A lunatic new year

February 17, 2026

At midnight last night I was woken up by fireworks which were apparently announcing the start of a new lunatic year. Perhaps they were celebrating the end of the lunatic year we’ve just had, but I have a premonition that the coming year is going to be significantly more lunatic than the last one. I have done my best to contribute to the lunacy, by trying to explain how the quantum gravity of the moon affects particle physics experiments in the Large Hadron Collider, and how this might explain why the CERN measurements of the W/Z mass ratio disagree with the Fermilab measurements, but the reality is that my lunacy is utterly feeble compared to the mainstream lunacy that runs the world these days.

As a fully certified lunatic, I know a thing or two about lunacy, and one thing I do know how to do is to distinguish between lunacy and stupidity. I may be daft, but I’m not stupid, as the saying goes. When I analyse the mathematics of particle physics, quantum mechanics, and/or general relativity, I do this as a professional mathematician. I am not stupid, and when I find mistakes in this mathematics, it is not because I have made a mistake (well, not always…). It is because there are genuine mistakes in the mathematics that has been taught to students for perhaps 100 years.

To give you an analogy, when I taught my first mathematics course as a junior lecturer in 1987, I inherited a set of notes that had been used by two or three previous lecturers on the course, for many years. At first, I didn’t question them, I just did as I was told and lectured from these notes. It wasn’t until I taught the course for the second time that, half way through the course, I came to something that I realised I didn’t understand. The method of calculation I was supposed to be teaching gave the wrong answer to a problem I knew how to solve. It was easy to check the answer – just substitute it back into the question – so there was absolutely no doubt about the fact that these lecture notes going back many years, probably decades, were actually wrong.

That was in fact a lecture course in mathematics for chemical engineers, so it was mainly a set of recipes for solving the kinds of differential equations you might encounter in chemical engineering. Not much emphasis on rigour, just a set of rules to get the right answer. But if the rules give you the wrong answer, there is a problem.

Much the same situation pertains in quantum mechanics and particle physics. The mathematics consists of a set of rules for getting the right answer. Not much emphasis on rigour. Not much emphasis on mathematical consistency. Not much emphasis on understanding the mathematics. So when the mathematics goes wrong, there is a problem.

And the reaction is the same – “it doesn’t really matter”. Well, I beg to differ. It does matter. If the mathematics is inconsistent (or wrong, which is the same thing), then we have to correct the mistake, not ignore it. That’s what I did with those lecture notes, of course. I corrected them. That is what I am trying to do with the inconsistent mathematics in particle physics.

And the reaction is the same – “don’t rock the boat! It works, don’t mess with it!” Rubbish. It works, up to a point. And then it stops working. It hits a brick wall, and cannot deal with the things that experiments reveal about the nature of reality – things like the three generations of elementary fermions (why?!), neutrino oscillations (how?!), the ~20 unexplained parameters (wtf?!), CP-violation (pardon?!) and so on. Just add epicycles and hope for the best. Yes, well, we know where that leads.

So when it comes to sorting out the mathematics of particle physics, I have no problem trying anything crazy, the madder the better, anything at all to resolve the inconsistencies that exist in the current mathematical theories. But the stupidity of blindly following recipes that we know are wrong – no, there I draw the line.

Happy lunatic new year!

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Adding spinors to SU(3,2)

February 16, 2026

One of the objections that people may have to my proposed SU(3,2) model is that it doesn’t have any spinors, whereas spinors are the cornerstone of quantum mechanics, and it would appear to be “obvious” that you can’t have a theory of quantum mechanics without spinors. Of course, I have been at pains to explain how the concept of spinors is a basic misunderstanding, and that they are quite unnecessary in a unified theory. But no-one will even read that argument, because they already “know” that it is “not even wrong”. So perhaps I need to add spinors to the model, even though they are unnecessary, just to provide a crutch for people to lean on, and perhaps discourage them from dismissing my model before even investigating it.

The point of using SU(3,2) is that it does the same job as the Georgi-Glashow 1974 model on SU(5), but with two obvious advantages: first it removes the non-existent new forces that cause proton decay, and second it includes the Lorentz group. But also in 1974 Georgi proposed extending SU(5) to SO(10), which had the benefit of unifying the fundamental particles into a single spinor representation, although it did not remove the problem of proton decay. So the obvious thing to do is to extend SU(3,2) to SO(6.4), so that we get the benefits of particle unification without the problem of proton decay.

As soon as we add in the spinors to SO(6,4), we’ve got an E6 model. This is an old idea, because the same is true for SO(10) – as soon as we add the spinors to SO(10), we’ve got an E6 model. The difference is just that we’ve got a different real form of E6, which is necessary in order to solve the two major objections (proton decay and no Lorentz group). Indeed, E6 models with various different real forms have been tried before, but somehow they seem to have gone astray, and not found the correct way to unify all the bits and pieces. That is understandable, because E6 is big, and the Standard Model will rattle around inside it if you’re not careful.

One of the issues is where to put the Lorentz group: is it SO(3,1) or SO(1,3)? Is the bit that’s left in SO(6,4) for the gauge group of something SO(3,3) or SO(5,1)? I think the consensus these days is that it has to be SO(3,1) – certainly the “octions” model uses SO(3,1), and SU(3,2) also requires SO(3,1), so we can probably regard that question as having been answered. Then the next question is how to interpret the gauge group SO(3,3) that is left. In the octions model, this is taken as a form of the strong (colour) gauge group SU(3), which is clearly wrong, or not even wrong, and is not worth discussing further.

The third question is how to extend this to the Dirac matrices and the Dirac equation. Do the coefficients in the equation generate SO(4,1) or SO(3,2)? Does the Dirac equation break the symmetry of SO(3,3) to SO(2,3) or SO(3,2)? I don’t think there is consensus about this, and you might think it doesn’t matter, because SO(2,3) and SO(3,2) are isomorphic groups. But it really does matter, because it matters whether the SO(2) is spacelike or timelike. The SU(3,2) model implies we need SO(3,2) x SO(3,2), and not SO(4,1) x SO(2,3). In other words SO(2) is timelike. In the octions model it is spacelike. Wrong again.

The people who believe in the octions model do not accept this argument, of course, because it relies on SU(3,2). It is impossible to argue against people who use guesswork to build their models, and blind faith in the results to justify their lack of interest in learning from their mistakes. But now they are stuck. They can’t go any further. They can’t predict anything, they can’t explain anything, they have a useless model that just sits there and looks pretty. They haven’t got three generations, they can’t explain symmetry-breaking, they haven’t even got a Dirac algebra.

Ah yes, the fourth question is how to build the Dirac algebra, which means how to build gamma_5. Now gamma_5 is a copy of SO(1,1) that commutes with the Lorentz group SO(3,1), so together with the other Dirac matrices generates SO(4,2). For this purpose it actually doesn’t matter whether we take the SO(4,1) version or the SO(3,2) version of the Dirac equation, both come with the same copy of gamma_5, which breaks the symmetry of SO(3,3) further to SO(2,2).

So the Dirac formalism breaks the symmetry of both copies of SO(3) down to SO(2). This is clearly what happens in electro-weak symmetry breaking, so that the correct interpretation of SO(3,3) is obviously a combination of electromagnetism, the weak force, and the three generations. Nothing to do with the strong force. The Dirac equation breaks the symmetry of the timelike SO(3), and defines mass, which is the only feature that distinguishes the three generations. So the timelike SO(3) is essentially a generation symmetry group.

What property does gamma_5 define, and what symmetry does it break? In the Feynman calculus, it appears only in the “vertex factors of neutral current weak interactions”, where it is attached to electric charge. So it defines charge, and it breaks the symmetry between the three charges, -1 for electrons, -1/3 for down quarks, and 2/3 for up quarks. So now we pretty much know exactly what SO(3,3) is doing – it has 15 degrees of freedom, six of which are compact, generating SO(3) x SO(3) for a gauge group of the weak force and the three generations, and nine of which are non-compact, and give us the nine masses of the nine fundamental fermions.

Of course, SO(3) x SO(3) is real overkill for describing the nine elementary particles, when only the group Z_3 x Z_3 is actually needed, and fits into SO(2) x SO(2) anyway. So there’s plenty of choice for how to choose a basis, and how to organise the calculations. The Standard Model changes basis all the time, and uses a set of nine “mixing angles” to organise these base changes. The most basic mixing angle is the Weinberg angle, which is defined by gamma_5, and breaks the symmetry of SO(3,3) to SO(2,2).

The other mixing angles are about the intermediate symmetry-breaking to SO(2,3) and SO(3,2). In both cases there is an overall phase in SO(2) and three flavour-changing angles in SO(3). In the SO(2,3) case the flavour-changing SO(3) mixes with the Dirac equation, so involves a change of mass, while in the SO(3,2) case the flavour-changing does not involve a change of mass. So the former is the CKM quark-mixing matrix, and the latter is the PMNS neutrino-mixing matrix.

In this way, we can get essentially all of the parameters of the Standard Model into SO(6,4), but we don’t get any explanations of the values of any of the parameters. The reason we don’t get any explanations is because we have extended the 24 degrees of freedom in SU(3,2) to 45 in SO(6,4), which is equivalent to adding 21 arbitrary parameters to the model. With SO(6,4) symmetry, you can put in whatever values of the parameters you like. With SU(3,2) symmetry, you have no choice – all the 21 parameters are defined for you, by the quantum gravitational field.

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Quaternion or dihedral?

February 15, 2026

When Hamilton discovered quaternions in 1845, he was so excited that he carved the equations in stone, and declared he wanted them on his gravestone. He was right to be excited, because he was then able to write all known physical laws, including the laws of electromagnetism, in terms of quaternions. And when quantum mechanics began to be developed 80 years later, quaternions were the key to understanding the mysterious properties of electron spin.

The equations are I^2=J^2=K^2=IJK=-1, from which it follows that IJ=K but JI=-K. In group-theoretical terms, we have a group of 8 elements, 1,I,J,K,-1,-I,-J,-K that multiply together amongst themselves, called the quaternion group (of order 8), written Q_8. But what Hamilton did was create a whole algebra out of this, by taking real numbers a, b, c, d and making things like a+bI+cJ+dK to calculate with. The real numbers enabled him to put in physical variables like position, momentum, energy and so on, so he could do the physical calculations.

There is exactly one other group of order 8 that has this non-commutative property (IJ is not the same as JI), called the dihedral group D_8, and the equations are R^2=RST=-1 and S^2=T^2=+1. This is the symmetry group of a square, if R is a rotation through 90 degrees, S is a sideways swap (of the left and right edges), and T is a triangular swap of two opposite corners. And you can make an algebra out of this in exactly the same way that Hamilton made an algebra out of the quaternion group. This algebra is usually called the “split quaternion” algebra, but I’m going to call it the dihedral algebra, because it is this dihedral (two-sided) property that I want to emphasise. The beautiful symmetry of I,J,K, all of which square to -1, is broken, so that only R squares to -1, and S and T square to +1.

When these algebras are used in quantum physics and particle physics, they are usually translated into Lie groups, whereby the unit quaternions (a^2+b^2+c^2+d^2=1) form the Lie group SU(2). In the dihedral case, we take a+bR+cS+dT with a^2+b^2-c^2-d^2=1, and get the Lie group SU(1,1) instead. The group SU(2) is called the “spin group”, and is used extensively in quantum mechanics to describe the spin of an electron (and, by extension, many other particles).

SU(2) is also used in particle physics to describe various kinds of “isospin” that distinguish between particles like the proton and the neutron. Or, archetypically, between the up and down quarks, whose names are modelled on the names of the up and down spin states of the electron. I have argued on many occasions that this is the wrong group to use for isospin, because isospin symmetry is “broken”, like the R,S,T symmetry, not unbroken like the I,J,K symmetry.

This argument is useless against a barrage of dogma, of course, but if you want an algebra to describe the actual physical symmetry that actually exists in the actual universe, instead of a hypothetical non-existent symmetry that you think “ought” to exist, because it’s so beautiful, then you will without a doubt follow me and use the dihedral algebra for isospin, and not the quaternion algebra.

I have sometimes also argued for using the dihedral algebra for ordinary spin as well, although I am not as confident that this is such a good idea. But I have recently found a new argument for it, coming from my attempts to persuade the E8-modellers that they’ve allocated the various groups (gauge groups and Lorentz group(s)) wrongly. However, I don’t need E8 to justify this approach, just the Dirac equation.

There are five terms in the Dirac equation. One is a real scalar mass term, and the other four are momentum-energy terms, which have some 4×4 complex matrices attached to them. The details are not important, but what is important is the Lie algebra that these matrices generate, which happens to be so(3,2). And, even more important, this algebra is the Lie algebra of 2×2 anti-Hermitian matrices over the dihedral algebra. Not the quaternion algebra.

For many years I tried to get the Dirac equation out of quaternion matrices instead. It almost works. You can easily fool yourself into thinking that it does work. But it doesn’t. Spin is a dihedral concept, not a quaternion concept. Not SU(2) as a double cover of SO(3), but SU(1,1) as a double cover of SO(2,1). It is not enough to “cover” the three dimensions of space, you have to involve time as well. Time is of the essence, when dealing with spin. I mean, time is defined by properties of electron spin. Of course time is of the essence.

The Dirac spinor, in other words, is not a pair of quaternions, as many people think, it is a pair of dihedrons (if I may be allowed to coin a new word). Or “split quaternions” if you’re allergic to neologisms. And the great thing about dihedrons is that they can be represented in two-dimensional real space (symmetries of a square), whereas the quaternions require two complex dimensions. So all those pesky complex numbers that smother quantum mechanics in a fog of incomprehensibility suddenly vanish. The sun comes out, and we can suddenly see what the world really looks like.

The Birmingham Spring is a wonderful thing.

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The answer is 5. What is the question?

January 29, 2026

If you watch “Mock the Week” you will be familiar with the format. In this case the discussion is about whether the question is “What is 2+3?” or “What is 4+1?” This discussion has been going on for at least a hundred years, with no resolution in sight. String theorists swear by 2+3. Woit says that is “not even wrong” and claims it is obviously 4+1. Normally in a conflict between Woit and string theory I find myself on the side of Woit. But not in this case. I am now convinced that in this instance Woit is not even wrong, and “not even wrong” is right.

You see, the main question about particle physics is “Why 2+3?”, or in technical terms, why is the gauge group of the nuclear forces SU(2) x SU(3)? There is a related question about the charge on the up quark: “Why 2/3?” There is no mention of 4+1 anywhere in particle physics. But it was only once I started to consider the question “How 2/3?” that I started to make progress in understanding what is really going on behind the scenes.

That is because I found the answer in the equation tan(2W) = 3/2, where W is the “weak mixing angle”, which is without a doubt the most important angle in the whole of particle physics. So after that I could turn my attention to the questions “Why 2/3?” and “Why 2+3?”. Or, to be more precise, to the question “What is 2+3”?

And I found that the answer is 5 Maxwell equations. The 3 Maxwell equations in here are the ones that relate the magnetic field to the current. These are the most important equations in the whole of pre-nuclear energy. They are the most important equations in the whole of pre-quantum technology. They are still the most important equations today, because you can’t run a wind farm or an electric car without these three equations.

If we write the equations in “natural” units, so that the vacuum permeability and the vacuum permittivity are both 1, then the speed of light is also 1, and the equations are written in vector form as curl(B) = J+dE/dt, where B is the magnetic field, J is the current, and E is the electric field. The rest of the equation is the mathematics that describes how the magnetic field varies in space (curl), and how the electric field varies in time (d/dt).

There is another companion equation that describes how the electric field varies in space, that is div(E)=Q, where Q is the electric charge (or technically the charge density). That gives you 1+3 Maxwell equations. Normally you would add another 1+3 equations div(B)=0 and curl(E)+dB/dt=0, but these are equivalent to Lorentz symmetry of the electromagnetic field, so if we assume the symmetry, then we don’t need the equations.

But we do need one more equation, that is conservation of charge. Electric charge cannot be created or destroyed, it can only move around, or cancel out. The equation that enforces this physical property is div(J)=-dQ/dt. Or is it dQ/dt=-div(J)? We know what the answer is, but what is the question? It shouldn’t matter, really, should it? Multiplying the equation through by -1, doesn’t change the equation, does it? Well, yes, actually it does. It doesn’t change the answer, but it does change the question. And therefore it is important. It is the same as the difference between the questions 4+1 and 2+3.

We know that the ultimate question of particle physics, the universe and everything is 2+3, so this is the question we must ask. And it tells us to write the equations in 5×5 matrix form with the differential operator (1, d/dt, del) acting on the columns (0, Q, J), (-Q, 0, E) and (J, E, -B) to produce the answer (0, 0, 0). That means that in the matrix form, Q and B are antisymmetric, while J and E are symmetric.

Now when I was taught Maxwell’s equations, back in the dark ages (1970s), I was taught that E is antisymmetric, not symmetric. This is wrong because if E is antisymmetric, then spacetime is Euclidean, not Lorentzian. Of course, I didn’t know that at the time, but I do now. The difference between 4+1 and 2+3 has nothing to do with this, it is the difference between Q being symmetric (and therefore J antisymmetric) or antisymmetric (and J symmetric). And the fact that charge can be quantised, but current cannot, implies that Q is mathematically a circle, not a straight line, which means it is antisymmetric, not symmetric.

So there we have it. The ultimate question is “What is 2+3?” The reason why Einstein failed to answer this question is because he thought the question was 4+1. So what I want to do now is explain what Einstein would have done in 1915 if he had realised the question was 2+3. To be fair to Einstein, there was no real reason to think the question was 2+3 until much later, when quantum theory was sufficiently developed to make it clear that the gauge group of a quantised entity had to be compact. It could in principle have been deduced from the Dirac model of the electron in 1928, but it wasn’t really until the 1950s that there was enough evidence to reconsider the question, by which time it was too late, as 4+1 had been set in stone.

As you know, the biggest obstacle to answering any question is asking the wrong question. And if you are fixated on answering 4+1, when the correct question is 2+3, you will just go round and round in circles, bashing your head against a brick wall, if that’s not a mixed metaphor. Once you know the question is 2+3, and the answer is 5 Maxwell equations, then it isn’t hard to imagine that to include momentum and energy as well as current and charge, you just have to extend the equations from real numbers to complex numbers, so that the imaginary part gives you five Einstein equations for gravity.

These equations are rather different from the Maxwell equations, because the symmetric/antisymmetric dichotomy is reversed by the extension to Hermitian/anti-Hermitian matrices. In particular we get some anti-Hermitian diagonal matrices, which make things more complicated, since there are no anti-symmetric diagonal matrices in the Maxwell case. The first equation to consider is conservation of energy, which is div(p) + d(p0)/dt = m, where p is the momentum, p0 in the energy (I can’t use E again, because that will cause confusion), and m is a type of mass. If this equation doesn’t make sense to you, then don’t try to think of m as being something you already know, think of this equation as being a definition of m, whose properties we then have to work out as we go along. It’s a Lorentz scalar, so mass is the obvious interpretation.

The second equation is div(g) = p0 + d(m0)/dt, where g is the Newtonian gravitational field. In Einstein’s language, E=mc^2, so that p0 replaces the Newtonian mass as the source term for gravity. But what on earth is this extra term d(m0)/dt doing here? It is an extra source of gravity that actually does exist in Einstein’s theory, although it is not easy to interpret it physically. It is not just a mass term, but a change-of-mass term. In standard gravitational theory, mass cannot change except by moving. But this is not true in reality, because mass is not conserved in the weak interaction. So the d(m0)/dt term cannot be ignored, and creates a genuine mixing between the weak force and gravity.

The importance of this remark cannot be overstated. I’ll pause a while to let it sink in.

The two diagonal terms m and m0 are two different types of mass, one of which is constant, while the other changes over time. Newtonian gravity assumes they are equal, which is of course logically impossible. Einsteinian gravity does not assume they are equal, but still assumes that there is only one type of mass, which is still logically impossible. Einsteinian gravity and the weak force are incompatible.

But if (as is obvious) we do not understand what type of “mass” m0 is, we can take the equation as a definition of m0 in terms of the gravitational field. That is, m0 is the time-integral of div(g)-p0, so it can be calculated from the gravitational field and the energy. That means it is a Machian type of inertia. But m is a Newtonian type of mass.

Now listen carefully – m0 is only defined up to a constant of integration. So you can choose your constant of integration to ensure that m=m0 and there is only one type of mass. However, m is constant (by definition) and m0 is not (also by definition), so over time m0 will drift away from m, and they won’t be equal any more. By reading the literature carefully I have worked out that the calibration m=m0 was carried out in the period between 1971 and 1973, and I have worked out how far m0 has drifted away from m in the past half-century or so.

And I also know why nobody has noticed this drift. It is because it only affects gravitational measurements, and does not affect the standard model of particle physics in any way. The Dirac equation edits out the gravitational field, and shunts this variability (which is physically real) off into some variables that cannot even in principle be measured. So it is logically impossible to detect this drift in particle physics experiments. It can only be detected in gravitational experiments.

And once you know this drift is physically real, you can detect it in the measurements if you look closely enough. You can detect it in Cavendish-type measurements of the Newtonian gravitational constant G. You can detect it in inconsistent measurements of the gravitational mass of copies of the International Prototype Kilogram. You can detect it in pre-1969 experiments to measure the electron/proton mass ratio, before the ruthless dictatorship of the Dirac equation came into force, and suppressed all opposition from the Machian gravitational party.

And just because I live in Birmingham I can’t resist mentioning the Lunar Society, which was a major force in the advancement of science, technology and philosophy in the second half of the 18th century. They regulated their activities by the phase of the moon, because it was only safe to walk around Birmingham at night if the moon was full. But they made progress in many endeavours by communicating across boundaries, sharing ideas, and thinking outside the box. They were not crackpots or lunatics, they were lunar luminaries.

Progress in science today also depends on communicating across boundaries, sharing ideas and thinking outside the box. Preferably by the light of the moon. And it was the light of the moon that shone on a very faint trace of a 19-year oscillation in the m0/m relationship. And it shone a gravitational light on the neutral kaon (weak) eigenstates. And it shone clearly on the charged/neutral kaon mass ratio. And on the charged/neutral pion mass ratio. By the light of the moon, walking fearlessly across boundaries and thinking outside the box, I have seen things you wouldn’t believe. I know, you don’t believe them. You think I am crackpot or a lunatic. But I am not. I am a lunar luminary.

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A new paradigm for unification

January 19, 2026

Last week I gave an online talk to the “causal fermion systems” group in Regensburg, as they had expressed some interest in my group-theoretical perspective on whether spacetime has signature (3,1) or (1,3). My perspective, in a nutshell, is that this is a meaningless question, because it is a question about Clifford algebras, and what is actually required for quantum physics is not Clifford algebras, but Lie algebras. Anyway, the slides are at https://robwilson1.wordpress.com/wp-content/uploads/2026/01/regensburg.pdf in case you want to look at them. They contain quite a bit more than I was able to get through in the time available.

At the same time, I was inspired to write a more detailed version, which, in an ideal world, I would have submitted to the arXiv. However, I know they would reject it, and probably use it as a pretext to cancel me completely. In the post-modern post-truth world, reasoned debate has been completely replaced by loud-mouthed idiots who want to be in charge but haven’t got the wherewithal to win a fair argument. Soon, I have no doubt, it will become the post-modern post-truth post-world.

Anyway, I’ve put the paper up here so you can read it at https://robwilson1.wordpress.com/wp-content/uploads/2026/01/spunif4.pdf, and judge for yourself whether you think it is post-modern, post-truth or just post-ed.

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A new paradigm for gravity

January 8, 2026

There have been two paradigm shifts in our understanding of the motions of the celestial bodies in the past 1000 years. Both were the culmination of long and arduous struggles to understand the data. Both resulted in a new mathematical model, based on new physical principles. The first was Newton’s theory of gravity, developed in the late 17th century. The second was Einstein’s theory of gravity, developed in the early 20th century.

To put Newton’s revolution into context, the ancient system was based on the assumption that circular motion was natural, so that all orbits had to be circular. The Ptolemaic system was a vast array of superimposed circular motions (called epicycles, meaning cycles on top (of other cycles)), which got more and more complicated as the data got more and more accurate. The Copernican revolution did not in fact change this one iota – it merely changed the interpretation of the same system of epicycles.

It was not until Kepler studied the incredible data collected by Tycho Brahe that it started to become clear that ellipses ruled the skies, not circles. But nothing really changed until Newton proved mathematically that an inverse square law of gravity gives rise to elliptical motion. At that point the paradigm shift became inevitable.

Newton’s theory of gravity held sway for more than two centuries, and explained practically everything incredibly well. But eventually some tiny details started to show that it wasn’t perfect. The data in question concerned the precession of the perihelion of Mercury. What this means is that the elliptical orbit of Mercury around the Sun does not stay in a fixed place, but the ellipse itself moves slowly backwards around the Sun. Newton’s theory predicts this phenomenon, so that’s not the problem.

The problem was that Newton’s theory got the wrong answer. There was more precession than Newton’s theory said there should be. Now it is quite difficult to build an entire new paradigm on one data point, but that is more or less what Einstein did. It helped that he had already built a new paradigm for electromagnetism, based on the physical principle of relativity – i.e. that physical reality is the same for all observers, even though different observers will naturally use different coordinate systems to describe this reality.

In Newton’s case, the principle says that the theory is the same in all places and at all times, and once you have picked a place and a time, the theory is the same in all directions. Mathematically, this says you can change your x,y,z coordinate system by any rotation, and these rotations form a group SO(3), which is called the point group of the theory. Einstein (aided and abetted by Lorentz, Minkowski etc.) had extended this point group to SO(3,1), which introduced three more transformations (called Lorentz transformations) that explained how moving observers measure time and space differently. By doing so, he ensured that the laws of physics remained the same for an observer moving at constant speed in a straight line.

What he wanted to do, but didn’t do, was to ensure that the laws of physics remain the same for observers that are rotating. For that, he would have had to base his theory on Mach’s principle, which in a nutshell says that physics looks different if you are rotating, but is fundamentally the same. To incorporate this principle into physics you need to consider two different observers, rotating in different ways relative to the phenomenon they are trying to observe. Each rotation requires three parameters for its description, so the transformation between them requires nine.

If you follow through the mathematics, then you can work out what these transformations look like from the point of view of the Lorentz group SO(3,1), and you can work out what the extended group (which obviously has 6+9=15 degrees of freedom in total) looks like. Actually the mathematics only tells you it is one of two possibilities. You have to use physics to tell you which is correct. Einstein picked one of them. I don’t know why he picked that one. I suspect it was because he relied on mathematicians to do the mathematics for him, and they weren’t thinking in terms of Mach’s Principle. Perhaps they didn’t realise that there were two possibilities. Perhaps they just assumed that because space and time are described by real numbers, the transformations must also be described by real numbers.

Anyway, he picked one, and used it to correct the precession of the perihelion of Mercury. And then he used it to predict that gravity bends light waves. Which was confirmed in 1919, by observing stars behind the Sun in a total eclipse. So the paradigm shift began.

This paradigm has lasted over 100 years, and has made a number of successful predictions, including a correction for atomic clocks running at a different rate in geostationary satellites and on the ground, which you rely on for your electronic gadgets to know where you are.

But by 1980, things weren’t looking so good. Galaxies weren’t behaving the way Einstein said they should. In the early 1980s, Mordehai Milgrom played the role of Kepler, by finding empirical laws that described the rotation of galaxies better than Einstein. But there is still no Newton to drive the paradigm shift. So although it is long overdue, Ptolemy’s epicycles (nowadays called “dark matter”) are still in charge. In fact, epicycles are a very good analogy for dark matter, because in both cases there is limitless scope for varying the geometry in order to fit observations. Dark matter isn’t a theory, it is a fudge factor to fit observations without having to deal with the fact (and fact it is) that the paradigm has been falsified.

We need a new paradigm, as Pavel Kroupa never tires of saying. Well, Pavel, I’ve got one right here. Do you want it? As I said, when Einstein built his theory of gravity, there were really two options. He chose one. If that one is falsified, as Pavel and many others believe, then we must choose the other one. All we have to do is multiply the Einstein tensor by a square root of -1. That’s it. You might think this is just a mathematical nicety that makes no practical difference, but that is not the case at all. It changes the point group from SL(4,R), which Einstein used, to SU(1,3), which I use.

The theoretical reason why we know SU(1,3) is correct is because it scales all the way down to the scale of a proton, and describes the behaviour of the quarks inside the proton in exactly the same way that the standard theory (quantum chromodynamics) does. Einstein’s SL(4,R) does not scale. We know it doesn’t scale, because it’s been tried, and it doesn’t work. Experimentally, we notice things going wrong even when we scale down to the kilogram scale, although it is still just about possible to put that down to experimental error.

Anyway, the SU(1,3) model scales from the mass of the solar system to the mass of the proton, which is 57 orders of magnitude. Can we also scale up another 23 orders of magnitude, to the mass of the whole observable universe? I believe we can. The paradigm shift is on its way.

Call me Isaac.

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The source of the “probabilities” in quantum mechanics

January 3, 2026

By re-writing the general theory of relativity in its logical form as a generalisation of special relativity, instead of the Einstein form which is a completely separate theory of gravity, I have achieved two things that Einstein failed to achieve. First of all, the new theory of gravity incorporates Mach’s Principle, and therefore has a complete theory of transformations between non-inertial frames of reference. Secondly, and relatedly, it unifies the electromagnetic and gravitational fields into a single mathematical construct, and shows how the dividing line between the two forces is observer-dependent, in exactly the same way that special relativity explains the movable boundary between electricity and magnetism.

In particular, the Lorentz group appears as an observer-dependent subgroup of SU(2,3) via SU(1,3) and U(1,3). Similarly, the local gravitational fields of the Earth, the Moon and the Sun, and the rotations and revolutions of these bodies, break the SU(3) symmetry of the spacelike coordinates down to nothing at all. In particular, the three coordinates x,y,z of macroscopic space cannot be chosen arbitrarily, because x,y,z symmetry does not respect the gravitational field. There is nothing you can do about this, because there is simply no way that you can shield your experiment from quantum gravity. In principle, your experiment could be contaminated by quantum gravity in multiple different ways, and you cannot ignore this possibility, so you should be alert to it.

Now in quantum mechanics a group SL(2,C) is used locally for a description of the “spin” of electrons, protons, neutrons and so on. It cannot be related to the Lorentz group SO(1,3) as a double cover, because this is mathematically impossible, so the only possibility is to embed it in SU(2,2) inside SU(2,3). That means that the three “directions” of spin are projected onto a two-dimensional space of macroscopic directions. The third macroscopic dimension has absolutely no meaning in quantum mechanics. But now it matters how the two dimensions that are actually used, relate to the three gravitational directions. Since this relationship is completely ignored in quantum mechanics, some important physical variables are simply thrown away, and replaced by a random number generator.

That is where the probabilities come from, and why they are there. Einstein was right – God does not play dice. But we do – we throw away one third of the information, and try to guess what happens based on the other two-thirds.

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CPR for physics

December 31, 2025

It can’t have escaped your notice that theoretical high energy physics is in crisis, and has been for decades. It is in desperate need of cardio-pulmonary resuscitation. The heart, or CORE (Classical Origin of Relativistic Electromagnetism, that is, Maxwell’s equations) has been injected with a Particle Source (PS) of electricity (i.e. an electron), and is in serious danger of becoming a CORPSE (Classical Origin of Relativistic Particle-Sourced Electricity).

Now as I explained in the previous post, the main problem in theoretical physics is to unify the four standard theories – Classical (C), Quantum (Q), and Particle (P) theories and Relativity (R). Quantum theory is always a problem: QC (foundations of quantum mechanics) is in a mess, to put it politely, QR (quantum gravity) shows no signs of progress, and QP (grand unified theories) has been in the doldrums for 50 years. So let’s forget about Q for now, and concentrate on the rest (CPR).

We start from C (Maxwell’s equations). These are differential equations with four (partial) differential operators, with respect to time t and three directions x,y,z in space, plus a constant “source” term for the electric charge and current. The equations therefore have five terms, so can be expressed in terms of 5×5 matrices. Usually only four of these five equations are called Maxwell’s equations, but the other one is equally important (conservation of electric charge). To remind you how it works, I’ll write the operator as (1,d/dt,del), where del is the usual shorthand for space derivatives, and use the charge (density) rho and current J. To avoid having to use the constants epsilon_0 and mu_0 I’ll use D = epsilon_0.E for the electric field and H = B/mu_0 for the magnetic field.

Conservation of charge is the equation d(rho)/dt + del.J = 0, that is d(rho)/dt + div(J) = 0, so I put (0,rho,J) in the first column of the matrix. The scalar equation is div D = rho, that is del D – rho = 0, so I put (-rho,0,D) in the second column. The vector equation is curl H = dD/dt + J, so I put (J,D,-H) in the remaining three columns, where -H here is a 3×3 anti-symmetric matrix with the three components of H in the off-diagonal entries.

Now you ought to be asking me the question, how do I know to put (-rho,0,D) in the second column, instead of (rho,0,-D)? And similarly for the last three columns: (J,D,-H) or (-J,-D,H)? How do I know that rho is antisymmetric, and J and D are symmetric? If I change the sign of the second column, I get D antisymmetric, and rho and J symmetric. If I change the sign of the first column, I get J antisymmetric and rho and D symmetric. If I change the sign of the last three columns, I get all of rho, J and D antisymmetric. These are the only possibilities, because if I change the sign of the whole matrix, the symmetry properties do not change.

  1. In the cases when D is antisymmetric, spacetime (i.e. the last four coordinates) is Euclidean, so we can rule out these two cases.
  2. In the case when rho is symmetric and J is antisymmetric, the Lie algebra is so(1,4), and we get the de Sitter model of spacetime which was used in the early days to unify electromagnetism and gravity.
  3. In the case when rho is anti-symmetric and J is symmetric, the Lie algebra is so(2,3), and we get the anti-de Sitter model of spacetime, giving an alternative model of unification of electromagnetism and gravity.

So we need a good reason to choose between de Sitter (dS) and anti-de Sitter (AdS). In Einstein’s day, only CR unification was being discussed (i.e. the foundations of relativity), because there wasn’t enough particle physics (P) known at that time. And in the CR regime, the only reason to choose one or the other is Einstein’s mass equation m^2c^4 = E^2 – p^2c^2, which points towards so(1,4) as being the correct signature. So have I got it wrong, or did they get it wrong?

Either way, the electric field must be a symmetric matrix, if it is to be united with the magnetic field (anti-symmetric) by the Lorentz group SO(3,1). The standard assumption that it is anti-symmetric is just plain wrong, I’m afraid. It is not just a “convention”, as physicists tend to argue when they get a sign wrong. The question is, whether the current should also be a symmetric matrix (AdS) or whether the standard assumption that it is antisymmetric (dS) is correct. This is not just a convention, it is a question of getting the sign right. We must not guess. This is a question of life or death. Either CPR works or it doesn’t. The evidence is that CPR using the standard (dS) signs does not work. So we had better try the other (AdS) signs before the patient becomes a CORPSE.

Either way, Einstein’s theory of gravity is obtained by adding energy and momentum to the charge/current 4-vector, and adding a “symmetric” tensor to the “anti-symmetric” D and H fields. I’ve put these words in quotes, because they are tensors of SO(3,1), not SO(4). The words symmetric and anti-symmetric, taken out of context, usually refer to matrices, which means SO(4) tensors, not SO(3,1). So what we need to do is generalise Maxwell’s equations to complex numbers, and replace “symmetric” and “anti-symmetric” by Hermitian and anti-Hermitian respectively (applied to matrices, not tensors). So now the Newtonian gravitational field is the imaginary part of D, and the Einstein correction is the imaginary part of H (three degrees of freedom), plus some imaginary diagonal matrices (four more degrees of freedom).

The Lie algebra then becomes su(1,4) in the dS case and su(2,3) in the AdS case. Either way, the spacetime part of the algebra (i.e. the vacuum, without any mass or charge, momentum or current, or energy) is su(1,3), splitting into a real part (electromagnetic field) and an imaginary part (gravitational field). The three “scalar” degrees of freedom (mass, charge and energy) are then related by a three-dimensional Lie algebra, which is compact su(2) in the AdS case, and split sl(2,R) in the dS case. Which is it? It is not just a convention. We must not guess. This is a matter of life or death. CPR either works or it doesn’t.

According to particle physics, the (weak) force that unites mass, energy and charge has gauge group SU(2). But can we trust them? Can we be sure that it really is compact and not split? In this instance, I think we can trust them, because they have found the three gauge bosons (W+, W- and Z), and bosons are always anti-Hermitian. This is a fundamental tenet of Yang-Mills theory also, that gauge bosons always live in a compact Lie group or Lie algebra. So I would say we can proceed with caution, on the assumption that SU(2) is correct. This means that CPR works with su(2,3), and does not work with su(1,4). We still need to test this hypothesis with experimental data, but it looks like the correct path to follow if we want full unified CPR.

So now we can look at the first two coordinates (1, d/dt) in isolation, and note that the charge (rho) is attached to the matrix (0,-1;1,0), which generates a U(1) gauge group for electromagnetism. The energy is attached to the gravitational analogue (0,i;i,0), and the mass to the diagonal matrix (i,0;0,-i). Together these generate the weak gauge group SU(2)_L, and it is chiral because it matters whether you define the mass in terms of energy x charge or charge x energy. If mass is always positive, then it must be one way round and not the other.

What about the space coordinates? If we treat them in isolation, without time, energy, mass or charge, we are left with SU(3). This group has no measurable properties of its own, but it is essential for the existence of the three-dimensional space we live in. In particle physics (P) it is the gauge group of the strong force, and the three things it acts on are called colours (and anti-colours, because they are complex variables, not real). But in our model of CR, it is a gauge group of space, that ensures the real (electromagnetic) definition of space is compatible with the imaginary (gravitational) definition of space. In order to do that, it has to ensure that the electromagnetic definitions of the electron and proton masses agree with the gravitational definition of the mass of the hydrogen atom.

At last! The pulmonary (P) part of CPR is coming to life! The patient is starting to breathe! The lungs are not just a vacuum, they are full of aether! Einstein gravity has gluons in it! And Newtonian gravity is made of neutrinos! (Or should I call them Newtrinos?) Now we can see how C and P work together to make R. It is no good having a heartbeat (Cardio) if you’re not breathing (Pulmonary). It is no good breathing if your heart has stopped. CPR needs a strong (spatial) force, and the timing (weak force) has to be right. Gravity needs massive protons and neutrons, and it is the strong force that generates that mass.

So, you see, I was right all along. The key is to unify C (Maxwell’s equations, with SO(2,3) symmetry) with P (Yang-Mills gauge groups SU(2), SU(3) and U(1)) into SU(2,3) for CP unification, and then interpret SU(2,3) as relativity (R). The patient lives. CPR is successful.

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Fifty shades of unification

December 30, 2025

Unification means different things to different physicists, but ideally we want to unify classical (C), relativistic (R), quantum (Q) and particle (P) theories of physics. There is a widespread view that CR unification was already achieved by Einstein in his general theory of relativity, but there is room for doubt on that score, because classical physics obeys Mach’s Principle, but relativistic physics does not. So there is a school of thought that this bit of unification needs more work, in order to incorporate Mach’s Principle into relativity theory. There is a widespread view that PQ unification was already achieved in the Standard Model of particle physics, but in reality P and Q have been just thrown together, without any attempt at unification beyond the cobbling together of electromagnetism and the weak force into an unintelligible mess. So there is a school of thought that this bit of unification needs more work, and a “Grand Unified Theory” (GUT) is required. In practice, such theories are far too Grand and not nearly Unified enough.

If, however, you really do believe that CR and PQ unification is already adequate, then you believe, as Einstein did, that QR unification is the issue. This union of quantum theory and relativity is also called Quantum Gravity, and there are various schools of thought as to how best to achieve this goal. Many people are working on this aspect of unification in one way or another. But if CR unification is really done and dusted, then it is enough to do CQ unification instead. This is where Einstein focussed his attack in the 1930s, because he did not believe that the Copenhagen School had done the CQ unification properly. Nor do I. The Danelaw that has been set up as a result of the Viking invasion is not a unification, it is an occupation. So if we can’t do QR unification, then the probable reason is that one or other of CR or CQ unification has been done wrong.

You will have noticed that I have ranted and raved about both CR and CQ unification many times on this blog. They cannot both be correct, because they are mathematically inconsistent (not to mention physically inconsistent). People are generally very tribal about which one they believe in, and which one they attack. But in reality they are both wrong, and it is impossible to construct a theory of quantum gravity without doing the whole CQR unification again from scratch. So that is what I am aiming to do, and I follow Einstein in thinking that the key is to repel the Viking invasion, and re-establish the theory of quantum physics on solid foundations. A Viking longship is not an adequate foundation for a castle. The “standard” view of CQ unification must be challenged.

Now if we approach the problems from a particle physics (P) perspective, QR unification is very far from their concerns. Not relevant to particle physics, they say. But if PQ unification has been achieved (which it hasn’t) then you might as well say that QR unification is equivalent to PR unification (which it isn’t, of course). From 2010 onwards, I was recruited to work on PQ unification (GUT), and not allowed to talk about QR or PR unification. But I could see that it was impossible to understand PQ unification without also discussing PR and QR unification. The whole triangle of PQR unification cannot be separated into separate pieces. So I looked at PR unification, and by 2015 I had found the evidence I needed that PR unification is actually the most interesting part of the whole jigsaw. That is where the experimental evidence is really bounteous, and unification is therefore tightly constrained.

Strangely enough, nobody is interested in PR unification. I find this utterly bizarre. I have talked to many people who are interested in PQ unification, and to many people who are interested in QR unification, but I have never met anyone who is interested in both at the same time. Neither group of people therefore has any interest in PR unification. As far as I can tell, I have the entire field to myself. (That probably isn’t true – there must be lots of other crackpots out there like me.) Ten years ago, it was all about trawling through the experimental evidence to find the links between P and R. Today, it is all about building the model of PR unification. That is what I have achieved in 2025. The model of PR unification is SU(2,3), whose compact part is the gauge group of P, and which splits into real (special) R and imaginary (general) R. Hence P mixes the real (electromagnetic) and imaginary (gravitational) parts of R (and C), while R mixes the different forces and particles in P.

The model therefore makes oodles and oodles of predictions. It fits the experimental data incredibly well. But because it is not in an established field like PQ, QR, CR or CQ it is ignored. Nobody but a crackpot would work in PR unification, would they?

Well, I’ve talked about five flavours of unification, that is CR, CQ, QR, PQ and PR, so what is missing? The “top” unification CP. I call it “top” because it is the top predator in this wilderness (landscape, swampland, whatever you want to call it). Classical physics (C) explains many things in terms of waves, which particle physics (P) explains in terms of particles. So, obviously, particles and waves must be the same thing, right? Particles must be like waves on a tiny vibrating string, right? So string theory was born out of an attempt at CP unification. It was a bold idea, to go straight from C to P without going through either Q or R on the way. Sidestep the problem that Q and R are incompatible by ignoring the problem altogether. Brilliant. Give that man a Nobel Prize for Nothing.

The trouble is, that because string theory completely ignores both Q and R, it is completely divorced from physical reality. It cannot, and never will, predict anything at all about real physics that we can measure in experiments. It is not, and never will be, the “only game in town”. There are, as I have shown, six games in this town. Each of them comes in various shades of grey. Let me remind you:

  1. CP – string theory
  2. QR – quantum gravity, e.g. loop quantum gravity, non-commutative geometry
  3. PQ – grand unified theories
  4. CQ – foundations of quantum mechanics
  5. CR – Machian gravity
  6. PR – my own private Idaho

I have listed them roughly in order on a scale from mainstream to crackpot – although you might order them slightly differently. The first four have been actively pursued by large numbers of people for a very long time, and between them they have absolutely nothing to show for it. No predictions, or else utterly wild predictions, none of which has ever been verified. Extremely complicated and ugly theories that make Ptolemy’s epicycles look like a model of simplicity. Landscapes and swamplands with no clue where to go. So isn’t it time to invest in the other two games?

Machian gravity has a few adherents, Unzicker and McCulloch for example, but they are regarded by the mainstream as crackpots. Revisiting the foundations of relativity (CR) is not regarded by the mainstream as a productive activity. Still, more productive than string theory, I would say. Revisiting the foundations of quantum mechanics (CQ) is more obviously necessary, but still ignored by the mainstream, as being too “philosophical” – what they really mean is, too difficult. But they can’t tell us what a “spinor” is, so where is the connection between quantum mechanics and reality? It’s all a bit too vague, if you ask me.

But even CR and CQ don’t make it as far on the crackpot scale as PR does. Only a lunatic would look seriously at PR. Well, why not, if all the other shades of unification have failed? Why not look at the only remaining possibility? You never know, you might find a link between:

  1. kaon decay CP-violation and curvature of the Earth’s gravitational field;
  2. kaon lifetimes and the gravity of the Moon;
  3. neutron lifetimes and the rotation of the Earth;
  4. the weak equivalence principle and the tilt of the Earth’s axis;
  5. the muon g-2 anomaly and the direction gravity;
  6. the W/Z mass anomaly and the tides;
  7. the weak force and the calibration of time;
  8. the strong force and the calibration of space;
  9. neutrino oscillations and curvature of spacetime;

and so on and so on. And when you’ve got that far, you may find that PR unification has helped you to understand what went wrong with CR unification in 1915, what went wrong with CQ unification in 1925, what went wrong with PQ unification in 1975, what went wrong with CP unification in 1985, and what went wrong with QR unification in 1995.

And put it right.

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Why is the Sun’s atmosphere so hot?

December 25, 2025

I have answered this question before, but I think I understand it a bit better now. The temperature of the Sun’s surface is about 6000 degrees (Kelvin, Celsius or centigrade), but the atmosphere is about 1000000K, which is the same as the surface of a neutron star. This is a clue that the physical processes that generate this extremely high temperature may be the same.

The surface of a neutron star literally boils at this temperature. The process is like nuclear fission, but with an atom that weighs as much as a star. You might think the individual neutrons would boil off one by one, but the process is more complicated than that. In nuclear fission, typically alpha particles (helium nuclei) are ejected, that is two protons and two neutrons stuck together. In a neutron star, the protons are generated by a quantum-gravitational equilibrium between the five-neutron-plus-one-neutrino state and the three-proton-plus-three-flavours-of-electrons state. These two states have exactly the same mass, so the change of state occurs adiabatically (without energy transfer).

But at the surface, this equilibrium is disturbed, because the electrons can boil off, leaving too many protons on the surface, which are then evicted for overcrowding, and then there’s an imbalance between the three electron flavours, which has to be corrected as well. Correcting this imbalance generates an enormous amount of energy, which keeps the temperature up.

Alternatively, one can consider a stable unit of 5n+3p+e+mu+tau getting separated off. It is trying to be an atom of lithium-8 (atomic number 3), but it is horribly unstable, both because lithium usually occurs as lithium-7, and because two of its electrons are in the wrong flavour (mass) eigenstates. Various things can happen, including beta decay to beryllium-8, which then captures a neutron (there are plenty around) to become beryllium-9, losing energy by muon and tau decay to electrons. Whatever happens, the energy is generated by muon and tau decay, and the protons and neutrons end up in some mixture of hydrogen, helium, lithium and beryllium nuclei (possibly including small amounts of heavier nuclei), which then boil off into outer space, together with all the electrons that got separated from them.

Basically, the same thing happens in the solar atmosphere. There’s plenty of hydrogen, helium, lithium and beryllium in the Sun, and it keeps boiling off into the atmosphere. You would think, of course, that all their electrons would be ordinary electrons, but you are forgetting two things. First, the nuclei are completely ionised – they have no electrons. The electrons are left entirely to their own devices. And second, solar neutrinos occur in equal numbers in all three generations. Therefore, so do the electrons.

Hence the solar atmosphere is exactly the same as the atmosphere of a neutron star. Hence it behaves in exactly the same way. Hence it is at the same temperature.

Basically, the same thing is supposed to have happened in the Big Bang. It is called BBN (Big Bang Nucleosynthesis), because it synthesises nuclei. But it doesn’t need a big bang. It happens all the time to neutron stars. And especially it happens when neutron stars collide. Which they often do.

So it turns out that that weird little mass equation that I stumbled upon back in April 2015 is the key to understanding both Big Bang Nucleosynthesis and Dark Matter, not to mention the coronal heating problem and the solar neutrino problem. Who woulda thunk it?

Oh, and by the way, this tells us that exploding neutron stars generate 5 neutrons for every 3 protons. Or you could say the atomic mass to atomic number ratio of a neutron star is 8/3 or about 2.667. Ordinary matter has a ratio of 1 (hydrogen), 2 (helium, carbon, nitrogen, oxygen, etc) rising to 238/92 or about 2.587 for uranium and 244/94 or about 2.596 for plutonium (96% neutron saturation). Anything more than this, and the nucleus will spontaneously decay under 5n -> 3p+e+mu+tau, i.e. the same process as neutron star evaporation and coronal heating.

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