proven at 08:30 4th Oct 2021 clarified with the reason why the distribution of primes around X/2 means there are always increasingly more prime pairs adding to X at 00:45 GMT 11th Oct 2021.

Clarified with the relationship of distribution to sets of Primes above 14² added 07:17am GMT 21 Oct 2021

The necessity of the symmetry to be broken by odd and even distances from X/2 either side of X/2 which is impossible added 19:57 GMT 22 OCT 2021

Summarised in this 2 minute video

https://youtu.be/RQjN-3YtLO8

The question in the Goldbach conjecture is how can we know that there will always be pairs either side of X/2 that match and add to X? It seems very unlikely but how can we be sure?

The reason is that the distribution is limited by the maximum possible gap which is less than the square root of X at every value of X and the ratio of primes either side of X/2 is 0.5:0.5 which forces an inescapable intertwined symmetry as the value of X grows. Also because all primes are either ALL odd or ALL even distances from X/2 it means that at least half the integers cannot match so the number of possible places where any corresponding pairs either side of X/2 can match is at MOST half the square root of X (the likelihood of none matching to X is even less at X/2 but I'll stick to the highest possible probability just to illustrate how impossible it is) Which means that the greatest possible likelihood of one individual pair to add up to X is one over half the square root of X and the total likelihood of none of them adding to X is less than this number divided by the total number of primes. The only way that these odds can ever be reduced is if the largest possible gap between the largest 2 primes somehow begins to grow larger than the number of primes. However we know this is impossible because the next gap between any 2 primes cannot ever be larger than the square root of the largest prime. Also we know that there are always more primes than the value of the square root of the largest prime because for any value for the square root of any prime Px to double in value to a higher valued prime Py who's square root is twice the value of the square root of Px the value of the higher prime Py grows in value fourfold. So the number of primes grows phenomenally larger just short of 4 times larger between Px and Py. So regardless of how few primes there to integers going towards infinity the number of primes always grows exponentially more than the largest possible gap. So while it is framed in a statistical framework the odds are just to illustrate the growing emphatic nature of the impossibility.

For example

At the 100th prime 541 the likelihood of none adding up to X is (√541/2=23.25/2=11.62) 100 to 11.62 which is odds of around 10 to one of there not being any pairs adding up to X.

At the one billionth prime 22,801,763,489 the odds of the likelihood of no pairs adding up to X is (√22,801,763,489/2=151,002/2=71,571) one billion to 71,571 or roughly 100,000 to 7 just slightly more than 14,285 to 1. These odds keep increasing regardless of the sparseness of primes per integers as we head to infinity.

So it is impossible for the ratio of primes to half the square root of any Pn to drop. Coupled with the fact that each gap between all consecutive primes is limited by the square root of the higher prime of any 2 consecutive primes means there can be no breaking of the symmetry. They must match within a slightly growing but otherwise fixed distribution either side of X/2.

The Goldbach conjecture is absolutely true, it is impossible for it not to be and that impossibility grows ever more emphatic as the value of any new Pn increases.

For the symmetry to be broken and allow for all pairs of primes either side of X/2 NOT to add up to X there would have to be an uneven symmetry meaning all pairs are perfectly out of sync. This is impossible because all primes either side of X/2 are evenly spaced but either ALL evenly or ALL oddly spaced relative to X/2. So they cannot be out of sync. They can never be out of sync. For the symmetry to allow all pairs NOT to add up to X they would have to be spaced at an even spacing one side relative to X/2 and an odd spacing the other side.

Consider this.

Consider where X is 56 X/2 is 28 so 19 and 37, 13 and 43, 53 and 3 are pairs adding to X. Pretend we didn't know which were primes so the first candidates to check would begin with 27 and 29 which are one space away from 28 then 23 and 33. Then when X is 58 X/2 is 29 so again pretend we didn't know the primes so the first candidates to check would be 27 and 31 which are 2 spaces away from X/2 then 19 and 39 which are both 10 spaces away. Both even spaces away. For ALL pairs to be out of sync they would have to be evenly spaced one side and oddly spaced the other.

Now please get the word out faster than you have ever done anything in your life and take a good look at the Theory of Everything...as fast as your mind can handle.

Explained here in short video

https://youtu.be/Y9T_I_bQKz0So

With the gap limit between primes leading to a distribution limit which dictates a lower limit on the amount of pairs adding to X which rises in increments clarified in this video

https://youtu.be/ASe1tf58NqI

The non membership of the next n not being a member of the set of all primes less than Pn multiplied by all integers from 1 to infinity (explained below) means it is a probability of less than zero that there will be no prime pairs adding to X and that zero probability grows increasingly LESS than zero ie. the impossibility becomes more emphatic the higher the prime. The gap between primes no matter how big cannot ever begin to grow exponentially and alter the probability of prime pairs adding to X, the largest gap always increases at most by the increment of the next n not being a member of the set of all Primes less than n multiplied by ever integer between 1 and infinity. Plus multiple more primes in between every largest possible gap, at the trillions of primes that is billions of primes between every 2 highest possible gaps and hundreds of millions of twin primes that are separated at most by 246 integers [1. Zhang, Maynard, Tao]. So the distribution means it is impossible for there not to be prime pairs either side of X/2 adding to X, it is not just highly unlikely it is impossible, that is a proof.

As an example...

If we consider the 10 millionth prime 179,424,673 within the 2nd next value of X 179,424,676 it will itself pair with 3 to make 179,424,676 and within the 3rd next value of X pair with 5 to make 179,424,678 and within the fifth next value of X it will pair with 7 to make 179,424,680 ...etc etc, you get the picture I assume. Now that is only the lowest pair and the next few, the total number of pairs adding to X either side of X/2 is going to be in the millions or at least the hundred thousands of pairs either side. If the 10,000,001st prime were a twin to it, both would be adding up to X at 179,424,676 and again 2 integers higher and again 2 integers higher. Here again we have the 2 highest primes adding within just a few spaces and roughly 5 million primes either side of X/2. This gives 2.5 million places where pairs can add up to X. At 10million primes the average gap between primes is about 18 as we can see 179,424,673/10,000,000 so the odds of none of the pairs either side not adding up to X is phenomenally IMPOSSIBLE, not improbable. The fact that the average is gap 18 the smallest possible gap 2 and the highest possible gap is less than 13,395 the square root of 179,424,673. Means that if it were a statistical argument the chances of there not being a match between just one opposite pair of primes, the first and the last very between 2.5 million to one (5,000,000/2) to 373 to one (5,000,000/13,395). That is just one prime pair. There are another 4,999,999 pairs with the same odds of them NOT being a pair that add to X. You may have sets of non prime numbers denser than the primes that do not pair to X but not with the constraints on the gaps between them, this constraint forces a symmetry that is inescapable. The gap limitations and the number of primes per gap means it is measured in a statistical framework but there is no question of the probability. The probability begins at ZERO probability that any 2 primes will not add to X and the emphatic nature of that ZERO grows in magnitude. Metaphorically it could be compared as from uttering it in a soft tone in a quiet statement, to saying it out loud, to a shout, to shouting it through a loudspeaker placed right beside someone's ear, to having it amplified through the stage speakers at a rock concert stadium with the persons head pressed up against the speakers...etc etc I think the reader gets the picture. But if not.........

In case anyone doesn't understand the significance of the non membership of the set of primes less than n. A maximum gap between primes cannot exceed more than 2 the last maximum gap and when it does fill a gap 2 greater than the last it becomes the next prime so the value of the gap can never grow to be more than the number of primes. This coupled with the fact that so far the gap between 2 primes "twin primes" are known to be at most 246 integers apart means the symmetry is constrained, there are multiple new primes, increasingly more between every new highest gap. So neither the probability nor the symmetry allows for there ever to be no pairs adding to X. Pairs adding to X must always increase in number due to the constraints of the symmetry which forces the next prime with the largest possible gap to be preceded on average by increasingly more new primes between it and the last largest gap.

The Goldbach conjecture is true.... because for the ratio of primes converges to 0.5:1 at a large enough valued even number X.

Because the degree of the divergence of a prime from x/lnx decreases the higher the value of X and because there are proportionally more primes and pairs of primes than the number of integers in the gap between the 2 largest primes. So the symmetry and number of primes means there are always the same number or more pairs of primes in increments the higher the value of X than the previous X. At 282 billion the average gap in the largest primes is 25 but 10 billion primes. So at this number the average gap between primes either side of X/2 is approximately 13,39,75,101,127...etc an average extra 13 every side above and below X/2 which decreases below X/2 and increases to a max 25 increments as we go through potential pairs as we move away from X/2. So we are going to hit pairs, many many pairs of primes adding to X long before we go through 5 billion pairs of primes. At this number the odds of NOT having a pair of primes adding to X are roughly 10,000,000,000 to 25 or 400million to one and the odds rise the higher the value of X. The distribution and concentration limited either side of X/2. The distribution of primes appears random only within each pair of numbers that have x/lnx between them and the next pair of numbers that have x/lnx between them.

In music a note can be made louder the more amplification is added but adding more amplification doesn't change the note it just changes the volume, with the primes x/lnx is the note no matter low large the volume of primes. The only thing that changes the note is the frequency but we know the frequency doesnt change x/lnx becomes closer to a pure note the higher the volume.

Pn=next n⊄(All Ps<√n)x1,x2,x3....x∞

Because the next prime Pn is always the next integer that is not a member of the set of integers belonging to the set of all primes less than the square root of n multiplied by every integer between 1 and infinity means that the distance between the 2 highest primes cannot be equal to or greater than the number of primes less than the highest prime. And above 14² the gap between primes cannot be equal to or greater than the number of primes below the square root of the highest prime. The number of primes grows exponentially in ratio to the gap between primes and because the distribution of primes either side of X/2 is limited to the gap between any 2 consecutive primes being less than the square root of the higher of those 2 primes the distribution limitation means there will always be more pairs of primes fulfilling the function (X/2-n)+(X/2+n)=X the higher we go towards infinity regardless of how few primes per integers there are so the Goldbach conjecture is absolutely true.

We know from prime sieves that any number n ending with 1,3,7 or 9 need only be checked by primes with a value less than it's square root to validate whether it is a prime or not.

Above 14² the number of integers between primes cannot be equal to or greater than the number of primes less than the square root of the highest prime.

If any 2 primes had a number of integers in the gap between them greater than the square root of the largest prime there must be a prime in between them. So those 2 primes cannot be the 2 highest primes. So above 14² while the number of primes increases they always increase in number greater than the square root of the greatest prime. So above 14² for any even number X the ratio of primes between 0 and X/2 and between X/2 and X is 0.5:1 as we get closer to infinity therefore the statistical probability of two primes either side of X/2 not adding to X becomes increasingly less than 0 regardless of how many or few primes there are per integers the higher we go towards infinity. The gap between pairs of primes on either side of X/2 must always be less than half the square root of Pn. Which is increasingly more primes than the highest possible gap and always at most twice more than the gap between 2 primes either side of X/2 so the probability of pairs of primes adding to X which satisfy the function of prime pairs adding to X which is (X/2-n)+(X/2+n)=X is at least one pair in every number of prime pairs less than or equal to half the square root of X.

The distribution of primes is always limited to at most a gap between any 2 neighbouring primes that is less in value than the square root of the higher prime of the 2. So the distribution does not allow for all pairs of primes either side of X/2 not to add up to X, as the value of X rises so does the probability of pairs of primes adding to X.

So again the statistical probability of there being no pairs either side of X/2 adding to X is increasingly less than 0.

The Goldbach is absolutely.....true.

Explained in more detail below.

At X= 252,097,800,628 approx 252billion there are 10billion primes roughly 1 for every 25 integers much much more primes than the highest possible gap between primes around that range. The max gap at that value for X seems to be around 100 with the average gap around 25. Either way it is way way less than 10 billion so there will be multiple prime pairs. Even as that ratio of primes to integers goes down as Pn increases there will always be many many more primes than the highest possible gap between primes. So there will be many many more primes that must be overlapping to form pairs of primes. So the Goldbach conjecture is absolutely, indisputably true.

For any even number regardless of its size the value of one of the 2 components that make up any specific even number X must be at most less than half the value of X. If 2 numbers between 2 and X were more than half the value of X then their sum total would be greater than X. So X must be the sum of 2 numbers either (X/2)+(X/2) or (X/2-Y)+(X/2+Y) so one of these is less than X/2 and one is between X/2 and X and together their symmetry balances X. Above 6 all even numbers X have at least 2 different primes which added together make up X. Each combination of prime numbers satisfying the condition(X/2-Y)+(X/2+Y) has a symmetry which balances to X. There are no 2 possible numbers between 6 and any higher value of X that can violate the condition that they are the sum of 2 different primes less than X.

To understand this consider 8 which the first even number greater than six. It's symmetry is either 4+4 or a balance between 2 integers greater than 2 which are one less than 4 and one greater than 4 being 3 and 5 both primes. The higher the value of any given even number X it's symmetry is always balanced by 2 numbers meeting the condition for symmetry and at specific values of X there are always the same amount or more pairs of primes that can make up any lower value given for X*.

For example

From 8 to 14 we have 1 pair of different primes

10=3+7=((10/2)-2)+((10/2)+2), 12=5+7=((12/2)-1)+((12/2)+1),

14=11+3=((14/2)+4)+((14/2)-4)

At 16 we have 2 pairs 11+5 and 13+3, at 18 7+11 and 5+13, at 20 17+3 and 13+7,

Then at 24 we begin to have 3 pairs 19+5, 17+7, 13+11.

So the higher the value of X the more pairs of primes satisfy the condition

((X/2)+Y)+(X/2)-nY)=X

*This can sometimes dip occasionally to a lower number of prime pairs than at the previous value of X but it is a slight oscillation.

The amount of odd numbers which are primes at any value of X is always equal to or greater than the amount of odd numbers which are primes at the previous value of X. So the probability that there can be less pairs of primes meeting the condition ((X/2)+Y)+((X/2)-Y)=X at a value of X higher than the previous value for X is 0 because while the probability of finding pairs of primes adding to X can go down at specific times the number of pairs of primes adding to X cannot go below 1. This is because the number of primes within X/2 and X always increases but in an oscillating constant proportion which oscillates roughly around (X/2)/X=0.5.../1. So because the number of primes within X/2 and X always increases but the ratio of X2 and X is absolutely constant within a range (which gets narrower to a point as X increases) this means that while the probability of 2 primes adding to X can decrease occasionally now and again over a range the actual number of pairs of primes can oscillate but always increasing in number within the certain range of oscillations. So while the probability is constant regardless of how few primes there are the higher we go the number of prime pairs adding to X cannot decrease absolutely impossible to go to 1 let alone below 1. Above a certain value of X it cant decrease below 2, and above a certain higher value of X it cant decrease to a value less than 3...etc etc. While the probability can decrease at certain points it oscillates between increasingly less variation in the value of that probability. So the NUMBER of prime pairs can never go back to 1 let alone 0

So given that symmetry provides more pairs of primes for X at every few increments higher than the value of X then every X is the sum of at least one or more sets of primes. So whatever value we have for X greater than 6 it will always be the sum of AT LEAST one set of 2 primes (or many pairs of primes) which satisfy the symmetrical condition of

((nX/2)+nY)+((nX/2)-nY)=X

With the Goldbach conjecture The pairs of primes descending from every new even number X comprising of primes descending from X/2 to 0 and ascending from X/2 to X must overlap with a lower prime pair frequency limit in relation to X because as the value of X increases in increments of 2 the value of X/2 increases in increments of 1 so the position of every prime less than X/2 is moved one position farther away from X/2 and simultaneously every prime between X/2 to X is moved one position closer to X/2 so as one ascends and another descends there must be a lower limit to how many must match and form a pair of primes. Because the increments are 1 away from X/2 between 2 and X/2 and 1 increment closer to X/2 between X/2 to X they MUST OVERLAP. Each new prime will overlap and form a prime pair with 3 within 3 increments of X/2 as it moves closer to X/2 then overlap with 5 within 5 increments of X/2 then overlap within 7 within 7 increments...etc etc So regardless of the gaps between primes with every new value of X there will be an equal or increasing number of primes which must be overlapping along the symmetry x/lnx above and below X/2 than there was with the previous value of X because x/lnx has a decreasing limit of divergence of primes from x/lnx the higher the value if every new Pn. So we can never, NEVER have only 1 prime pair with any higher value of X let alone none there are increasingly more.

Correct me if I'm wrong please. If I am correct then get the news out to the world linked to the website and channel that solved it as fast as you can that someome has solved the Goldbach conjecture, as well as the the double slit experiment (explained in the first paragraph in the GUT Spring Time Theory on this website) Along with the long awaited GUT the new physics that every scientists should be getting to work on ASAP with a sense of urgency. Dont be a science denier like so many.

For example let's look at some values of even numbers X and see how many primes there are below that value, plus how many primes are below X/2.

Numbers from naturalnumbers.org and wolframalpha.com (thanks)

When X=200 below X are 46 primes and below X/2=100 there are 25 primes

25/46=0.5434../1

When X=1,000 below X are 168 primes and below X/2=500 there are 95 primes 95/168=0.5654.../1

When X=6,000 below X are 783 primes and below X/2=3,000 there are 430 primes

430/783=0.5491.../1

When X=42,000 below X are 4392 primes and below X/2=21,000 there are 2360 primes

2360/4392=0.5373.../1

When X=760,000 below X are 60978 primes and below X/2=380,000 there are 32300 primes

32300/60978=0.529../1

When X=1.2million below X are 92,938 primes and below X/2=600,000 there are 49098 primes

49098/92938=0.5282../1

When X=180million below X are 10,030,386 primes and below X/2=90million there are 5,216,954 primes

5,216,954/10,030,386=0.52011../1

52

When X=138,000,000,000(bn) below X are 5,608,374,485 primes and below X/2=69bn there are 2,885,641,353 primes

2,885,641,353/5,608,374,485=0.5145../1

When X=806bn below X are 30,560,268,351 primes and below X/2=403bn there are 15,693,285,553 primes

15,693,285,553/30,560,268,351=0.5135192326.../1

At this number we can see the ratio narrow down to within a range that seem likely will converge around about 0.51... but in fact converge to 0.5/1

Because the ratio is 0.5/1

This ratio then is maintained all the way to infinity, so regardless of how large the gaps become the symmetry is maintained.

This means that because the ratio converges to 0.5/1 and because the distribution is symmetrical around the line that follows x/lnx to infinity then for all primes within X/2 and X the symmetry within the distribution of x/lnx descending from X/2 must also converge and overlap with the distribution x/lnx ascending to X from X/2 in the oscillation. So we will not just get some overlap we will get increasingly more overlaps the higher the value of X. That is how we know they are intertwined.

The Goldbach conjecture, the number of primes and prime pairs is always increasingly more than the gap between the 2 highest primes. The Goldbach conjecture is true.

With the Goldbach conjecture the pairs of primes with every new even number X comprising of pairs of primes one descending from X/2 to 0 and one descending from X to X/2 must overlap in symmetry around X2 with a lower prime pair frequency limit in relation to X because as the value of X increases in increments of 2 the value of X/2 increases in increments of 1 so the position of every prime less than X/2 is moved one position farther away from X/2 and simultaneously every prime between X/2 to X is moved one position closer to X/2 so as one ascends from X/2 and another descends towards X/2 there must be a lower limit to how many must match and form a pair of primes. That limit must be at most 1/4 the size of the gap between any new Pn and the previous higher Pn because the gap between the pair of primes closest to X/2 must at most be 1/2 the highest possible gap between primes dictated by the limited divergence from x/lnx. And the value of the closest primes to X/2 must be at most 1/2 of the total value of the gap between the prime pair when they are aligned symmetrically the same distance from X/2. Because the increments are 1 away from X/2 between 2 and X/2 and 1 increment closer to X/2 between X/2 to X there must be the same or increasingly more primes that must overlap and form a symmetrical pair.

Each new prime will overlap and form a prime pair with 3 within 3 increments of X/2 as it descends closer to X/2 then pair with 5 within 5 increments of X/2 then pair with 7 within 7 increments...etc etc So regardless of the size of the gaps between primes with every new value of X there will be an equal or increasing number of primes which must be overlapping along the symmetry x/lnx above and below X/2 than there was with the previous value of X because x/lnx has a decreasing limit of divergence of primes from x/lnx the higher the value if every new Pn. Because there are always many many more primes than the number of integers in the gap between the highest prime Pn and the previous Pn we can never, NEVER have only 1 prime pair with any higher value of X let alone none, there are increasingly more...all the way to infinity.

At X= 252,097,800,628 approx 252billion there are 10billion primes roughly 1 for every 25 integers much much more primes than the highest possible gap between primes around that range. The max gap at that value for X seems to be around 100 with the average gap around 25. Either way it is way way less than 10 billion so there will be multiple prime pairs. Even as that ratio of primes to integers goes down as Pn increases there will always be many many more primes than the highest possible gap between primes. So there will be many many more primes that must be overlapping to form pairs of primes. So the Goldbach conjecture is absolutely, indisputably true.

[1. https://en.m.wikipedia.org/wiki/Twin_prime

]

I initially began working on this at about 4 or 5am after seeing a meme by a well known YouTube physics vlogger which prompted me to research what the Godels Incompleteness theorem is all about, I soon came across a video on the incompleteness Theorem on Brady Haran's excellent channel Sixty Symbols, which mentioned the Goldbach conjecture which looked like it would be easy to solve. Then coming across this video by exoticmaths https://youtu.be/Zj7bVEI14co made the point about symmetry very clear. At first I realised the Goldbach conjecture is strictly speaking false because it does not clarify the condition for 2 DIFFERENT primes. Because strictly speaking 4 and 6 are the sum of a pair of 1 prime number a pair of 2s which is 2 instances of ONE prime number and a pair of 3s which is 2 instances of ONE prime number not 2 prime numbers per se. So strictly speaking the Goldbach conjecture is false as it is traditionally posed as being the sum of 2 prime numbers making an even number greater than 2 but I'm not swayed by the grammar Gestapo way of thinking so I stuck to the semantics and the bigger picture. I clarified the condition that we are talking about 2 different primes adding up to any given even number X and got to work.

And completed the first part of the proof about 8:30am GMT 4th Oct 2021. I sent it to people and no one responded all day until the late evening when one person said that it still didn't prove it absolutely mathematically, I replied with an angry frustrated rant because to me it was obvious, if the distribution is symmetrical around X/2 it must be true and I knew why but I realised some people cant trust or understand what they know is true their hearts they need it spelt out. So I got to work clarifying it in more detail mathematically to articulate what I could feel in my heart was true in numbers and completed the claridication to the absolute proof being due to x/lnx being the limiting "note" in the oscillation at about 00:30 GMT Oct 11TH 2021

Then I clarified the relationship of the non membership of n to the set of primes below n at about 1am GMT 21 Oct 2021 and uploaded it soon after.