# Locklin on science

## BTC bubbles

Posted in econophysics by Scott Locklin on April 17, 2013

Not surprisingly, Bitcoin prices are well described by the  log periodic power laws describing the dynamics of bubbles. A reminder of what a LPPL model looks like; here is a simple one:

$p(t) = A + B(t_c - t)^\beta + C(t_c - t)^\beta \cos( \omega \log(t_c-t)+\phi)$

I didn’t profit from this. I thought of applying LPPL to the BTC bubble well before the crash during a bullshit session with a friend, but I didn’t run the analysis until after. I have better things to do with my time than play with weird monopoly money, and the “exchanges” presently offering shorts are not even close to useful. I also think anyone who trades on LPPL is basically gambling. The most interesting parameter, $t_c$ is hardest to fit, and, well, with all those parameters I could fit a whole lot of elephants. Just the same it is a useful enough concept to justify further research. No, I won’t be telling the world about that research on my blog. A man’s got to eat, after all. Doing bubble physics costs money.

If you don’t know about LPPL models, click on these two helpful links. The “hand wavey” idea is, if the price is formed by market participants looking at what other market participants are doing, as with Dutch tulips, pets.com, and market prices in various eras, the price is an irrational bubble which will eventually burst. This isn’t an original idea: Charles Mackay was talking about it 180 years ago. The original idea is mapping this behavior onto an Ising model,  running some renormalization group theory on it, and fitting to the result to get a forecast of bubble burstings.  Sornette, Ledoit,  Johanson, Bouchaud and Freund did it and told the world about it; may the eternal void bless them with healthy returns for being kind enough to share this interesting idea with us.

Here’s a plot of BTC close prices from MtGox (via quandl), with the LPPL model fit 10 days before the bubble pop. I wasn’t real careful with the fit; no unit root tests were done, no probabilistic estimates were made and no Ornstein Uhlenbeck processes were taken into account. This is just curve fitting. The result is compelling enough to talk about. As you can see, with these parameters, the out of sample top is fit fairly well. Amusingly, so is the decline.

What can we learn from this? You can see a “fair value” of around $20/BTC due to be hit in a few weeks, with perhaps a full mean reversion to$10/BTC.  BTC doesn’t seem to have a helpful “anti-bubble” decay; if anything, it is decaying faster than expected so far (it is possible I mis-fit the $\omega$). The fit parameters for this version of the model tell us a few interesting things about the herding behavior which you can read about in Sornette’s book.

I don’t have any strong opinions about using BTC as a currency. I think most of its enthusiasts  are naive and do not understand the nature of money and what it is good for. I do think BTC would work a lot better as a store of value with a properly functioning foreign exchange futures market. There are no properly functioning BTC futures exchanges at present; just an assortment of dreamers and borderline crooks cashing in on hype. This is more of an engineering and legal problem than it is an inherent problem with using BTC as a currency. The way things are presently set up, without shorts, any extra media attention will result only in people buying the damn things. Without the ability to easily short them, price discovery is impossible, and herding behavior is the rule. It ain’t a market without shorts. It’s a bubble maker. Shorts don’t guarantee there will be no bubbles; we see plenty in shortable markets, but a lack of shorts will virtually guarantee future BTC bubbles.

## The enigma of the Ford paradox

Posted in chaos, physics by Scott Locklin on March 7, 2013

“God plays dice with the Universe. But they’re loaded dice. And the main objective of physics now is to ﬁnd out what rules were they and how we can use them for our own ends.” -Joe Ford

Joe Ford was one of the greats of “Chaos Theory.” He is largely responsible for turning this into a topic of interest in the West (the Soviets invented much of it independently) through his founding of the journal Physica D. It is one of the indignities of physics history that he isn’t more widely recognized for his contributions. I never met the guy, as he died around the time I began studying his ideas, but my former colleagues sing his praises as a great scientist and a fine man. One of his lost ideas, working with student Matthias Ilg and coworker Giorgio Mantica, is the “Ford paradox.” The Ford paradox is so obscure, a google search on it only turns up comments by me. This is a bloody shame, as it is extremely interesting.

Definitions: In dynamical systems theory, we call the motion of a constrained system an “orbit.” No need to think of planets here; they are associated with the word “orbit” because they were the first orbital systems formally studied. It’s obvious what an orbit is if you look at the Hamiltonian, but for now, just consider an orbit to be some kind of constrained motion.

In most nontrivial dynamical systems theory, we also define something called the phase space.” The phase space is that which fully defines the dynamical state of the system. In mechanics, the general convention is to define it by position and momentum of the objects under study. If the object is constrained to travel in a plane and its mass doesn’t change, like, say, a pendulum, you only have two variables; angular position, and its time derivative, and you can easily visualize the phase space:

For my last definition, I will define the spectrum for the purposes of this exposition. The spectrum is the Fourier transform with respect to time of the orbits. Effectively, it is the energy levels of the dynamical system. If you know the energy and the structure of the phase space, classically speaking, you know what the motion is.

Consider a chaotic system, such as the double pendulum. Double pendulums, as you might expect, have two moving parts, so the phase space is four dimensional, but we can just look at the angle of the bottom most pendulum with respect to the upper pendulum:

If you break down the phase space into regions, and assign a string to each region, one can characterize  chaos by the length of the string in bits. If it is a repeated string, the system is non-chaotic. Chaotic systems are random number generators. They generate random strings. This is one of the fundamental results of modern dynamical systems theory.  A periodic orbit can be reduced to simple sequences, like: {1 0 1 0 1 0}, {1 1 0 1 1 0 1 1 0}. Effectively, periodic orbits are integers. Chaotic orbits have no simple repeating sequences. Chaotic orbits look like real numbers. Not floats which can be represented in a couple of bytes: actual real numbers, like  base of the natural log $e$ or $\pi$ or the golden ratio $\phi$. In a very real sense, chaotic orbits generate new information. Chaotic randomness sounds like the opposite of information, but noisy signals contain lots of information. Otherwise, qua information theory, you could represent the noise with a simple string, identify it, and remove it.  People have invented  mechanical computers that work on this principle. This fact also underlies the workings of many machine learning algorithms. Joe Ford had an extremely witty quoteable about this: “Evolution is chaos with feedback.”

This is all immediately obvious when you view the phase space for a chaotic system, versus a non-chaotic system. Here is a phase space for the end pendulum of a double pendulum at a non-chaotic set of parameters: it behaves more or less like a simple pendulum. My plots are in radians (unlike the above one for a normal pendulum, which I found somewhere else), but otherwise, you should see some familiar features:

It looks squished because, well, it is a bipendulum. The bottom which looks like  lines instead of distorted ellipses  is where the lower pendulum flips over the upper pendulum. The important thing to notice is, the orbits are all closed paths. If you divided the phase space into two regions, the path defined string would reduce to something like {1 0 1 0 1 0…} (or in the lower case { 0 0 0 0…}) forever.

Next, we examine a partially chaotic regime. The chaotic parts of the phase space look like fuzz, because we don’t know where the pendulum will be on the phase space at any given instant. There are still some periodic orbits here. Some look reminiscent of the non-chaotic orbits. Others would require longer strings to describe fully.  What you should get from this; the orbits in the chaotic regions are random. Maybe the next point in time will be a 1. Maybe a 0. So, we’re generating new information here. The chaotic parts and not so chaotic parts are defined on a manifold. Studying the geometry of these manifolds is much of the business of dynamical systems theory. Non-chaotic systems always fall on a torus shaped manifold. You can see in the phase space that they even look like slices of a torus. Chaotic systems are, by definition, not on a torus. They’re on a really weird manifold.

Finally: a really chaotic double pendulum. There are almost no periodic orbits left here; it’s all motion in chaotic, and the path the double pendulum follows generates random bits on virtually any path available to it in the phase space:

Now, consider quantum mechanics. In QM, we can’t observe the position and momentum of an object with infinite precision, so the phase space is “fuzzy.” I don’t feel like plotting this out using Husimi functions, but the ultimate result of it is the chaotic regions are smoothed over. Since the universe can’t know the exact trajectory of the object, it must remain agnostic as to the  path taken. The spectrum of a quantum mechanical orbital system looks like … a bunch of periodic orbits. The quantum spectrum vaguely resembles the parts of the classical phase space that look like slices of a torus. I believe it was W.P. Reinhardt who waggishly called this the “vague tori.” He also said, “the vague tori, being of too indistinct a character to object, are then heavily exploited…” Quantum chaologists are damn funny.

This may seem subtle, but according to quantum mechanics, the “motion” is completely defined by periodic orbits. There are no chaotic orbits in quantum mechanics.  In other words, you have a small set of  periodic orbits which completely define the quantum system. If the orbits are all periodic, there is  less information content than orbits which are chaotic. If this sort of thing is true in general, it indicates that classical physics could be a more fundamental theory than quantum mechanics.

As an interesting aside: we can see neat things in the statistics of the quantum spectrum when the classical equivalent is chaotic; the spectrum looks like the eigenvalues of a random matrix. Since quantum mechanics can be studied as matrix theory, this was a somewhat expected result. Eigenvalues of a random matrix were studied at great length by people interested in the spectra of nuclei, though the nuclear randomness comes from the complexity of the nucleus (aka, all the many protons and neutrons), rather than the complexity of the underlying classical dynamics.  Still, it was pretty interesting when folks first noticed it in simple atomic systems with classically chaotic dynamics. The quantum spectra of a classically non-chaotic system are more or less near neighbor Poisson distributed. Quantum spectra repulse one another. You know something is up when near neighbor spectral distribution starts to look like this:

Random matrix theory is now used by folks in acoustics. Since sound is wave mechanics, and since wave mechanics can be approximated in the short wavelength regime by particles, the same spectral properties apply.  One can design better concert hall acoustics by making the “short wavelength” regime chaotic. This way there are no dead spots or resonances in the concert hall. Same thing applies to acoustically invisible submarines. I may expand upon this, and its relationship to financial and machine learning problems in a later blog post. Spectral analysis is important everywhere.

Returning from the aside to the Ford paradox. Our chaotic pendulum is happily chugging along producing random bits we can use to, I dunno, encrypt stuff or otherwise perform computations. But, QM orbits behave like classical periodic orbits, albeit ones that don’t like standing too close to one another. If quantum mechanics is the ultimate theory of the universe: where do the long strings of random bits come from in a classically chaotic system? Since people believe that QM is the ultimate  law of the universe, somehow we must be able to recover all of classical physics from quantum mechanics. This includes information generating systems like the paths of chaotic orbits. If we can’t derive such chaotic orbits from a QM model, that indicates that QM might not be the ultimate law of nature. Either that, or our understanding of QM  is incomplete. Is there a point where the fuzzy QM picture turn into the classical bit generating picture? If so, what does it look like in the transition?

I’ve had  physicists tell me that this is “trivial,” and that the “correspondence principle” handles this case. The problem is, classically chaotic systems egregiously violate the correspondence principle. Classically chaotic systems generate  information over time. Quantum mechanical systems are completely defined by stationary periodic orbits. To say the “correspondence principle handles this” is to merely assert that we’ll always get the correct answer, when, in fact, there are two different answers. The Ford paradox is asking the question: if QM is the ultimate theory of nature, where do the long bit strings in a classically chaotic dynamical system come from? How is the classical chaotic manifold  constructed from quantum mechanical fundamentals?

Joe Ford was a scientist’s scientist who understood that “the true method of knowledge is experiment.” He suggested we go build one of these crazy things and see what happens, rather than simply yakking about it. Why not  build a set of small and precise double pendulums and see what happens? The double pendulum is pretty good, in that its classical mechanics has been exhaustively studied. If you make a small enough one, and study it on the right time scales, quantum mechanics should apply. In principle, you can make a bunch of them of various sizes, excite them to the chaotic manifold, and watch the dynamics unfold.  You should also do this in simulation, of course. My pal Luca made some steps in that direction.  This experiment could also be done with other kinds of classically chaotic systems; perhaps the stadium problem is the right approach. Nobody, to my knowledge, is thinking of doing this experiment, though there are many potential ways to do it.

It’s possible Joe Ford and I have misunderstood things. It is possible that spectral theory and the idea of the “quantum break time” answers the question sufficiently. But the question has not to my knowledge been rigorously answered. It seems to me much a more interesting question than the ones posed by cosmology and high energy physics. For one thing, it is an answerable question with available experimental tests. For another, it probably has real-world consequences in all kinds of places. Finally, it is probably a  productive approach to unifying information theory with quantum mechanics, which many people agree is worth doing. More so than playing  games postulating quantum computers. Even if you are a quantum computing enthusiast, this should be an interesting question. Do the bits in the long chaotic string exist in a superposition of states, only made actual by observation? If that is so, does the measurement produce the randomness? What if I measure differently?

But alas, until someone answers the question, I’ll have to ponder it myself.

For people with a background in physics who want to understand the information theory behind this idea, the following paper is useful:

“The Arnol’d Cat: Failure of the Correspondence Principle” J. Ford, G. Mantica, G. H. Ristow, Physica D, Volume 50, Issue 3, July 1991, Pages 493–520

## Nano on the pink sheets: anatomy of a pump and dump

Posted in fraud, nanotech by Scott Locklin on December 8, 2012

I don’t remember how this got into my browser window, but it got there somehow. Nanotech. Someone wants me to invest, like, cash money into Nanotech. They want this so much they registered a domain name dedicated to … some obscure company called “Nano labs” or CTLE on the OTC pink sheets. Please click on this link for some high comedy. Here’s a snapshot for posterity, since I’m pretty sure this website and the half dozen others I found will be nuked after the pump and dump.

They’re not even clever about it. I’m not the only person to notice: someone over at SeekingAlpha has noticed. Someone who specializes in pinks at Boston.com was also suspicious. What boggles me is, this outfit is apparently worth $230.6 million dollars. The post-November publicity pump has added more than$200m notional to their market value. They have no assets. They recently issued 100 million shares to their new CEO, Dr. Victor M. Castrano in exchange for “new nanotechnology.” Since lying to the SEC is a serious crime, they, to their credit, do  say that it “involves a coating that can be applied to almost any surface, has low thermal conductivity and protects surfaces from water leaks, corrosion and rust. Nanotechnology involves mixing microscopic particles into paints, coatings and films that can be applied to most surfaces to provide temperature resistance and increased structural integrity.” Yes, in fact, later press releases reveal they’re talking about the nano-invention known as “house paint.”

Dr. Castrano is  a graduate and professor of physics at a Mexican university whose physics department webpage is broken. A literature search turns up some actual references written by this guy, published in perfectly respectable journals. Though it is the usual dreary “nanotech” stuff which cheeses me off so much; one of those links involves goop made of chicken feathers.  What Doc Castrano does is considered respectable “nanoscience.” The journal titles are legitimate and even impressive.  Yet this company is an obvious fraud based on the sheerest nonsense. “Nano Labs” was originally founded in 1995 … to “sell and install stone, tile and marble products used in residential and commercial buildings.” Yet, somehow, in spring of 2012, they reorganized, , sold themselves for … $500, issued a zillion shares, and renamed themselves “Nano Labs Corp.” Why did they do this? I haven’t the slightest idea. I am guessing the CTLE ticker was already listed on the pink sheets; OTCQB is almost respectable; only one step away (via OTCQX) from NASDAQ, as they do some limited SEC compliance. These clowns aren’t even bothering to change the wording of the marketing bullshit in their 10-Q. In the most recent 10-Q, they use the same line as in the above screenshot, “The Company is pursuing opportunities for global market leadership in the field of nanotechnology, a sector with the prospect of$2.6 trillion in global revenues – representing 15 per cent of all projected global manufacturing – by 2014.” It is my considered opinion that the field of nanotechnology will not be worth $2.6 trillion globally in 2014; not even if the dollar were to experience Zimbabwe-style inflation between now and then. I think it very close to an established fact that the only way nanotech will represent 15 per cent of all projected global manufacturing by 2014 is if we are nuked into the stone ages by nanotech-weilding space invaders. They mention in their 10-Q that they only have one part time and one full time employee … yet their notional is$230.6 million dollars on the pink sheets market. And get a load of their 10-K statement, listing cash flows, debts and assets in the tens of thousands of dollars. Their company financial records look somewhat like my credit card statements on a bad year.

The way I see it, there are two and a half possibilities here. One is that the principals are frauds. Considering the nature of nanotechnology and “nanoscience” and the type of clown involved in hyping their “nano” research, and considering the preposterous websites and press releases pumping this stock, this seems a very strong possibility. The other possibility  is that the principals have been kidnapped by the Mexican Mafia for some other purpose; perhaps this company is being used for money laundering, or perhaps it is outright fraud perpetrated by mobsters rather than “respectable” nanotechnologists. I don’t know much about Mexico, but these guys are based there, and some very scary news reports come from that part of the world. If that is the case, I call on Doc Castrano’s “respectable” nanotech colleagues to send General Pershing  to his rescue. Finally, I count it as a half possibility that Doc Castrano and company are in earnest,  and are not familiar with the SEC regulations on fraud, or are not connected in any way with the publicity campaign going on in their name. This only seems like half a possibility, as their website, complete with links to 10-K and 10-Q financials, seems completely bonkers. I mean, what on earth is a “Nano mellon?”

I  proclaim myself an agnostic as to which of the two and a half possibilities is the actuality, though I will vehemently maintain that these are the only two and a half possibilities which exist in reality. Their wave function  has only two and a half states. Unless you’re  long their stock or are involved in law enforcement, it doesn’t matter which possibility is true. What really matters is, this is a completely shady situation involving a seemingly legitimate nanotech researcher. A  legitimate nanotech news aggregation website, run by members of the Foresight institute, touts a “Nano labs” press release as something worthy of attention. This is a “respectable” website on the subject of nanotechnology, with actual venture capital experts  advising it. Yet, somehow these nanotech experts and big dollar VC types cannot see what I find completely obvious: this “Nano labs” CTLE company is an egregious and embarrassing fraud.

I think “nanotechnology” is a fraudulent concept, over sold by snake oil salesmen who should be reading comic books in their parents basements instead of bothering sensible people. Despite my horse laughs at the pretensions of “nanotechnologists,” I never expected to see something this preposterously brazen.  It is actually much worse than the imbeciles attempting to convince the world they have a working quantum computer, as at least some of the quantum computing community laughs at them in public. This is flim-flammery on a  significant scale, and abetted by the Nanotech hype machine. Other than  a few watchers of penny stocks and my own bad self, nobody seems to have noticed.  Shouldn’t real-life nanotechnologists care that their precious ideas are being perverted by frauds? Shouldn’t they be policing their own?

Naaaah! Why dull the nano fireworks?

I’ve never tried to short a pink sheet. I’m not sure what the downside risks are of that if the SEC shows up and legally notices that this is being run out of someone’s boiler room. If they were traded on a real market, I’d short the shit out of them.

Edit add: this morning I notice the very same professor made the news a few years ago for allegedly turning Tequila into Diamonds.  I give up. Humanity is a pack of credulous numskulls without hope. That was linked everywhere from Arxiv to the BBC. I am picturing this guy with his bottle of Tequila,  a bucket full of chicken feathers and sporting a preposterous Speedy Gonzales sombrero counting stacks of greenbacks.

## 30 open questions in physics and astronomy

Posted in physics, physics anomalies by Scott Locklin on August 2, 2012

A friend of mine asked me if I thought there were actual open questions in physics, ones that individuals or small groups could make a contribution to (as opposed to things like the Higgs boson which require 4000 people and billions of dollars to suss out). Here is a list I came up with. I don’t think it is definitive, and for all I know, some of these problems may no longer be open questions as of today, but I didn’t find anything better on the internets. It may be of interest to young researchers wishing to make a real contribution to human knowledge. Or maybe it’s just something to bullshit about.

Unlike other such lists, there are no silly cosmological or quantum gravitic types of questions on it. I think these are unanswerable questions, and not presently solvable by Baconian science. Essentially, such questions are metaphysical. They can’t presently be solved even in concept by making observations about reality. We’d still like to know the answers to such questions as how to unify gravity with the other forces, but it’s effectively a sort of mathematical philosophic enquiry, rather than normative science.

The other aspect of my “open questions” is they could conceivably be solved by an individual or a small team. I had to use my judgement on that, such as it is. I think these are all interesting and worthy mysteries; ones which could be of great import to the human race. I suppose they vary quite a bit in importance, but all of ‘em are interesting.

1. High Tc superconductors: they cost nothing, and liquid nitrogen is cheap. Nobody knows how they work, or if they could make one at room temperature. The consequences would be tremendous if we could! IMO, every barnyard physicist who is worth two shits should have some perskovites and liquid nitrogen kicking around the lab, just for fiddlin’ with.
2. Turbulence and Navier-stokes is still little understood: this is pencil and paper physics which stumped Heisenberg. If you think liquids are important, this is huge.
3. Why is life chiral? When you make amino acids using chemistry, it isn’t chiral. How come life is chiral?
4. Quantum mechanics is still mysterious, particularly in the classical limit: pencil and paper contributions and relatively cheap (though carefully done) experiments are possible. This is one of the biggest open questions for it’s philosophical and technological (quantum computing?) implications.
5. Cosmic ray physics still has plenty of unusual phenomena. Detectors are cheap. You do have to wait for things to happen. What’s up with the giant cosmic rays for example?
6. Solid to glass phase transitions are poorly understood and very interesting.
7.   Fractional quantum Hall effect: simple and cheap experiments, and pencil and paper theory which could help us understand lots of other things in nature.
8.   Catalysis is fairly mysterious and potentially revolutionary for novel technologies. The models I have seen are pretty hand-wavey, and not very useful for inventing new catalysts or predicting the properties of old ones.
9.  Entropy and the arrow of time; this is at least as important as Ernst Mach’s philosophical ideas on relativistic things, which eventually helped lead to relativistic physics. Pencil and paper and thought experiments will suffice here. This is a very important philosophical question. Probably more important than understanding quantum mechanics.
10.   What is life? Nobody knows.
11.   How do brains work?
12.   Properties of metallic hydrogen -I think you can do these experiments in diamond anvils. Or you could send a lab to Jupiter.
13.   Is there a physics analogue for solving NP-hard problems? OK, this is cheating and stealing from computer science, but there may be physical algorithms you could use as proofs here, just like certain spectra could theoretically calculate Reimann’s Zeta. I’m not the only one to have looked in physical systems for answers here.
14.   Is there a knowable physics of granular materials? How do singing sands work?
15.   How does non-equilibrium thermodynamics work? We know there is order here; you can see the order with your eyes, but we don’t know what the rules are. This is potentially bigger than understanding quantum mechanics.
16.   Are there more new weird properties of matter? We keep discovering poorly understood stuff like high Tc superconductors and the fractional quantum Hall effect. Material science is vast and potentially technologically revolutionary.
17.   How does water work? The large heat capacity of water is an enormous physical mystery. Water should be vapor at STP. It ain’t.  People wave their hands and talk about hydrogen bonding, but hand waving doesn’t do much. This is also potentially huge. It’s freaking water: doesn’t get much cheaper than a jar of water.
18.   WTF is going on inside the earth? Whence comes the magnetic field? Why doesn’t Venus have one? Why is it so damn hot in there? Yes, I know there are theories: they don’t even pass a sniff test.
19.   What is the story with the Pioneer anomaly? If it’s accurate and not from something silly like an outgassing thumb print, this could throw everything we know about physics and astronomy into utter chaos. There is a way of answering this which would cost a half billion or so; shoot something into interstellar space on purpose and see what happens in 20-40 years. This is arguably far more important than anything in particle physics. The “flyby anomalies” turned out to be dipshits not understanding special relativity… I’m assuming that dipshits have done enough special relativity on Pioneer to rule this out. Otherwise: undergraduates: get to work! Edit add: I thought the flyby anomaly was resolved with SR, but googling further; it ain’t!
20. The atmosphere is filled with anomalies: ball lightning, sprites, ELVES, blue jets, TIGERs, green flashes & etc. I have a fat book by William Corliss catalogueing mysteries from the 60s; there are even more now.
21. There was a gamma ray burst where the high energy gamma rays got here before the low energy ones. Looks like highly anomalous physics.
22. GEO600 has produced some bizarro gravity results.
23. Corona physics makes no sense. Why is the corona hotter than the sun’s surface?
24. What are diffuse interstellar bands? Nobody has a clue as to what is absorbing light at those wavelengths, yet … I’m supposed to believe the standard model explains everything? Chyeah, right.
25. Is dark matter real, or is it the same thing that makes orbital mechanics fuck up? Something is weirding up the rotations of galaxies. I’m very tempted to put this in with the Pioneer anomaly and say, “gravity is largely untested using experiments; we should change this.”
26. Horizon problem;  why does the universe look homogeneous? It shouldn’t be.
27. For that matter, why is cosmic microwave background anisotropic, when everything else is isotropic?
28. What are magnetars?
29. Long delayed echos? This is a seemingly science fictional level of WTF. I recall some science fiction type speculated this was a sign of alien intelligence in the Solar System.
30. Why is there more matter than antimatter? I threw that one in for high energy types. You already have a result: if you’re so smart -figure it out.