Update on the 4 Sphere Model and Upcoming Publications

[clmarchesan]

Hi Louis, Hi everyone,

I am writing following a period of pause, due partly to some initial difficulties I faced in publishing my material on A Cosmology Group. At the time, I only had a very large PDF and preferred not to generate multiple intermediate versions of it—especially since those additional revisions were entirely predictable, and I was concerned that the files might take up too much of the group's storage space.

Secondly, but no less importantly, my hyperspherical model still felt incomplete to me. I was not satisfied with how I had addressed:

• the metric tensor issue
• the interplay between gravity and expansion

So, I preferred to wait until I had a clearer, more coherent interpretation—a point I believe I have now reached, even if it took me much longer than I had originally planned.

The model is not exhaustive, and I do not expect to develop it much further, but I consider its current formulation accurate and mature. I also maintain a public OSF space—still under construction— used solely to store the Python source code and the Excel worksheet that provide the complete computational demonstration of the SN1995K distance validation.

At this stage, I think that the latest development of my “4 Sphere” (S^3) hyperspherical cosmology can be described as clearly and accurately as possible through a sequence of focused articles. For this reason, I would be glad to share an update on A Cosmology Group summarizing the current status of the model and indicating where the corrected and definitive versions of the preprints can be found.

For completeness, I would like to mention that some intermediate algebraic and numerical steps were assisted by general purpose computational tools such as the AI. Their role was limited to symbolic manipulation and the solution of differential equations that would otherwise require dedicated software. In addition, I also used AI to cross check the standard FLRW framework against the structure of my model, ensuring that my interpretation of the differences was accurate and well founded.

Whatever the use of AI was, all physical interpretations, modelling choices, and conclusions are entirely my own, and every result has been checked for internal consistency within the framework of the model.

If this is acceptable to you, I will proceed gladly by replacing the original PDF with a concise update note that outlines the model and provides direct links to the definitive versions hosted on Zenodo and viXra. As always, I welcome any feedback, technical critiques, or comments from the group—especially constructive ones that can help test or further refine the consistency of the model.

Claudio

PS: My idea is to replace the single, bulky essay previously uploaded with a concise update note embedded with direct links to the specific papers. This ensures clarity and immediate access to the relevant sections of the model.
Here is the text I plan to post in the .md file (the old PDF must be deleted):

THE 4-SPHERE MODEL

INTRODUCTION

Recent JWST detections — including the JADES GS z14 0 galaxy at z ≈ 14.3— place unexpected pressure on FLRW based distance and age estimates, as such systems appear markedly more massive and more evolved than the cosmic ages predicted by the standard metric would allow. This motivates a re examination of the conceptual foundations linking expansion, redshift, and cosmological geometry. Here I explore an alternative framework in which the Universe is modeled as the three dimensional surface of a hypersphere (S^3) expanding at a constant rate, with radius _r=ct_. In this scenario, matter appears as a discontinuity within a geometry governed by the CMB, preserving Hubble’s law while offering a finite yet unbounded Universe without invoking dark matter or dark energy. The model introduces a distinct interpretation of galactic recession and provides testable predictions through supernova distance measurements, including those accessible with JWST.

Its broader implications concern global energy, the relation between entropy conservation and reversibility, the dynamical role of BAO and the CMB, the status of the cosmological constant and its correlation with quantum field theoretic results in the hyperspherical context, and the gravity–expansion balance encoded in the Hyperspherical Expansion Acceleration (HEA).

The model further introduces a General Methodology—applicable in both Special Relativity and FLRW contexts—that enables the validation or falsification of its predictions through supernova distance measurements. This includes the use of JWST photometric filters, converted into rest frame Johnson B and V bands via transmission curve analysis, allowing a direct comparison with established supernova datasets.

From a macroscopic perspective, the model relies on the physical laws and principles that govern the observable Universe; from a microscopic viewpoint, it considers only the particles currently known to the Standard Model. To provide a clear and organized roadmap of the research, the relevant literature has been structured across Zenodo and viXra. The theoretical core and recent advancements are detailed in the Zenodo preprints, while the standalone empirical validations and observational case studies are hosted on viXra.

Consolidated Publications & Core Framework

I am pleased to share the latest developments of my S^3 hyperspherical cosmology, hereafter, the 4-Sphere model. Although the treatment is not exhaustive, I have aimed to present the framework as clearly and accurately as possible through a sequence of focused articles.

The definitive and updated versions of my research have now been consolidated on Zenodo and viXra.

For readers interested in the broader context and in the most scientifically accurate formulation of the model, the relevant repository links are provided below.

Note on Links: All the links below point directly to the Zenodo and viXra versioning pages, ensuring you always have access to the latest, corrected revisions.

Theoretical Framework (Hosted on Zenodo): The list below represents the recommended reading order:

1. Dynamics on an Expanding Hypersphere: Reassessing the Cosmological Principle in light of the CMB)
2. The Vacuum Catastrophe revisited: when the quantum vacuum does not contribute to cosmological equilibrium
3. The Aporia of Reversibility in Cosmological Expansion
4. Ancient orbital structures: simulating Local Group evolution under full Hyperspherical Expansion Acceleration
5. A new perspective on Hubble's law through a four-dimensional spatial model
6. Reevaluating the Necessity of Dark Matter and Dark Energy within Cosmological Models
7. Non Keplerian MW–M31 dynamics under Hyperspherical Expansion Acceleration: a bound solution without Dark matter

The broader implications of this speculative framework rest on the empirical robustness of Hubble’s law and the independent validation of stellar distances, but also on a set of necessary assumptions—discussed in the cited works—that remain conjectural. These include the predominant dynamical role of BAO and the CMB, a reconsideration of the Cosmological Principle, the status of the cosmological constant in its relation to Quantum Field Theory, the restoration of the link between entropy conservation and reversibility. Matter as a discontinuity implies the model’s global energy conservation associated with the CMB alone. Conversely, the gravity–expansion balance encoded in the newly introduced Hyperspherical Expansion Acceleration (HEA) imposes the non-conservative energy E_HEA of matter.

A separate line of investigation addresses the consequences of this framework for the interpretation of Dark Matter and Dark Energy, as developed in the dedicated study listed above.

Case studies and data validations (Hosted on viXra): In addition to the updated framework available on Zenodo, the following earlier publications on this platform represent foundational case studies and data validations for the model's development:

1. Star Distance Validation from Data of a High-Z Supernova Ia in the Special Relativity Context
2. Concerning the Apparent Magnitude
3. Concerning the Time Dilation of the Supernovae
4. On the Inaugural Observations of the James Webb Space Telescope: A Short Study of the Stephan's Quintet

While the papers on apparent magnitude and supernovae time dilation lay the necessary theoretical foundations, the study of a High-Z Supernova Ia serves as a crucial empirical validation of the model's ability to accurately calculate cosmological distances [1]. Complementing this, the examination of Stephan’s Quintet is not intended as a new astrophysical analysis of the system, but rather as a focused illustration of how the framework’s formulas operate in practice. In this way, the 4 Sphere model is shown to successfully account for the groundbreaking inaugural observations of the James Webb Space Telescope.

[1] ‒ See the section “On the significant validation of the model and more” in A new perspective on Hubble's law through a four-dimensional spatial model, where the validation of the Type Ia Supernova distance holds a pivotal importance.

Philosophical and Geometric Foundations

Here, the correctness of the FLRW model is not in question; however, it is built upon specific foundational assumptions. My work does not dispute its internal consistency, but explores the consequences of relaxing one of these assumptions, while preserving the Big Bang scenario. Within this modified framework, an alternative cosmological model naturally emerges.

This model is fundamentally simple, rooted in the geometry of the 3-sphere. It belongs to the field of alternative cosmology, though the core concept of the hypersphere has an illustrious origin, being attributed to Einstein himself.

From a macroscopic perspective, the model relies on the physical laws and principles that we apply to the real world; from a microscopic viewpoint, it considers only the particles currently known to the Standard Model.

I have termed it the '4-Sphere' model. This choice was made partly to distinguish it from other mathematical treatments, but also because, while setting up a stellar map for the model, I encountered 2D, 3D, and 4D geometric figures intricately intertwined. In that moment, I realized that the mathematical term 3-sphere—though technically correct for the spatial boundary—did not fully capture the profound 4-dimensional spatial reality of the object I was observing.

[RedshiftDrift]

@clmarchesan you've certainly done a lot of work!
AI is fine as a tool to check text, equations, etc. as long as the author performs some tests to eliminate errors made by robots.

My idea is to replace the single, bulky essay previously uploaded with a concise update note embedded with direct links to the specific papers. - @clmarchesan

That's an excellent idea! I've reinstated your write access to the repository, you should be able to modify and delete files, and update the .md file when the references change.

---

As always, I welcome any feedback, technical critiques, or comments from the group - @clmarchesan

I had a quick look at "A NEW PERSPECTIVE ON HUBBLE'S LAW THROUGH A FOUR-DIMENSIONAL SPATIAL MODEL". In the manuscript you mention $r = ct$ but I can't find a Hubble Law in the form $d = f(z)$, did I miss it? Or it's in another paper?

Do you know about F. Melia $R_h = ct$ cosmology?
Here's one of his papers about it “The Rh = ct universe without inflation,” A&A, vol. 553, p. A76, May 2013, doi: 10.1051/0004-6361/201220447.

[HanDeBruijn]

@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.

[clmarchesan]

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.

[ericcosmos]

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

[HanDeBruijn]

@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.

[HanDeBruijn]

@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 !!

[HanDeBruijn]

@clmarchesan .

assisted by general purpose computational tools such as the AI [ .. ] I also used AI to cross check [ .. ]

How can you be sure that AI is not hallucinating while "assisting" you ?

[clmarchesan]

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]

Regarding AI, the two diagrams on page 22 of DOI: 10.5281/zenodo.14986654 (showing the expected and observed BAO correlation) are a good example of how I used it. I asked AI to generate these plots simply to gain a clearer sense of the magnitude of the BAO signal. My model already excluded BAO because of the viscosity assumed at recombination, and the BAO peak disappears when my stellar distances are used. Therefore, the diagrams did not support my model; rather, they helped me appreciate that the observed BAO peak—although small—is nevertheless a genuine strength of the FLRW framework.

[RedshiftDrift]

@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.

[clmarchesan]

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.

[clmarchesan]

Appendix 3 has been revised.

[clmarchesan]

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.

[clmarchesan]

There is no issue with AI as a tool itself; the problem arises only when it is treated as a scientific authority.

[RedshiftDrift]

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.

[clmarchesan]

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.

[HanDeBruijn]

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 ?!

[clmarchesan]

For anyone interested, the section MODEL'S FORMULAS AND CONCEPTUAL SIMPLICITY in DOI: 10.5281/zenodo.17797831 summarizes the main quantities of the model, especially the distance definitions, whose correspondence with FLRW quantities is not always straightforward.

[azouhr]

@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.

[HanDeBruijn]

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

[Francois-Zinserling]

@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.

[azouhr]

@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.

[HanDeBruijn]

@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]

A brief comment on why I abandoned the FLRW metric.

My motivation was not simply to propose a different geometrical framework. It also stems from observational and conceptual issues discussed in Dynamics on an Expanding Hypersphere: Reassessing the Cosmological Principle in light of the CMB.

Recent observational studies suggest that the dipole inferred from large-scale radio source counts may have an amplitude significantly larger than expected from the kinematic interpretation of the CMB dipole. If confirmed, this raises important questions about applying the cosmological principle to the large-scale distribution of matter and about identifying the CMB rest frame with the matter rest frame.
Together with the recent JWST observations motivating a re-examination of cosmological distances and ages, these issues led me to explore an alternative geometrical framework rather than a modification of the FLRW metric.

[clmarchesan]

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.

[ExpEarth]

@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.

[clmarchesan]

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.

[Francois-Zinserling]

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.

[ExpEarth]

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.

[luisenrique-arch]

@clmarchesan

Okay, if you'd rather keep the discussion public, I'll share with you the cosmological models I find interesting here on the forum instead of by email.

One cosmological model I've always found particularly interesting is Curvature Cosmology, developed by David Crawford in the 1990s and refined throughout the 2000s. Unfortunately, Crawford passed away in November 2024. However, he continued working on and updating the model until shortly before his death. In fact, the last paper he published, which provides an overview of the model, came out in 2023. Here it is:

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://vixra.org/pdf/2207.0121v4.pdf&ved=2ahUKEwjaw721jqyUAxWQg%5C_0HHcwWBwYQFnoECB8QAQ&usg=AOvVaw1aEn81rf5UELFvc9KgCeNT

Here's the abstract:

Curvature-cosmology is a tired-light cosmology that predicts a well-defined, static, and stable universe. Since it is a complete challenge to the Big Bang paradigm, it can only be judged by its agreement with direct cosmological observations. It predicts a universe of hydrogen plasma with a temperature of 2.456 × 10⁹ K [observed: 2.62 × 10⁹ K] and a cosmic background radiation temperature of 2.736 K [observed: 2.725 K]. It has only one parameter, which is the density of the cosmic plasma. In addition, this paper provides a new, simpler raw data analysis for Type Ia supernovae, which provides excellent predictions for the redshift variation of Type Ia supernova light curve width and magnitude. A new discovery is the intrinsic magnitude distribution. The analysis of 746,922 quasars provides important cosmological information on the distribution of intrinsic magnitudes and the density distribution of quasars. Other major observations that are shown to be consistent with Curvature-cosmology are Tolman surface density, galaxy clusters, angular size, galaxy distributions, X-ray background radiation, and quasar variability. It does not need inflation, dark matter, or dark energy.

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.

I defended and supported this model for a long time. 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. However, I still wanted to share it with you (@clmarchesan) because I'm quite fond of it. I think it's one of the best-developed, most elegant and most comprehensive alternative cosmological models.

Let me know what you think!

[ericcosmos]

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?

[HanDeBruijn]

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.

[luisenrique-arch]

@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.

[ericcosmos]

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.

[luisenrique-arch]

@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.

[clmarchesan]

I must start by saying that any opinion I offer on this may have limited value because 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.

Comparing the two from a technical standpoint is fundamentally problematic. To properly evaluate Crawford's work, I would need to study a model that is fundamentally different from my own.

[HanDeBruijn]

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.

[HanDeBruijn]

Let's have a closer look into the paper named Dynamics on an Expanding Hypersphere: Reassessing the Cosmological Principle in light of the CMB by Claudio Marchesan. Quote:

The considerations developed so far have led to the simplified equation that defines the balance between pressure and gravity:

$$ \frac{\partial(c^2\rho\delta V)}{\partial t}\cdot dt = \delta w - p \frac{\partial(\delta V)}{\partial t}\cdot dt $$

where $p$ and $\rho$ represent the pressure and density of the CMB, and $\delta V$ denotes a small proper volume element defined in the local frame of the observer, while $\delta w$ is the work done to keep the universe in its shape. [ .. ] The condition $\partial_t(c^2\rho\delta V)=0$ ensures that the CMB remains in isoentropic equilibrium on the expanding 4d-sphere.

And that's it. There is no further mathematics to be considered. The above equation simplifies to

$$ \delta w = p \frac{\partial(\delta V)}{\partial t}\cdot dt = p\cdot\delta V $$

where the right hand side is indeed a well known expression for work done in Thermodynamics. I would not be surprised that Claudio, being a Chemical Engineer (Retired) has decent knowledge about that discipline (even without the help of Artificial Inteligence). But wait, the product rule for (partial) differentiation also says another thing.

$$ \frac{\partial(c^2\rho\delta V)}{\partial t} = \delta V\cdot\frac{\partial(c^2\rho)}{\partial t} + c^2\rho\cdot \frac{\partial(\delta V)}{\partial t} $$

So it seems that $c^2\rho=-p$, which is dimensionally correct. With $E=mc^2$ the interpretation is a radiation energy density $u$. And that is correct as well, because for the CMB/blackbody radiation field

$$ p = \frac{u}{3} \quad \mbox{where} \quad u = \frac{4\sigma}{c}T^4 $$

where $\sigma=$ Stefan-Boltzmann constant, $T=$ absolute temperature.

What now is the net result of all this?
I have the uneasy feeling that Claudio's formulas are merely sort of AI embellishement, their main purpose being for the prose in his paper to look scientifically more impressive. The mathematics does not serve much progress. That's the bad news.
The good news - at least for me - is that the Cosmic Microwave Background might indeed deliver the work done to keep the universe in its shape.

[clmarchesan]

@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.

[azouhr]

@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.

[HanDeBruijn]

@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 !

[HanDeBruijn]

@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 ?

[azouhr]

@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.