Validating that which cannot be directly measured

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“Prove it to me!”
by Bruce E. Camber
a working draft

Background. In December 2021 a friend asked me, ‘How can we begin to validate some part of this theory?’ It is a tough question and one that speculative people throughout time have asked about new concepts. We have a special problem because the “things” of our first 64 of 202 base-2 notations are below the thresholds of measurement.* We all really need to understand the “things” of those 64 notations out of the 202 notations that literally encapsulate everything, everywhere for all time. -BEC

Numbers, Geometries, Equations. Unlike so many theories, this one — our most-simple, base-2 mathematics from Planck’s base units to the age and size of the universe today* — is all numbers, geometries, and equations.

There are well over a thousand numbers to explore. All are generated from that base-2 equation. More importantly, it all started with geometries; and, those geometries stay with us no matter how small or large our numbers become. Also, we’ve had the landmark calculations of Max Planck and George Johnstone Stoney. Although Max Planck’s numbers are disputed by some scholars (John Ralston, 2012), Stoney’s numbers provide a comparative analysis. A few of us consider such numbers to be a symbolic description of the first instant of the universe. The most critical consider those extrapolated numbers — the 202 notations from Planck Time to the current time — to be “just numbers.”

For this article, we simply ask you defer judgment in order to explore this model further.

I think these continuity equations generate quite a bit more than just numbers. These are living numbers. This extrapolation of numbers from the first instant to this day so closely approximates actual measurements, it loudly calls for more scrutiny of this grid.

The Speed of Light. From the infinitesimals to the largest possible numbers, one of the best numbers to study is the currently-accepted value for the speed of light: Particularly, I point to Notations 143-and-144:

On February 1, 2022 I had a wonderful exchange with Viktor Toth, a Max Planck scholar. He is a frequent co-author with John Moffat (Perimeter Institute for Theoretical Physics) and others. I had written right here within an earlier draft of this paragraph, “In 1899 when Max Planck was calculating his base units, the speed of light had not been determined strictly through mathematics.” I just didn’t dig deeply enough. Toth gave me the precise references for Max’s calculations!** Max used his base units to divide length by time to render the speed of light at 300,000 meters/second. I had found references to Planck Length — 1.616255×10-35 meters — which when divided by Planck’s Time — 5.391247×10-44 seconds — renders 299,792.4228 kilometers per second. The provenance of those calculations is not yet clear. Notwithstanding, whether 300,000 or 299,792, the results are close enough and worth pondering. Planck Length divided by Planck Time: Is it always equal to the speed of light? Showing up within Notation-143 right about at the distance light travels in a second, is that meaningful? Does that base-2 chart truly encapsulate the universe within 202 doublings of those Planck units?

In that light, I ask you, “Is this a validation that the chart has some cogency?” I think so.

Earlier work, other references within this area:
Speed of light 1:
Speed of light 2:
Just a second:

Variable Speed of Light (VSL). [1] There is a line of numbers within our chart, each a multiple of Planck Length divided its multiple of Planck Time. Yes, line 10 is the value of the Planck Length within a given notation divided by its corresponding multiple of Planck Time. As one might anticipate, the results are a variable-speed of light (VSL) [1]. It was a new study for us, and it had an acronym, VSL, and it had scholars — a rather substantial group — who had concluded that the speed of light is not constant. To our knowledge, our line 10 is the first time this concept of a variable had been defined as a simple function of mathematics.

Could it really be the first mathematical confirmation of an hypothesis of others? As a result, the works of scholars advocating VSL are of interest to us. This could become a significant validation of this base-2 model; it appears to be heading in that direction.

Please note: It seems that the top-level numbers within Notation-202 approximate numbers currently determined by actual measurements by our academic and scientific communities of today’s existing realities. Such coincidences are also perfunctory validations. Yet, I’ll be the first to admit, that as these numbers within this chart are more carefully studied, adjustments will be necessary. I would not be surprised to see shifts of an entire column, left or right!

Finite-Infinite. Identified within the footnotes of several prior articles are three master keys to the big questions about our universe. The first is a Finite-Infinite key. The second key is for the Integrative Systems that Structure the Universe. The third is the Redefinitions of Space-Time (Mass-Charge) key. The longest running of the three is the finite-infinite key. You can well-imagine that it goes back throughout most of our human history.

Since around 1974, I have been following the work of the John Templeton Foundation. [2] The founder of that foundation, one of our world’s early billionaires, knew that religion could inform science about the universal, and that science could inform religion with their universals, and that the two groups rarely truly listened to each other. I shared that sentiment profoundly and still do.

The Templeton Foundation helped to start the Foundational Questions Institute (FQXi) [3] with physicists, Max Tegmark and Anthony Aguirre. Those two are very sophisticated scholars working at the edges of scientific inquiry. FQXi has supported many fine scholars and any one of them could help us clarify our issues and craft ways to test our model. Of several hundred scholars, at the time of this writing, I had asked over 60 of them for a critical review of our base-2 model. I recognize our presuppositions are a stretch and I appreciate why most are quite cautious with their responses.

Notwithstanding, I will continue to engage FQXi (my 2021 paper for them) and their scholars regarding the finite-infinite relation. Can it be distilled to a simple discussion of those aspects of pi that are infinite? Going forward, that will be my focus with the FQXi people, i.e. an email to Tim Palmer of Oxford.

Redefinitions of Space-Time: The first three seconds. [4] I am hoping that we might slowly build working relations with a number of scholars who have made the infinitesimally small their primary focus. Among them are the scholars who have been analyzing the first three seconds of the universe. All 27 contributing scholars of the article, The First Three Seconds, were asked:
• Is there a mathematical continuum from the Planck base units to the current time and size of the universe?
• Is there any possibility that our universe is exponential and that our base-2 chart describes a most-basic manifestation of exponentiality in our universe?
• Is an infinitesimal sphere the first manifestation of the Planck units (and light, pi and other probable dimensionless constants)?
• Is pi one of our earliest examples of continuity (never-ending, never repeating numbers), symmetry, and harmony within the physical universe?
• Is this hypostatization a key finite-infinite relation?
• Is sphere stacking a fundamental action of the universe starting with the Planck base units?
• Is cubic-close packing of equal spheres a fundamental action for the expansion of the universe? Does it open up the Fourier transform and Euclidean geometries?

I think such questions are worth asking. As I continue to read their most current work, I will engage these scholars further.

Physics of Quantum Electronics. [5] There are other efforts that are equally ambitious. The scholars within this most-select, highly-creative scientific group, Physics of Quantum Electronics (PQE), have been writing and presenting papers since 1971. They are motivated to reduce quantum theory to practice, that is, practical applications and devices. To do so, they take chances with new ideas. We will continue to ask questions of these creatives to engage our model because it should open many new ways the quantum world can be reduced to practice. We have redefined that world; it only goes as small as its quantum fluctuations and then there are domains of perfection where quantum physics no longer abides. It is such a radical stretch, most will shut us down. For those who know we have been going in circles for well-over 100 years searching for a new paradigm, I say, “Thank you and welcome. There has got to be what we might call technologies of perfection that will demonstrate a reality of this hypostatic domain.”

Christodoulou, Di Biagio & Martin-Dussaud. [6] The infinitesimal (or smallest-scale universe) cannot be measured directly, yet there are scholars who are working to validate its presence by measuring its direct effects and possible affects. In 2019 I enjoyed discovering the work of post-docs, Andrea Di Biagio (La Sapienza), Marios Christodoulou (Oxford), Pierre Martin-Dussaud (Aix-Marseille Univ., Université de Toulon). They are trying to formulate an experimental environment to determine time sequences as short as the PlanckSecond. One of their goals is to determine if time is discrete-and-digital or continuous-and-qualitative.

We will continue to encourage their work. I’ve suggested to them that they need to add a third category: both digital-and-continuous. We will attempt to help as they inch closer to a reduction in practice to measure the unmeasurable.

Institute for Particle Physics Phenomenology (IPPP). [7] The IPPP of Durham University is entirely focused on the range from the Planck scale to the electroweak scale. Their leadership has not yet engaged the logic-and-mathematics of our first 64 (out of 202 notations); it’s still open for discussion. All 64 notations are well below the thresholds of measurement. It would be good to have one or two within this group to study our model. Perhaps the prior homepage, The First Particle, could be dressed up to become an invited paper for their 2022 annual conference.

Routledge Handbook of Emergence [8]. Also at Durham University, three faculty members, Sophie Gibb, Robin Findlay Hendry, and Tom Lancaster, edited a landmark book, Routledge Handbook of Emergence, as part of the Durham Emergence Project. Many other people have also been involved; among them scholars from the Durham University of Institute for Advanced Studies. Many at Durham University have struggled for years to clarify the unsolved mysteries and tensions within science. All those involved might consider our novel approach to be of interest.

Independent scholars around the world.[9] There are over 10,000 thoughtful people listed by Jean de Climont in his 2022 publication, The Worldwide List of Alternative Theories and Critics (PDF). The editor, Jean de Climont, listed himself (page 475) — he’s written many books that explore the edges of physics. Our work has been referenced on page 311. With over 10,000 people listed, there are mistakes, yet de Climont updates pages very quickly. Most of the people listed have poked holes into the fabric of our most commonly-accepted scientific concepts and theories. Sorting it all out is time-consuming yet it is a place to check to see what is said about a scholar whose work is intriguing. When given, the contact information listed is generally accurate. It is also a good place to learn names of others asking similar questions. There are, for example, 81 people listed who investigate the cosmological constant. There are 71 listings that use the concept, perfect. There are twelve who use the word, continuity. One-hundred use the word, symmetry. Nine use the word, harmony. For me, each is a reference of interest.

There are many more facets to add. [10] We all seem to be working in the same direction. We’re in search of some fundamental truth that will make all our disparities begin to cohere, plus cut down on the number of arguments and lay foundations for new constructions. We have identified eight such disciplines that can create new bridges and close some gaps. Each could spark a revolution in the way we think about things.

Shell particle. There are groups of people working to define a shell particle that could help us understand our many hypothetical particles. A better definition could spark a revolution in the way we think about things.

Spin. There are groups of people attempting to identify the nature of spin, both particle spin and that spin defined by the Fourier transform. The very nature and reason for particle spin is a mystery for those who study it. The spin within the Fourier transform is perhaps more settled. The relations between the two have begun to be explored.

Cosmological constant. If Planck Time gives us a de facto rate of expansion of the universe — that is one Planck sphere per Planck unit of time — it is a simple computation, 539 tredecillion spheres per second. Somebody will figure out how to test that rate within our time. Given Ralston‘s misgivings about the Planck Constant, as a symbolic gesture, perhaps we should include Stoney’s number and suggest a symbolic range whereby between 539-to-4605.4 tredecillion infinitesimal spheres — first particles — are generated every second and that rate defines the look-and-feel of the expansion of the universe.

I believe that such a rate of expansion may eventually be considered a cosmological constant.

Perfected States Within Space-Time. There is a thrust and compactification, whereby what manifests within the earliest notations reflect what defines a perfected-state within space-time. It is hypothesized that quantum indeterminacy first begins to manifest with a five-tetrahedral structure sharing a common edge. Hold five tetrahedrons tight to a common edge and there is a 7.356103+ degree gap. That gap is also natural within expressions of dodecahedral or icosahedral structures. Within our work, those spatial dynamics are currently generally classified as quantum fluctuations.

Continuity-Symmetry-Harmony (CSH). I have saved the best for last. Many scholars and scientists are vehemently opposed to the idea that the foundations of science should be the foundations of theology and these should be the foundations of ethics and values. That this is true within this construction is a key validation that it is on the right path. These foundations — CSH — do give rise to our ethics and values. It is a very important part of this very holistic approach to mathematics, physics, the sciences, ethics, and values.


Conclusion. Nothing is ever easy. Yet, this model, even with its many moving parts, is simple. It is not easy, but it is simple. Some of our elementary school children (6th graders) appreciated it. So what do you think?

Thank you very much. -BEC

Of course, your comments are most welcome.



* This work emerged from concepts started in a high school in 2011. The chart of 202 base-2 notations from the Planck scale to the current time is here: Quantum indeterminacy appears to be measurable from about Notation-67. Notation-0 to Notation-64 are well below the scale of quantum physics. Also, even though this study is young — it began in December 2011 — we’ve started to address the very nature of Planck Temperature.

** See: Von Kirchoff bis Planck: Theorie der Wärmestrahlung in historisch-kritischer Darstellung, Hans-Georg Schöpf, Springer, 1978. Also, see Planck’s 1899 article, “Über irreversible Strahlungsvorgänge“, Max Planck, 1900 (which is included within the von Kirchoff book.


[1] VSL. There is a wide range of leading scholars who have been writing about VSL for many years. Perhaps the most foundational is by Robert Dicke in 1957. John Moffat (Wikipedia) (1992), Andreas Albrecht (Wikipedia) and João Magueijo (Wikipedia) (1998) kept it alive. The chart of 202 notations, particularly line 10, is one possible mathematical validation of VSL.

Ellis, George F R, Note on Varying Speed of Light Cosmologies (PDF), in General Relativity and Gravitation39 (4): 511–520. 2007
Magueijo, João, New varying speed of light theories (PDF), Reports on Progress in Physics, 2003, 66 (11): 2025–2068. arXiv:astro-ph/0305457
Moffat, John W, J. Magueijo), “Comments on ‘Note on varying speed of light theories”,” 2008
Yves-Henri Sanejouand, Empirical evidences in favor of a varying-speed-of-light, ArXiv (PDF), 2009


[2] Templeton. Many online discussions about the finite-infinite relation can trace their roots back to the John Templeton Foundation. The most simple definition of that relation is found within our understanding of the three faces of pi (π) which is the focus of this article. There is no guessing about the nature of the infinite; pi gives it away.

Harvey Friedman, Foundational Investigations Into The Infinite/Finite In Mathematics, 2012:
Sy Friedman, The Hyperuniverse: Laboratory Of The Infinite, 2015
Heather Templeton Dill, The Purpose of the John Templeton Foundation, 2021
Wikipedia’s listing about the John Templeton Foundation


[3] FQXi. The Templeton Foundation was the founding sponsor of the Foundational Questions Institute (FQXI). The institute has empowered studies of the most foundational questions such as:
What is the finite-infinite relation? Mathematicians Bridge Finite-Infinite Divide, N. Wolchover, Quanta, 2016
Theodore Slaman, University of California, Berkeley
Ulrich Kohlenbach, Technical University, Darmstad
 Keita Yokoyama, Japan Advanced Institute of Science and Technology, Nomi, Asahidai (Japan)
 Ludovic Patey, Diderot University, Paris
 Andreas Weiermann, Ghent University, Ghent
• What is the nature of space/time?
• What are the most basic systems that structure our universe?

Footnote: Craig Callender, Can Time Be Saved From Physics? (FQXi), 2019


[4] Three seconds. Twenty-seven scholars wrote The First Three Seconds: a Review of Possible Expansion Histories of the Early Universe (PDF), June 2020. They used the concept of Steven Weinberg’s 1977 book, The First Three Minutes to begin to consider the first three seconds. In our chart, the first three minutes places us in between Notations-149-150. The first three seconds is between Notations-143 to-145. This group needs to look at the first three Planck Seconds, Notations 1-2-3.

Our standards people (NIST, ISO, etc.) do not even have a name for something of such short durations. Our goal is to follow up our first analysis with all 27 scholars and to encourage them to explore the first three Planckseconds. Bernal and Hooper were among the 27 contributors.
Nicolás Bernal, Moira Venegas et al. New opportunities for axion dark matter searches in nonstandard cosmological models (PDF), 2021
Dan Hooper, TASI Lectures on Indirect Searches For Dark Matter (PDF), 2018
Adrienne L. Erickcek, Formation of microhalos (PDF-page 299ff), 2020
Tommi Tenkanen, Catarina Cosme, Spectator dark matter in non-standard cosmologies, (PDF) 2021
More to come. The next up is the other corresponding author, Ville Vaskonen, and then everybody else!


[5] PQE. This 2018 article opened the door for me on the Physics of Quantum Electronics folks. There are about 30 people listed for each annual conference. Our appeal will be with no less than five of these “practical-but-most-esoteric” folks and that will be based on recognizing their name from other groups and articles. (PDF)
Light, the universe, and everything — 12 Herculean tasks for quantum cowboys and black diamond skiers (PDF), 2018
Frank Wilczek, Are there new quantum phases of matter away from equilibrium that can be found and exploited – such as the time crystal?, page 23, 2018 (PDF), ArXiv, Light, the Universe and everything


[6] Authors. Those graduate students and postdocs who collaborate (especially from various fine schools) and who ask questions and make suggestions that may open new grounds to be explored, are to be encouraged.
•  Andrea Di Biagio (La Sapienza), Marios Christodoulou (Oxford), Pierre Martin-Dussaud (Aix-Marseille Univ, Université de Toulon), An experiment to test the discreteness of time, 2020 (revised 2021) (PDF) .
Arno KeppensEmergent Quantum Mechanics – David Bohm Centennial Perspectives, Physics Today, July 22-26, 2020. Keppens says, “Important attempts to devise an emergent quantum (gravity) theory require space-time to be discretized at the Planck scale.”
•   Randyn Charles Bartholomew, Let’s Use Tau–It’s Easier Than Pi, Scientific American, June 25, 2014


[7] Durham University IPPP. The Institute for Particle Physics Phenomenology IPPP of Durham University is entirely focused on the range from the Planck scale to the electroweak scale. Although their leadership has not yet affirmed our model with those first 64 notations well below measurement, this has been their domain for years and years. Perhaps they can tell us where we are going wrong. Perhaps they can tell us why those 202 mathematically-defined notations that encapsulate everything-everywhere-for-all-time the universe do not qualify as a reasonable STEM tool for teaching-learning. I will continue to follow this group. At some point they may have people who can suggest further tests of the actual numbers and inherent geometries within our little model. Of course, a dream would be to see invited papers for their 2022 annual conference.
• Seven IPPP graduate students, What is the smallest thing in the universe?, 2021
•  Isabel Garcia Garcia, Bounce of Nothing, June 2021 (PDF) Also see; ArXiv, May 2021


[8] Routledge. The first collection of articles to attempt to discern the essence of emergence, from the Routledge Handbook of Emergence, I started with these four articles:
• Kerry McKenzie, Fundamentality, Chapter 3, 11 pages, (video)
George Ellis, Evolution, Information and Emergence, Chapter 30, 18 pages
Christian Wüthrich, The Emergence of Space and Time, Chapter 25, 12 pages
• Robin Findlay Hendry, Sophie Gibb, Tom Lancaster, Introduction, 19 pages


[9] The Worldwide List of Alternative Theories and Critics (WLATAC). If Jean de Climont continues to work on these listings, this publication could become an even more valuable tool. The Worldwide List of Alternative Theories and Critics (PDF) will become most helpful in putting people together. Those collaborations will help to clarify issues and we may finally begin to breakthrough the impasses of our times. Among all the unaffiliated, boldly speculative thinkers, there are leading scholars like Gerald Holton (Harvard) and Andrei Linde (Stanford).
•   Gerald James Holton, Thematic Origins of Scientific Thought, HUP, 1988, WLATAC, page 894
•   Andrei Linde, (wiki), 2014 Kavli Prize in Astrophysics for pioneering the theory of cosmic inflation with Alan Guth (wiki) and Alexei Starobinsky (wiki) WLATAC, page 1195


[10] Facets of physics opening many questions: (1) Langlands programs, (2) String theory and Mtheory, and (3) SUSY (including work Beyond the Standard Model), (4) Causal sets and causal set theory, (5) Loop Quantum Gravity, (6) Scalar Field Theory, (7) Spectral Standard Model and (8) Causal Dynamical Triangulation.


Validations to date

A domain of perfection: Logically-inferred in December 2021
Planck Temperature at Notation-0: Scientific-orthodoxy force fit to work here, December 2021
Continuity-symmetry-harmony (CSH): Logically validated, circa 2021
The speed of light using Planck Natural Units: Mathematically validated, circa 2016
The Variable Speed of Light (line 10): Mathematically-and-logically validated, circa 2016
The geometries of quantum fluctuations (gap): Logically-and-geometrically validated, circa 2013
Base-2 continuity from Planck Time to the Now: Mathematically-and-logically validated, circa 2013

The presuppositions: (as a checklist)


Open Questions about Continuity-Symmetry-Harmony

•  Does continuity-symmetry-harmony define the infinite as well as the very first instant?
•  Are the Planck units or the Stoney units the best approximations of the first dimensions?
•  Are we at the limits of measurement of time with a trillionth of a billionth of a second (or a decimal point followed by 20 zeroes-and-a-1)? It is confirmed to be the shortest scientific measurement of a unit of time. It is within Notation-74.
•   Is there just too little time from the zeptosecond to the Planck second for imperfections?
•   When might a special combination of geometries and equations begin to manifest such that a quantum fluctuation could begin? … years? The first year manifests within Notation-169. Perhaps the patterns and thrusts of perfection are so great that it takes thousands if not a million years (Notation 189) before there is a quantum fluctuation.
•  Is the infinite qualitative and the finite is quantitative?
Are infinitesimal circles-spheres-tetrahedrons-and-octahedrons invariant within each notation?
•  Does cubic-close packing (ccp) of equal spheres define the first functional activity within space-time?
•  Might the expansion rate of the universe be set by PlanckTime (or possibly Stoney Time) and might that number (between 539 tredecillion spheres to 4605.4 tredecillion spheres per second) be a cosmological constant?
Is there an initial thrust for perfection for homogeneity and isotropy and a smooth beginning?
•  Are the best possible studies to answer these questions Langlands programs,.string theory and Mtheory, SUSY (including Beyond the Standard Model work), .Causal Sets (and Causal Dynamical Triangulation),.Loop Quantum Gravity, .Scalar Field Theory, and the Spectral Standard Model?


References / Resources  ________ Prior / Next

References from within this website:
1. This work began in 1970 within the study of the 1935 EPR paradox.
2. It was part of a conference at MIT in 1979 in search of first principles.
3. There are many pages that consider the first instants of the universe.
4. There are also these presuppositions and assumptions.



  1. George Scialabba, Cambridge, Massachusetts
  2. Prof. Dr. Jenann Ismael, Columbia University, NYC
  3. Viktor Toth, Papers and publications, Ottawa, Ontario
  4. Adrienne L. Erickcek, UNC, Chapel Hill, North Carolina
  5. Prof. Dr. Barends Mons, CODATA, Leiden, Netherlands



@KyendeKimeu Martin – We all need an expansive grounding. Our little worldviews are too small, solipsistic and often nihilistic. A highly-integrated mathematical view of the universe opens a whole new dialogue. A start is here: Technologies are derivative.

@brithume If you want to have clarity in your comments, you’ll have to let go of that limited worldview and begin working on a highly-integrated, mathematical view of the universe. It’s simple, but not easy: is a start. So much of DC has adopted nonsense!

@Acosta Please become an advocate to encourage us all to break free of little worldviews and to adopt a highly-integrated view of the universe. What happens is infinity is bridged by continuity-symmetry-harmony It’s a start.

@georgesoros I’d like to recruit you to become an advocate to encourage us all to break free of little worldviews and to adopt a highly-integrated view of the universe. What happens is infinity is bridged by continuity-symmetry-harmony It’s a start.

To many others: If there is ever going to be a little harmony in this world, we’ll need to break out of our little worldviews for an integrated view of the universe. Ours is a simple start: And, that it provides a foundation for ethics was gratifying:


Invitations and Collaborations

With whom do we collaborate? Of the hundreds of people who visit this site every month, who among them might want to extend a right hand and say, “Let’s work together.” Our only thrust is that the foundations of this universe and life itself be seen in light of infinity and the continuity-symmetry-harmony that the infinite engenders. Please, talk to us. Thank you. -Bruce


Key dates for this document, validate.


Afterthoughts: A Personal Addendum. The concept of a worldview was formalized in the 1700s first by Immanuel Kant and then by the general philosophical community. Yet, the intellectual and religious communities have had worldviews dating back to ancients like Plato. As the world began shrinking, we learned that there are many different worldviews and there are tensions among them.

Even when the German word is used, Weltanschauung, it’s still not big and comprehensive enough. Although Sir Isaac Newton and Immanuel Kant captured the world, the mysterium of a German word does not encompass the universe. As a concept, it is dated, incomplete, and not quite big enough. We need to move on to a integrated, mathematical view of the universe. -BEC


+ About the two dates at the top of the home page Close to 6 AM (TZ-19 or USA CST) each day, the days listed at the top of this page get advanced by one digit. It should be a relatively easy program to write, yet I rationalize that I do it manually just to remember our granular (sun-to-earth) sense of time. TZ-19 is time zone #19 assuming that the International Date Line is #1 and Greenwich Mean Time falls within Time Zone 13. Notwithstanding, we all know the only time is the Now (not based on our solar system and includes everything, everywhere for all time).