Validating that which cannot be directly measured.

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“Prove it to me!”
by Bruce E. Camber

Background. In December 2022 a friend asked me, ‘How can we begin to validate some part of this theory?’ It is a tough question and one that many very fine scholars have been asking and seeking to answer for hundreds of years, “How do we validate this theory?” We especially ask here because the “things” of our first 64 notations out of the 202 in our chart are just too small to measure.* -BEC

Numbers, Geometries, Equations. Unlike so many theories, this one — our most-simple, base-2 mathematics from Planck’s base units to 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 (John Ralston, 2012), Stoney’s numbers provide a comparative analysis. Some consider such numbers to be a 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. 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: In 1899 Max Planck did not know the speed of light. Although a simple calculation using his base units, it appears that calculation was never done (or, at least never acknowledged within the literature). The results are worth pondering. That calculation, using 1899’s Planck Length divided by Planck Time, is within .001% of the NIST/ISO value for the speed of light which was not set until 2019.

It was our first validation that the chart may have some cogency. Additional analyses emerged:
Overview: and
Speed of light:
Just a second:

Variable Speed of Light (VSL). There is an entire line of 202 numbers, each a multiple of Planck Length divided by Planck Time; it is being more closely studied. Line 10 is the result of the value of Planck Length within a given notation divided by the 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 of scholars 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 has 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 certainly appears to be heading in the right direction.

Note: The top-level numbers within Notation 202 so-well approximate numbers currently determined by careful measurements of our existing realities, these too have become 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, up or down, or 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 universals, and that science could inform religion about universals, and that the two groups rarely truly listen to each other.

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 who work on 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 and do not respond.

Notwithstanding, I will continue to engage FQXi (2021 paper) 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.

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. We have contacted all 27 contributing scholars to their article, The First Three Seconds. The several key questions are:
• 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 base-2 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? Our primary reference: and
• 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? Studying how it opens up the Fourier transform and Euclidean geometries, again it seems obvious that the question is worth pursuing.

Physics of Quantum Electronics. [5] There are other efforts that are equally ambitious. The scholars within this most-selective, 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 these creative to engage our model because it opens many new ways the quantum world can be reduced to practice.

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 inch them closer to a reduction in practice to measure the unmeasurable.

IPPPP. [7] The Institute for Particle Physics Phenomenology 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 is open for discussion. All 64 notations are well below the thresholds of measurement. It would be good to have one of two within this group to study our model. They will have invited papers 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 for clarity about the unsolved mysteries and tensions within science and all those involved might eventually consider our very different approach.

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 list are people who have poked holes into the fabric of our most commonly-accepted scientific concepts. Sorting it all out is too 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. Also, it is a good place to learn the names of others. There are 81 people listed who investigate the cosmological constant. There are 71 listings that use the word, perfect. There are twelve who use the word, continuity. 100 use the word, symmetry. Nine use the word, harmony. Each is a reference of interest.

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 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 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 every expression of dodecahedral or icosahedral structures. Within our work, those spatial dynamics are currently generally classified as quantum fluctuations.

Continuity-Symmetry-Harmony. Also important is to see how these foundations 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. Elementary school children (6th graders) understand it and so can you.

Thank you very much. -BEC

Of course, your comments are most welcome.



* This work emerged from these 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:


[1] VSL. There is a wide range of leading scholars who have been writing about VSL for many years. Perhaps the most substantial is Robert Dicke’s work of 1957. John Moffat  (1992),[16] and Andreas Albrecht (Wikipedia) and João Magueijo (Wikipedia) (1998) have kept it alive. The chart of 202 notations, particularly line 10, is the first actual mathematical validation of the VSL.

Magueijo, João, New varying speed of light theories (PDF), Reports on Progress in Physics, 2003, 66 (11): 2025–2068. arXiv:astro-ph/0305457

Ellis, George F R, Note on Varying Speed of Light Cosmologies (PDF), in General Relativity and Gravitation39 (4): 511–520. 2007


[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
• What is the nature of space/time?
• What are the most basic systems that structure our universe?

See: 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 BernalMoira 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


[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 many 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


Validations to date

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 validated, circa 2014
Base-2 continuity from Planck Time to the Now: Mathematically-and-logically validated, circa 2013


Open Questions about Continuity-Symmetry-Harmony

•   Do these concepts define the infinite as well as the very first instance?
•  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 1971 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. RE: Investments in Educator Development
    TO: Sara Leikin, Ed.D., STEM Solutions and Partnerships
    National Math and Science Initiative, Dallas
  2. Ngo Bao Chau, University of Chicago on Langlands Programs
  3. Siska De Baerdemaeker, Stockholm University
  4. Jeremey Nicholas Butterfield, Cambridge University



@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 high-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 high-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. Here’s a simple start:


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