Most recent update: February 2018
A working draft for the scientific – academic communities. Or a shorter version for the general public.
Stephen Hawking rhetorically asked, “Where did the universe come from?” He continued, “The answer, as most people can tell you, is the big bang. Everything in existence, expanding exponentially in every direction, from an infinitely small, infinitely hot, infinitely dense point, creating a cosmos filled with energy and matter. But what does that really mean and where did it all begin?” Ref.  May 2016 PBS-TV series, Genius
Within this post, we focus on a very simple model and nascent theory that has only been explored by a relatively small group of high school people and others within our extended community. We ask, “Is it possible that the universe began with an infinitesimally small length and time, and a relatively small charge and mass? Yes, we use those numbers defined by Max Planck in 1899, the Planck base units, to begin. In this model the universe is most simple and the opposite of infinitely hot and infinitely dense. Also, this structure becomes part of the ongoing structure of the universe. It supports the claim of Neil Turok and his colleagues that this universe is in a state of perpetually starting.
Our simple journey started in December 2011. In 2012 we began asking more-pointed questions about the first 67 of 202 notations. In 2013 we began our rather informal studies of the big bang theory. In 2014 we began to question it. Today, we are hoping the experts can tell us why using the powers of 2, doublings from the Planck scale to the Age of the Universe is not a proper outline for a model and theory. Having learned how idiosyncratic it is, we have many questions.
Might the mechanism for the doubling of a cell be a rather limited metaphor for the expansion of the universe? Could our universe be functionally based on the simplest mathematics, doubling both in size and in number up to this current notation and each step of the way? Is our universe, in fact, exponential (as in Euler’s equation), highly-ordered, and totally relational?
First principles. We postulate that the Planck scale is the unification of the four forces of nature with the unification of the Planck base units with those constants that define each unit, and that this unification, all defined as working ratios, is uniquely differentiated within each doubling throughout the entire 200+ steps from the first moment of creation to this moment, the current time and present day. It appears that all 202± steps are dynamic, actively participating in the current time and this definition of our universe. This postulation provides a working environment by which we hope to build a diversity of bridges from the Planck scale to all existing physical theories whereby each notation creates a very unique environment for predictive values.
We call this model the Quiet Expansion of the Universe, hereinafter, abbreviated QE.
We begin this study with the Planck Epoch, then attempt to justify reinterpreting the Grand Unification and Inflationary Epochs. The Electroweak Epoch begins the crossover which continues through the Quark Epoch and into the Hadron, Lepton and Electron Epochs. Thereafter, the QE will have so many bridges up from the Planck Epoch, it should become an expressway to the remaining epochs and all the useful definitions developed over the years within the big bang theory (hereinafter, abbreviated bbt).
The Planck base units. Further defined by the speed of light (or special relativity), the gravitational constant (or general relativity), the reduced Planck constant (or ħ or quantum mechanics), the Coulomb constant (or ε0 or electric charge or electromagnetism), and the Boltzmann constant (or kB or of temperature), all are bound within this Planck scale; and, herein it is proposed to be the foundations for a highly-ordered, totally-relational universe. The key to our model is multiplication by 2, starting with the Planck base units. A nexus of transformation between the finite and the infinite is defined by the crossing lines at “0” within the images on the right (just above Frank Wilczek).
We still have many questions. We have hopes and dreams. If the QE numbers can withstand the scrutiny of the academic and scientific communities, and we can begin to grasp the finite nature of space and time, and we can open a larger discussion about the nature of the finite-infinite relation, just maybe the bbt will recede and take a new role as an important chapter in academic as well as human history.
Our hope is that this posting will be able to stand the test of time; however, we full acknowledge that it is a “very rough first draft.” Given the depth and breadth of the foundations upon which the big bang theory (bbt) currently rests, your comments while this posting is being refined, are most welcomed. If this embedded link does not open your email browser, my address is camber (at) 81018 (dot) com (or click on Contact (always within the navigation bar at the top).
Those pivotal Planck calculations were done between 1899 and 1905 by Max Planck. In December 2011 we were just beginning to learn about Planck and his calculations. We sought out experts and quickly found the work of Prof. Dr. Frank Wilczek (MIT).
Frank Wilczek. With very few exceptions, it was not until Wilczek began writing a series of articles in 2001, Scaling Mt. Planck, (Physics Today), did anybody think those Planck numbers amounted to anything more than numerology. It would take another ten years before we would come along, naively doing our thing with our geometries and simple doublings.
Though most academics are familiar with Kees Boeke’s 1957 work (Cosmic View) using base-10, we were not. Most all our academic contacts made quick reference to it, yet they were still surprised to see our chart from the Planck Length to the Observable Universe. Some asked, “Why haven’t we’ve seen this before now?” The others just thought it was more quantum mysticism and numerology akin to Paul Dirac’s work with large numbers (link goes to a YouTube audio where Dirac explains it in his own words).
This simple work of multiplying the Planck units by 2, and then each result by 2, over and over and over again is a bit tedious. If you were to do it, in just over 200 steps you would emerge at the Age of the Universe and the Observable Universe. You can follow the progression in any one of several charts. A simple doubling is what cells do. Other processes like chemical bonding and bifurcation theory have analogous dynamics.
These 200+ doublings have at various points been called: (1) archetypes, (2) clusters, (3) containers, (4) domains, (5) groups, (6) layers, (7) notations, (8) ratios, (9) sets or (10) steps. We believe that each captures a face of the functionality within the notation. We recognize that these Planck base units can be computed in many different ways. Eventually, in order to refine results, the reduced Planck constant may be used. The various values of gravity (G) can be tested. Important at this time is consistency and equivalence of methodologies across all calculations within all 200+ notations. Our initial goal is to create a simple working model that outlines the general working parameters and boundary conditions to give us a platform. Now we begin looking at the key critical ratios throughout the model with a hope that we may discern natural groups and patterns that might help us to judge the veracity of the model itself.
The web page, Big Board-little universe, provides more background of our rather brief history.
Big-bang Theory Drowns Out Discussions
To learn as much as possible as quickly as possible, we’ve used Wikipedia’s summaries. Wikipedia’s goal is to represent the best current thinking of the thought leaders within the relevant scientific communities. The scientists who are most often quoted have lived within this theory throughout their professional careers. It is part of their intellectual being. Notwithstanding, we believe most all of their work can be absorbed within the QE. Questions are primarily raised about the Planck Epoch, the Grand Unification Epoch, the Inflationary Epoch and the Electroweak Epochs. Taken together, the first three “epochs” represent less than a fraction of a fraction of a second within the QE model. And, with just a few tweaks, we believe some of this work and all the work within the subsequent epochs can be readily integrated.
The writers within the Wikipedia community overlap with those within these scientific communities. Wikipedia, constantly in the process of refining their writing, provides several summaries of the History of the Universe. All work based on observations and measurements has a place within the QE model. Our guess is that the interpretation of those observations and measurements will become richer and more informative when the QE parameters and boundary conditions are engaged.
In 1970 there were truly competing theories about the beginning of the universe. By 1990 the bbt had become dominant. In 2011 our little group of high school geometry people began to explore the interior structures of the tetrahedron and octahedron. Then we found within our tilings and tessellations, and then all the simple doublings from the Planck base units to the Age of the Universe and to the Observable Universe. That continuum appeared so simple, we first engaged it as an excellent STEM (Science-Technology-Engineering-Mathematics) tool. Yet, with further study and thought, it also seemed to challenge some of our basic commonsense assumptions about nature (the back story). As we studied our new little model, the bbt continued to solidify its dominance within the general culture at the same time we started to question it. We began to believe that the actual physics of the first moments of creation might be better defined by the simple mathematics of a quiet expansion, especially those first 67 notations. These 67 have never been recognized as such and certainly have not been discussed within academia. The great minds throughout the ages have been unaware of the 202 doublings, especially those first 67 steps. So mysterious are the 67, we began more actively to think about them and to make some postulations about their place and purpose.
This is our first posting about this quiet expansion. It is a result of our naive, informal, and often idiosyncratic studies of the Planck base units, simple doublings, and an inherent geometry assumed (hypothesized, hypostatized, and/or imputed) to be within every scale (doubling, layer, notation, step, etc) throughout the universe. We have moved slowly. Having backed into the Planck base units from our simple exercises in geometry class, we were not at all sure of ourselves. So, after observing our results for a couple of years, we began asking the question, “Could this be a more-simple, more-inclusive model of the universe than the big bang theory?” Because we only have the beginnings of an outline of a model, we continued our quest and continued to ask more questions:
Who: The history of the Big Bang Theory (bbt) is highly documented. It is an intellectual cornerstone within experimental and theoretical physics, cosmology, and astrophysics.
What: To challenge the bbt appears foolhardy at best. Yet, there are many, many reasons to challenge it, but most of all because (1) it is overly complex and confusing, (2) it is not very good philosophy, and (3) it is very poor psychology.
Why: The first three key parts of the bbt, involving substantially less than a trillionth of a second, are based on hunches and a need to shoehorn data to support the model.
We say that the Planck scale is the starting point for the initial six notations (de facto defined by the bbt) and that these notations are shared by everything, everywhere in the universe. Painfully aware of the limitations of our vocabulary, these first notations are considered to be archetypal forms, structure and substance. Archetypal is used in the sense of the original pattern or model by which all things of the same type are representations, the prototype, or a perfect example. For more, see all of 67 encapsulating notations (opens in a new window or tab).
This “Singularity” Is a Meeting Place of Converging Formulae, perhaps also known as a Modulus-or-Nexus of Transformation
Keys to this Quiet Expansion: More than the big bang theory‘s four forces of nature — gravity, electromagnetism, the strong force, and the weak force — within this Planck scale we assume these four are encapsulated within all four Planck base units and the constants that define them, and some manifestation of this unification is carried throughout all 202 notations. And, as we have noted, the Planck base units are defined by length, time, mass, and charge; and, these are further defined by the speed of light (or special relativity), the gravitational constant (or general relativity), the reduced Planck constant (or ħ or quantum mechanics), the Coulomb constant (or ε0 or electric charge or electromagnetism), and the Boltzmann constant (or kB or of temperature).
The Planck scale is not beyond logic, numbers, and conceptual integrity. Homogeneity, isotropy and simple logic rule. Yet, within the Quiet Expansion (QE) model, we have applied that simple logic somewhat arbitrarily by placing Planck Temperature at the top of the scale, just beyond the 202nd notation and then dividing by 2, it goes down approaching Absolute Zero. We are ready to adjust it at any time when a more integrative logic prevails! Also, we are increasingly finding a simple relational logic between the four original Planck base units. Notwithstanding, this logic will be constantly revisited throughout our ever-so-slow development of QE model.
Within the QE model, the Planck Charge, a Coulombs value, is taken as given. Within the big bang theory (bbt), the Planck Charge is ignored. The bbt value is as large as possible. Their measurement is given in GeV units, one billion electron volts. Add 1016 zeroes and you have quite a charge.
To begin to understand all these numbers and their correlations, questions are asked, “Are these all non-repeating, never-ending numbers like Pi? Are all numbers that are non-repeating and never-ending somehow part of the infinite yet also the beginning of quantum mechanics?” The suggestion has been made that we carry out each of the Planck numbers at least 10 decimal places, and if need be, 100 decimal places, and possibly even 1000 decimal places, to see if patterns can be discerned. We recognize that relative to other units of measurement, such as the SI base quantities, the values of the Planck units are approximations mostly due to uncertainty in the value of the gravitational constant (G).
The QE model holds that things are simple before complex and everything is related to everything. Imputed, hypostatized and/or hypothesized are pointfree vertices and simple geometries as the deep infrastructure that gives rise to the work on combinatorics, cellular automaton, cubic close packing, bifurcation theory (and the Feigenbaum’s constants), Langlands program, mereotopology (point-free geometry), the 80-known binary operations, and scalar field theory. Here are people working on theories and constructions of the simple, yet their concepts are anything but simple.
The exacting nature of the correlations between the multiples of the Planck base units is just being explored for the first time. But, to say the least, within the QE everything everywhere is related through simple mathematics.
When: In the very beginning…
Wikipedia says that the Planck epoch requires speculative proposals, a “New Physics” such as “…the Hartle–Hawking initial state, string landscape, string gas cosmology, and the ekpyrotic universe.” Each is a conceptually-rich, dense jungle of ideas. Cutting through that entanglement is only for the highly-motivated and academically astute. Most of us will just go on to the grand unification epoch, in search of a logical system that builds consistently upon itself.
About the bbt model, Wikipedia simply says, “The three forces of the Standard Model are unified.” Of course, the QE goes much further, however, first consider a bbt problem. Electromagnetism, gravitation, weak nuclear interaction, and strong nuclear interaction are most often related to relations defined above the 65th notation.
Wikipedia says, “Cosmic inflation expands space by a factor of the order of 1026 over a time of the order of 10−33 to 10−32 seconds. The universe is supercooled from about 1027 down to 1022 kelvins. The Strong Nuclear Force becomes distinct from the Electroweak Force.”  (Our emphasis) First, consider that the Planck Temperature is 1.41683×1032 Kelvin. The bbt appears to skip the cooling from 1032 to 1027 Kelvin and it uses bubbly magic to address what causes the cooling to 1022 Kelvin. Also, consider the amount of expansion and the short duration assumed in their statement above. To create that much space in that short of an interval would require light to travel so far beyond its normal speed; it would constitute the penultimate anomaly (pardon us, Sean Carroll fans).
Also, because the bbt begins at the Planck Temperature, they truly need a supercooled concept. Within the Quiet Expansion model the temperatures from notations 1 through 102 are all superconducting, being well below the superconducting transition temperatures. Perhaps the very concept of temperature will become better understood as a result of our struggles to define a different model of the universe.
About this inflationary epoch, Wikipedia says, “The forces of the Standard Model have separated, but energies are too high for quarks to coalesce into hadrons, instead forming a quark-gluon plasma. These are the highest energies directly observable in experiment in the Large Hadron Collider.”
Within the QE, the quark-gluon plasma which requires 1012 Kelvin, is between notation 135 and 136, 9.6008×1011 Kelvin to 1.92016×1012 Kelvin respectively. Notation 136 is 4.6965×10-3 seconds from their space-time “singularity.” One second is between Notations 143 and 144. Also, the Kelvin scale is counter-intuitive in many ways. The temperature of the Sun is about 5,778 K. Within the QE, that is expressed between Notations 107 (3.5765×103 K ) and 108 (7153.178 K). The human temperature at 98.6 degrees Fahrenheit is 310.15 Kelvin which is between Notations 103 and 104 (447.073 K). Also, at Notation 103 the Planck Length is now .163902142 millimeters or 1.63902142×10-4 meters or about the size of a human egg.
In Wikipedia, their category experts say, “The physics of the electroweak epoch is less speculative and much better understood than the physics of previous periods of the early universe. The existence of W and Z bosons has been demonstrated, and other predictions of electroweak theory have been experimentally verified.”
Finally the the bbt gives us something that isn’t incomplete or highly speculative. Yet, even with such assurance, the logic of the bbt is difficult to follow. Again, within the QE model the only duration that would allow for W and Z bosons is somewhere around notation 65. There is just not enough “conceptual” space and time for elementary particles and their effects.
Within QE progression, the first measurement with a visceral meaning is at Notation 31; the mass of the universe is 46.79 kilograms or about 103 pounds. By Notation 40 it is up to 2.39×104 kilograms (52,758.8 lbs or 27 tons. The universe is bulking up quickly and it is creating space and time as it goes. Though we have some ideas about this mass, it should become more clear as we begin experimenting with the calculations of the Planck base units. Beyond SI: We may also extend all the decimals out to at least ten places and begin to calculate more carefully each ratio within each notation to do in-depth ratio analyses of each progression.
Consider this unusual concept. Within every notation, the QE model aggregates base-8 pointfree vertices using scaling laws and dimensional analysis (recommended by Prof. Dr. Freeman Dyson). Base-2 and base-8 progressions are single line entries within the horizontally-scrolled chart.
There are 10,633,823,966,279,326,983,230,456,482,242,756,608 pointfree vertices at the 41st notation. These simple doublings could be aggregating structure as groups or sets. Defined by the Planck base units, in the range 41-to-60, we hypothesize that these domains are archetypal relations and systems: 549,755,813,888 base-2 pointfree vertices (Notation 41) and 5,070,602,400,912,917,605,986,812,821,504 (Notation 104).
The possibilities for mathematical constructions. Here ratios manifest as the real reality of the universe. The entitive nature of things (above the 67th notations) is derivative; the ratio is primarily real. In our world of subject-object thinking, the hyphen represents that ratio.
Here is the deep infrastructure of the universe where the simple mathematics of ratios between space, time, charge, mass and temperature create real realities within every notation. We postulate that these ratios are the “really real.” Within the continuum of charge here is the so-called dark energy within notations 185 to 200 and with the continuum of mass there is the dark matter. If this model is ever validated, perhaps it’ll be seen that both are deep energy and deep matter of the universe, the manifestations of really real mathematical ratios.
The bbt’s Quark Epoch generalizes 63 of the QE notations, from 41 to 104. These notations within the QE model are foundational so perhaps this comparison to Quark Epoch is a key. Consider the estimated requirement for temperature. The bbt epochs can not begin until the temperature is cool enough. Given that temperature requirement, within the QE model, the Quark Epoch would not begin until up-and-around Notation 136 where the temperature has finally risen to 1.9201×1012 Kelvin. If that is the right range, as suggested by proponents of the bbt, less than a second has transpired, the universe has a diameter of about 874 square miles and a mass of about 1.896×1032 kilograms.
Within the QE model from around Notations 65 to 69 is the transition from the small scale to the human scale. This “human scale” is the middle third of the 201 notations, i.e. 67-to-134. Even though two-thirds of the way through the 201 doublings, less than a second has transpired from the start.
In the Quark Epoch the bbt and QE begin to cross paths and overlap. Wikipedia says, “Quarks are bound into hadrons. Over the hadron epoch, the process of baryogenesis results in an elimination of anti-hadrons (baryon asymmetry).” As noted within Wikipedia, some of these perceptions come directly out of the laboratory, such as CERN in Geneva, where this phenomenon has been observed. So, other than the improbable placement within the time/temperature curve, all processes herein after become readily integrated within the QE model.
Let us take stock of where we are. Even though the Quark Epoch of the bbt seems to overlap and begin to become simpatico within the QE, there are fundamental logic and conceptual problems ahead.
A key question within the QE model is, “What is a notation?” All 200+ are also known as an archetype, cluster, doubling, group, layer, set, and/or step. Each word is perspectival. Each notation is dynamic, always in the process of being defined, right up to the current time within our current notation. Space and time are local per notation and all “past” is an imprint on the universe that literally defines it beingness right now, thus no time asymmetry.
What does that mean? Each notation has an active role right now in defining who we are and what this universe is here and now. Each notation has an active role in defining all other notations.
Today, right now, all of these notations are actively defining the now. We are imprinting on the universe right now. The past is not past; it is an imprint on the universe. There is only the Now, only right now, only today.
Each notation has an active role in defining who we are and what this universe is; and, each notation has an active role in defining all other notations. Today, right now, all of these notations actively define humanity or the human scale (67-to-134), must therefore be something like the archetypes of forms and functions (notations 1-to-67) that define our deeper beingness. The notations from 134-to-200 define our planetary and galactic systems and this is where most of the work of those physicists, cosmologists, and astrophysicists have worked.
In just a few more notations, between 142 and 143, the universe is at the one second mark. This measurement is most often used to determine the speed of light. Yet, as noted in earlier postings, within every notation, the Planck length divided by the multiple of the Planck Time renders an approximation of the speed of light. It is just commonsense when we see that the speed of light plays prominently in the definitions of Planck Length and Planck Time.
The question to be answered, “What is the meaning of temperature? …within the bbt? Within the QE model, we impute that it is the total temperature throughout the area defined by the notation (or cluster, container, domain, doubling, group, layer, or step). This measurement within the Hadron Epoch within the bbt is lower than it is within the QE. There is a natural correlation between all these numbers within the QE simply because they start with the same definitional characteristics (the Planck base units) and are the evolution of those numbers. The ratio of length to temperature renders 7.3322+ ratio. [Editor’s note: Please double-check this figure. Then check it again.] That result is currently being analyzed, space-to-temperature or kelvin per meters.
In 1972 George Ellis and Stephen Hawking began to explore the boundary conditions that define our universe between 10-13 centimeters (elementary particles) and 1028 cm, the assumed radius of the universe. They did not approach the Planck base units which would have expanded their range to 1.616199×10−35 meters (Planck Length) and then it would have tucked them in at about 5.1942×1025 meters according to current best guesses regarding the Age of the Universe.
Earlier it was observed that the big bang is not good philosophy and it is bad psychology. Philosophy is taken as a study of first principles and systems, the universals and constants that create the boundary conditions as well as the continuity equations that bind our universe together. Since 1972, especially with the very key question about the very nature of the first microseconds, the bbt has not progressed very far. Their Planck epoch is still mysterious. It is bad psychology for that very reason. It is so disjointed, so out of touch with anything human, it de facto encourages various forms of nihilism.
Theories should have elegance, beauty, coherence, and simplicity. Children should be able to begin to understand. And with the QE, children quickly begin to understand 2 times 2. We just have to carry it out a few more places for them.
Conclusions: What does it all mean?
What are the implications if the Quiet Expansion is true?
For us all:
1. The finite-infinite relation is the key and requires more study.
2. The universe is finite, quantized, and derivative. Space and time are also finite, quantized, and derivative.
3. The infinite is continuity, symmetry and harmony giving rise to order, relations and harmony.
4. There are, therefore, natural laws, ethics, and values.
For the big bang theory:
1. All the actual measurements and observational work that have gone into the big bang theory (bbt) are supported by the Quiet Expansion.
2. All the major theoretical constructs of the bbt including and after the Quark Epoch are supported by the Quiet Expansion. There are adjustments of the time scale in which things occur, yet these are minor.
For the Quiet Expansion (QE):
1. The continued expansion of the universe is fully supported within the Quiet Expansion.
2. There are just over 200 notations that define the universe.
3. These notations are all active, functional, and necessarily build on each other.
4. These notations will also be defined as:
5. As an archetype, each notation serves specific purposes in defining the textures and substance of the universe determined by the ranges within the Planck base units.
The future, both short-term and long-term: Our “To Do” List.
1. Run the ratios: There is a meaningful ratio between each of the five Planck base units within each of the notations. Volunteers? Want to help?
2. Double-check the numbers. Just yesterday there was a question about the Coulombs doublings. The simple mathematics of every doubling has to be correct.
3. Study the progression of Planck Mass from 0 to 202. We will be trying to intuit the meaning of mass given by just the pointfree vertices throughout the twenty notations from 31 to 70. Formulate key critical questions? What are these ratios saying?
Disclaimer: Our charts and discussion are our first time to make a comparative analysis between the big bang theory and our Quiet Expansion. Silly errors are inevitable. We are neophytes, not scholars, within these fields, so please point out any of our failures with logic, math, and physics. We will be most grateful.
This ends the first story about two very different models of the universe. Of course, it is a story that is to be continued.
Footnotes and endnotes:
The first working title of this posting was “Can A Quiet Expansion Challenge the Big Bang?” which was deemed too confrontational. The more important question was, “How did it all begin and what does it mean?” That change was made on Friday morning, June 17, 2016, then later, it was changed again to “Quiet Expansion of the Universe.” And just for fun, http://QuietExpansion.com, that web-URL was initiated (comes right back to this page).
Reference 1 Big bang theory: The world-renown Cambridge University physicist, Stephen Hawking, was the leading spokesperson for the big bang. He became a rock star among scientists because he was been so successful as its primary advocate. Within his May 2016 PBS-TV series, Genius, he asks, “Where did the universe come from? The answer, as most people can tell you, is the big bang. Everything in existence, expanding exponentially in every direction,from an infinitely small, infinitely hot, infinitely dense point, creating a cosmos filled with energy and matter. But what does that really mean and where did it all begin?” His confidence also exudes from his 1988, best-selling book, A Brief History of Time: From the Big Bang to Black Holes, and even from his foundational writing in 1973 (co-authored with Cambridge colleague, George F. R. Ellis) the highly-technical book, The Large Scale Structure of Space-Time.
Are space-and-time unbounded or bounded? If bounded, is our universe a container universe? Are the Planck base units and all the dimensionless constants part of the definitions of the boundaries between the finite and the infinite?
Within the current bbt analysis gravitational waves arise from within their inflationary period. The bbt thought leaders ascribe a much faster-than-light expansion just after the big bang. And, that begs the question: What are the preconditions of superluminal events and motion? There haven’t been any answers since 1902 when Jacobus Kapteyn made his initial observations, since the 1983 “superluminal workshop” at Jodrell Bank Observatory, and since the subsequent studies of microquasars, their accretion disks and such phenomenon as magnetorotational instability. It is all a very special language, logic and reality; the observational results are well-defined; yet, the most-penetrating conclusions are pending.