You are right.

In my previous versions, I assumed that Hubble’s Law in the form $d=f(z)$ was implicitly covered by the use of the Doppler effect. It is not, and I plan to rewrite the section ‘4-SPHERE IN A NUTSHELL: DOPPLER REDSHIFT IN GALACTIC RECESSION’.In the meantime, from the relation $\beta = \theta = \frac{v_r}{c} = \frac{(1+z)^2-1}{(1+z)^2+1}$, we obtain the proper distance as $d = ct_{now}\theta$. However, it must be noted that this ‘proper distance’ is not particularly useful in this model because, given the absence of light-wavelength stretching, the distance that truly matters is the light-travel distance $d_l$ (the actual path covered by the light). These points need to be stated more clearly, especially in the sections introducing the model as a ‘Hubble Law’.“

Regarding inflation, I have not developed this aspect yet, as I have hypothesized that it occurs within the four spatial dimensions. This is an intriguing topic, and I plan to look into Fulvio Melia’s work on the matter for further insight.

I have revised the unclear part, but I am holding off on republication until I receive further corrections. The new text is provided below for those interested:

4-SPHERE IN A NUTSHELL: HUBBLE’S LAW AND DOPPLER REDSHIFT IN GALACTIC RECESSION

The model, named 4 Sphere, bases its physics on the expansion of a four dimensional spatial manifold, simplifying the dynamical description by considering the CMB in the absence of matter. Its definition of the Hubble constant arises from a linear relationship between galactic recession velocity and the arc angle (rather than the arc length). In this framework, Hubble’s law becomes: $$v_r=cƟ$$. In this framework, the redshift is not a consequence of metric space expansion (stretching of the wavelength), but rather a geometric result of light propagation within the growing 4-Sphere. This leads naturally to the Doppler formula: $$β=((1+z)^2-1)/((1+z)^2+1)$$ where β=v/c The remaining relations follow directly from the identification Ɵ= β. While the Doppler effect describes the recession velocity $$v_r$$, the observed redshift z must be reconciled with the light-travel distance. Consequently, the standard “proper distance” $$d_P=ct$$ Ɵ is a geometric property of the 4-sphere, but the relevant distance for observations is the path length covered by light: $$d_L=c∆t$$ (defined above). This model treats the redshift as either Doppler-like (for galactic recession) or gravitational (for the CMB), as both mechanisms preserve the blackbody spectrum, merely altering the temperature scale. The key to this speculation lies in the interpretation of Hubble’s Law and distance validation: Were JWST’s observations to suggest that there is no observable maximum limit in redshift—and that our inability to see farther stems solely from telescope sensitivity—then an asymptote is clearly the solution to the cosmological distance problem. The Doppler effect inherent in this angular-based recession naturally presents this asymptote, providing a framework consistent with galactic formation timescales.

@clmarchesan

I’m in the tired-light camp, so I think many of the problems you’re referring to can be resolved simply by assuming that the universe is Euclidean, infinite in all directions, static and eternally existent. However, I think your model is very elegant and I respect it, just as I respect @ExpEarth’s model.

I am not familiar with the ‘tired-light’ models, as my research has been focused primarily on analyzing existing $R=ct$ frameworks to see if anyone had already reached the same conclusions as mine. E.g., I found that Melia’s $R_h=ct$ model differs significantly from my own approach. Melia relies on the standard FLRW formalism, forcing the total energy density (matter plus radiation) to satisfy the $\rho+3p=0$ condition to achieve linearity. 4‑Sphere departs from the FLRW framework.

I appreciate your kind words about the elegance of my model. I believe we all have a ‘common adversary’ in the ΛCDM paradigm—which often forces us into complex ‘fixes’ to reconcile theory with observation—so it is refreshing to discuss alternative frameworks that aim for a more fundamental simplicity, whether they are static, tired-light, or geometric in origin.“

@clmarchesan . Let’s taste the pudding by eating it. You write: >In this framework, Hubble’s law becomes: $v_r=c\Theta$.

Then you write: >This leads naturally to the Doppler formula: $\beta = ((1+z)^2-1)/(((1+z)^2+1)$ where $\beta$ = v/c

Is that correct? Wikipedia says: for motion solely in the line of sight [ .(!). ], this equation reduces to:

$$ 1+z = \sqrt{\frac{1+v/c}{1-v/c}} $$

Patiently solving for $(v/c)$, step by step, and by hand:

$$ (1+z)^2 = \frac{1+v/c}{1-v/c} $$

$$ (1-v/c)(1+z)^2 = 1+v/c $$

$$ (1+z)^2-1 = (v/c)(1+z)^2+(v/c) = ((1+z)^2+1)(v/c) $$

$$ \beta = \frac{v}{c} = \frac{(1+z)^2-1}{(1+z)^2+1} $$

Sigh .. hope that hasn’t been too fast for you. (Of course not, Han: AI has it derived for him)

The remaining relations follow directly from the identification $\Theta=\beta$.

Sure, because then we certainly have that the velocity is the recessional velocity:

$$ v_r=c\Theta = c\beta = c\frac{v}{c} = v $$

the standard “proper distance” $d_P=ct\mbox{ }\Theta$ is

$$ d_P=ct\mbox{ }\Theta = v.t $$

Conclusion: distance = velocity x time. But wait !! Didn’t we know that already? For a couple of centuries or so?

If AI has lead you to these insights, then it has made NO errors. But I’m afraid that’s all. You are getting nowhere.

We agree that $$d_p = r Θ = ct Θ$$ where t is cosmic time. However, it does not automatically follow that the cosmological redshift must be of Doppler origin. The mathematical form may be identical, while the physical interpretation is different. That distinction is precisely what my model explores.

@clmarchesan . >it does not automatically follow that the cosmological redshift must be of Doppler origin.

Correct. However, looking up in Wikipedia we see that the formula as employed by you is in the table - quote - for Relativistic Doppler, Minkowski Space (flat spacetime), For motion completely in the radial or line-of-sight direction. Which is clearly different from Cosmological redshift, FLRW (expanding Big Bang universe), Hubble’s law in the next row of the same table. According to another Wikipedia page and/or Google Gemini too, Hubble’s law is mathematically expressed as $v = H_0\times d$, where $v$ is the recessional velocity, $d$ is the distance, and $H_0$ is the Hubble constant. Especially the Hubble constant is nowhere in the above writeup by @clmarchesan !!

I am routinely skipping all sort of eloquent prose - called “physics” by some individuals here. And go straight to the math. My English may be good enough for concatenating formulas, but Dutch is my own beautiful mother’s tongue. Any renaming or different labeling of poor mathematics does not help you to be convincing. Because it’s a simple fact that elementary algebra is not lying.

The key point is that my work is not a reformulation of the FLRW metric. It is an alternative cosmological model that abandons the FLRW metric and develops its own geometrical framework. Therefore, comparing individual equations directly with their FLRW counterparts, without first adopting the model’s definitions, can easily lead to misunderstandings.

Hey @clmarchesan, did you use AI to write this message? https://github.com/orgs/a-cosmology-group/discussions/379#discussioncomment-17467284. It has em dashes, and there’s a stray quotation mark at the end 👀

Also, I agree with @luisenrique-arch about tired light. If you’re not, it’s probably worth reading up on it before cooking up another theory.

Also, this part doesn’t really make sense to me: “This model is fundamentally simple, rooted in the geometry of the 3-sphere. I have termed it the ‘4-Sphere’ model.” “If my Grandmother had wheels she would have been a bike” https://www.youtube.com/watch?v=A-RfHC91Ewc

@clmarchesan writes: >4‑Sphere departs from the FLRW framework.

Really? Isn’t the FLRW universe often depicted too as a four-dimensional balloon expanding in a fifth dimension sorry: nothingness.

@ericcosmos writes: >about tired light. If you’re not, it’s probably worth reading up

Tired about tired light? Then Variable Mass is probably worth reading up !!

The same question applies to any secondary source, software package, or human colleague. Ultimately, scientific responsibility rests with the author, not with the tool.

@clmarchesan On p. 14 of DOI: 10.5281/zenodo.14986654, you write

There is a vast literature on methodologies that consider the anisotropy of elliptical galaxies. This is what ChatGPT, the AI (https://openai.com/), says about it

The text that follows cites no references, so the reader cannot know if it is reliable or only hallucinating AI slop. If there is a vast litterature, cite it!

The same applies to your Appendix 3, no reference is cited in the text, so it’s considered AI hallucination until you show that it’s not.

That is unacceptable since your claims cannot be verified independently.

Scientific responsibility rests with the author who has to show that reliable tools have been used; that’s not what you do.

Thanks, I’ve now added the most relevant references from the literature in both points discussed, to improve clarity and traceability. I will review the rest of the manuscript to ensure consistency in the same regard.

Appendix 3 has been revised.

Appendix 3 has been revised. The AI-generated summary is now supported by bibliographic references and is intended only as a qualitative summary of selected aspects of the standard ΛCDM framework, providing the background for the subsequent qualitative discussion of the 4-Sphere model.

There is an issue with AI tools. Read the disclaimer on ChatGPT:

Output may not always be accurate. You should not rely on Output… Our Services may provide incomplete, incorrect, or offensive Output that does not represent OpenAI’s views. If Output references any third party products or services, it doesn’t mean the third party endorses or is affiliated with OpenAI.

Yes, I am aware of the limitations of AI tools. It has been used as a support tool for drafting and calculations, with consistency checked within the assumptions of the model.

Yes, I am aware of the limitations of AI tools.

It’s far more dangerous than AI having “limitations”. Read The Claude Delusion. BTW, Claude, Claudia, Claudius and … Claudio ?!

I also created my own framework and speculation about the origin of the universe etc. However, I really only thought that this was just a fun project without any serious background, until I found a possibility to calculate the gravitational constant G which turned out to match the latest Schlamminger results from April this year. Since the same mechanism also provided other universal constants, it obviously is not numerology.

Still, this is highly speculative, and I would not claim that this is how the universe works. If you are interested, have a look at DOI 10.5281/zenodo.20306533. The geometric models within are from me, claude ai helped a lot with text and calculations.

@azouhr . If you only would be so kind to explain the math symbols in this copy from your paper:

And where can I find §C.1.7.5. ??

@HanDeBruijn The companion is on the same site like the original PDF, just have a look below the PDF display. It is not directly displayed, but available for download or preview.

• l_P (ell-sub-P) is the Planck length, about 1.6 × 10⁻³⁵ metres — the fundamental length quantum of the gravitational condensation.
• m_P is the Planck mass, about 2.2 × 10⁻⁸ kilograms.
• a₀ is the Bohr radius, about 5.3 × 10⁻¹¹ metres — the natural size of the hydrogen atom.
• α (alpha) is the fine structure constant, approximately 1/137 — the dimensionless measure of how strongly electromagnetism couples.
• m_e is the electron mass, about 9.1 × 10⁻³¹ kilograms.
• ħ (h-bar) is the reduced Planck constant.
• c is the speed of light.
• λ_C, the reduced Compton wavelength
• F_EM, the electromagnetic force scale
• G₀, the leading-order gravitational constant

@azouhr I am still reading through which seems to be rather abstract. If someone is ever ready to prove that we live on the surface of a hard-disk then I do not want to know that! :)

Schlamminger is not happy with his 2026 result for G but I think you might know that. Just stating it here for others to consider.

Your equations are not numbered but you have $$m_e → G₀ = ħc/(m_e² F_{EM}²)$$ with $$F_{EM}$$ = 2⁶⁸ × 3⁴. From which you claim to calculate the value of G. The value of $$F_{EM}$$ = 2⁶⁸ × 3⁴ equates to $$M_P/m_e$$ which makes the equation above to be the known Planck relation for G $$G = ħc/m_P² $$ ($$m_e$$ cancels out). Seems to be circular reasoning unless you show $$F_{EM}$$ did not come from inserting $$m_e$$.

I am interested though to how you got to the value $$F_{EM}$$ = 2⁶⁸ × 3⁴ without inserting $$m_e$$ if that is as you claim.

If you agree that The value of $$F_{EM}$$ = 2⁶⁸ × 3⁴ equates to $$M_P/m_e$$, … Does your derived ladder scale reveal similar relations for other fundamental particles, e.g. muon and tau? (again - without inserting their mass in a $$M_P$$ relation).

@Francois-Zinserling Thanks for pushing on this - you’re asking exactly the right questions.

First, the Schlamminger point: yes, I’m aware he’s not fully happy with his own 2026 number. Good to have it stated here for others reading along.

On the circularity: you’re right that F_EM = 2⁶⁸×3⁴ is close to M_P/m_e - about 0.06% off. Let me separate two things I was blurring together before: which note a condensation sits at, and which octave.

The note (which of the five positions - C, G, D, A, E - an optimization occupies) comes from a genuinely independent argument: it’s fixed by the quint circle, geometrically forced, the same mechanism that gives the Wyler fine-structure constant its geometric home. That holds for every condensation, gravity included, and doesn’t insert m_e or G anywhere.

The octave is where it gets more mixed. For gravity specifically, the octave is independently forced too - gravity is the one condensation that’s both-sided (reads both faces of the boundary), and that structurally requires it to sit on the one rung whose ratio to the Planck mode is a pure power of 2, no mass needed. That argument is special to gravity, though - it doesn’t carry over to EM, QCD, or the weak sector, which are inner-sided and free to carry additional structure. For EM specifically, the octave is currently fixed by matching to the measured electron mass, not derived the way gravity’s is - and when I traced that back, it leans on a length scale whose own history uses the measured Planck length and Hubble constant. So: no, I can’t currently show you F_EM without inserting m_e, and further back, without something that traces to G. That’s an open problem, not a solved one, and I was overstating things by implying otherwise. What I can say in its favor: it’s not an arbitrary fit - it lands exactly on the same rational comb (powers of 2 and 3) the rest of the ladder is built from, not a free real number. Weaker than a derivation, but not nothing.

On muon/tau: the answer right now is I don’t know yet, and I’d rather say that than claim a generalization I haven’t checked. My own muon numbers are still being reconciled internally - I have two calculations that don’t yet agree with each other, so I’m not going to claim they support a clean M_P-relation until that’s sorted out. If the same test fails for muon/tau, that’s real information, and I’d rather find that out and say so than assume it works.

Thanks for the reply @azouhr In my gravity model I have a significant parameter η (not mine initially but produced by Rueda and Haisch as a vacuum interaction parameter for particles of mass). I have since derived that this η equates to e.g. $$m_e/m_P$$ for the electron, and similar for muon, tau and other fundamental particles. As you can see $$η<<1$$ (Rueda and Haisch constrained it to $$0≤η<1$$) with an immediate conclusion that the mass off all fundamental particles << Planck mass, iow $$m_{fi}<<m_P$$.

The Planck mass relation turned out crucial but since my mission is about understanding nature I would like to find more about η and what makes it different for the particles. In relation to your work - for the electron - $$η_e = 1/F_{EM}$$. I still can’t wrap my head around your abstract model but the geometry might lead to a deeper intuition if you find something.

@azouhr writes:

it lands exactly on the same rational comb (powers of 2 and 3)

Don’t know if it is related but Can any positive real be approximated as 2^m/3^n with (m,n) large enough? Quote from bottom answer: FWIW, $log_23$ is rather important in the mathematical music theory of equally-tempered scales. The standard 12 tone scale uses the convergent 19/12.

Now I wish I had paid more attention in music class because it is logical that the scaling of fundamental particle masses might be in resonance patterns.

@HanDeBruijn writes:

Don’t know if it is related but Can any positive real be approximated as 2^m/3^n with (m,n) large enough? Quote from bottom answer: FWIW, l o g 2 3 is rather important in the mathematical music theory of equally-tempered scales. The standard 12 tone scale uses the convergent 19/12.

While you found a general pattern, the values for m and n are not that variable in the framework. n must follow the quint circle - with just 12 steps, and m is limited by the 103 octaves.

I see however that this was not really clearly readable in the article, and I just posted an update that should make things clear.

@azouhr . As a musician I have no problems with the quint circle - with just 12 steps. But where does the limited by the 103 octaves come from?

@azouhr . As a musician I have no problems with the quint circle - with just 12 steps. But where does the limited by the 103 octaves come from?

@HanDeBruijn This comes originally from the megafith concept - and it really was fitted. If you look at 12 megafith - 2^8x(3/2), which basically gives each quint 8 octaves space, you get exactly to the 103 octaves: 12 × log_2(384) = 103.01955 octaves with the overshoot being exactly the pythagorean comma. I do have a new, independent confirmation of the 103 in the meantime, and I’m preparing a journal article that lays that out, but the result itself doesn’t change.

@azouhr . When asking Google (Gemini) what “megafith” means, I get no answer. I see that your calculation is alright but I cannot make sense of it.

Op@azouhr . The following copy is again from On Pythagoras within the Universe.

Three remarks.

1. The expression for the fine-structure constant $\alpha$ is certainly interesting. Google says: In 1969, physicist Armand Wyler proposed a controversial geometric formula for the fine-structure constant ($\alpha$), a fundamental physical constant defining the strength of the electromagnetic interaction. By utilizing the Euclidean volumes of bounded spaces, Wyler derived the famous approximation:

$$ \alpha = \frac{9}{8\pi^4}\left(\frac{\pi^5}{2^4 5!}\right)^{1/4} $$

      But there are other derivations, such as the one by Hans de Vries in The fine structure constant.

2. The gravitational constant $G$ has been subject of a reply by @Francois-Zinserling and that’s it for me (& for the moment being).

3. Quantity: MOND acceleration scale $a_0$. Derived: $cH_0/(2\pi)$. Measured: exact. Agreement: Verlinde.

• Derived here and here for the ($2\pi$ vs. $6$) problem: IMHO it is neither of these two !!

$$ a_0 = Hc \times \begin{cases} 1/6 & \approx 0.16667 \ 1/(2\pi) & \approx 0.15915 \ 1/(2W) & \approx 0.14777 & \mbox{<== !!} \end{cases} $$

      $a_0=$ MOND acceleration, $H=$ Hubble parameter, $c=$ speed of light in empty space, $W\approx 3.3836$ (typical for VM).

• Verlinde: Entropic gravity is a rather baroque construction [ .. ] to [ .. ] reproduce [ .. ] Newtonian gravity.

@HanDeBruijn sorry, I still have to do a full-time job and I am often slower than I like myself with answering. Still, I have to admit, that quite some of the statements are more confident than appropriate. I also understand, that the motivation for the lattice wasn’t formulated good, and since the whole project is still in flow, the reasoning for facts change over time or are even falsified. Still, the G derivation is one of the pieces that did not change since I published it - only the reasoning about things like 103 Octaves. So, here is a current version:

The 103 octaves can be deduced from the following:

Twelve fifths overshoot seven octaves by the Pythagorean comma, PC = 3¹²/2¹⁹ ≈ 1.01364 - expressed as an octave-distance, ε = log₂(PC) ≈ 0.01955 octaves, the finest feature this lattice contains. By Nyquist–Shannon, resolving a feature that fine (rather than letting it alias away) needs a sampling depth above 2/ε ≈ 102.3.

N > 2/ε ≈ 102.3.

The smallest integer satisfying that is N = 103. At N = 102 the comma spans 1.995 rungs, just below threshold.

This means, it is a requirement from information density, and with less octaves, the pythagorean comma would not be resolved correctly.

This is what Google AI says:

        : the Nyquist-Shannon sampling theorem.

@azouhr . >the 103-octave ladder from the Planck scale down through the electron [ .. ]

From what I understand you have merely devised a numerology (yes!) that can be replaced by a tempered scale eventually :

$$ m_{EP} = m_P \times 2^{-(n/12)} \quad \mbox{with} \quad n = 0,1,2, \cdots , 12\times 103 $$

where $m_{EP}=$ mass of elementary particle, $m_P=$ Planck mass.

The tempered scale (base 2) in music is essentially the same as the familiar logarithmic scale (base 10) in physics. @RedshiftDrift has pictured the difference between the latter and a linear scale:

Summarizing. I think that your overly complicated theory is more or less equivalent to matching and stretching numbers on a discrete logarithmic scale wih sufficient refinement in the upper “notes” as to correspond with physical quantities.

claude ai helped a lot with text and calculations. [ .. ] (btw, also supported by claude code),

Numerology. Am I right? Who knows, because your way of publicing with Claude is a mess. Such “help” of AI makes that people cannot judge if somebody is a genious or an idiot. Anyway it is not fair to ask from a human being to wade through the verbosity it generates.

So here we go again: Bye bye Claude !!

P.S. Another terrific example from Zenodo. The author has made use of AI. Geometric Theory of Physics: The $\large ^6\Pi_4$ Model - Technical Compendium & Empirical Evidence

@HanDeBruijn are you missing a bracket there somewhere? Not sure if I can try some calculations from there, or have you tried to find mass of other fundamental particles? For n=0 the result ~ $$m_{ep}=~m_P/9$$ and there is no fundamental particle that massive. (even if bracket is missing since $$2^0=1$$) I find the $$1/m_e$$ problematic because it scales everything from electron mass. Not saying it can’t but it seems a stretch.

Interesting that $$2^{103}$$ is almost $$1/m_e$$ but not quite, and as before the combo was ‘almost’ the Planck mass.

Sorry @Francois-Zinserling , I’ ve made a mistake. Can you please check if the reply has been improved now? I have become really tired of this, you know…

Update

Okay, let’s proceed:

$$ m_e = m_P \times 2^{(-n/12)} $$

$$ -\ln(m_P/m_e) = -(n/12).\ln(2) $$

$$ n = 12\frac{\ln(m_P/m_e)}{\ln(2)} $$

Calculation:

m_e := 9.1093837*10^(-31);
m_P := 2.176434*10^(-8);
n := evalf(12*ln(m_P/m_e)/ln(2));

Outcome: n = 892 = 74*12 + 4 = 74 octaves out of 103 and note E in octave 75. Check:

m_e := evalf(m_P*2^(-892/12));

Giving $m_e \approx 9.145 \times 10^{-31} \mbox{ } kg$.

@HanDeBruijn Let me ask: do you actually agree that the very model could explain where G comes from? Let me summarize:

• A boundery is a surface, where every entropy place is fully set
• The boundery itself has only imaginary time - and thus cannot be modified in real time
• Looking at a black hole boundery, we find that only gravitational constraints are visible from outside, the rest is also visible from the inside
• The gravitational constraints from outside led to the idea that the boundery is just a redirection from some outermost boundery
• Applying Verlinde gravitational entropy leads to the assumption that the outermost boundery is a mirror - it reflects the gravitational entropy back into real.
• Also from Verlinde: next to the universal boundery, there is an imaginary boundery with different entropy
• And that is the first main result: G comes from the entropy difference between a real and imaginary boundery

The thing that you were looking at, and that obviously has to contain assumptions is the attempt to build a model that explains how those two bounderies were created in the first place. I only found that one explanation with the caley-dickson doubling along the Hurwitz theorem - however, that very model can be used to then run through all the different optimization condensations like Weak W/Z, QCD, EM and finally Gravity.

The thing that you are struggeling with seems to be only the very last step: What relations does the Cayley Dickson doubling follow. And the reason is, that if you get that right, you can determine all of those constants from a theoretical standpoint.

There is lots more to the theory in the meantime, but I accept the responsibility to first clarify all the bits and pieces and since I am limited with time, and I also have to learn a lot, this takes time. So, if someone wanted to help me with that, fine - but as it stands now, all the remaining parts will take considerably more time - and that includes Muon, Tau and other fundamental particles that would in principle already be solved.

Ahh .. and before I forget: I originally found the pythagorean scale with the constraints and just reapplied to the forces. And no, there is no equal tempered scale that allows to match the energies of the different constraints there. All the others - and I only know the german words like gleichschwebend or wohltemperiert - would not work, and I also would not have expected that the universe works with a relation that is not rational.

@Francois-Zinserling . Just to let you know that I have made significant updates of my last two replies. They were both wrong again. (Sigh !!)

@HanDeBruijn once more into the fray … I replaced the n in your $$2^{-n/12}$$ with ‘a’ in the table below so as not to confuse with the η of Rueda and Haisch below. From what I can see it’s the same thing because I have it as $$m=ηm_P$$ thus $$η=2^{-n/12}$$ according to our hacks. Excuse the attempt at a table below.

The a Table (Coupling Exponent) Particle…..Mass (kg)………………………..η=m/m_P……………….. $$a=-12*{log}_2*η$$

Electron…. $$9.11×10^{−31}$$ ……………… $$4.18×10^{−23}$$ ……………..~893.7 Muon…….. $$1.88×10^{−28}$$ ……………… $$8.64×10^{−21}$$ ……………..~800.4 Tau…………. $$3.17×10^{−27}$$ ……………… $$1.46×10^{−19}$$ ……………..~746.0

The results are not true integers but there could also be a small error. As for the notes and octaves I don’t have a clue how to convert, nor if there is a pattern to be seen there.

@Francois-Zinserling For notes, you can create a template from a quint circle, octaves are actually just powers of 2 (exact) that relate to the respective quint. IMHO, the octaves are not interesting from a harmonics perspective, and so they are also not interesting here.

@Francois-Zinserling .

I think you mean   $a=-12.\log_2 \eta$   but OK, minor nitpicking: thanks for the table !!

As for the notes and octaves I don’t have a clue how to convert, nor if there is a pattern to be seen there.

Muon : 800.4 $\approx$ 800 = 66 x 12 + 8 = octave 66 note 8 in { C , C# , D , D# , E , F , F# , G , G# , A , A# , B } = G# Tau : 746 = 62 x 12 + 2 = octave 62 note 2 = D in the above set.

Important: here all counting starts with 0. Hope you have a clue how to convert now.   See the pattern ? I don’t.

@clmarchesan

Are you Italian? I’m asking because Claudio is an Italian name.

I know this comment is off topic, but I promise I won’t derail the discussion too much. I was just curious.

Yes, I am Italian. I use AI-assisted tools for language support when writing in English.

@clmarchesan

I’m Italian too! I’m from Naples.

Do you have an email address? I’d like to write you in private to share some alternative cosmological models you might find interesting.

My email is available on my papers. I prefer to keep the discussion here in the public thread, so that it remains open and transparent for everyone interested. These groups are meant exactly for this kind of open exchange.

@clmarchesan

The paper you first posted seemed unclear to me, but I can understand a bit more in your paper “Dynamics…:” There is some slight overlap with my own model of a shell black hole universe #256 in that you have an exterior surface being supported by the CMB.

But there are some problems with what you are saying. First off, I don’t think the CMB is supplying “surface tension”. Instead, it should be supplying the outward pressure to keep your surface inflated. Secondly, I think you are using the local 2.7 K CMB, though you have not specified. A 2.7 K CMB has so little energy in it that it could not support anything. Instead, I suggest you use the idea that the CMB temperature increases as you move outwards to the surface, following the (1 + z) relation. In my model, the CMB temperature is 29 K near the shell, which supplies 10^4 the energy and pressure.

Even still there is a last problem which still needs solving. In general relativity, radiation and pressure are considered sources of extra gravity. The CMB, in this view, would actually cause the surface to collapse even more. I am not sure GR is correct on this point and maybe people here can comment on that. So actually you need some other energy source to prevent the collapse in GR - some form of dark energy or Einstein’s cosmological constant.

Thank you for your observations. I would like to clarify a fundamental aspect of my framework that likely explains the difference in interpretation: my model is based on a closed $$S^3$$ manifold, i.e. a compact space without boundary or external embedding.

In this context, there is no physically meaningful “exterior” region, and therefore no external radiation pressure acting on the system. The dynamics must be described entirely in terms of intrinsic curvature and the internal energy–momentum content. For this reason, the role of the CMB in my framework is not related to an external pressure or surrounding environment, but is instead incorporated into the internal geometric-energy balance, as detailed in the paper “Dynamics…: you referenced.

“Surface tension” is used purely as an analogy for an effective internal balancing mechanism and does not correspond to a physical interface or an external pressure in an embedding space. All dynamics are intrinsic to the closed S^3 geometry.

A term δW is introduced as the effective work associated with deviations from the equilibrium configuration of the closed $$S^3$$ universe. This contribution vanishes in equilibrium, where the geometry is self-sustained by the intrinsic curvature–energy balance.

Even still there is a last problem which still needs solving. In general relativity, radiation and pressure are considered sources of extra gravity. The CMB, in this view, would actually cause the surface to collapse even more. I am not sure GR is correct on this point and maybe people here can comment on that. So actually you need some other energy source to prevent the collapse in GR - some form of dark energy or Einstein’s cosmological constant.

This part of GR does not sit well with me either. Before I go blabbing I should probably check if it had been experimentally verified - which I think must be difficult to achieve since gravity is already so weak. No, ‘expanding space’ is not the evidence I’m looking for.

I think we can already agree that GR is slightly faulty or at best incomplete. It requires dark matter and dark energy for the geometric model to work. These are not minor flaws; it’s actually huge. Unless we find DM or DE, which we haven’t for 50 years, GR has a flaw. Not that it doesn’t deserve some of its god-mode status, but what’s happening now is the rest of the universe and physics have to adapt to GR instead of the other way round. Hence the cosmological constant. That’s what we do - we fix a flaw not by taking a step back, but by adding another flaw.

Back to the self-gravitating radiation. I think this is a flaw of the GR model too, and also the main reason QFT and GR cannot be unified. QFT wants the vacuum at $$10^{113} J/m^3$$ and GR wants it, well, almost zero. Talk about incompatible! In GR anything a little bit above zero must gravitationally self-collapse. Since that is not happening the consensus is that space must be ‘zero’ while the consensus should be ‘GR is wrong about this too’.

I sometimes wonder if Einstein had intended for it to be interpreted in that way, or if it just ran away wtih him. It would have made sense to include ‘energy absorbed in matter’ into the metric, but free radiation - nope.

Now I am not shy to announce my bias toward aether solutions, which cannot have an energy density ~0. After all, push-gravity cannot work in a zero vacuum.

No surprise, but we are definitely on the same page here! It’s a mystery to me how one goes from Einstein’s initial conception of GR to the idea that radiation and even radiation pressure act as positive terms in gravity. In my own gravity model it is the CMB (at 29 K) which causes gravity. It can’t be completely compatible with GR. But in a sense it is line with GR, since radiation energy and gravity are linked. But in GR radiation pressure adds a separate contribution to gravity apart from energy. I don’t see it. Radiation pressure has to counter gravity. On the vacuum density GR has it right, at least according to my model. The vacuum density is just the same as the CMB density at each point in space.

@clmarchesan , @luisenrique-arch .

Thanks, Luis !! I have saved that paper in a safe place.

This model is very similar to the static universe originally proposed by Einstein in 1917. It describes the universe as a four-dimensional hypersphere that is closed in on itself but has no boundaries.

It is very similar to the model by Claudio too, except for the CMB:

@luisenrique-arch The model is complex. There are others in the group with a much deeper expertise on the ‘Tired Light’ hypothesis and static universe models. While I find these alternative perspectives intellectually stimulating, I don’t feel I have the necessary background to take a firm stance. When it comes to analyzing an alternative cosmological model, you might get a much more comprehensive answer by reading Challenges to the Lambda CDM Cosmology (From a Talk presented at the Royal Society Discussion Meeting ‘Challenges to the standard cosmological model)’, which I mentioned in DOI:10.5281/zenodo.18458109.

@clmarchesan

Most people on this forum already know Crawford’s model, or at least they’re familiar with it. So I don’t really need to ask for their opinions. I brought up the model because I wanted to know what YOU think about it, especially since Crawford’s model has a few things in common with your own.

I know it’s a complex model and I know you’re not a theoretical cosmologist, so I’m not expecting a detailed technical analysis or anything like that. I only wanted to hear your general opinion.

@luisenrique-arch “I don’t support it anymore because, in my opinion, it conflicts with some of the more recent observations made by the James Webb Space Telescope.” Could you elaborate on that part? What seemed to conflict with what the James Webb Space Telescope has been seeing more recently?

@ericcosmos

As I already explained in the comment I posted above, Curvature Cosmology describes a relativistic, hyperspherical, closed universe very similar to the one originally proposed by Einstein in 1917. In this framework, just like in Einstein’s static universe, a beam of light emitted from a given point could, in principle, travel around the universe indefinitely due to its global geometry.

In the paper I shared earlier, Crawford himself states that, given the geometric structure of his model, the maximum observable redshift is z = 22.141. In other words, within Curvature Cosmology, redshifts higher than z = 22.141 are not physically possible, and the observation of any object with a redshift above this limit would falsify the model.

Now, JWST has observed what are known as “Little Red Dots.” In standard Big Bang cosmology, these objects are typically interpreted as extremely compact and highly massive systems, because their inferred properties depend heavily on distances and angular sizes derived from the expansion-based cosmological model.

However, if we instead adopt distance and angular size estimates based on a tired-light framework, these Little Red Dots wouldn’t necessarily need to be interpreted as unusually compact or extreme objects. They could, in principle, be fairly normal galaxies with extremely high redshift values.

The issue is that, in order for them to be interpreted as ordinary galaxies within a tired-light model, their redshifts would have to be significantly higher than z = 22.141, likely on the order of z = 28 or even z = 30. Since Curvature Cosmology doesn’t allow redshifts above z = 22.141, it can’t accommodate the existence of the Little Red Dots.

On this topic, I looked up the idea of light traveling around the universe. Oddly enough, Einstein wasn’t the first to suggest it. A William Kingdon Clifford brought it up before he did. Crawford should’ve given Clifford a shout-out.

Yeah, I agree with your point about “fairly normal galaxies with extremely high redshift values.” LRD are FNG. The F can stand for whatever you like.

You said the redshifts are impossible, but when I checked the numbers, the highest confirmed redshift so far is 14 point something. Where did the 28 or 30 numbers come from? Why isn’t Google showing 28 or 30 is the highest?

when I checked the numbers, the highest confirmed redshift so far is 14 point something. Where did the 28 or 30 numbers come from?

With the Variable Mass theory: It turns out that there is an absolute maximum of 28.5 for the intrinsic cosmological redshift, as has been concluded with formula (12) in the accompanying paper. Nothing I can ever “improve” on that number. There is no truth without risk.

@ericcosmos

According to @RedshiftDrift, within a tired-light model, Little Red Dots can be interpreted as ordinary galaxies with redshifts at least four times higher than the currently estimated values. He explained his reasoning here. Personally, I consider him to be an expert in tired-light and therefore trust his judgment.

Since most Little Red Dots have an estimated redshift of z = 5 or z = 6, multiplying that by four would give z = 20 or z = 24. The most distant Little Red Dot discovered so far has an estimated redshift of z = 9, which would correspond to z = 36 under this interpretation. So yes, we’d be talking about redshifts starting at around z = 24 and extending beyond that. And that doesn’t sit well with Crawford’s model.

Can you help me understand this a bit better? Also, is z = 9 really the upper limit, when we’ve already seen a galaxy at around z = 14? If that’s the case, your idea could even scale up by a factor of four or so, like getting to 56, if you really wanted to push it.

What I don’t understand is @RedshiftDrift’s correction to multiply the redshift by four. The redshift is a measurement and you can’t just change it. Was the idea to multiply the distance by four? But I’m still not clear on how that step is being justified. Saying we should just multiply all redshifts by four feels like a pretty big leap. I looked into the Balmer break and Lyman break, and as far as I can tell, if hydrogen gas is adding some extra redshift on top of the cosmological one, wouldn’t that mean the observed redshift has been inflated? In that case, it feels like you’d need to correct for the hydrogen gas, divide rather than multiply, to recover the true cosmological distance. That actually puts the galaxy a lot closer to us, not farther, meaning the universe would be smaller than what the Crawford model suggested.

@ericcosmos

if you re-read my comment, you’ll notice I never said the highest redshift ever detected is z = 9. What I said was that the Little Red Dot with the highest redshift has a value of z = 9. That’s a different claim. Also, @RedshiftDrift never said that all redshifts should be multiplied by four. He was only talking about the Little Red Dots.

I’m just an interested amateur, so there are many things I don’t know or fully understand. In some respects, I tend to place a lot of trust in other people. Since @RedshiftDrift is the leading proponent of tired-light in this forum, and has even built a cosmology calculator to compare the distance predictions of tired-light with those of the Big Bang, I tend to trust what he says. If he says it’s reasonable to multiply the redshifts of the Little Red Dots by four, I trust him. He’s the one with the PhD in physics, not me. I’m just a random guy on the internet who happens to love these topics.

@ericcosmos I just found this unanswered post:

What I don’t understand is @RedshiftDrift’s correction to multiply the redshift by four. The redshift is a measurement and you can’t just change it. - @ericcosmos

The redshift measurement doesn’t change, but the value changes if there is a mistake on the identification of the spectrum. The Balmer break ($\lambda = 364\mbox{ nm}$ shifted by z = 7 looks similar to the Lyman break ($\lambda = 91\mbox{ nm}$) shifted by z = 31.

See also https://github.com/orgs/a-cosmology-group/discussions/367#discussioncomment-16950550

if hydrogen gas is adding some extra redshift on top of the cosmological one, wouldn’t that mean the observed redshift has been inflated? - @ericcosmos

The hydrogen-redshift can only be there if the LRDs have a Lyman-alpha forest, i.e. they are seen through atomic hydrogen. I don’t think this has been demonstrated yet, so there’s not much we can say with certainty in this case.

Speculating: with a galaxy at STz-electron redshift $z_G = 24.6$ seen through hydrogen that adds a STz-hydrogen redshift of $z_H = 0.25$, the law of redshift composition gives

$$ 4 \times (1 + z_P) = (1 + z_G) \times (1 + z_H) $$

with a factor 4 due to the Balmer/Lyman misinterpretation. This gives a published redshift value $z_P = 7$. Because the spectrum has been distorted by STz-hydrogen redshift, the Lyman break is difficult to recognize and misinterpreted as a Balmer break shifted by z = 7.

The factor of ~4 is due to the Balmer/Lyman misinterpretation, the error in recognizing the spectrum is a distortion due to hydrogen (if present). Now astronomers are recognizing that LRDs may be galaxies!

Crawford’s model is explicitly static—it denies the expansion of the universe entirely—whereas my model accepts the core premise of the Big Bang and an expanding universe. Because of this, our models occupy completely different ‘universes’ of thought; they aren’t just different theories, they are built on contradictory foundations.

Contradictory foundations ?? Not necessarily so !! With the Variable Mass theory, section 2 of the paper, quoting from the Abstract: Length Contraction is a feature of Variable Mass. It will be argued that Length Contraction with VM is observationally equivalent with the expansion of Empty Space. In this way a couple of mathematical tools that are useful for our theory can be outsourced to the (nowadays common) expanding universe model.

In a closed $$S^3$$ manifold there is no boundary and no external space, so the notion of work against an external pressure is not physically defined in this model.

The explicit physical mechanism underlying δw, introduced as the work required to maintain the universe in its equilibrium configuration, is not specified within the model and is not required for the subsequent derivation. The equilibrium condition is instead used to describe the isentropic evolution of the CMB within the proposed framework, as discussed in The Aporia of Reversibility in Cosmological Expansion, rather than within the FLRW picture.

The resulting relation is further referenced in the section “An approximate solution for the observable universe from Einstein’s weak fields” (DOI: 10.5281/zenodo.17797831), where a consistency check based on weak-field assumptions shows that CMB curvature effects remain negligible even for high-redshift sources (e.g. GN-z11, as considered in the 2020 analysis).

This weak-field formulation was originally used as a consistency test of the Doppler interpretation of galactic recession. For readers interested in the core structure of the model, the following discussion introduces the proposed metric tensor formulation.

AI systems are truly intelligent. So I am unable to prove that I am talking with a machine. But my gut tells me so - augmented by trivialities I have observed in its mathematics. This is below my dignity. Therefore I have decided to withdraw from this discussion. Bye bye Claude.

I think that’s Claudio talking and he already says he is assisted by an AI. I’m not sure what language the AI is working in, as I have seen the word isentropic and also isotropic, where perhaps isotropic was meant. Or am I the one not getting isentropic? Anyway, Claudio is on the right track in complaining about the fate of CMB energy in the standard model. That has been one of my major topics for years. It forms the basis for my shell model also. For a more general account of the problem, here is an old paper of mine: https://link.springer.com/article/10.1007/s10509-012-0985-x https://arxiv.org/abs/1111.2331

The CMB in my model is the agent driving gravity. So in a way it’s contributing to the forces causing the universe to collapse (and in this way is similar to GR). But there is a flipside. Claudio, you said there is no way to generate the CMB in your “Aporia” paper, but there is. If you’re interested you can read up in my shell universe discussion #256

I have also wondered at times whether everyone here in ACG is alive or bottish. What can we do to enhance our DNA credentials?

Claudio is definitely the one speaking. As he said in another comment in this thread, English is not his native language. He’s Italian, just like me. So when he writes his comments, he uses AI to translate them from Italian into English, which is exactly what I do as well.

I also use AI for translations, because let’s face it, ChatGPT and Gemini are way better than Google Translate. They can take a bit more freedom with the translation when you ask them to, and that means they can produce texts that sound much more natural and fluid in the target language you’re trying to write in.

For those interested: to the formula underlying my weak fields equation, I have applied—with the necessary modifications for my model—the principles discussed in Tolman’s classic text, Relativity, Thermodynamics and Cosmology (Section 86). I find it a very beautiful book especially for thermodynamics, while for cosmology I think it now has only historical value.

I’m currently unable to dedicate time to examining alternative models due to my ongoing work. Furthermode, these models are too complex for a superficial reading.

@ExpEarth You can consider the term ‘isentropic’ as the correct translation wherever I have used it. About ‘Aporia…’, here, I challenge the FLRW model on the grounds that it derives the conservation of CMB entropy (as a perfect fluid) while assuming an absence of reversibility. Regarding energy—a topic within your area of expertise as well—, I opened a discussion regarding the potential conflict between Noether’s theorem and Gauss’s theorem within my model.

@luisenrique-arch I agree with you. While I sometimes use Google Translate, I prefer AI because it allows for better contextualization. Even though Google has improved significantly, it often lacks the nuance required for technical texts; for instance, without the proper context, a standard translator might struggle to decide between isoentropic and isentropic.

@ExpEarth I read your arXiv paper. Congratulations!

I think we are both trying to understand the same fundamental issue. I’m not sure how far your approach can demonstrate quantitatively that energy is conserved in FLRW, but the apparent loss of energy still seems difficult to reconcile with the implications of Gauss’s theorem and Noether’s theorem when applied to the Universe as a whole.

Perhaps this is also what you meant, but, simplifying to the extreme, you seem to have identified a relationship between expansion and gravity.

In my own model, once I move away from the cosmic void and consider perturbative matter “islands”, I encounter a similar issue with my $$E_{HEA}$$. It represents the energy lost when matter is deflected by gravity from its unperturbed recession trajectory.

Unlike your approach, however, I do not expect to quantify this effect, it’s not possible for me! My conjecture is summarized as follows:

“One possible interpretation is that the decrease of $$E_{HEA}$$ reflects the energetic cost associated with the deviation of matter trajectories from the purely radial expansion. Matter concentrations may locally distort the hyperspherical background, and the corresponding geometric response could compensate the apparent loss of effective mechanical energy without violating the global conservation implied by the closed geometry. Although this interpretation is speculative and is not derived from the present formalism, it provides a possible physical picture linking the effective HEA dynamics with the global hyperspherical framework.”

Good luck with your model!

@clmarchesan Your claim about the expanding hypersphere is a result that I also found in my model. However I had to distinguish between gravitational entropy and the remaining entropy - and while the gravitational entropy just stays as it is, the remaining really increases, and allows for r = ct - although you have to use a different definition of time.

The geometric model that leads up directly to that is, if you have a observer that is outside of a Black Hole, and another that is within. the outside observer just sees an object with gravitational properties that is largely invariant. The internal observer does not see an edge - any photon that ist sent to the edge never reaches it, because there is no time at the edge. This means, the universe looks as if it was infinitly big, and it also shows an increasing size.

The origin is not gravity though, but the need of the universe to allow for updating a readonly information boundery - else everything would be static.

This was one of the original models that brought me to the universal model that is described at DOI 10.5281/zenodo.20306533. This used to be a fun project only (btw, also supported by claude code), until I found a model with two bounderies - a real and an imaginary that had different temperature. And together with the gravitation from entropy idea from Verlinde from 2011, this could explain the gravitational constant. Only after G could be calculated and turned out to be a valid value, I published this for others to have a look.

@azouhr I find it interesting that your model also leads to an expanding hyperspherical geometry with $$r=ct$$.

In my approach, $$r=ct$$ is a geometrical consequence of assuming that the universe extends over the surface of an expanding hyperspherical structure. The challenge is then to understand how local gravitational systems can exist within this global expansion and how the corresponding energy balance should be interpreted. My idea is intentionally very simple. I take as a starting point the geometry of an expanding $$S^3$$ hypersphere, with $$r=ct$$, without attempting to explain its origin.

From your description, your approach instead seems to start from entropy and information arguments, with expansion emerging from the need to update an information boundary. From what I understand, you refer to the entire hyperspherical structure, where the absence of a boundary results from informational constraints at the “edge”. In my approach, instead, the physical space is the $$S^3$$ surface itself, and its lack of a boundary is a purely geometric property, not the consequence of any additional mechanism.

By distinguishing between two entropies, you propose the ‘Residual entropy’ as the driver of expansion—a component that I have not been able to develop, since my idea would be to explain the $$S^3$$ expanding geometry as the final configuration resulting from an inflation that occurred in four spatial dimensions, whose physics I do not yet know.

This is a very different perspective, but it appears to address a similar question: how can a globally evolving geometry be compatible with local gravitational structures?

Regarding the analogy with an observer inside a black hole, I understand the idea that the absence of an observable boundary can lead to an apparently infinite and expanding universe. It is an interesting connection with the role of horizons and information in cosmology.

Your result about obtaining a calculable value of G is noteworthy, because it suggests that your framework may capture a deeper physical mechanism behind gravitational coupling.

@clmarchesan The approach I took is basically just with geometric observations. One of the important parts is, that there is kind of an information boundary at a black hole - designated by Schwarzschildradius. The condition that I found possible were that

• the boundary is a fully set entropy boundary
• there is no time there
• even though gravitation must be inherent involved, the gravitational constant G isn’t fixed by the entropy condition alone. You have to provide it externally to determine the horizon
• when a black hole forms, the inner volume does not change - it just stays what it is at least for a moment - there is no sudden deflation to a singularity (which would be required in standard models)

After pondering over this, I found a number of things that result from there:

• there is an inner face and an outer face of the boundary. Gravitational entropy is both sided, the rest only inner sided
• since G is not found at that boundary, and the gravitational entropy is still exposed, it is probably an indirection of an outer boundary
• there cannot be infinite outer boundaries, because G has to be defined somewhere

and at that point, I had the idea of a mirror for the constraints, which turned out to be possible with a conjugated boundary adjacent to the real boundary that we live in. Since the refresh mechanism allows to cross those boundaries, the entropy difference forms the gradient needed for the gravitational constant G.

Now, all of that obviously was speculation, and even if we know about the origin of G now, that is not explaining the value. And that is, when I had the idea of a S4 null-surface and a complex oscillator - and that turned out to be interesting. During cooling down, the dimensions walked through a Cayley-Dickson tower with adding dimensions over time:

real -> complex -> quaternion -> octonion

The octonion split really is the gravitational split and it causes the real and conjugated boundary to be separated - and stay there for all time. Since the Cayley-Dickson tower does not just build up randomly, but in defined steps, all of the condensations created their own constants, and especially the fine-structure constant and the gravitational coupling constant could be used to finally calculate G with just the electron mass as input.

The fact that the calculation of G resulted in a value, that was within the error of Schlamminger’s latest measurements is the reason that I am writing this at all. In fact, I wasn’t expecting anything useful anymore until that calculation.

So much for a very condensed walkthrough with lots of omissions. There were many geometric models that I used to build the framework, and that is also documented on zenodo. If you like the approach “how did I get there” more than “what does the framework look like”, you are probably better off with the geometric models.

@azouhr . I tried hard to reproduce the calculation of G as displayed from your paper in my previous post but didn’t succeed in doing so. Please help !

@azouhr . Here comes an analysis of that G computation, working from the end towards the beginning.

$$ G = G_0(1+\delta_G)^2 $$

$\Longrightarrow \quad \delta_G$ is a dimensionless number.

$$ F_{EM} = \lambda_C \times \frac{\alpha}{2\pi} $$

$\Longrightarrow \quad F_{EM}$ has the dimension of length.

$$ F_G = 2^{103} $$

$\Longrightarrow \quad F_G$ is a dimensionless number.

$$ \delta_G =\left(\frac{F_{EM}}{F_G}\right)^{(3/8)} $$

How can there be such a power of a quantity with the dimension of length ?

$$ G_0 = \frac{\hbar c}{m_e^2 F_{EM}^2} $$

The dimensionality of $G$ must the same as the dimensionality of $G_0$. So we only have to check for the latter.

$$ m^3.kg^{-1}.s^{-1} =?= \frac{(J.s)(m.s^{-1})}{kg^2.m^2} = \frac{(kg.m.s^{-2}.m.s).m.s^{-1}}{kg^2.m^2} = m.kg^{-1}.s^{-3} \quad \Longrightarrow \quad m^2=s^{-2} $$

Which is nonsense. Or did I make a mistake somewhere ?

@HanDeBruijn you actually found an error in the main document - the companion already had the fix in there. I am sorry, this is purely my fault. I will update the article accordingly.

The current line reads

m_e → λ_C = ħ/(m_e c) → F_EM = λ_C × α/(4π) → G₀ = ħc/(m_e² F_EM²)

This is wrong as written: it makes F_EM a length, which breaks the units of G₀ (G₀ = ħc/(m_e² F_EM²) only has the correct units, m³ kg⁻¹ s⁻², if F_EM is dimensionless).

The correct formula - already stated correctly in the companion document - is:

F_EM = 2⁶⁸ × 3⁴ (exact integer; the EM condensation ladder step)

F_EM is a pure number, not a length. It never carried units - the earlier phrasing was a leftover from an intermediate derivation step and should not have made it into the final chain. No numerical results are affected: the G value (6.673820×10⁻¹¹, −0.13σ vs. Schlamminger 2026) was always computed with the correct integer form; only the main article’s prose statement of the formula was wrong.

Thanks for finding this, and sorry for the hazzle.