Theoretical Physics

At Multiverse Impasse, a New Theory of Scale

Mass and length may not be fundamental properties of nature, according to new ideas bubbling out of the multiverse.

Mass and length scales.

Andy Gilmore for Quanta Magazine

In a radical new theory, mass and length scales arise from interactions between particles.

Though galaxies look larger than atoms and elephants appear to outweigh ants, some physicists have begun to suspect that size differences are illusory. Perhaps the fundamental description of the universe does not include the concepts of “mass” and “length,” implying that at its core, nature lacks a sense of scale.

This little-explored idea, known as scale symmetry, constitutes a radical departure from long-standing assumptions about how elementary particles acquire their properties. But it has recently emerged as a common theme of numerous talks and papers by respected particle physicists. With their field stuck at a nasty impasse, the researchers have returned to the master equations that describe the known particles and their interactions, and are asking: What happens when you erase the terms in the equations having to do with mass and length?

Nature, at the deepest level, may not differentiate between scales. With scale symmetry, physicists start with a basic equation that sets forth a massless collection of particles, each a unique confluence of characteristics such as whether it is matter or antimatter and has positive or negative electric charge. As these particles attract and repel one another and the effects of their interactions cascade like dominoes through the calculations, scale symmetry “breaks,” and masses and lengths spontaneously arise.

Similar dynamical effects generate 99 percent of the mass in the visible universe. Protons and neutrons are amalgams — each one a trio of lightweight elementary particles called quarks. The energy used to hold these quarks together gives them a combined mass that is around 100 times more than the sum of the parts. “Most of the mass that we see is generated in this way, so we are interested in seeing if it’s possible to generate all mass in this way,” said Alberto Salvio, a particle physicist at the Autonomous University of Madrid and the co-author of a recent paper on a scale-symmetric theory of nature.

In the equations of the “Standard Model” of particle physics, only a particle discovered in 2012, called the Higgs boson, comes equipped with mass from the get-go. According to a theory developed 50 years ago by the British physicist Peter Higgs and associates, it doles out mass to other elementary particles through its interactions with them. Electrons, W and Z bosons, individual quarks and so on: All their masses are believed to derive from the Higgs boson — and, in a feedback effect, they simultaneously dial the Higgs mass up or down, too.

The new scale symmetry approach rewrites the beginning of that story.

Thomas Lin/Quanta Magazine

Alessandro Strumia of the University of Pisa, pictured speaking at a conference in 2013, has co-developed a scale-symmetric theory of particle physics called “agravity.”

“The idea is that maybe even the Higgs mass is not really there,” said Alessandro Strumia, a particle physicist at the University of Pisa in Italy. “It can be understood with some dynamics.”

The concept seems far-fetched, but it is garnering interest at a time of widespread soul-searching in the field. When the Large Hadron Collider at CERN Laboratory in Geneva closed down for upgrades in early 2013, its collisions had failed to yield any of dozens of particles that many theorists had included in their equations for more than 30 years. The grand flop suggests that researchers may have taken a wrong turn decades ago in their understanding of how to calculate the masses of particles.

“We’re not in a position where we can afford to be particularly arrogant about our understanding of what the laws of nature must look like,” said Michael Dine, a professor of physics at the University of California, Santa Cruz, who has been following the new work on scale symmetry. “Things that I might have been skeptical about before, I’m willing to entertain.”

The Giant Higgs Problem

The scale symmetry approach traces back to 1995, when William Bardeen, a theoretical physicist at Fermi National Accelerator Laboratory in Batavia, Ill., showed that the mass of the Higgs boson and the other Standard Model particles could be calculated as consequences of spontaneous scale-symmetry breaking. But at the time, Bardeen’s approach failed to catch on. The delicate balance of his calculations seemed easy to spoil when researchers attempted to incorporate new, undiscovered particles, like those that have been posited to explain the mysteries of dark matter and gravity.

Instead, researchers gravitated toward another approach called “supersymmetry” that naturally predicted dozens of new particles. One or more of these particles could account for dark matter. And supersymmetry also provided a straightforward solution to a bookkeeping problem that has bedeviled researchers since the early days of the Standard Model.

In the standard approach to doing calculations, the Higgs boson’s interactions with other particles tend to elevate its mass toward the highest scales present in the equations, dragging the other particle masses up with it. “Quantum mechanics tries to make everybody democratic,” explained theoretical physicist Joe Lykken, deputy director of Fermilab and a collaborator of Bardeen’s. “Particles will even each other out through quantum mechanical effects.”

This democratic tendency wouldn’t matter if the Standard Model particles were the end of the story. But physicists surmise that far beyond the Standard Model, at a scale about a billion billion times heavier known as the “Planck mass,” there exist unknown giants associated with gravity. These heavyweights would be expected to fatten up the Higgs boson — a process that would pull the mass of every other elementary particle up to the Planck scale. This hasn’t happened; instead, an unnatural hierarchy seems to separate the lightweight Standard Model particles and the Planck mass.


With his scale symmetry approach, Bardeen calculated the Standard Model masses in a novel way that did not involve them smearing toward the highest scales. From his perspective, the lightweight Higgs seemed perfectly natural. Still, it wasn’t clear how he could incorporate Planck-scale gravitational effects into his calculations.

Meanwhile, supersymmetry used standard mathematical techniques, and dealt with the hierarchy between the Standard Model and the Planck scale directly. Supersymmetry posits the existence of a missing twin particle for every particle found in nature. If for each particle the Higgs boson encounters (such as an electron) it also meets that particle’s slightly heavier twin (the hypothetical “selectron”), the combined effects would nearly cancel out, preventing the Higgs mass from ballooning toward the highest scales. Like the physical equivalent of x + (–x) ≈ 0, supersymmetry would protect the small but non-zero mass of the Higgs boson. The theory seemed like the perfect missing ingredient to explain the masses of the Standard Model — so perfect that without it, some theorists say the universe simply doesn’t make sense.

Yet decades after their prediction, none of the supersymmetric particles have been found. “That’s what the Large Hadron Collider has been looking for, but it hasn’t seen anything,” said Savas Dimopoulos, a professor of particle physics at Stanford University who helped develop the supersymmetry hypothesis in the early 1980s. “Somehow, the Higgs is not protected.”

The LHC will continue probing for convoluted versions of supersymmetry when it switches back on next year, but many physicists have grown increasingly convinced that the theory has failed. Just last month at the International Conference of High-Energy Physics in Valencia, Spain, researchers analyzing the largest data set yet from the LHC found no evidence of supersymmetric particles. (The data also strongly disfavors an alternative proposal called “technicolor.”)

The multiverse hypothesis has surged in begrudging popularity in recent years. But the argument feels like a cop-out to many, or at least a huge letdown.

The implications are enormous. Without supersymmetry, the Higgs boson mass seems as if it is reduced not by mirror-image effects but by random and improbable cancellations between unrelated numbers — essentially, the initial mass of the Higgs seems to exactly counterbalance the huge contributions to its mass from gluons, quarks, gravitational states and all the rest. And if the universe is improbable, then many physicists argue that it must be one universe of many: just a rare bubble in an endless, foaming “multiverse.” We observe this particular bubble, the reasoning goes, not because its properties make sense, but because its peculiar Higgs boson is conducive to the formation of atoms and, thus, the rise of life. More typical bubbles, with their Planck-size Higgs bosons, are uninhabitable.

“It’s not a very satisfying explanation, but there’s not a lot out there,” Dine said.

As the logical conclusion of prevailing assumptions, the multiverse hypothesis has surged in begrudging popularity in recent years. But the argument feels like a cop-out to many, or at least a huge letdown. A universe shaped by chance cancellations eludes understanding, and the existence of unreachable, alien universes may be impossible to prove. “And it’s pretty unsatisfactory to use the multiverse hypothesis to explain only things we don’t understand,” said Graham Ross, an emeritus professor of theoretical physics at the University of Oxford.

The multiverse ennui can’t last forever.

“People are forced to adjust,” said Manfred Lindner, a professor of physics and director of the Max Planck Institute for Nuclear Physics in Heidelberg who has co-authored several new papers on the scale symmetry approach. The basic equations of particle physics need something extra to rein in the Higgs boson, and supersymmetry may not be it. Theorists like Lindner have started asking, “Is there another symmetry that could do the job, without creating this huge amount of particles we didn’t see?”

Wrestling Ghosts

Picking up where Bardeen left off, researchers like Salvio, Strumia and Lindner now think scale symmetry may be the best hope for explaining the small mass of the Higgs boson. “For me, doing real computations is more interesting than doing philosophy of multiverse,” said Strumia, “even if it is possible that this multiverse could be right.”

For a scale-symmetric theory to work, it must account for both the small masses of the Standard Model and the gargantuan masses associated with gravity. In the ordinary approach to doing the calculations, both scales are put in by hand at the beginning, and when they connect in the equations, they try to even each other out. But in the new approach, both scales must arise dynamically — and separately — starting from nothing.

“The statement that gravity might not affect the Higgs mass is very revolutionary,” Dimopoulos said.

A theory called “agravity” (for “adimensional gravity”) developed by Salvio and Strumia may be the most concrete realization of the scale symmetry idea thus far. Agravity weaves the laws of physics at all scales into a single, cohesive picture in which the Higgs mass and the Planck mass both arise through separate dynamical effects. As detailed in June in the Journal of High-Energy Physics, agravity also offers an explanation for why the universe inflated into existence in the first place. According to the theory, scale-symmetry breaking would have caused an exponential expansion in the size of space-time during the Big Bang.

However, the theory has what most experts consider a serious flaw: It requires the existence of strange particle-like entities called “ghosts.” Ghosts either have negative energies or negative probabilities of existing — both of which wreak havoc on the equations of the quantum world.

“Negative probabilities rule out the probabilistic interpretation of quantum mechanics, so that’s a dreadful option,” said Kelly Stelle, a theoretical particle physicist at Imperial College, London, who first showed in 1977 that certain gravity theories give rise to ghosts. Such theories can only work, Stelle said, if the ghosts somehow decouple from the other particles and keep to themselves. “Many attempts have been made along these lines; it’s not a dead subject, just rather technical and without much joy,” he said.

Courtesy of Marcela Carena

Marcela Carena, a senior scientist at Fermi National Accelerator Laboratory in Batavia, Ill.

Strumia and Salvio think that, given all the advantages of agravity, ghosts deserve a second chance. “When antimatter particles were first considered in equations, they seemed like negative energy,” Strumia said. “They seemed nonsense. Maybe these ghosts seem nonsense but one can find some sensible interpretation.”

Meanwhile, other groups are crafting their own scale-symmetric theories. Lindner and colleagues have proposed a model with a new “hidden sector” of particles, while Bardeen, Lykken, Marcela Carena and Martin Bauer of Fermilab and Wolfgang Altmannshofer of the Perimeter Institute for Theoretical Physics in Waterloo, Canada, argue in an Aug. 14 paper that the scales of the Standard Model and gravity are separated as if by a phase transition. The researchers have identified a mass scale where the Higgs boson stops interacting with other particles, causing their masses to drop to zero. It is at this scale-free point that a phase change-like crossover occurs. And just as water behaves differently than ice, different sets of self-contained laws operate above and below this critical point.

To get around the lack of scales, the new models require a calculation technique that some experts consider mathematically dubious, and in general, few will say what they really think of the whole approach. It is too different, too new. But agravity and the other scale symmetric models each predict the existence of new particles beyond the Standard Model, and so future collisions at the upgraded LHC will help test the ideas.

In the meantime, there’s a sense of rekindling hope.

“Maybe our mathematics is wrong,” Dine said. “If the alternative is the multiverse landscape, that is a pretty drastic step, so, sure — let’s see what else might be.”

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  • When I was a little boy I used to stare at ants and study how organized and sophisticated they were- especially for being so tiny.
    One day I realized that the ants couldn’t see me; they couldn’t even acknowledge my presence much less imagine my existence.
    That’s the first time I realized there’s a trap to being a human and having a human-quality brain. We are fundamentally limited in capacity to understand the fully abstract, completely unknown, and universe-scaled giant.
    I’m not just saying we don’t know what we don’t know. I’m saying that for some observations we have the relative processing capacity of ants.
    The paradox is that even though we eventually we hit a ceiling in our capacity to grasp or conceptualize things far more profound than ourselves it’s almost impossible to realize it.
    What scholarly articles and scientific studies that attempt to grasp the entire mechanics of the universe consistently fail to articulate is the possibility that our own limitations block our ability to fully grasp and formulate our existence.

  • Well said Marcos Rodriguez. I agree, and your post made me think of Michio Kaku’s discussion of the Kardashev Scale in “Hyperspace.” We are still very much a primitive civilization type according to this scale, and until we can transcend concepts like money and placing artificial value on known resources, we unfortunately will never move beyond our current state. Venturing deeper into the sciences takes capital, and the majority with the capital see no gains in funding these explorations. Sad sad.

  • “The universe is not only stranger than we imagine, it’s stranger than we can imagine.”
    J. B. S. Haldane

  • Natalie, thanks for this great article.

    Marcos, this is a great point. Maybe we will never be able to fully understand the universe. However, maybe we will. It is not at all clear to me which it is. What I do know is that we have made, and continue to make, a lot of progress. And although it seems that we are at an impasse, that doesn’t mean we are anywhere close to hitting the wall of what we can learn. There have been many similar impasses in the last century and every one has been followed by a profound new understanding of nature. So, back to your point that there “is the possibility that our own limitations block our ability to fully grasp and formulate our existence.” Yes, that is a possibility, but it is not certain. As long as there is a chance that we can learn more, we should keep trying. 🙂

  • I would be so bold as to call this scalar relativity. The implications are manifold. Perhaps a harmonization of general relativity and quantum mechanics is now within reach …?

  • It would seem to follow if spacetime is an emergent phenoumen0n then mass length would be as well. Maybe the heiarchal Fermi-Planck scale (Standard Model to high energy gap) itself should be treated as a mathematical object.

  • According to Thomas Henry Huxley:
    “Agnosticism is not a creed but a method, the essence of which lies in the vigorous application of a single principle. Positively, the principle may be expressed as in matters of intellect, follow your reason as far as it can take you without other considerations. And negatively, in matters of the intellect, do not pretend that matters are certain that are not demonstrated or demonstrable…..
    That it is wrong for a man to say he is certain of the objective truth of a proposition unless he can provide evidence which logically justifies that certainty. This is what agnosticism asserts and in my opinion, is all that is essential to agnosticism.”

  • Eliminate scale values like mass and length, and you eliminate experiments to measure them. And if you eliminate experiments, what is left of science? Little more than theoretical mathematics—and in this case some very messy math.

    Heck, you can’t even make a ham sandwich without taking into account mass and length. Am I getting this wrong? This sounds like a suicide pact for cosmology and physics, although I image the other sciences could linger on, pretending that mass and length aren’t illusions.

  • @Michael
    How would you expect to measure something immaterial with method that is material? Science is stuck in matter and there in lies the limitation. Theoretical physics exercises the imagination. Mathematics helps us visualize it. Why again are we so attached to measuring everything when we know that existence is infinite and timeless?

  • These steps are not surprising; the fact that only mathematics is left of physics is precisely because physics was never truly physics in the sense of being about empirical reality. The reality that physics describes was from the very beginning a purely theoretical model based on Galileo’s adoption of Euclidean geometry and applying its ideal concepts to real objects. But objects of physics are not at all the same as real objects of experience. A physics based on metaphysics will collapse at some point. What is difference between the multiverse theory and just saying that god made this all. It is a pretty embarrassing situation and will remain so until physics saves itself from superstition and becomes objective and rational for the first time. It should be understood that a working model is not necessary the same as the corresponding reality. Model is an abstraction made for utiliterian purposes, and the universe of physics is such model with no basis in experiential reality. And now they are finding out that even mass or length too may be gone; of course, because a pure theoretical model never has a mas or length to begin with. It is as if you take a movie projected on the screen to be real and then try to study its laws. You soon will find out that there is no reality behind it. The picture offered by physics has no reality behind it. Now they are going to understand this and we all see its signs. From no mass and length or holographic universe, to laws of QM, all point to the fact that nothing lies underneath this picture.

  • Michael, I don’t know if you have heard of the Holographic principle, that we exist simultaneously as a hologram on the cosmic event horizons some 15 BLY away, and in 3-D space with discrete X,Y,Z values. In a hologram, distance does not exist, because everything is everywhere at the same time, only time exists, the time for patterns to change to reflect the new relationships between particles. Think of the lack of distance in this way, imagine a kaleidoscope pattern representing a holographic you, and imagine you have a coworker who you want to visit as also being a kaleidoscope pattern that intersects with thin lines to your coworker who is ten miles away. As you move closer in the 3-D sense, the holographic pattern would change, think of it as showing connecting lines indicating stronger particle interactions. You coworker is right next to you in the hologram, so your travel can only be explained by a change in the amplitude of information, and possibly shape, you share, but not the position, meaning, distance is an illusion from a holographic perspective.

    Mass can be relative, if you are stationary, and you see an astronaut fly by at 99.999999% the speed of light, they would from a relativistic standpoint, appear to weigh far more than you. It could be that you are actually the one flying almost the speed of light and the astronaut is standing still, but to you it still appears that the other guy is the one who weighs so much more, and the astronaut thinks the same thing about you, so mass can be relative in that sense. The trick is to know which equations you want to remove energy and distance in order to simplify and hopefully obtain and answer, and which equations you will need them.

  • Like the physics of ants, our physics can tell us something meaningful only about such things we can, in principle, perceive. But we know that in principle we cannot perceive anything smaller than Planck-size. So for us we must take space-time as granular. But this granularity must be isotropic, which implies that the granules of space-time are in a constant (up to Planck dt) state of deformation, which in turn implies that since there must be a non-zero probability any granule for a time greater than Planck dt may remain static or at least deform periodically such granules will be anisotropic and hence define mass. From this it may be possible to understand dark matter and also an isotropic flow resulting in gravitational phenomenon.

  • When I was a little boy I used to stare at ants and study how organized and sophisticated they were- especially for being so tiny. One day I realized that the ants couldn’t see me; they couldn’t even acknowledge my presence much less imagine my existence. That’s the first time I realized there’s a trap to being a human and having a human-quality brain. We are fundamentally limited in capacity to understand the fully abstract, completely unknown, and universe-scaled giant. I’m not just saying we don’t know what we don’t know. I’m saying that for some observations we have the relative processing capacity of ants. The paradox is that even though we eventually we hit a ceiling in our capacity to grasp or conceptualize things far more profound than ourselves it’s almost impossible to realize it. What scholarly articles and scientific studies that attempt to grasp the entire mechanics of the universe consistently fail to articulate is the possibility that our own limitations block our ability to fully grasp and formulate our existence.

    If you were born 50 years erlier you may had a talk with Schrodinger and if you had persuade him his equation may had never been discovered..
    You are ok with not knowing everything?I dont know where our limits are if they exist but i prefer to die trying to know everything.

  • William Bardeen is the son of John Bardeen – Nobel prize winner & co-inventor of the transistor.
    (I thought Bardeen sounded familiar — so – to Wikipedia!)
    John studied with Eugene Wigner – who “laid the foundation for the theory of symmetries in quantum mechanics.” (Wikipedia again!).
    Circle of life etc.

  • Yet another empty theoretical speculation with no experimental backing. There is absolutely zero evidence for agravity and fully conformal field theories. Quite the contrary, Nature is consistently telling us that scale invariance is broken!

  • This is a wonderful development. Changing perspectives on what the universe is about in this way opens the door to a great deal of what metaphysical thought has explored, and to some of the ideas presented in what had been thought of as mystical fiction. By thinking of these two realms as being directly related in this way, we can better appreciate the magickal formulation of ‘as above, so below’. It’s a major reframing of how we think the world works, and can make it possible for people to accept the possibility of many previously unthinkable thoughts. Thank you.

  • >Phil Fogle says:
    >August 20, 2014 at 1:36 am
    >What is the conservation law that is associated with Scale Symmetry?

    Correct. This is a good question.
    And where can one get more info on these ghost objects? The author did
    a fine job introducing subjects.

  • No wonder god did not want us to eat the apple from the forbidden tree. Every article I read ends up with a “we really don’t know” I relate this article to the one in which “things are everywhere at the same time” and they only come into existence when measured which is the same thing they are saying here. All this thinking is taking away from my sleep. Can you publish something that we know for sure instead of ending always with “we really don’t know”

  • Physics and economics are two disciplines becoming more alike in that much of what they opine is not even wrong.

  • My definition of properties of nature are factors which shape nature in all its appearances endogenously, being generic and not independent, their proportions being fully interelated, continuously changing each other’s properties.
    So I agree that length is not a property of nature. Mass however is considered a property of nature by me.
    Furthermore I am happy to inform you that multiverses exist. Two of my favourites are the Iliad and the Odyssey.

  • Has the energy or temperature at which the phase transition occurs been calculated? What is it?

    Someone asked about the conserved quantity for scale invariance. AFAIK, there is none as the what is invariance under scaling is not something that we can measure. The WiKi:

    “In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves.”

    Ghosts need not be “particles” per say. They could be emergent or second order effects that need not have any actual direct measurable properties. For example, we explain the accelerating expansion of the physical universe with something called “dark energy” and such is treated as if it where a real substance. It need not be that at all. Dark energy would have “a strong negative pressure (acting repulsively)” and not be capable of being diluted or concentrated as the space it creates changes in size. if it where a measurable substance.

  • the notion of that the very small and the very big are somehow linked goes back to earliest human thought: e.g. totem, magic, etc… Taken together these practices give us a concept of Universe as “syntagm” (rather than “system”…which is scale dependent). Roger Penrose recently contributed to this idea with his CCC Theory in which the ultimate “Heat Death” of Universe is mathematically equivalent to the “Big Bang”>

  • Rhadames Fortuna wrote:

    “Physics and economics are two disciplines becoming more alike in that much of what they opine is not even wrong.”

    Yes. And perhaps less like “disciplines” than industries, industries that have high-budget carnival-like marketing departments.

    Only a slight exaggeration to say: “It’s all showbiz.” Step right up.

  • Dear Natalie,

    To me, you may not be the composer but you are the orchestra. Not being a physicist, neither a mathematician, not knowing how to read the music, I could never hope to access its beauty unless you played it for us. This is what your articles are to me. I feel for you the same mixture of admiration and gratefulness that music lovers profess to their favourite orchestras. It is unspeakable admiration, gratefulness, and sympathy, to the extent that you operate a miracle of sorts. Most of your readers will have the background required to stay up-dated reading the original papers, just keeping an eye on a small number of journals, or participating in online discussion groups, attending events, etc., but many will simply rely on accounts like yours to garner information on the latest developments in physics/mathematics outside their own field (s). These guys will surely apreciate the quality of your reports, but can never feel the same as we, mere sightseers, that could never hope to enjoy the views unless you took us there and told us what there is to see, as we are blind to it. Therefore, this is the miracle you operate: you enable the blind to see, and when this happens the blind just cries, speechless.

    The first article of yours I read was that on the amplituhedron, I think, at which I marveled. I have only read very few of them, not more than three or four I would guess. I like to read about physics, which fascinates me. Mathematics as well. Other authors like you have always allowed me to satisfy my curiosity. I am a biologist but feel really privileged to have received reasonable level classes of physics and maths, which made me about just sufficiently familiar with the language and nearly able to follow, guess at things, or conjure up low contrast visions of them. A bit like a blind person might perceive things, blurred and shades of gray, through some brain implant linked to a camera.

    Your work is unvaluable to those like myself, lay but wishing to steal a glimpse of the vicinity of our field of vision, of what lies just behind our hedge. Thank you so much Natalie!

  • @Natalie Wolchover, Whew!… I’m glad that @Artur_Sixto went over the deep-end with his welcome (not to mention well-‘penned’) fit of honorific praises, already before me, so I don’t have to say anything more than “My thoughts exactly!” 😉

    I’ve only ran through a small handful of your articles, (Natalie,) and all here, but I’m compelled to state again that your writing has a special lucidity and centering ‘soul’ to it, and I make no exaggeration in saying that I’m familiar with_N0_other author (in your field), out there today, that can do all the things that you can do in composing these (really!) incredible articles—BOTH in terms of scientific competency and that they’re a joy to read. The first ‘something similar’ that pops up in my head is Sagan(!), so mine is high praise indeed!

    To illustrate: I had read dozens of popular-level physics books, plus a couple years going beyond that in my own self-directed furthering education and amateur participation in online communities dedicated to scholarly discourse. Yet, somehow, through all that, I never quite “got” the hierarchy problem and it’s relation to the higgs particle, and what exactly was so “suspicious” to physicist about the ‘coincidental’ particle masses, etc. Well one ‘article-read later’ and suddenly everything I’d heard throughout years ‘clicks’ into place, and all is clear! It’s a kind of thing that you ‘know’ you’ll never forget, just because, in the final synthesis, the end product I so intuitive…. So whatever magic you sow into your articles is clearly doing the job!

    Keep up the great work, and Thanks!

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