I Think you could be Right! These Research findings Support your idea Preprint The Universal Fractal Chirality of Riemann Zeta-Function: On...

It's not a good idea to assemble all the possible buzzwords, without trying to understand the subject. The answer to the question is that it doesn't make sense.

Not all questions make sense and learning physics and mathematics involves learning which questions make sense and which questions don't. This question simply doesn't make sense.

Quantum mechanics doesn't have anything to do with general relativity: The former describes how quantum fluctuations affect physical systems, in general, the latter describes the classical dynamics of manifolds, eventually, taking into account the classical properties of matter.

How the-known-quantum properties of matter can affect the spacetime geometry isn't known, even in principle, in general. Nor how to describe the quantum fluctuations of classical spacetime geometries. One reason is that classical gravity is a gauge theory with a non-compact gauge group and what is known is how to describe the quantum fluctuations of gauge theories with compact gauge groups.

The only property of quantum gravity that is known is the dimensionality of the space of states of a certain class of black hole spacetimes, in the presence of quantum matter. This does allow to deduce some properties of the dynamics of the probes and of the response of the spacetime geometry. But the results aren't expressed in terms of buzzwords. Finding the buzzwords in the technical papers doesn't mean anything, the content matters.

Stam Nicolis Stam, I respect your experience, but I firmly reject the claim that this question "doesn't make sense".

The connection between discrete structures, prime numbers, and the emergent continuum is a serious topic of discussion in modern mathematics (see fractal theory, Riemann-Zeta distribution, non-commutative geometries of Connes).

If the differentiable formalism that we use in relativity and quantum mechanics is only an effective approximation of a deeper structure, discrete or fractal, then the question posed is not only sensible, but necessary.

It is easy to dismiss what goes beyond the traditional schemes as "buzzwords" and then between us you are not the scientific community, your opinion is only valid for your opinion. It is more difficult to deal with new hypotheses, even if still in the process of formalization, otherwise you would still be smoothing inclined planes.

If physics had always ignored "uncomfortable" questions, neither quantum mechanics (which, between us, has large margins of error, and I say this as a complex systems engineer) nor general relativity would exist today.

Then, between us Stam, given the tone of your observations, I sincerely ask you: have you ever directly dealt with fundamental physics, understood as problems of quantum gravity, the structure of space-time, or the foundations of quantum mechanics?

Reading your publications on ResearchGate, most of your work concerns collective dynamics, complex systems and biological models (you could also be a troll accustomed to machine learning). Very respectable fields, but quite distant from the heart of the problems I am discussing.

Precisely for this reason I believe that dismissing deep questions on the discrete or continuous nature of time and space as "nonsense" is an error of perspective. In theoretical physics, the exploration of hypotheses is as important as experimental verification.

With all due respect, it would perhaps be more useful for each of us to intervene mainly in the fields in which we have real expertise, without blocking discussions that, perhaps, have interesting implications for those who work in the field of physics.

I would like to thank you for making one point even clearer: it is precisely the current inability to integrate the discrete into the continuous that represents the real problem, not the questions that try to anticipate it.

The question   “Is it possible that the continuum emerges from a discrete topology related to the distribution of Riemann zeros ζ, attributable to bifurcation theory?” looks  as is rather questionable,  since  the cardinality of continuum is much larger then the cardinality of yet  rational numbers set, and, of course much larger then the cardinality of any discrete set.

And, besides, in the question at least this [as is in the question context]

“.…problems of quantum gravity, the structure of space-time, or the foundations of quantum mechanics.”

- really has  no relation to physics.    The utmost universal Matter’s spacetime, as that rigorously scientifically shown in the Shevchenko-Tokarevsky’s Planck scale informational physical model, two main papers are https://www.researchgate.net/publication/391209088_The_Informational_Conception_and_Basic_Physics, [ in the paper secton 2.9 “Mediation of the forces in complex systems” can be passed since this is more comprehensively  given  in section 6.“Mediation of the fundamental forces in complex systems” in other paper below] , and

https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics ,

- is  fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, [5]4D spacetime with metrics (cτ,X,Y,Z,ct)  [whole spacetime with space dimensions that relate to fundamental Nature Gravity, Weak,  Electric, and Nuclear/Strong forces has (at least) [4+4+1 metrics (cτ,X,Y,Z, g,w,e,s,ct),],

-  where cτ,X,Y,Z, are 4 utmost universal  space dimensions;   g,w,e,s dimensions relate to the Forces; ct is the [absolutely infinite]  fundamentally unique  time dimension.

At that all space dimensions are fundamentally infinite, and so, say, if there can be infinite number of other  “Matters” that have the same design as the tribal one, all they would be placed in the spacetime above, while distances between “Matters” could be arbitrary, including infinite.

Cheers

Sergey Shevchenko

Dear Sergey,

Thank you for sharing your work and for pointing out the informational physical model with its strong assumptions about a fundamentally continuous and Cartesian spacetime.

However, I would like to propose a constructive comparison from a different perspective. The Electron Approach Theory that I have developed (see the attached articles) addresses several conceptual and structural gaps present in both classical and quantum theories, including the rigidity of those models that assume spacetime as flat and continuous from the beginning.

It is true that discrete sets (such as prime numbers or natural numbers) have a smaller cardinality than the continuum ( Method Discussion on the method - The mathematical basis of the theory

The Approach Theory proposes a dynamical model where the oscillation of the electron between energy levels is governed by an exponential decay law:

E(t) = E_0 e^{-\frac{t}{RC}} \cos(\omega t)

Assuming flat continuity as a starting point, your model excludes the possibility of nonlinear transitions and feedback mechanisms—mechanisms that are fundamental to describing observable phenomena such as irreversible transitions, decoherence, or quantization.

In the fractal extension I propose, space and time are constructed from coordinate triplets indexed by primes, and the depth of the temporal layers is described algebraically, where the one-to-one map between absolute (discrete) and relative (continuous) spacetime follows the relation:

T_{absolute} \longrightarrow T_{relative}, \quad \text{where the iteration depth } d \in \mathbb{N} \text{ governs the observable coherence.}

So in my view the approach theory naturally unifies classical, quantum and relativistic insights, without violating the fundamental postulates, but instead showing how quantization can emerge without being postulated. This distinguishes it from models that postulate, but do not derive, the nature of transitions.

In summary, I find your model coherent and interesting, but, as it were, it does not make it clear to me how to accommodate those emergent, oscillatory or fractal mechanisms that could explain quantization, time asymmetry, and even quantum coherence as natural outcomes.

With respect,

Douglas Ruffini

Thought experiment...

The idea is to be able to construct "perfect" geometric figures in an ideal space, being able to arbitrarily decide the unit of measurement, equal to one, and use it for our purpose, also having an infinite magnifying glass at our disposal. So, wanting to see if, in principle, it was possible to create a square with side one: "Imagine being in an infinite space (you can also represent it with the same color in a uniform way), without reference points, nor units of measurement. Imagine considering a linear segment of arbitrary length and considering it as a basic unit of length equal to one. Now, let's say that you manage to construct a square with side one and that you can travel one of its diagonals starting from edge A to edge B. Since edge B is at √2, would you, like others, be able to reach it?"

At this point we have come up against a fundamental problem: the nature of irrational numbers. We saw that the diagonal of a “perfect” square is related to √2, the circumference of a circle to π, and so on. These numbers cannot be expressed exactly with a fraction, which means that we can never “fix” their position with absolute precision. So even if we used our ideal lens we would not have been able to find the end of the path between A and B.

The difficulty perhaps is that in the space defined by the thought experiment we had no way of finding or measuring 90-degree angles with certainty, because the thought experiment does not provide for the presence of measuring instruments (such as a protractor) or references (such as a predefined coordinate system). Making things a little simpler we introduce other factors: “I want to make it easier for you by inserting gravity and a planet into this space. Would you have been able to construct a 90-degree angle?”

Then we tried to make it more concrete by introducing gravity and a planet for references. But even then, we have concluded that constructing a perfect right angle, which is essential for a square, turns out to be impossible. We can get as close as we want, but we will never reach the mathematical ideality of being able to isolate a point on a circle, which incidentally is defined as 2πr.

“There is an infinite regress in my reasoning. If I need an infinite magnifying glass to "see" points with arbitrary precision, how can I construct a circle (which requires precision in the definition of the radius and circumference). So even with an instrument of unlimited precision such as an infinite magnifying glass, there would be fundamental limitations that would prevent us from arriving at the construction of a perfect square, circle, cube or sphere in the ideal mathematical sense.”

Finally, let's consider how modern physics, with relativity and quantum mechanics, describes space and matter. : Irrational numbers are essential to describe many physical phenomena. General relativity describes spacetime as a curved manifold, whose geometry is described by equations involving irrational numbers. Quantum mechanics uses wave functions to describe the behavior of particles, and these functions can take on irrational values.

In other words, physical reality, when we look closely, does not correspond perfectly to our ideas of perfect squares and circles: “From what we have discussed, we can say that we have a mathematical but not physical counterpart of irrationality. Irrationality, which is a mathematical concept, manifests itself in the limitations we encounter when trying to describe and construct perfect geometric figures in physical space, especially when we consider the nature of spacetime. In other words, the impossibility of constructing a perfect square or circle is not just a problem of practical precision, but reflects a fundamental property of space itself.”

So, the logical inference we have arrived at is that the ideal geometric figures of mathematics are not found in nature in their mathematical ideality. This does not mean that we cannot use geometry to describe the world, but that we must be able to rationalize (such as saying that all squares are parallelograms).

What follows

Every real measure is defined by a discrete set of finite operations and finite instruments.

Every physical geometric construction is subject to structural limits and cannot realize instances of irrational numbers.

conjecture: The structure of real space is such that every definable physical process can be mapped onto a countable set (finite counting).

The use of irrational numbers in physics is a mathematical modeling without a direct ontological counterpart.

Dear Douglas ,

- sorry, but I comment  as a rule only conventional physics; and don’t comment alternative approaches, besides  if that is in the question of a thread. So sorry, but I cannot to add something to what is in my post above, note only again - the actualizations “space” and  “time” dimensions of the  absolutely fundamental [“Logos” set elements] phenomena/notions “Space” and “Time ”  are fundamentally continuous and have no any intrinsic own measures. So if there exist only one object it is senseless  to state what is its size, besides that it is more zero, if only one change happened it is senseless  to state in what time interval that  happened, besides that the size and the interval  are more zero.

So  the distances/lengths  between/of some objects in a space  and changes’ time intervals can be compared  only relatively i.e. only if there are more one objects, and more one change happened -  lager/lesser/equal,

-  including measured by comparison with some space and time units  standards

Cheers

Douglas Ruffini

Dear Douglas,

As you can gather from the preceding post, no mainstream theoretician is authorized by fear of peer opinion and potential dire professional consequences, as was the case for Paul Marmet, to even admit reading out of the box approaches, let alone publicly discuss them.

By the way, you may be interested to know that QM for one was grounded on an involontary phase wave velocity calculation error, something else that mainstream theoreticians are forbiden to look at. Mathematical proof published last year:

Article Critical Analysis of the Origins of Heisenberg's Uncertainty Principle

Regarding your interesting ontological idea, maybe you would be interested also in this different approach proposed by Noel Coughlin:

Preprint Correlation of the electromagnetic mechanics of elementary p...

Curious as to what happened to Marmet, here is a link to the site of his estate, not mentioned on wikipedia, all of whose entries about physics are orthodox mainstrean controled and compliant:

https://www.newtonphysics.on.ca/info/author.html

I note that the treatment that was dealt out to him by the University of Ottawa has already been stricken out from the site of his estate, but not before I was authorized by his estate to quote the critical parts in Appendix A of this book chaprer published in 2021:

Article De Broglie’s Double-Particle Photon (Expanded republication PI)

André Michaud

Dear André,

I read your article "Critical Analysis of the Origins of Heisenberg's Uncertainty Principle". Your work offers an intriguing perspective on the origin of the uncertainty principle, highlighting how an initial misinterpretation may have profoundly influenced the development of quantum mechanics.

Your position André is unconventional, but not necessarily unfounded. In fact, your criticism touches a delicate point in the history of theoretical physics: many conceptual pillars such as uncertainty or wave-particle dualism were born in a period of enormous ferment, where they consolidated rapidly and often on the basis of partial or even simplified models. But this is history, it is not clear why if someone does a historical analysis it should imply the possibility of being blacklisted as a heretic.

It is true that the derivation of the de Broglie wave relies on an empirical relation, and that the phase frequency is conceptually problematic (it is superluminal). Of course, this leaves room for other interpretations, besides yours, since it implies velocities greater than c², a consequence that remains open to different conceptual readings. The idea that an initial conceptual error could have directed a theory on a specific track is not far-fetched at all. It happens in all scientific fields.

Of course, one must be careful because any revision must recover (at least) the successes of the standard formalism.

Having said that, one must reflect on the role of those who do science: "in the attempt to continually refine existing theories, one risks neglecting alternative approaches that could offer new insights". It is like incessantly smoothing an inclined plane in the hope of eliminating friction, without considering that perhaps it would be necessary to change paradigm.

I appreciate your courage in questioning consolidated concepts and in promoting an open dialogue on fundamental questions of physics.

Sincerely

Douglas

Douglas Ruffini

Dear Douglas,

Thank you so much for your appreciation of my paper.

Note that I mentioned Marmet's fate apparently out of context here, because he was instrumental in my rediscovery of electromagnetic mechanics, whose foundation was established by Heaviside, Searle, Wien and Lorentz in the 1890's, experimentally confirmed by the Kaufmann's experiments in the early 1900's, but whose absence of referencing from 1907 onwards made it so in the 1920's that it prevented de Broglie and the whole upcoming generation of theoreticians from becoming aware of the correct frequency that would have allowed calculating the correct velocity for the phase wave, which would have made useless the use of the theoretical supra-luminal beat frequency for the electron momentum energy to catch up with it.

Marmet's contribution was his rediscovery in 2003 that the electromagnetic mass of the electron can only increase with velocity, that was initially theoretically discovered by Searle in the late 1890's and experimentally confirmed soon afterwards by Kaufmann, then confirmed by Abraham and very clearly by Lorentz in his 1904 paper, then by all major physicists between 1905 and 1907; to be voluntarily discarded as the Special Relativity theory was adopted, due to common agreement that this never was observed for macroscopic ponderable masses. Revealed only in 1982 on page 159 of Abraham Pais' biography of Einstein.

As you have read in Appendix A, this trend of Marmet, the most advanced experimentalist of the second half of the 20th century, to do fundamental research, cost him dearly career wise towards the end of his life, but not before he could publish this so important fundamental re-discovery about the accelerating electron.

Best Regards, André

André Michaud

Dear André,

you refer to a series of historical developments in theoretical physics between the end of the 19th and the beginning of the 20th century, focusing on the concept of electromagnetic mass of the electron and the implications for quantum mechanics and relativity.

Yes, this revision is also necessary in my opinion.

I appreciate very much your historical summary, which brings to light an often overlooked theme: the critical revision of the concept of electromagnetic mass and its implications in early 20th century physics. Your perspective is a point of view that can be shared.

In my work — the Approach Theory — I address an alternative interpretation of the temporal evolution of the electron. In fact, I propose that its motion is the result of a space-time oscillation between two poles: a relativistic future, where the electron accelerates and “gives up” mass and energy to the field, and an absolute past, in which these values ​​are recovered. This configures a closed loop of energy feedback, which provides dynamic stability without having to postulate instantaneous conservation.

This view may, in fact, reflect some of the phenomena that the Lorentz-Searle formalism had anticipated, but that were abandoned with the advent of special relativity, as you recall. I was struck, for example, by your mention of Kaufmann's experiments and the systematic removal of the concept of electromagnetic mass from standard treatments (already in the Annalen der Physik, 1905–1907).

Of course, the main difficulty today remains experimental validation. Finding a setup that distinguishes between equivalent predictions of competing theories — especially in a landscape where the current formalism “works” — is far from trivial. Not to mention the weight of the academic infrastructure in determining who receives attention, funding, and peer-review validation.

A scientific revolution, between us, I see as difficult:

"why should we change to something that needs experimentation and fine-tuning when we already have something else that works more or less well?"

Yet, without opening the space for “heretical” proposals, the great scientific revolutions of the past — from quantum mechanics to relativity itself — would never have happened.

This is a very important topic, thanks again for your contribution and good work André.

Sincerely

Douglas

Douglas Ruffini

Dear Douglas,

You wrote: "you refer to a series of historical developments in theoretical physics between the end of the 19th and the beginning of the 20th century, focusing on the concept of electromagnetic mass of the electron and the implications for quantum mechanics and relativity."

Indeed I do. Actually, I spent the past 30 years searching through readings and discussing with physicists online for any bit of info that I progressively collected and synthesized about the origins of electromagnetic mechanics, that seemed to never be mentioned anywhere in the current literature.

The whole case seemed to have revolved about whether or not the confirmed electromagnetic mass increase of electrons with velocity (Kaufmann's experiments) also applied to macroscopic ponderable bodies.

The community chose that it did not in 1907 as SR was adopted without this discovery being taken account of, since SR was meant to mainly address the relative motion of macroscopic ponderable bodies whose masses at any velocity was assumed to remain invariant.

I eventually retraced the ontological origin of electromagnetic mechanics to the historical research work of Heaviside, Searle, Wien, Lorentz, Abraham and Kaufmann, who first established the foundation of the electromagnetic mechanics of elementary particles, before the community stopped referencing and teaching it in 1907.

If specifically interested, all their papers are now in the public domain, and direct links to them are available in the Reference section of this synthesis that now synchronizes electromagnetic mechanics with kinematic mechanics, according to Wien's project as formulated in 1901:

Article Electromagnetic and Kinematic Mechanics Synchronized in thei...

You wrote: " In my work — the Approach Theory — I address an alternative interpretation of the temporal evolution of the electron. In fact, I propose that its motion is the result of a space-time oscillation between two poles: a relativistic future, where the electron accelerates and “gives up” mass and energy to the field, and an absolute past, in which these values ​​are recovered. This configures a closed loop of energy feedback, which provides dynamic stability without having to postulate instantaneous conservation."

I have not yet had time to look at your model, but I always had a view that no stone should remain unturned in our search to understand the physically existing universe in which we exist. This is why I manage an informal journal initially set up by a dear friend of mine in 2001, Walter Babin now passed away, in which all ideas are allowed as preprints for public exposure and discussion, including ideas that run contrary to my own views.

One never knows out of which proposition well or even less well presented, the next major progress will emerge from, so we accept all preprints, and of course all papers that discouraged authors can't se the day that they could be published in some mainstream approved journal.

https://www.gsjournal.net/

You wrote: " I was struck, for example, by your mention of Kaufmann's experiments and the systematic removal of the concept of electromagnetic mass from standard treatments (already in the Annalen der Physik, 1905–1907)."

Yes. I had the same reaction at becoming aware of this back in the 1990's.

You wrote: " Of course, the main difficulty today remains experimental validation. Finding a setup that distinguishes between equivalent predictions of competing theories — especially in a landscape where the current formalism “works” — is far from trivial. "

This may come as a surprise to you, but such a set up has been in constant use in the engineering community ever since Kaufmann established the method. All cathode ray tubes (CRT screens, such as our old style TVs) reproduce exactly the method used by Kaufmann in his bubble chamber.

Propelling electron beams on utterly precise curved trajectories by energizing them with a calibrated E-field, and deflecting them by an added precisely calibrated B-field. The detailed Lorentz force equation required is presented in the previously referred synthesis and is available although less detailed in all high energy accelerators reference works, and of course in all engineering reference works on cathode ray tubes.

Given that Kaufmann did this with the rudimentary means then available to him, this should be no big feat to recollect the related data and analyze it in light of the more extensive pool of knowledge that we now have at our disposal.

You wrote: "Not to mention the weight of the academic infrastructure in determining who receives attention, funding, and peer-review validation. A scientific revolution, between us, I see as difficult: "why should we change to something that needs experimentation and fine-tuning when we already have something else that works more or less well?""

Indeed, this seems to be the ultimate hurdle. Note that I was aware of this problem from the get go. It seems obvious that the fear inducing peer-pressure driven mental straightjacket that the Copenhagen interpretation forces all practicing mainstream theoreticians to restrict their thinking within cannot be directly breached.

Actually, my articles were never meant to catch the attention of mainstream.

Not possible since peer-pressure induced fear for their careers and reputation irretrievably prevents public discussion of any idea not already rubber stamped by their forced collective consensus.

You may have noticed that all equations in my articles are first level, meaning that they all are immediately resolvable on a simple pocket scientific calculator, involving only known constants plus a single variable, the wavelength of the electromagnetic energy of the particle considered. This is to put all my analyses within easy reach of even beginners in physics, typically physics students of the upcoming generation who are not yet captive of the mainstream peer-pressure induced mental straightjacket, and who feel totally free to read whatever catches their attention in their domain of interest.

From the RG site alone, my papers have been downloaded at a rate exceeding 500 per month for the past 7 years, the period that I know of. So these recovered discoveries are now progressively re-entering general awareness in the community, and I expect that at some point, discussions of these issues will naturally start among these new theoreticians as they integrate mainstream. When in sufficient numbers, they will eventually run the show, and theoretical progress will resume.

This about summarizes what my project was all along.

Luckily for us all, the engineering community kept the flame alive by never stopping their successful use of the Lorentz force equation, first tested by Kaufmann.

You wrote: " Yet, without opening the space for “heretical” proposals, the great scientific revolutions of the past — from quantum mechanics to relativity itself — would never have happened."

I completely agree. Hopefully, this will happen again.

Have a nice day and Best Regards, André

“.…Marmet's contribution was his rediscovery in 2003 that the electromagnetic mass of the electron can only increase with velocity, that was initially theoretically discovered by Searle in the late 1890's and experimentally confirmed soon afterwards by Kaufmann, then confirmed by Abraham and very clearly by Lorentz in his 1904 paper, then by all major physicists between 1905 and 1907; to be voluntarily discarded as the Special Relativity theory was adopted, …”

- yeah, in  first 1900s the entire physics beau monde, here we can add Poincare, and not  only.,  thought that everything in Matter was determined only by electromagnetism, and, since   in those times and   till now in the both   - classical and quantum - electrodynamicses   it was/is postulated that EM fields, which are constantly radiating by electric charges,  contain energy; so unique in those time known particle “electron” has “electromagnetic mass”, which  is in a few ways proportional to E/c2.

What  really was/is  fundamentally wrong since  rather evidently violates the energy conservation law, and can be true only if charges by some mystic way have infinite enrgy sources.

Really - as that is rigorously scientifically shown in the SS&VT Planck scale informational physical model, more see the 2-nd linked paper in SS post  on page 1, section 6 “Mediation of the fundamental forces in complex systems” , the fundamental Nature Electric fields [and at least also Gravity and Nuclear/Strong Forces fields] at least at statics,  doesn’t contain  energy, and so  there cannot be fundamentally no any “electromagnetic masses of electron”, etc..

Though in this case  it is necessary  also to understand - so what are really fundamental phenomena/notions “Energy” and “Inertia”  - so further what is “inertial mass”? -, which are fundamentally transcendent in  official physics, and so such as the above postulates in the physics exist.

The rigorouls scientific ckarification of the both items above can be,  and is, again given only in the pointed  SS&VT model, more see first  section  in  the link above.

Nonetheless  just in 1905  in “Does the inertia of a body depend on its energy content?” Einstein, in this case  quite correctly,  proposed  that energy  and inertia of every material object are   relate as E=mc2 independently on has or not some object electric charge;  and that was indeed  one of utmost fundamental findings in 20-th century.

Cheers

PS this thread is indexed  in its “Science topic" sections

Sergey Shevchenko

Dear Sergey,

I did a little analysis of your theory, for information purposes.

And I noticed that your SS&VT model is built on a Euclidean spatial framework, that is, it assumes the validity of Euclid's fifth postulate, the one that states the uniqueness of the parallel passing through a point external to a line.

But what would happen if we changed geometry (Euclid's fifth postulate is indeducible), for example:

Let's say that no line is parallel (all lines meet).

Would the discrete lattice infrastructure be deformed? Would the cyclic propagation of particles (the algorithms closed on FLEs) lose linearity/invariance over long distances?

Would fixed minimum distances (such as Planck time/length) be modified by the curvature?

Could the information propagation trajectories close, causing self-interference in the information loops?

Let's say for every point outside a line, there are infinitely many parallels.

Would the density of the FLE lattice increase exponentially towards the "outside", creating a non-uniform information network?

Would the "bits" of information (flips of the FLEs) no longer be isotropic and the propagation would depend on the direction?

Would this directly affect the uniformity of the physical laws in the model, requiring a dynamic compensation mechanism?

If the fifth postulate falters, the risk is that the geometric coherence underlying your formalism will break down. Have you ever considered these implications?

Perhaps it could be modified to incorporate dynamic curvatures, for example variable curvature as in GR or fractal topologies as proposed in my Approach Theory.

Sincerely

Douglas Ruffini

Sergey Shevchenko

You quoted: "“.…Marmet's contribution was his rediscovery in 2003 that the electromagnetic mass of the electron can only increase with velocity, that was initially theoretically discovered by Searle in the late 1890's and experimentally confirmed soon afterwards by Kaufmann, then confirmed by Abraham and very clearly by Lorentz in his 1904 paper, then by all major physicists between 1905 and 1907; to be voluntarily discarded as the Special Relativity theory was adopted, …”"

Since this quote is from me, I suppose that you are addressing me, and are possibly expecting an anwer?

You commented: "- yeah, in first 1900s the entire physics beau monde, here we can add Poincare, and not only., thought that everything in Matter was determined only by electromagnetism, and, since in those times and till now in the both - classical and quantum - electrodynamicses it was/is postulated that EM fields, which are constantly radiating by electric charges, contain energy; so unique in those time known particle “electron” has “electromagnetic mass”, which is in a few ways proportional to E/c2."

Not a bad assessment of the state of affairs in the early 1900's. But since from your own often repeated admission, you never read anything but conventional physics, you can have no clue as to the progress that was made in fundamental understanding of electromagnetic mechanics away from this antiquated worldview. All this material is completely out of reach for you for this reason.

Speaking of Larmer hypothesis that accelerating electron constantly radiate energy, you wrote:

"What really was/is fundamentally wrong since rather evidently violates the energy conservation law, and can be true only if charges by some mystic way have infinite enrgy sources."

You are right of course. This makes no logical sense.

You wrote: " Nonetheless just in 1905 in “Does the inertia of a body depend on its energy content?” Einstein, in this case quite correctly, proposed that energy and inertia of every material object are relate as E=mc2 independently on has or not some object electric charge; and that was indeed one of utmost fundamental findings in 20-th century."

I agree. He was right.

[André Michaud, sorry, but that is for all RG members, and so doesn’t require your comment]

In the passage

“…Speaking of Larmer hypothesis that accelerating electron constantly radiate energy, you wrote: [SS quote]

"….What really was/is fundamentally wrong since rather evidently violates the energy conservation law, and can be true only if charges by some mystic way have infinite enrgy sources."

You are right of course. This makes no logical sense.….”

That in the SS quote above about the energy conservation law has no relation to “Larmer hypothesis that accelerating electron constantly radiate energy” , that relates to [in the SS post]

“…..in those times and till now in the both - classical and quantum - electrodynamicses it was/is postulated that EM fields, which are constantly radiating by electric charges, contain energy;….”

- i.e. that in the standard now - classical and quantum - electrodynamicses - charged particles radiate energy [in the 4D Minkowski space with metrics (ict,X,Y,,) ] always, including when are at rest or move with constant velocity, in CED that are containg energy fields, in QED that are containing enetgy virtual photons. What, again, evidently violates the energy conservation law.

However if a charged particle is accelerated, it radiates energy, say, if an electrically charged one is accelerfated , as real photons – obtaining for this energy from the external source that accelerates the charge.

Really, again see the SS&VT initial Planck scale models Of Gravity, Electric, and Nuclear/Strong Forces in https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics, section 6 “Mediation of the fundamental forces in complex systems”, the Forces nediators aren’t particles, and so don’t carry/transmit [at rest and unertial cases] energy in 3DXYZ space, but carry/transmit specific momentums, for which the g,e,s, dimensions in the real Matter’s spacetime with metrics (at least) [4+4+1 metrics (cτ,X,Y,Z, g,w,e,s,ct) are used, and the Forces. interactions in closed systems happen with while sysstem’s energy be constant.

More see the SS posts and linlks in the posts.

Cheers

It is theoretically possible that the continuum emerges from a discrete topology connected to the distribution of Riemann zeros of the zeta function, particularly through frameworks like bifurcation theory, which studies how small parameter changes can lead to sudden qualitative changes in system behavior. The zeros' statistical resemblance to random matrices' eigenvalues and their hypothesized link to quantum chaos suggests an underlying structured, possibly discrete, order that might be interpreted topologically. While speculative, such ideas align with ongoing research into the deep connections between number theory, dynamical systems, and quantum mechanics, where bifurcations could metaphorically or physically represent transitions leading to continuous structures from discrete ones.

Muhammad Bura Garba

Dear Muhammad Bura Garba

Thank you for your thoughtful comment. Your reference to bifurcation theory and its connection to the emergence of continuity from discrete structures touches directly on what I am trying to explore in my paper. I fully agree that the statistical similarity of ζ zeros to the eigenvalues ​​of random matrices may indicate not randomness, but a hidden discrete order. In my approach, each zero corresponds to a transition point – a topological attractor – that guides the “motion” (or informational transition) between discrete states. This motion is modeled as a kind of recursive symmetry breaking, where the temporal and spatial dimensions are seen as emergent consequences of these discrete bifurcation points. I am particularly intrigued by your reference to quantum chaos, which could serve as a bridge between number theory and the dynamical behavior of physical systems. If attractors can be reinterpreted as logical invariants in discrete time, we may be one step closer to a unified informational basis for physics. In the same context I was analyzing some statements: "if the concept of attractors were extended as points distributed on a curve at the limit called time, one could agree that all the points corresponding to irrational numbers, undergo the same geometric evaluation described for decimal distributions", or more precisely: "the succession of their fractional parts, would correspond to the Kronecker distribution". The idea is that irrational numbers, not being periodic in their decimal expansion, never repeat themselves, but fill the space uniformly if considered modulo 1.

The succession of their fractional parts (i.e. the decimal digits after the decimal point, or more formally {nα} for n ∈ ℕ, where α is irrational) is not only non-periodic, but also dense in the interval [0,1].

So In this context: "if we imagine time as a limit curve in which events are arranged, and if the time points were represented by sequences of numbers (like irrational fractions), then the oscillation itself can be described by a quasi-periodic system (like the Kronecker sequence)."

I would be very happy to know your point of view, on this connection.

Most of this discussion is over my head but a simpler (I think) question that seems to be somewhat related to the question that started this thread is whether a discrete set can generate a continuum. It can in at least one special case. Consider a continuous function of a single real number (or multiple real numbers but I say single just to simplify the vocabulary). Math will show that this continuous function is uniquely determined at all arguments if the function values are given at each rational number. The set of rational numbers is countable, so the number of degrees of freedom of a continuous function is countable. This explains why such a function can be expanded in a series (e.g., a Fourier series) with a countable number of terms. So, by specifying that a function be continuous, or just sectionally continuous, the uncountable infinity of function values can be generated by a discrete set.

L.D. Edmonds

Dear L.D. Edmonds,

Thank you for your contribution. You are absolutely right. This principle underlies much of classical analysis and Fourier theory, and shows how an “apparent continuum” can be described by a discrete set of coefficients.

In my original question, I was exploring a more foundational (and speculative) path: the possibility that physical spacetime, as modeled in general relativity and quantum field theory – being differentiable and continuous – could itself emerge as a statistical average or projection from a deeper, discrete, topological substrate.

In that context, I invoked the zeros of the Riemann ζ function not as simple analytical tools, but as physical indicators of transition states (or attractors), possibly corresponding to a form of discrete time evolution.

To adequately understand this distinction, even without formalizing it in a rigorous way, I will say that on the contrary, the set of Rational numbers (R) can be described as dense (representative of the points of a line); that is, given two rational numbers A and B, there will certainly be another one included between them. Note, however, that in this case too we will speak of potential infinity, because the definition does not state that between two points there are infinitely many, but only that we can go on as long as we want and always find one more than those already found up to that moment. Therefore, if we assume the exact definition of a dense set:

{m/n|m∈R ∪ n∈R ∪ n≠0},

it will still require that this potential infinity be contained in a limited area, where, however, it will not be possible to perform divisions by zero, since the writing m/0 will have no meaning, thus obtaining that each number R will not be able to verify its (fundamental) successor.

So, if the problem is to understand what relationship there may be between the Before and the After, then I ask you, since you intervened in a pertinent way, whether a similar emergent behavior can occur at the physical level, possibly mediated by bifurcations, chaotic transitions or informational constraints.

Thanks again for your thoughtful note: I would be happy to hear more from your point of view on this topic.

Kind regards,

Douglas Ruffini

Dear all, to delve deeper into the theoretical framework I presented in The Electron Approach Theory (https://www.researchgate.net/publication/389871123_Electron_Approach_Theory_A_Damped_Oscillation_Model_Based_on_Relativistic_Effects_and_Space-Time_Feedback) and in the questions and answers presented in these chats, I invite you to read the entire text of my book, The Mechanics of Time, now available in its entirety in the "Book" section of my profile (https://www.researchgate.net/publication/394013969_The_Mechanics_of_Time_A_Damped_Oscillation_Model_Based_on_Relativistic_Effects_and_Space-Time_Feedback). Thank you for your time, and I remain available for any questions on the topics covered.