1 hour ago, Rincewind said:

The speed of light as defined in SI units was never any different in the past, according to the FTS model. Speed is the property of AT. However, there was a moment in the distant past where the speed of light was relatively twice as fast, even though it was still the same measure in SI units, because fractal scaling depends on the duration of an ET second, that was second was twice as long where galaxies have a redshift of Z= 1. When the light left that galaxy, call it galaxy A, they would have seen our galaxy with a redshift of just Z = 0.5.

The speed of light was never different in SI units, but over time, the duration of the ET second evolved. Therefore, at galaxy A, at Z = 1, an ET second was twice as long, meaning light could cover more space per ET second. From that galaxy's perspective, our Milky Way would have appeared at Z = 0.5. Beyond our Milky Way, at Z = 1 galaxy B, it would be within its Hubble sphere, whereas now it is not. In the standard model, light would no longer of had time for the light to have ever reached it. The FTS model resolves such horizon problems as any light that reaches us (or anything else) was within the emitters' Hubble sphere when the light left.

So are you not answering the question to just be difficult, because you can’t, or because you realize it will reveal that your conjecture is false?

Just now, Rincewind said:

because fractal scaling

There you go again invoking fractals, without answering my question.

On 5/9/2025 at 2:10 PM, studiot said:

Finally, how do Fractals have tangents ?

Fractals cannot have tangents.

If I am wrong about this I would expect to see a few mathematical lines starting with a recognised mathematical statement such as Derrick's theorem and ending with

something like

Therefore vector PQ is a tengent of fractal AB

41 minutes ago, studiot said:

There you go again invoking fractals, without answering my question.

Pretty clear that it’s just a bit of word salad, with no legitimate meaning behind it.

2 hours ago, swansont said:

So are you not answering the question to just be difficult, because you can’t, or because you realize it will reveal that your conjecture is false?

I’m not avoiding the question; I'm clarifying that the fundamental constants remain invariant locally in SI-defined measures, but their observable effects shift due to fractal scaling over cosmic Ephemeris Time (ET).

What physically changes? The observed relationships between constants and space-time.

At Z = 1, an ET second was twice as long, meaning light traversed more space in one ET second, at the same constant speed AT.

-From that galaxy’s perspective, our Milky Way would have been at Z = 0.5, not at today’s observed redshift.

The Hubble sphere evolves in ET, resolving the horizon problems without requiring inflation.

If you're asking for specific numerical changes in constants like ( \varepsilon_0 ), ( \mu_0 ), ( h ), and ( e ), my argument is that their numerical values remain fixed in SI, but their effective influence scales over cosmic ET, meaning interactions between matter and radiation evolve over time.

This isn’t an evasion, it’s precisely how FTS avoids the pitfalls of VSL models while explaining apparent changes in cosmic causality without breaking established physics.

Just now, Rincewind said:

fractal scaling..

........without breaking established physics.

Just breaking established mathematics .

28 minutes ago, Rincewind said:

If you're asking for specific numerical changes in constants like ( \varepsilon_0 ), ( \mu_0 ), ( h ), and ( e ), my argument is that their numerical values remain fixed in SI, but their effective influence scales over cosmic ET, meaning interactions between matter and radiation evolve over time.

And I am asking you to quantify this “scaling” (assuming c scales by a factor of 2) which you still have not done.

This isn’t an evasion, it’s precisely how FTS avoids the pitfalls of VSL models while explaining apparent changes in cosmic causality without breaking established physics.

It’s not at all clear you’ve not broken physics, which is pretty big on quantifying things.

19 hours ago, Rincewind said:

This is not a modification of fundamental interactions

That’s exactly the problem. You’re assuming that these fundamental interactions don’t change, but at the same time you’re saying that atoms “shrink” over time relative to some absolute background. This doesn’t work, since the interactions don’t scale - if you try to shrink atoms, you break the physics in the process.

I want to thank you all for your insightful remarks. I am beginning to understand the challenges of this model for your modus operandi.

Clarifying the Conceptual Challenges of the FTS in Formal Mathematics

The Fractal Topology of Space-time (FTS) model reinterprets foundational aspects of time, space and cosmic expansion, making the conventional mathematical formalisms, such as the Friedmann equations, applicable within this framework. This is because:

1.      Metric-based spatial expansion, whereas FTS describes an evolving scaling transformation of matter within a static spatial background.

2.      Fixed unit relationships in SI time measures, while FTS distinguishes between Atomic Time (AT) and Ephemeris Time (ET), showing how the apparent evolution of cosmic phenomena is governed by time-scaling effects.

3.      Constancy of physical constants within SI, whereas in FTS, constants remain numerically invariant, but their observable effects shift over cosmic ET.

Why Standard Formalism Cannot Fully Capture FTS Yet

Formal mathematical frameworks developed in conventional cosmology assume that time and space evolve homogeneously together in a metric-expanding universe. However, in FTS:

- Time is fractionalized into infinitely many "now" moments, shifting the meaning of distance and velocity.

- A light-year in the past was longer, fundamentally altering how cosmic age, expansion rates, and the Hubble parameter should be understood.

- The Hubble sphere shrinks relative to the spatial background, changing causal connectivity without requiring inflation.

These concepts do not break mathematics but require extensions beyond standard differential equations, such as:

- Non-Euclidean fractal scaling models to express matter shrinkage without violating locality.

- Fractional calculus to quantify evolving scaling effects within an inverse AT-ET framework.

- Measure-theoretic tangents to describe local differentiability properties within fractal topology.

Future Directions for Formalisation

To bridge the gap between philosophical reasoning and formal mathematics, FTS requires:

1.      An alternative metric framework accounting for the inverse scaling of AT and ET rather than assuming a homogeneous expansion.

2.      A revised formulation of speed and age, expressing instantaneous rate changes in fractionalized time.

3.      A new mathematical structure to rigorously define horizon dynamics without relying on standard inflationary assumptions.

I recognise that the way FTS treats time, space, and scaling isn’t easily expressed in conventional mathematical structures. I’d like to clarify why established models' use of the Friedmann equation and other similar conventions loses conceptual relevance in this framework, and how different approaches may be necessary to formalise their principles.

On 5/12/2025 at 6:04 AM, Markus Hanke said:

That’s exactly the problem. You’re assuming that these fundamental interactions don’t change, but at the same time you’re saying that atoms “shrink” over time relative to some absolute background. This doesn’t work, since the interactions don’t scale - if you try to shrink atoms, you break the physics in the process.

The apparent shrinkage of atoms in the FTS model does not break physics because all interactions scale uniformly within their local frame. Since atomic forces, such as electromagnetic binding, depend on relative distances rather than absolute sizes, the perceived contraction is relative to the evolving cosmic background, not an intrinsic modification of atomic forces.

This aligns with the idea that fundamental laws remain invariant within atomic scales but appear different over cosmic time due to fractal scaling in ET. Much like scale-invariant theories that preserve ratios, FTS reinterprets the spatial contraction of atoms without requiring a change in fundamental interactions.

Redshift-to-distance observations indicate that the scaling effects are gradual over deep cosmic time. Space within gravity-bound systems, such as our spiral galaxy, will show no evidence of expanding space as the atomic time-based distances scale with the atoms. This fits with observations of other galaxies.

Standard FRW models assume uniform expansion across all regions, yet observations show no evidence of expansion within gravity-bound systems. What mechanism within the standard model accounts for this discrepancy?

25 minutes ago, Rincewind said:

Standard FRW models assume uniform expansion across all regions, yet observations show no evidence of expansion within gravity-bound systems. What mechanism within the standard model accounts for this discrepancy?

Two things:

(1): Over cosmological distances, gravity-bound systems are but mere specks.

(2): The standard FLRW model is just that... a model. It is a simplified approximation of the actual universe. However, it should be noted that it is superior to consider deviations from a simple model than to use a complicated model from the outset.

14 hours ago, Rincewind said:

Since atomic forces, such as electromagnetic binding, depend on relative distances rather than absolute sizes

No, here’s where the issue is - specifically the strong and weak interactions do operate on an absolute size scale. For example, the upper limit for binding between quarks is at around 0.8 femtometers, and the residual strong force between nucleons operates within ~3 femtometers. These are very much absolute values at ordinary energies. This is why the rescaling of atoms does not work.

18 hours ago, KJW said:

Two things:

(1): Over cosmological distances, gravity-bound systems are but mere specks.

(2): The standard FLRW model is just that... a model. It is a simplified approximation of the actual universe. However, it should be noted that it is superior to consider deviations from a simple model than to use a complicated model from the outset.

The assumption that gravity-bound systems are negligible at cosmological scales is problematic; galaxies and large structures are not just 'mere specks'; they occupy significant portions of space. If FLRW truly models uniform expansion, we should see measurable effects even within galaxies, yet we don’t.

While it's true that FLRW is a simplified model, science should prioritise constructive deviations based on observations, rather than dismissing inconsistencies as secondary effects. If standard cosmology has mechanisms to reconcile the apparent lack of internal expansion within galaxies, what are they? Otherwise, the need for an alternative framework like FTS remains valid.

4 hours ago, Markus Hanke said:

No, here’s where the issue is - specifically the strong and weak interactions do operate on an absolute size scale. For example, the upper limit for binding between quarks is at around 0.8 femtometers, and the residual strong force between nucleons operates within ~3 femtometers. These are very much absolute values at ordinary energies. This is why the rescaling of atoms does not work.

You're absolutely right that strong and weak nuclear interactions operate at defined distance scales (quark confinement at ~0.8 fm, nucleon binding at ~3 fm). However, this does not contradict the FTS framework, because the scaling effect applies at cosmic scales, not local atomic physics.

This is analogous to how Conformal Field Theory (CFT) preserves fundamental interactions under scale transformations, certain quantum field effects remain invariant, even if the overall scale of spacetime shifts In FTS:

- Atomic physics remains internally stable, with interactions following fixed femtometer distances.

- The rescaling is cosmic, meaning atomic structures appear smaller relative to large-scale space evolution.

- Just as CFT retains symmetry under scaling, FTS preserves local interactions while adjusting their cosmic positioning.

This is why the model does not 'break physics', it applies scale invariance principles similarly to those seen in field theories, just over cosmic time.

49 minutes ago, Rincewind said:

The assumption that gravity-bound systems are negligible at cosmological scales is problematic; galaxies and large structures are not just 'mere specks'; they occupy significant portions of space.

How can you know? Since we don't know the size of space...

Or to put it another way, I'm at the centre of the universe, as is every 'me' part of the universe; it seems to me that even my observable universe could be seen as a speck, in that context.

Edited May 16May 16 by dimreepr

19 hours ago, Rincewind said:

However, this does not contradict the FTS framework, because the scaling effect applies at cosmic scales, not local atomic physics.

This is not what you said earlier - you explicitly stated that matter shrinks relative to some fixed absolute background. That is a rescaling of atoms.

21 hours ago, dimreepr said:

How can you know? Since we don't know the size of space...

Or to put it another way, I'm at the centre of the universe, as is every 'me' part of the universe; it seems to me that even my observable universe could be seen as a speck, in that context.

That’s a good point. I like the way Blaise Pascal put it: “ Nature is an infinite sphere of which the centre is everywhere and the circumference nowhere."

To make my argument more valid, I should have framed it differently. Galaxies and large structures occupy significant portions of space relative to the observable Hubble sphere. This means their gravitational effects and stability must be accounted for in cosmic models, regardless of whether the universe itself is infinite or not."

4 hours ago, Markus Hanke said:

This is not what you said earlier - you explicitly stated that matter shrinks relative to some fixed absolute background. That is a rescaling of atoms.

I understand your concern regarding the distinction between rescaling at atomic levels versus cosmic scaling. The FTS model does not claim that atomic structures physically compress in a way that alters interactions like quark binding or electromagnetic forces. They do not shrink relative to the way we define distances in SI measures.

Instead, atomic scales remain invariant within their own reference frames; what changes is their observed size relative to cosmic evolution. The ‘shrinking’ effect is not a direct rescaling of atomic interactions, but a shift in how space and time evolve at cosmological distances.

This aligns with scale-invariant field theories like CFT, where interactions remain stable while their global interpretation changes over time. Fundamentally, FTS preserves local physics while refining cosmic-scale relationships.

2 hours ago, Rincewind said:

That’s a good point. I like the way Blaise Pascal put it: “ Nature is an infinite sphere of which the centre is everywhere and the circumference nowhere."

To make my argument more valid, I should have framed it differently. Galaxies and large structures occupy significant portions of space relative to the observable Hubble sphere. This means their gravitational effects and stability must be accounted for in cosmic models, regardless of whether the universe itself is infinite or not."

Then we're back to 'local', and there's a definite minimum size of 'local' as described by Max Planck.

1 hour ago, dimreepr said:

Then we're back to 'local', and there's a definite minimum size of 'local' as described by Max Planck.

Surely the Planck scale is fundamentally defined using SI units, meaning it operates within our standard measurement system rather than being an absolute cosmic limit.

A useful distinction here is that Planck length (≈1.6 × 10⁻³⁵ meters) is a theoretical construct based on quantum gravity constraints, not necessarily an intrinsic boundary beyond which space cannot exist or be meaningful. It’s derived from fundamental constants (ℏ, G, c), but those constants themselves exist within our defined measurement system rather than some absolute framework independent of human conventions.

"Local" is relative to the Hubble sphere and not constrained by Planck scales; we cannot assume that Planck units impose universal limits on cosmic structures.

The Standard Model allows for equivalent interpretations, whether as expanding space or as scaling matter, depending on the reference framework.

7 hours ago, Rincewind said:

Galaxies and large structures occupy significant portions of space relative to the observable Hubble sphere

The Hubble sphere has a radius of ~14 billion LY. The Milky Way has a diameter has a diameter of ~100k LY. So less than a part in 10,000. That’s “significant”?

You seem to be allergic to doing any actual math as part of this.

On 5/16/2025 at 7:53 PM, Rincewind said:

If standard cosmology has mechanisms to reconcile the apparent lack of internal expansion within galaxies, what are they?

Gravity.

19 hours ago, Rincewind said:

what changes is their observed size relative to cosmic evolution. The ‘shrinking’ effect is not a direct rescaling of atomic interactions, but a shift in how space and time evolve at cosmological distances.

Sounds like ordinary metric expansion to me, then.

17 hours ago, swansont said:

The Hubble sphere has a radius of ~14 billion LY. The Milky Way has a diameter has a diameter of ~100k LY. So less than a part in 10,000. That’s “significant”?

You seem to be allergic to doing any actual math as part of this.

The radius of the Hubble sphere (HS) depends on which model is applied, SM or FTS. Standard cosmology (SM) accounts for comoving distances due to metric expansion.

FTS, however, uses D ≈ (z × c) / H₀, where distance is determined by scaling effects relative to Ephemeris Time (ET) rather than space expanding. This means that in FTS, HS is not a static comoving boundary but a positional reference that evolves based on the rate of cosmic scaling. The observer moves through ET in the frame of the measuring instrument, marking the HS as the horizon of future causality, a point where, in the past, light-speed rulers were twice as long relative to the background. It is the horizon of future causality, and a place where, in the past, light speed rulers were twice as long relative to the background.  

So, while the Milky Way's diameter is indeed a fraction of the Hubble sphere, significance isn’t just about scale, it’s about gravitational influence and cosmic structure. Large-scale formations like galaxy clusters, filaments, and voids shape cosmic evolution across hundreds of millions of light-years.

If metric expansion were truly homogeneous, why don’t galaxies expand internally? Standard cosmology acknowledges that gravitational binding overrides expansion within certain regions, yet lacks a clear mechanism explaining exactly how this transition occurs.

Regarding math, cosmology relies on both quantitative and conceptual reasoning, and the issue here is not just numbers, but the physical implications of large-scale structure in cosmic models.

5 hours ago, Markus Hanke said:

Sounds like ordinary metric expansion to me, then.

While metric expansion describes space itself stretching between objects, FTS treats scaling as a function of matter evolving relative to cosmic time rather than an inherent change in spatial separation.

The distinction is important: standard cosmology assumes space expands while objects retain constant size, whereas in FTS, matter itself scales over Ephemeris Time (ET) while spatial positioning is preserved.

Thus, while both frameworks address cosmological evolution, FTS does not equate directly to metric expansion; it offers an alternative interpretation where the scaling of matter itself drives observational effects, not the stretching of space.

Additionally, the FTS model requires no insertion of extra energy in the form of an arbitrary parameter, unlike certain standard models that rely on unexplained forces such as dark energy. Instead, FTS operates through inherent scaling effects, maintaining consistency without requiring an external energy source whose causal mechanism remains unknown.

17 minutes ago, Rincewind said:

While metric expansion describes space itself stretching between objects, FTS treats scaling as a function of matter evolving relative to cosmic time rather than an inherent change in spatial separation.

The distinction is important: standard cosmology assumes space expands while objects retain constant size, whereas in FTS, matter itself scales over Ephemeris Time (ET) while spatial positioning is preserved.

Thus, while both frameworks address cosmological evolution, FTS does not equate directly to metric expansion; it offers an alternative interpretation where the scaling of matter itself drives observational effects, not the stretching of space.

Additionally, the FTS model requires no insertion of extra energy in the form of an arbitrary parameter, unlike certain standard models that rely on unexplained forces such as dark energy. Instead, FTS operates through inherent scaling effects, maintaining consistency without requiring an external energy source whose causal mechanism remains unknown.

I think the fundamental problem with FTS is, it's not using the correct language to describe the physics/reality; I know this to be true bc I can understand it.

46 minutes ago, dimreepr said:

I think the fundamental problem with FTS is, it's not using the correct language to describe the physics/reality; I know this to be true bc I can understand it.

I think you might be right, because I wrote it :)