Experimentalist approach works poorly in mathematics. Mathematics (formal science) and natural sciences (physics, biology, chemistry etc) demand different methodologies.

 An experimentalist approach is one of the known science approaches in physics (in very crude terms it may be called trial and error method). Another well known approach dominated in the second half of the 20th century is a theoretician approach (new phenomenon is predicted from the analysis of already existing theory and merely verified by experimentalist by using the exactly set experiment - like top quark or Higgs boson discovery). The present day huge crisis in fundamental physics may need for resolution the return back to experimentalist approach [1-3], however, it is the interesting question about methodology - may this method (experimentalist approach) be applied to other disciplines (for example math)? This publication describes an attempt and demonstrates that this is not a good idea.

    I tried "experiments" with well known Fibonacci sequence (already 900 years old and well researched). May be something new found through tinkering around (like experiment with electronic devices made from newly synthesized semiconductors or synthesis of new ferromagnetic or potential room temperature "superconductors" [4]). In physics it works well, not like in [4] but like discovery of high-temperature superconductors in 1986 [5]. 

    The original Fibonacci sequence is a linear recurrence relation:

F(n)=F(n-1)+F(n-2)

what if the sign is minus instead of plus? Kind of experiment without looking for general theory, in horizontal direction of knowledge search, not in vertical, without generalizations? And limiting research for whole numbers only?

F(n)=F(n-1)-F(n-2)

The result is unusual - it will generate a simple cycle. 

F(n-2)=0

F(n-1)=1

F(n)=F(n-1)-F(n-2)=1-0=1

F(n+1)=F(n)-F(n-1)=1-1=0

F(n+2)=F(n+1)-F(n)=0-1=-1

F(n+3)=F(n+2)-F(n+1)=-1-0=-1

F(n+4)=F(n+3)-F(n+2)=-1-(-1)=-1+1=0

F(n+5)=F(n+4)-F(n+3)=0-(-1)=0+1=1

F(n+6)=F(n+5)-F(n+4)=1-0=1

F(n+7)=F(n+6)-F(n+5)=1-1=0

F(n+8)=F(n+7)-F(n+6)=0-1=-1

F(n+9)=F(n+8)-F(n+7)=-1-0=-1

............

The sequence is obviously a simple cycle: 0,1,1,0,-1,-1,0,1,1,0,-1,-1,.......

This simple cycle is easily proved for any original numbers:

F(n-2)=a

F(n-1)=b

F(n)=F(n-1)-F(n-2)=b-a

F(n+1)=F(n)-F(n-1)=b-a-b=-a

F(n+2)=F(n+1)-F(n)=-a-(b-a)=-a-b+a=-b

F(n+3)=F(n+2)-F(n+1)=-b-(-a)=-b+a=a-b

F(n+4)=F(n+3)-F(n+2)=a-b-(-b)=a-b+b=a

F(n+5)=F(n+4)-F(n+3)=a-(a-b)=a-a+b=b

F(n+6)=F(n+5)-F(n+4)=b-a

F(n+7)=F(n+6)-F(n+5)=b-a-(b)=-a

F(n+8)=F(n+7)-F(n+6)=-a-(b-a)=-a-b+a=-b

F(n+9)=F(n+8)-F(n+7)=-b-(-a)=-b+a=a-b

The sequence is a cycle for any input numbers: a;b;b-a;-a;-b;a-b;a;b;b-a;-a;-b;a-b.......

So at first glance the experiment leads to something completely new, not associated with Fibonacci sequence at all. 

However, quick glance into theoretical approach (checking for general theory using calculus, differential equations, discrete mathematics) quickly explains that this is actually well known result, but not for Fibonacci sequence but for more general linear recurrence relations [6]. Actually there is a big branch of mathematics devoted to recurrence relations. In the simple case of linear recurrence relations of kind 

a_rx(n+r) + a_r-1x(n+r-1)+….+a₀x(n)=f

if f=0 and modality r=2 the second modality recurrence relation becomes:

a₂x(n+2) + a₁x(n+1) + a₀x(n)=0

And if  a₂=1, a₁=a₀=-1 it becomes: x(n+2)-x(n+1)-x(n)=0 or F(n)=F(n-1)+F(n-2) - Fibonacci sequence. And for a₁=-1, a₂=a₀=1 it becomes x(n+2)-x(n+1)+x(n)=0 or F(n)=F(n-1)-F(n-2) cyclic Fibonacci sequence with unusual properties obtained by experiment.

The general solution of recurrence relations of second modality a₂x(n+2) + a₁x(n+1) + a₀x(n)=0

is determined by the characteristic equation [6]: 

a₂λ²+a₁λ+a₀=0

(for the other modalities that would be a_rλ^r+a_r-1λ^r-1+…. + a₀=0 )

For Fibonacci sequence the characteristic equation is λ²-λ-1=0 with both roots are real - it will follow the solution described in [6].

For cyclic Fibonacci the equation is λ²-λ+1=0 and both roots are complex numbers - similar to linear differential equations the solution is oscillating (periodic). If the roots are degenerate (for characteristic equation λ²-2λ+1=0 both roots are equal to 1 the Fibonacci sequence becomes:

F(n)=2F(n-1)-F(n-2)

And this is the third type of "Fibonacci" sequence - neither exponential growth nor periodic but linear: 0,1,2,3,4,5,6,7,8,9

Strictly speaking only original sequence F(n)=F(n-1)+F(n-2) may be called Fibonacci, the rest is simply linear recurrence relations [6], there are infinite number of them and analysis is very easy if characteristic equation is used - if you use theoreticians approach. You may choose what period you need, figure out the necessary roots and arrive to linear recurrence relation without "trial and error" method. The difference may be illustrated by the scheme:

So in the case of mathematics the experimentalist approach is kind of useless and theoretician (mathematician) approach is the best. What is the problem with physics in this consideration? Why it is not working perfectly exactly like described?

The answer is that mathematics is  a formal science, not natural science. The general solution found on upper level works for any object on lower level without limitations. But physics is different [7,8] - the nature is unpredictable and impossible to guess. It is always "orthogonal" to what may be expected. 

Suppose Einstein never appeared with General Relativity. From experimental observations of Tycho Brahe   Kepler and Newton developed upper level of planets motion theory - Newton physics. It predicted discovery of planets Neptune, Pluto successfully and thus was considered as a general theory that must work everywhere (like general theory of recurrence relations in mathematics). Since Mercury has a problem with orbit, the Vulcan planet was predicted [9]. It was searched for decades with zero success but since the general Newton theory tells it must be present and in the absence of Einstein what would be the final conclusion? This is a dark planet! Because it must be present and because it is so close to Sun it is somehow invisible, only reveal itself by gravity (because the general Newton theory must hold). That sounds very familiar in the beginning of 21st century because it is exactly what modern physics is looking for: dark matter. This is where the physics deviates so much from mathematics - mathematical laws are absolute, works for any numbers and objects, physical laws are merely approximations, they have limitations sooner or later discovered. 

If mathematics would be also non-exact, the experimentalist approach would be vindicated. For example, the extensive search of linear recurrence relations may eventually lead to discovery of still "Fibonacci" sequence like constant or logarithmic "Fibonacci" sequences or whatever it turned out. It would be like mathematics established (greatly exaggerated)  that 1*1=1 and 2*2=4 but 3*3=8.999 and 10*10=99.3 or even sometimes 99.3, sometimes 100.2. Impossible for mathematics but sometimes happened in physics. 

Possibly the huge crisis in modern fundamental physics is exactly because of this - modern fundamental physics is theoretician (mathematicians) approach, while real physics needs both theoreticians and experimentalists approaches to move forward.

References.

1. Dmitriy S. Tipikin “The quest for new physics. An experimentalist approach. Where to find new physics?”// LAP Lambert Academic Publishing, 2021.

2011.0172v1.pdf (vixra.org) or https://www.researchgate.net/publication/353523212_The_quest_for_new_physics_An_experimentalist_approach

2. Dmitriy S. Tipikin “The quest for new physics. An experimentalist approach. Vol.2 Reflections on tired light hypothesis versus Big Bang” // LAP Lambert Academic Publishing, 2022.

2212.0058v1.pdf (vixra.org) or

https://www.researchgate.net/publication/366067523_The_quest_for_new_physics_An_experimentalist_approach_Vol2_The_second_book_on_the_topic_with_emphasis_on_certain_ideas#fullTextFileContent

3.Dmitriy S. Tipikin "The quest for new physics. An experimentalist approach. Vol.3 The new cosmology" // Published on Internet on January 7 2026.

2601.0030v1.pdf

https://vixra.org/pdf/2601.0030v1.pdf

4.LK-99 - Wikipedia

5.High-temperature superconductivity - Wikipedia

6.4recurrence.pdf

https://www.math.uci.edu/~ndonalds/math180b/4recurrence.pdf

7.Tipikin: Relation between theoreticians and experimentalists from historical perspective. Why so frequently the "small" experiment precedes the theory.

8.Tipikin: Superstring theory has in foundations tremendous "leap of faith" - why it is dangerous and why other approximations may be work better.

9.Vulcan (hypothetical planet) - Wikipedia

Fibonacci simple cycle. What if the sign is minus instead of plus?

 Fibonacci sequence is a marvel of mathematics and thoroughly investigated [1]. There are numerous generalizations of Fibonacci numbers (tribonacci, tetrabonacci etc), but I found no mention of simplifications of Fibonacci sequence, mainly Fibonacci cycle.

        The classical Fibonacci sequence is generated as follows:

F(n)=F(n-1)+F(n-2)

Now let's play around and replace sign "plus" to sign "minus". The result is unusual - it will generate a simple cycle. 

F(n-2)=0

F(n-1)=1

F(n)=F(n-1)-F(n-2)=1-0=1

F(n+1)=F(n)-F(n-1)=1-1=0

F(n+2)=F(n+1)-F(n)=0-1=-1

F(n+3)=F(n+2)-F(n+1)=-1-0=-1

F(n+4)=F(n+3)-F(n+2)=-1-(-1)=-1+1=0

F(n+5)=F(n+4)-F(n+3)=0-(-1)=0+1=1

F(n+6)=F(n+5)-F(n+4)=1-0=1

F(n+7)=F(n+6)-F(n+5)=1-1=0

F(n+8)=F(n+7)-F(n+6)=0-1=-1

F(n+9)=F(n+8)-F(n+7)=-1-0=-1

............

The sequence is obviously a simple cycle: 0,1,1,0,-1,-1,0,1,1,0,-1,-1,.......

This simple cycle is easily proved for any original numbers:

F(n-2)=a

F(n-1)=b

F(n)=F(n-1)-F(n-2)=b-a

F(n+1)=F(n)-F(n-1)=b-a-b=-a

F(n+2)=F(n+1)-F(n)=-a-(b-a)=-a-b+a=-b

F(n+3)=F(n+2)-F(n+1)=-b-(-a)=-b+a=a-b

F(n+4)=F(n+3)-F(n+2)=a-b-(-b)=a-b+b=a

F(n+5)=F(n+4)-F(n+3)=a-(a-b)=a-a+b=b

F(n+6)=F(n+5)-F(n+4)=b-a

F(n+7)=F(n+6)-F(n+5)=b-a-(b)=-a

F(n+8)=F(n+7)-F(n+6)=-a-(b-a)=-a-b+a=-b

F(n+9)=F(n+8)-F(n+7)=-b-(-a)=-b+a=a-b

The sequence is a cycle for any input numbers: a;b;b-a;-a;-b;a-b;a;b;b-a;-a;-b;a-b.......

The original Fibonacci sequence has a lot of implications in physical world [1]: golden ratio, spirals of pine cones, honeybees etc.

The cycle Fibonacci sequence has even more implications for physical world - oscillations are everywhere and time itself is based on the presence of repeating sequences (cycles). Yet the cycle sequence is very easy to obtain - the great question on the exam in elementary school to illustrate how small change may lead to great difference in behavior.

References.

1.Fibonacci sequence - Wikipedia

The quest for new physics. An experimentalist approach. Vol.3 The new cosmology.

 Vol.3 of my series devoted to the experimentalist approach for the search of New Physics is published on Research Gate:

(PDF) The quest for new physics. An experimentalist approach. Vol.3 The new cosmology

https://www.researchgate.net/publication/399529971_The_quest_for_new_physics_An_experimentalist_approach_Vol3_The_new_cosmology

Supernova GRB 250314A at z=7.3 clearly demonstrates angular size larger than resolution limit of JWST. Smaller objects at smaller z are clearly seen.

             Thanks to the NASA publication of the original image of the supernova at high Z (GRB 250314A) [1] everybody may demonstrate the new physics using simple manipulations with Paint program. After downloading the full image into Paint, I greatly multiplied it so the original image of supernova would be easy to see (it is merely a dot on the low-resolution image). Then I cut and copied part of the greatly amplified image into second Paint picture. On the original image (greatly expanded) I easily found a well resolved galaxy at low z (only low-z galaxy are possible to resolve even by JWST) which is not bright (no projectile-looking artefacts) and has the objects with visibly much smaller angular size (those objects at low z, where the light scattering is small are representing the real optical resolution of JWST). The result is here:

      In the second figure multiplication is even larger:

In this figure the galaxy on the left is well resolved (clearly below z~0.1) so the expected blurring is minimum, some objects (possibly bright star clusters or star associations inside the galaxy) are having small angular size (approximately the diffraction limit of telescope). The supernova at z=7.3 is clearly blurred and looks fuzzy due to light scattering present. It can not have any resolved angular size (at z=7.3 merely impossible, to resolve it the telescope must have the mirror of a size of a thousand of kilometers). Yet this very small in astrophysical sense object (just ~20 times larger than Pluto orbit) clearly demonstrates the angular size - the only possible explanation is that this is problem with light (almost certainly tired light theory is valid) and such observation is a strong objection against Big Bang.

There is one more figure with the participation of the largest galaxy (lowest Z) from original image:

          This exercise is similar to already published before [2-4] and demonstrates the presence of light scattering at high z directly. Galaxies may be big or small but supernova is a very compact object. It is only visible at the around maximum brightness and at that time (24-30 days after explosion) is merely 20 times larger than Pluto orbit. At the enormous distance of tens of billions of light years away it must be at the diffraction limit of telescope (be one of the smallest dots visible on image). It can not be resolved under any circumstances for Z>0.001 even by JWST. Unless the "primordial" supernovae (in Big Bang cosmology) are expanding at speed of around 100000 times speed of light they can not reached the size necessary for resolution by JWST. Observation of such objects having real angular size (clearly larger than diffraction limit of telescope) creates enormous stress on Big Bang theory and may be only reasonably explained by tired light theory (obviously after billions and trillions of small events of scattering the light is not only reddened due to energy loss but also a little scattered). 

References.

1.GRB 250314A Pull-out (NIRCam Image) - NASA Science

2.Tipikin: Little red dots and brown dwarfs – demonstration of the light scattering by point-like objects.

3.Tipikin: Supernova's large angular size due to light scattering for high z is clearly seen at multiple JWST images.

4.Tipikin: Supernova at Z=2.83 - large angular size, smaller objects on the same image, relatively weak to completely exclude detector saturation - one more confirmation of light scattering

Revealing the internal structure of electron - energy of 0.8 to 1.6 TeV (at least) and easier for muons

             In the previous post [1] the idea of the internal structure of the electron is discussed (based on the already discovered for quarks fractional charges, what makes the idea of electron (muon) holding primal, most basic integer charge unit somewhat improbable. But what would be the energy necessary to see such an internal structure of the electron? Is it easier for muon (obviously the electron and muon would have similar structures with merely different "electronic quarks" being present, muon merely having heavier "electronic quarks"). This is the discussion topic here.

            Obviously the constituents of the electron or muon can not be the same quarks as in proton or pion - they are way too heavy for this small (only 0.5 MeV for electron) particle. Let's call them "electronic quarks" to emphasize analogy to already discovered quarks - they most probably have the same fractional charges of 1/3, 2/3 as well. This also goes from pion known structure (consists of two quarks) and may go through W- boson to either muon or electron. Inside this W- boson the known quarks are converted into "electronic quarks" while keeping theirs primal basic charges thus transferring this integer charge (in reality combination of -2/3+(-1/3) or -2/3+(-2/3)+(1/3) or (-1/3)+(-1/3)+(-1/3) further to "elementary" particles like electron and/or muon). It is possible, of course that nature is so bizarre that after all it have two types of primal, most basic values of charges - both fractional and integer. As described in another blog, it is impossible to "outsmart" the nature [2] and every fundamental theoretical work only takes place after some small experimental result (like a hint in the correct direction), including the Standard Model, which was only possible after the high-energy particle experiments demonstrated some wrong results. For now the presence of fractional charges in quarks I consider as a hint that electron and muon are not elementary particles and may be split (at least the scattering experiments may reveal the complex nature of them because most probably the "electronic quarks" may not be isolated pretty much like already known quarks). 

            In order to make any evaluations of the energy and probability of splitting involved it would be necessary to switch to empirical consideration of the structure of electron. Instead of further development of Standard Model to include new types of "electronic quarks" (what would theoretician prefer) I would go to the concept of surface tension (similar to Aage Bohr [3]) (similar to liquid drop model [4]), because I am an experimentalist and very rough estimate is OK for me. The "surface tension" in composite (non-elementary) particles plays more and more important role as size becomes smaller and total energy larger since the smaller the size of the particle (and correspondingly the energy is larger) the more virtual particles may it excite from quantum vacuum (both discovered and undiscovered, either small energy or large energy, on all possible levels). In certain sense the quarks can not be separated because the force between them becomes larger with distance but because if somebody pumps the proton with energy in an attempt to split it the "surface layer" is becoming larger and larger and sorbing the energy inside, creating the bubble of virtual particles around those quarks preventing energy being delivered directly to quarks. In essence the properties of quantum vacuum as we know it preventing this event. It may be possible however, if the quantum vacuum around the proton is modified, if the event of strike takes place inside quark-gluon plasma, for example [5]. The importance of pure empirical "surface term" as size goes down may be demonstrated as follows:

A. Rydberg hydrogen atom considered as obtaining more energy through "surface term": diameter of around 1 micrometer [6] and added energy is 13.6 eV (ionization energy of hydrogen), the "surface term" energy would be E= - α*S, α=1.8*10exp(13) eV/m²

B. Nucleus of iron: radius is 4.6*10exp(-15) m and "surface term" from [4] is 17.23MeV*A^2/3

what yields E= - β*S, β=1.44*10exp(37) eV/m²

^{C. W- boson being considered as non-elementary particles: energy is 80 GeV (for simplicity it is assumed all energy now is from "surface term", the other terms are neglected for evaluation). Radius may be only estimated as 10exp(-17) m [7] and the "surface term" would be:}

E= - γ*S, γ=2.5*10exp(44) eV/m²

My way to explain why those "electronic quarks" are capable to be inside the particle like electron with mass only 0.5 MeV or muon with mass of 105.7 MeV is that they stay so tightly together, that the surface term due to minimization of size becomes very small too. Thus the size of "electron nucleus" may be estimated (from formula E= - γ*S) as 2.5*10exp(-20) m for electron and 3.7*10exp(-19) m for muon. Since electron is not considered here as elementary particle it may now have two sizes, similar to any atom which has Compton size of ~2.43*10exp(-12) m and Compton size of nucleus of ~1.32*10exp(-15) m. Now electron has the electric field determined Compton size of ~2.43*10exp(-12) m and "electronic quarks" determined size of 2.5*10exp(-20) m.

            The only way to evaluate how much energy would be necessary to see the internal structure of electron and muon (since they both have one progenitor W- boson) is to use analogy: quarks were visible for proton when accelerators reached around 10-20 GeV, W- boson is 80 times heavier, so the lowest level would be 0.8-1.6 TeV for electron-positron, muon-muon, electron-electron etc. beams energy. The predecessor of BAC was only able to reach 200 GeV for electron-positron beam, way below what is necessary. 

            Another way to evaluate energy is to find at what energy ultra-relativistic particle would have the De-Broglie wavelength of 2.5*10exp(-20) m (for electron internal structure) or 3.7*10exp(-19) m (for muon). From formula λ=h/p=h*c/p*c=h*c/E, where h is Plank's constant, c is speed of light and  E is energy of  the ultra-relativistic particle it follows: for electron energy is 7.956*10exp(-6) J and for muon it is 5.38*10exp(-7) J. Corresponding energy in TeV would be 50 TeV for electron and "merely" 3.36 TeV for muon. Again for muon it seems to be easier to be split apart (sense internal structure),  electron is by far tougher particle. 

            It is also enormously small probability of correct interaction in the beam - since the only way to see the structure is to see the scattering of nearly head to head collision, with size of only 2.5*10exp(-20) the scattering event would be very rare. Those accelerators will mainly generating the same stuff which was discovered on predecessor of BAC, possibly decades of experiment would be necessary to see such scattering, which would revealed the internal structure of electron and muon. It seems millions of times easier for muon, of course (since it is heavier, this energy may be reached in circular accelerator, much less energy loss due to synchrotronic radiation [1]; plus it is bigger, so the probability of correct strike is much higher). For electron the special linear accelerator is to be built, the expenses are expected to be enormous and such discovery may only prognosed for the next century (or may be 23d century). 

            As a conclusion - the hint from fractional charges of quarks means the shortest way for new physics in the higher energy direction includes electron or muon accelerator for 0.8-1.6 TeV (another evaluation 3.4-50 TeV) capable of working for decades in order to catch enormously rare event of scattering revealing the internal structure of those "elementary" particles. While it is possible from engineering point of view, such breakthrough would demand so enormous money that may be safely considered as sci-fi for now. Repeating [1] - new physics in the opposite direction (ultra-low energy particles, search for fifth force weaker than gravitational or in the gap between gravity and electromagnetism) would be much cheaper to reach.

References.

1.Tipikin: The quest for new physics. An experimentalist approach. The quest for new physics in high energy direction.

2.Tipikin: Relation between theoreticians and experimentalists from historical perspective. Why so frequently the "small" experiment precedes the theory.

3.https://gymarkiv.sdu.dk/MFM/kdvs/mfm%2020-29/mfm-26-14.pdf

4.https://www.personal.soton.ac.uk/ab1u06/teaching/phys3002/course/04_liquiddrop.pdf

5.Tipikin: Increase of the yield of the elementary particles created at collider through manipulations with quantum vacuum. A chemical approach to the elementary particle physics.

6.Rydberg atom - Wikipedia

7.W and Z bosons - Wikipedia

The quest for new physics. An experimentalist approach. The quest for new physics in high energy direction.

     The previous ideas were devoted mainly to the search of interactions weaker than gravitational or in the gap between the electromagnetic and gravitational force [1-3]. That seems like the cheapest and most under-investigated direction of research (for ultra low weight particles, ultra weak forces which may only reveal itself on the cosmic scales [4]). 

    However, there is a second more common direction of research in fundamental physics - towards the higher energies, as high as possible. The recent success of Standard Model instantly posed a question - what is next, where to go further? I already generated couple of ideas in this direction [5,6] and now it is time for the next one. 

    Assuming the Standard Model is valid in the areas the predictions coincide with experiment (on quark level) the fundamental electric charge (one of the oldest discoveries in physics) is not actually integer of charge of electron, but rather 1/3 (because quarks have charges 2/3, 1/3, positive and negative). Even artificial intelligence agrees with me, if asked through Google [7]:

"Fractional quark charges (like +2/3e or -1/3e) are more fundamental in describing particle substructure..."

They can not be seen outside the elementary particle like proton, this is true, but the presence of such fractional charges even inside the proton means that 1/3 and 2/3 charges are more primal compare to integer and possibly any integer is in essence combination of those fractions. That instantly means that electron is not elementary particle (because most probably it consists of some parts with 1/3 or 2/3 charge, despite most probably those constituents can not be separated similar to quarks). Of course, it is necessary to emphasize that the known quarks can not be inside the electrons (they have much higher mass and the whole Standard Model would fell apart if this is possible). That idea, however, is exactly New Physics - it is obviously goes beyond the known fundamental physics, does not contradict it (electron and muon are considered as elementary particles merely as a fit, as an approximation toward more general theory exactly like proton and neutron were considered as elementary particles 100 years ago). For example the neutron decay involves the conversion in charge from fractional number (in quarks) to the whole number (in electron) through the virtual W^- boson, which is very heavy (so may easily consists of many particles inside) and responsible for conversion down quark to up quark (manipulation with fractional charges) so it is a strong hint that in reality it is also a composite particles with constituents having charge 1/3, 2/3 and it is not split yet merely because it demands too much energy. The particle  W^- boson is electron  and most probably the fractional charges inside this boson is transferred to electron creating the unified charge of -1, being actually presented inside electrons like combination of -1/3, -1/3, -1/3 or -2/3, -2/3, +1/3 or even more complex. While nature is bizarre in it's laws and may have after all both fractional (1/3, 2/3) and integer numbers (-1, +1) being fundamental, there is a strong hint from history of science that only one variant predominates and electron, muon and other charged "elementary" particles are not really elementary, they only look elementary (because the fractional charge is more fundamental than integer).

     Why such discovery is not yet made? The answer is in experiment - during the rotation of charged particles in the synchrotron the energy loss due to synchrotron radiation is inversely proportional to non-relativistic  mass in the power of four [8]. Since proton is 1800 times more heavy it is dramatically easier to accelerate it in a classical circle accelerator compare to electron. While electrons are successfully accelerated in linear accelerators they can not yet reach the energies of TeV already reached for protons. 

    There is a serious push for even larger accelerator in the high energy community now and this hint toward discovery of internal structure of electron (muon actually should be easier to crack because of higher mass and possibly less "rigid" structure inside) may stir community interest toward linear accelerators of enormous power instead of more classical circles. In any case the money involved are huge and from my perspective much cheaper way to find New Physics stays in the opposite direction (toward as low energies as possible, see [1-4]

References.

1.THE QUEST FOR NEW PHYSICS An experimentalist approach: Where to find new physics?: Tipikin, Dmitriy: 9786204731735: Amazon.com: Books

2.https://www.morebooks.de/shop-ui/shop/product/9786205529867

3.https://www.researchgate.net/publication/375517684_Tired_light_hypothesis_possibly_got_confirmation_by_direct_observation_of_light_scattering

2406.0162v1.pdf

4.Tipikin: Axionic dark matter possibility from light scattering demonstrated by JWST for high z objects.

5.Tipikin: Quantum vacuum application to gravity: the Higgs boson antigravitational particle predicted

6.Tipikin: Increase of the yield of the elementary particles created at collider through manipulations with quantum vacuum. A chemical approach to the elementary particle physics.

7.is fractional charges of quarks are more fundamental than the integer charge or electron - Google Search

8.Lecture 9a Synchrotron Radiation.ppt

Galaxy as an open system with thermonuclear engine causing rotation in spiral galaxy.

          This idea was already considered in this blog [1]. Indeed, why we consider the rotation of stars in the galaxy being equivalent to planets? The stars are generating a lot of energy, and if the mechanism is present transforming only ~1% of this energy into rotation, it may easily explain the rotation curve of any galaxy. The hint is actually Tully-Fisher relation [2] - brighter galaxies are rotating faster compare to dimmer one. What if indeed such mechanism is present (and indeed the original relation was for brightness of the galaxies - amount of energy they are generating-  not about mass, it is merely assumed that brighter galaxy has higher mass, which after all may be not true). In[1] it was hypothesized that such idea is possible because of "heavy light" - light inside the stars is moving much slower (the effective refraction coefficient is >100) and thus gravitating strongly, but the direct evaluation of this effect using binaries demonstrated that this effect even if present is way too weak to be responsible for the too fast rotation of galaxies [3].

          However, recent observation of the very large sizes of supernovae at high Z [4] leads to the idea of the dark matter being represented by some ultralight particles (femto to pico eV total mass) which do interact with light somehow (not "dark matter" after all) [5]. The present day search for extremely light particles interacting with light and predicted by Standard Model (axions) is well justified in this content - it is approximately what the so-called "dark matter" most probably looks like. I am expecting they will be discovered relatively soon (unfortunately they are enormously light in mass and thus discovery is enormously difficult). And those particles interacting with light indeed easily explain the so-called cusp problem [6] (why in the center of galaxy like Andromeda galaxy the concentration of "dark matter" is so small, if this matter is interacting only gravitationally [7]).

          The other property of such "light matter" halo around the galaxy is that it may create the mechanical feedback loop necessary to transform energy of light into the rotation of the galaxy:

            It is obvious that in such a dynamic system due to the slightest always present fluctuations the static equilibrium will be broken and dynamic equilibrium will corresponds to mixture of rotation and oscillations. Because of the presence of stars generating energy the positive feedback will quickly transform the galaxy into the open system similar to vortex in the bathtub. If the energy  supply is high enough - it will generate the similar pattern and will rotate approximately as a solid body starting from certain distance from the center - exactly what is observed. If the energy is not enough - no rotation, the attrition between stars will work like  a friction [8] and prevent rotation like in an elliptical galaxy.

            Afterall the galaxy rotation being similar to the bathtub water swirl may be due to the fact that both of them are open systems and not because of strangely looking "dark matter" which in reality is "light matter" and responsible for the transformation of the energy of thermonuclear synthesis into the mechanical energy of rotation.

References.

1.Tipikin: Light matter attraction as a driving force for Galaxy rotation - the energy transformed from thermonuclear to mechanical

2.Tully–Fisher relation - Wikipedia

3.2007.0195v1.pdf

https://vixra.org/pdf/2007.0195v1.pdf

Tipikin: Weak equivalence principle check for non-barionic matter using eclipsing spectrometric binaries. No evidence for dark matter.

4.Tipikin: The higher Z, the stronger the effect of light scattering present in the supernova images. Supernova at Z=3.6 looks gigantic.

5.Tipikin: Static Universe and cosmological constant - back to original Einstein equation with empirical cosmological constant. Light matter absorbs light, heats, generates pressure and holds galaxies from coalescence.

6.Cuspy halo problem - Wikipedia

7.Dark matter fraction derived from the M31 rotation curve | Astronomy & Astrophysics (A&A)

https://www.aanda.org/articles/aa/full_html/2025/02/aa52753-24/aa52753-24.html

8.An introduction to dynamical friction in stellar systems | American Journal of Physics | AIP Publishing

https://pubs.aip.org/aapt/ajp/article-abstract/75/2/139/1056314/An-introduction-to-dynamical-friction-in-stellar?redirectedFrom=fulltext