a point can be written r(p). We can separate r and p (not divide as this is a separation not a division) if we take p out, the lower concept can be removed. We can place a different kind of lower concept which can have freedom of motion in the higher concept, while the original lower concept remains fixed. Then the two lower concepts are also sharing space and don’t coincide.
One entity of no extent is able to host the guest entity of no extent. So that the two exist together without combining. The guest entity doesn’t host so it is different from the host entity.
We can remove the guest entity of no extent ‘p’ and replace with ‘e’ a new entity of no extent which is capable of moving through an extent of hosts. Also we could put p back in and e would exist together with p and r. All these would be different. P is fixed while e is mobile.
So the host is a next level place of e’s, hosting guest places p or e. Where hosting is defined as r and p or e sharing ( together but not combining) position, with p or e capable of being removed from r and e moving in r. R is the hosting space coexisting with p and e. Briefly (r[(p(e)].
I can place a point on another point, but let’s not have them coincide but let the new point be a point where points could be. So I am thinking of an e placed on a different concept of e. Rather than just a plain point (1st level). This would be a second level, possible since the new point and the original point carry the same basic notion, that of a place of no extent, so that the upper concept becomes a place of places of no extent-containing the lower places of no extent. It fits exactly with the lower concept. These concepts are co-existing. This is concept building.
Since e is not p, e can have parts. But since it is of zero-dimension it has to have zero-dimensional parts which add to itself. These can’t be p’s since p’s add to a single p.
But must be on the same level as e’s and p’s, so must be e’s. Therefore e can have parts of itself so that e(1)=(e(1)(e(1)).
So e is able to self-replicate and split up or recombine in a space of r.
In concept sharing we have the same basic notion but two different ideas. So two ideas are sharing the same space, as they are only concepts, this is possible. It must be that I can take one away from the other. So one exists at the next level from the other, as there is no other option at the beginning.
That there is a next level is granted because any concept can be continued. Hosting can be considered as exact containment. The concept can fit exactly into the next level of itself. Since this is an exact fit we can move into the next level.
In geometry the host becomes the place of a place at the position of the place. The shared concept is “no extent”
In numbers the host becomes the number of a number at the position of the number. The shared concept is a natural number.
Since points and numbers both have positions these ideas can be combined into a new plane.
We must have more than one place at this place of lower places so that one lower place may be fixed to keep contact with the first level and the other places would be mobile in the new dimension as I can have multiple places of lower places of no extent.
Then a new number dimension is co-created. Number the lower places (1(1) and (1(2) or (2(1)-for two lower places. These are two new higher numbers where the number of lower numbers is 2 and not 1.
We can think of a jigsaw puzzle of a landscape being taken apart.
This is how it would be for the concept of a point and a number. In general then we can regard any mathematical concept as capable of this sharing. Coincider a concept A and related concept B.(Shown as circles in the diagrams below).
We can have an axiom of concept sharing if I introduce the notation ((). This is showing that the combined concepts are not in coincidence. Like (<—() The left bracket is moved away showing the revealed concepts. The extra structure is shown as a revelation of one labeled concept away from the combination which does not coincide as the concepts are different in some way to be revealed, it is an expanded notation and makes a sentence. So for example (A(B) means A is not in coincidence with B, but sharing the same space.
So the axiom is expressed as (A’[(A(B)] or for a specific concept (A’(1)[(A(1)(B(1)}. A(1) is a specific, fixed instance of a concept. B(1) is another instance of the concept yet different in some way. In order for the two not to be in coincidence another level of the concept is opened up, overlying but co-existing, so that A(1) and B(1) can be in the same lower space, able to separate.[,] is showing containment. The spaces are co-existing so that they can be in the new space to begin with. This creates a concept space with the next level concept denoted A’(1). Otherwise there is no “room” and A(1) and B(1) just coincide.
To have (A(B) I have to take out A. A(1) can only be taken out if there is an overlying concept A’ which copies the notion of the original concept A but exists at a higher level such that A’ is the concept of the concept A taken to a new expanded level. Such as for points a “ new place of lower places B”. This creates a leveled concept space …(A’’[(A[’(A(B)]] with A’’ even higher than A’. All these A’, A’’, … coexist with A as the (() sentence shows.
The concept B can move in the expanded concept level as B can overlap with A’ since A’ is the expanded notion of A and B is shared with A. Making the logic of A’ and B consistent as B must be different from A, being able to move in A’ while A is fixed.
The next level concept carries the same basic notion of the original concept, so that it can overlap and connect with the original concept; this is how a concept can build upon itself. Yet it is overlying and co-exists with the original concept.
Then if I have another instance of this concept say A(2) I can have (A’(2)[(A(2)(B(2))]
Then (A’(1)[(A(1)(B(1))] and (A’(2)[(A(2)B(2))] allows (A’(1)[(A(1))] and (A’(2)[(A(2)B(1)(B(2))].
This is the movement of B(1) making B(1) different from A(1) as A(1) is fixed. Then the axiom is self-consistent as A(1) is different from B(1) as required.
I have to take out A to have (A(B) so this necessitates A’ Which then A and B are defined to exist in. A’ has to have the same notion as A and B, but extended.
B has to travel in A’ so that A’ has to be a new dimension of concept A, an extension of concept A. The only way it can be extended is to have another level of concept A a concept of concept. So for example in points a place of places or in numbers a number of numbers.
If we take out a place, we have a place of lower places, where more than one place can be. This has to be built first. If we take out a number we have a number of lower numbers, a number which indicates how many numbers there are present.
These can number the places in the place of lower places.
We can have more than one number or place because we are opening up another dimension of the concept. The concept can now move away from its singularity.
There has to be more than one in order to move the other type of concept along the new dimension. Then the new concept dimension has to copy the notion of the original concept but be at its next level higher, so that the original concept doesn’t change when moved along the new concept dimension.
So A’ is an overlying , coexisting space. A new level of concept A.
In standard math, we are taught that a “point” is the only thing that can exist at a specific co-ordinate. But think about the pixels on your screen. A single pixel isn’t just one “thing”-it is actually a shared space where Red, Green and Blue exist together.
When these colors share the pixel, they create a new result (like white light), but they don’t lose their individual identities. They are nested within that single coordinate. My notation (A’[(A(B)] works the same way:
The pixel is the higher concept level (A’). The Red is the fixed anchor(A) and the Blue is the second entity(B) that shares that space
The magic hapens in the separation. In a standard co-ordinate system, if you move the “point” the whole pixel moves. But in this new language we can “unfold” the pixel We can keep the Red fixed while moving the Blue to a new placement. This allows us to see how complex structures-like prime numbers or intricate knots-are actually built from multiple “colors” sharing a single origin.-This analogy was created in collaboration with an AI assistant.
In the case of two ordinary points we would have two of the same item, leading to the same item.
In order to have the two different items together we have to be able to remove the concept of a point as the only item of no extent and the removal of the point itself. It requires the construction of an overlying space of higher places of lower places where points can be, coexisting with the space of points.
I’m deleting the concept of a point as the only item of no extent in geometry. As well, I can delete it physically as I now have an overlying level.
This then allows the creation of other new entities of no extent.
In particular, we can think of removing a point from space, necessitating an overlying, co-existing plane of new places of original places, by extending the concept of a place or a point, into a plane. This also gives room for a new entity ‘e’.
We can create another new entity called e, which can move in this new dimension making it different from p which can still be fixed in this new overlying, co-existing dimension. This keeps the connection with the geometry which already exists.
This creates a mixed space of e and p in an overlying co-existing space of r.
We can think of e’s as having the ability to move, unchanging. An analogy is that of a jigsaw puzzle of a landscape being taken apart. The e’s can be thought of as pieces of the puzzle, moving but unchanging.
Here is then another dimension where e can move off into making it different from p as required. See the diagram below:
Since in mathematics we wish to have this concept of no parts or no extent, so too can we have this further notion. The further notion is that two items could be together as one, if they both had no extent. They would have to be two different items to no extent. But how do we number these? We can think of removing the number one and replacing it with 1(1) and 1(2). Similar to removing a point to make room for e.
Then the two items touch in their entirety both being of no extent. But they do not combine being different entities.
They could not both be points as we know them now, as they would just become one point as Aristotle thought. Aristotle thought that two points could not be contiguous since there would have to be no line in between them and we would have no extent.-Aristotle : Physics- “as there are an infinite number of points between two selected points”. The idea is that the two points would combine into one. They would touch in their entirety. But he only had the idea that there could only be one item of no extent.
The new entity e(1) has a location but e(1)1 and e(1)2 if we start off with two e’s together also sharing with p) can also have new locations of their initial locations, if they move in the new dimension of new locations of original locations opened up by the creation of the new level.
E(1) can be fixed for now, but could move, so its new location of its original location is the same as its initial location. This is the difference between e(1)1 and e(1)2. This also makes it different from a point which we can say still has its location fixed in r.
This finds application in the theory of knots:
Online Tutoring Services Ontario Canada » the knottedness and chirality of the trefoil
Reference:
Aristotle. Aristotle’s Physics Books 1&2. Oxford:
Extension to other concepts:
Furthermore, all math concepts are point-like in that they are exact, universal ideas which have no existence in physical reality. They are mental constructions which are not available to the senses. They are in thought only.
We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.
Math concepts are universal so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.
So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!
The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.
The idea is to go forwards into the idea. To create higher spaces.
Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.
Then the extended entity has the location of an initial location, which makes it different from a point, having only location.
The concept sharing of a number:
We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:
The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 3 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The three numbers 1, 1(1) and 1(2) are sharing the concept of 1. We have the usual number line on the midline of the new plane.
Think about position and consider a race where 2 runners are tied for 10th place.
We can say the runners are tied for tenth place giving the number 10 to both runners. Yet we can now describe this situation better by taking out the number 10 as the only item of 10th position and replacing by it by (10(10), (10)(x2) = (10<….(10) =(10(10). here the 10th position is shared, similar to the concept sharing of a point, this is the concept sharing of a number. We need this idea to come together with the idea of the concept sharing of a point for two e’s to share the concept of a point (that which has no extent). Here then is the extension of the number 2.
But suppose instead of a foot race, two points are together in a race along the number line. Then in 10th place two points could be together as has been seen. Then two number 10’s can be given one to each point. These numbers could go with the points as they are mapped into the new plane which has been mentioned.
A circle which is the 10th circle to be formed could contain the two e’s with the two number 10’s. Then 10(1) and 10(2) could also show an amount of points, 2. As well these could be the labels we are giving to these points, or the position in a number line, 10.
If sets are made of points, the sets could have concept spaces. If sets were made of numbers, the numbers could be associated with points and we could have concept spaces. Similarly if groups are made of numbers or diagrams which are made of points they too could have lower concept spaces.
The concept sharing of a number:
A number is an amount, as in a counting number, or a position on a number line, or a label.
It is point-like in that it has no existence in physical reality, it is a mathematical object, not a physical object. Therefore we can make a correspondence between the idea of sharing in points and an idea of sharing in numbers.
So that means the concept of a number can be extended , similar to what we have seen in points, so that we have a number of original numbers space and this number of sharing numbers after we take out the original number. So, for example, with the number 1; we have a number of numbers space, let the number of original numbers be 2, instead of 1. Take out the number 1, then we can have 1(1) and 1(2) sharing the position of the number 1. This can be notated (1(1)(1(2)). We can show that there are really 2, 1’s there by calling it (1(1) like so:
(1)(x2) = (1<….(1) =(1(1). We remove the original 1 or 2 or 3.. and replace by (1(1), (2(2), (3(3) etc. as shown below:
The left 1 is really together with the right 1, yet I need to work with both of these so both of these are shown. These numbers can move because there exists an underlying plane as mentioned above. The numbers are associated with moving e’s. We have 1 dot or 2 dots or 3 dots as shown below:
The twin prime conjecture:
There is a lower level of prime numbers using the concept sharing as it applies to numbers. For examples: (2(2), (3(3), (5(5),…I drop the (1), (2) notation for clarity and brevity.
Yet there is now a plane of numbers as shown below. With entries such as (1(2), (2(3), (3(2). These are partially shared prime numbers. We can make squares as shown, these are the primes and composites as seen with the extra added dimension.
There is a split between the natural numbers and the primes at (3(3). After this, larger squares with partially shared prime numbers at the corners appear. These are built from the smaller squares.
We might also make copies of these squares. exe can be repeated indefinitely. Yet there is an uncountable infinity of these possible, also a countable infinity within this. Think of the first square, this is the model we use to build the structures. There must be an uncountable number of these available at the beginning. We can think of “folding-up” these new squares. there can be an indefinite amount of copies. Yet we can balance this infinity by stretching the copies out to the other infinity that is available to us.
The first, primary square is used in repetition, to build the entire structure.
It must be repeated indefinitely, further along the diagonal as well as filling in space in the larger squares.
Since the larger square exists it must also be repeated indefinitely.
It ends, yet the structure must somehow be repeated as the lines which make up the edges of the square can be multiple.
The first time I encounter a new square type, that is a larger square, there is no rule for how many copies exe=exe I have present. The only way I can have it exist is if there are an infinite uncountable number of copies present. Then I can move these copies into the other uncountable infinity I have present. I do this because it is necessary to have only one copy of the shared numbers present in order to recover the usual numbers. This means that the countable infinity must also exist.
The next such larger square is similar so that I do the same with this. This larger square is at a different, “lower”, level than the smaller square, above it.
Then this leaves one copy of each square as we climb higher. Then I can recover the usual numbers by replacing exe with pxp=p ie. (1(1) with 1.
The first small square I encounter, I spread out three prime squares as shown above and after that only composite numbers. The new definition of a prime square in the plane, is to have a prime sharing at every corner. Here is seen what happens when we have a prime gap of 4.
The second square is the new construction of what comes before a prime in the plane, with a prime shared number at every corner. There must be an infinite uncountable number of copies here as well, when you consider the folded up case. I have to be able to spread this out too, so I have an infinite number of twin primes, primes that differ by 2, and an infinite number that differ by 4, etc.
As there is an infinite number of naturals (composites considered), these go inward to build the larger prime squares and an infinite number of primes (larger squares considered)
The composites and primes can’t be anywhere else in the plane but on the midline. Since we need to recover the ordinary composites and ordinary prime’s, there can only be one copy of these. The rest of the copies are moved out into the infinity that is available, that is, the rest of the number plane. Then in this way we are able to recover the composite and prime numbers. Then (1(1),(2(2),(3(3),(4,4),(5,5),(6,6),(7(7) can become 1,2,3,4,5,6,7…
Since I can make an indefinite amount of copies, the same gap must be possible infinitely as we go further up. Therefore the twin prime conjecture is seen, as well as we see other gaps repeating. I can make squares of any size by accessing the next lower levels. All levels must exist, one after the other, to create the structure which leads back to all the prime and natural numbers.
I can fold up, like a two-dimesional accordion, all of the squares with partially shared prime numbers at the corners. Also the smallest squares with the composites sharing the corners can be also folded up.
Then we obtain the squares of length 1,2,4,6,8 as infinitely uncountable. The squares of lengths 3,5,7 come along as well. We must be able to access all the different levels. All folded up the squares look like below:
All the squares of side length 1 fold up into the first unit square, all the squares of side length 2 fold up into the second unit square, etc. The notion is that the squares with an uncountable number of copies is more basic than the squares all spread out. This is given at the very beginning. When folded up the squares are on different levels, when spread out they are on the same level.
