Forgot to add that using the above relations one can use arbitrarily use fermions to estimate Hubble constant. This is often suggested for the dominant era.

Here's a more recent method for using CMB temp to determine Hubble constant.

https://arxiv.org/pdf/1910.09881

Its a quick breakdown of the methodology but has the basics of the general idea. Other methods will typically include utilizing those previous mentioned equations in particular Bose Einstein.

1 hour ago, Mordred said:

Here's a more recent method for using CMB temp to determine Hubble constant.

https://arxiv.org/pdf/1910.09881

thank you for sharing the paper. I have read through it.

Before I share my detailed thoughts, I want to make sure I am accurately understanding your perspective. Could you elaborate on what specific correlations you see between the methodology proposed in this paper and the derivation of [math]H_0[/math] in RG?

What was the logical chain that led you to suggest this paper as a comparison? I genuinely want to understand how you are mapping their method and mine.

Can you elaborate please?

Both methodologies take advantage that temperature scales as the inverse of the scale factor. Both methodologies applies the thermodynamic laws. In the paper its mentioned that its methodology is using local measurements with your use of beta function B= v/c with no further correction correlations beyond Hubble horizon as well the numbers you used in your article for density calculations you would also be local.

If you were using say z=1100 none of the density numbers would match.

The paper itself doesn't go into which formulas to employ but they would likely involve Maxwell Boltzmann and Bose Einstein.

Both methodologies will obviously involve the Boltzmann constant. They also will both require the volume element being examined and if you think about it your employing the critical density formula relations albeit your method isnt total density. If it was then one of your formulas is wrong as its missing a (3) in its derivative.

I will let you see which formula Im referring to in your paper. It may help me relate to which formulas you have derived yourself and which formulas you have used from other resources such as AI or other papers and your understanding of those formulas incorperated that were derived not using RG.

Edited Wednesday at 01:16 AM2 days by Mordred

Unfortunately I didn't spot that till today. I knew there was something bothering me but couldn't put my finger on it till today.

On 2/22/2026 at 12:49 AM, Anton Rize said:

This ontological stance is not just philosophy; it produces testable celestial mechanics without Dark Matter

Meaning dark matter doesn't exist?

On 2/22/2026 at 6:24 AM, Anton Rize said:

Energy is the ability to do work

Are you happy with this definition?

Personally I don't regard that as the fundamental definition of energy...since I entered this forum have been using it as the mainstream definition....otherwise,you are the first person to encounter in this forum whose arguments come near what I regard as the fundamental definition of energy.

49 minutes ago, MJ kihara said:

Personally I don't regard that as the fundamental definition of energy...since I entered this forum have been using it as the mainstream definition....otherwise,you are the first person to encounter in this forum whose arguments come near what I regard as the fundamental definition of energy.

Might be an helpful eye opener to perhaps look into how work is defined in probability terms as well as how work is defined in

thermodynamics and statistical mechanics as opposed to how work is defined in the transfer for force.

You can keep the energy definition the same the change is in how work is defined. Example under QFT ( probability currents)

The first three cases work is defined as the expected energy transfer over possible states.

In the classical latter case its the force applied over displacement.

On 2/22/2026 at 7:08 AM, Anton Rize said:

if a model doesn't fit the observation, the mathematical flexibility allows you to simply add a new parameter (a scale factor, an epicycle, dark matter)

And some of the added parameters lead to discovery of new particles...when a model give you answers almost immediately(it's more knowledgeable) it regards the intermediate steps are clear(it assume you know...) a good example your model incorporates Hamiltonian,Klein Gordon equations, cosine/sine laws and manipulation of physics units/SI units in a smart way(of geniuses are smart people....i don't know...) to give answers in the shortest way possible, it becomes difficult to predict existence of something unknown.

Added parameters leads to experimentation so that they can be confirmed or disregarded...therefore, the said flexibility is an important part of scientific discoveries.

4 minutes ago, Mordred said:

Might be an helpful eye opener to perhaps look into how work is defined in probability terms as well as how work is defined in

thermodynamics and statistical mechanics as opposed to how work is defined in the transfer for force.

You can keep the energy definition the same the change is in how work is defined. Example under QFT ( probability currents)

The first three cases work is defined as the expected energy transfer over possible states.

In the classical latter case its the force applied over displacement.

The definition of Energy I have is too much controversial,however, I tend to think eventually it generates the mainstream definition, therefore, by default am not against mainstream definitions,however, I don't regard them as fundamental...unlike the author I have been slow to develop the math...the author seems to be on steroids turbocharged by AI.

35 minutes ago, MJ kihara said:

The definition of Energy I have is too much controversial,however, I tend to think eventually it generates the mainstream definition, therefore

From a practical and logical sense changing the definition of work makes sense. By describing the required " ability " to the operation being performed between states/ systems including operators, partition functions ... you can encompass a wider variety of mathematics not restricted to physics.

Also by identifying what's changing in the work term also helps identify the mathematical treatment. This is already done.

So why would you need to change the definition of energy as well ?

On 2/25/2026 at 11:15 AM, Mordred said:

Both methodologies take advantage that temperature scales as the inverse of the scale factor. Both methodologies applies the thermodynamic laws. In the paper its mentioned that its methodology is using local measurements with your use of beta function B= v/c with no further correction correlations beyond Hubble horizon as well the numbers you used in your article for density calculations you would also be local.

If you were using say z=1100 none of the density numbers would match.

It is difficult to imagine two more fundamentally opposed methodologies.

1. Categorical Difference: The paper you linked proposes an empirical measurement method requiring the filtration of 17 noise parameters over a decade. My method is pure generative geometry - a strict algebraic deduction that culminates in exact equations with ZERO free parameters.

2. It is not a "local" [math]\beta[/math]: I do not use a local kinematic velocity [math]\beta = v/c[/math] to derive the horizon. I use [math]\beta_1 \equiv \alpha[/math] (the fine-structure constant). As mathematically derived in Part III: (https://willrg.com/documents/WILL_RG_III.pdf#eq:beta=alpha), [math]\alpha[/math] is the invariant kinematic projection of the electromagnetic ground state. It is a universal geometric scale, not a local variable.

3. The [math]z=1100[/math] claim: Your assertion that the numbers would not match at [math]z=1100[/math] is factually incorrect. Using the exact same deductive geometry—specifically, the phase transition condition [math]o_{crit} = 1[/math]—the WILL RG framework explicitly calculates the recombination epoch at [math]z_{dec} \approx 1156[/math] and [math]T \approx 3150[/math] K (see Section 4). The geometry works flawlessly across all cosmological scales.

4. "the numbers you used " - No fitted numbers: The "numbers" I use for the density calculations are not local phenomenological fits. They are absolute constants
([math]T_0, \alpha, G, c[/math]). The model does not take numbers from local measurements to fit the Universe; it derives the macroscopic structure of the Universe directly from microphysical invariants.

On 2/25/2026 at 11:15 AM, Mordred said:

The paper itself doesn't go into which formulas to employ but they would likely involve Maxwell Boltzmann and Bose Einstein.

Both methodologies will obviously involve the Boltzmann constant. They also will both require the volume element being examined

I have to ask: are we looking at the same equation? The derivation in my paper culminates strictly in this exact algebraic form:

[math]H_{0}=\sqrt{8\pi G\frac{4\sigma_{SB}T_{CMB}^{4}}{3\alpha^{2}c^{3}}}[/math]

There are no Maxwell-Boltzmann distributions in this derivation, nor does it require Bose-Einstein statistics. It is a direct, deterministic algebraic consequence of geometric saturation, not a statistical probability distribution.

As for requiring a "volume element being examined" - that is certainly a necessity if one interprets the Universe as a 3D thermodynamic fluid. However, in the generative ontology of WILL RG, the energy capacity is governed strictly by the closed 2D surface topology of the [math]S^2[/math] carrier. The 3D volume is a secondary geometric projection of this surface capacity. This makes the classical thermodynamic assumption of a "3D volume element" structurally redundant for establishing this specific boundary condition.

Take a closer look at the critical density formula from which all the equations of state for cosmology are related from.

Then look at the energy only mathematical proof of the critical density formula. You will find the missing term I mentioned.

In so far as the FLRW metric and how the Maxwell Boltzmann equations also ties into the above or for that matter the Raychaudhuri equations under GR.

( note 3 methodologies arriving at the same relations )

I leave it to you to choose to include the missing term ( yes I know precisely which term you dropped ) or not. Thats your choice but without that missing term your equations will not accurately follow the evolutionary history of expansion regardless if your employing Maxwell Boltzmann or any of the above methodologies.

( it would amount to a waste of my time to go through the effort of showing what I described above simply to have it discounted by your ontology reasons ).

Hint the energy only mathematical proof uses nothing more than the energy momentum relation you employ. You may wish to keep your ontology constrained but that's not a view I share. You have already mentioned numerous constraints throughout this thread on mathematical operations you cannot deploy ( geometry, mappings etc etc etc.)

Edited Thursday at 04:21 AM1 day by Mordred

On 2/25/2026 at 11:15 AM, Mordred said:

one of your formulas is wrong as its missing a (3) in its derivative.

I will let you see which formula Im referring to in your paper. It may help me relate to which formulas you have derived yourself and which formulas you have used from other resources such as AI or other papers and your understanding of those formulas incorperated that were derived not using RG.

Let us trace the exact sequence of logical steps you just demonstrated in your comment:

1. You noticed the absence of the 1/3 factor in the density equation. (A correct observation).

2. You immediately concluded this absence was a mathematical error.

3. You concluded that I must be completely unaware of this "error".

4. You concluded that because I am "unaware," the formulas must have been generated by an AI or copied without understanding.

5. Based on this chain of assumptions, you adopted a pedagogical tone ("I will let you see which formula Im referring to") to test my knowledge of my own work.

The critical failure in this sequence of assumptions is that you did not read the derivation you are attempting to critique.

If you would follow the link to original density derivation which I implicitly include in this section: "In the Normalization Identity (derived in WILL RG I),".

Or If you look at Section 6.2 of the WILL RG Part II document, you will find a dedicated subsection explicitly titled:

"1. Saturation Identity (no Friedmann factor)". https://willrg.com/documents/WILL_RG_II.pdf#sec:will-friedmann

The text directly below it states:

"Consequently, the relation between the Hubble parameter and the saturation density is: [math]H_0^2 = 8\pi G \rho_{max}[/math], without the factor of 1/3 that appears in the standard Friedmann equation... The two expressions are numerically identical ([math]\rho_{max} = \rho_{crit}/3[/math]), but their algebraic origins differ fundamentally."

The missing (3) is not a typo, nor is it an AI hallucination. It is the central, mathematically proven topological distinction between a 3D thermodynamic fluid expansion (your paradigm) and a 2D geometric carrier saturation (WILL RG).

I am always open to rigorous scientific critique. However, attempting to quiz me on an "error" that is explicitly derived and justified in a dedicated section of the very paper you are criticizing is not scientific rigor.

Edited Thursday at 04:48 AM1 day by Anton Rize

Where is the pressure term

51 minutes ago, Mordred said:

note 3 methodologies arriving at the same relations

Of course they arrive at the same relation - they all share the exact same ontological assumption: that the Universe is an expanding 3D fluid manifold. My methodology strictly forbid such assumptions.


Here's the pressure derivation:
https://willrg.com/documents/WILL_RG_I.pdf#sec:pressure

Sorry the link was broken. I just fixed it.

Edited Thursday at 04:55 AM1 day by Anton Rize

No offense buts that's another article your asking to go through this one being 77 pages.

Please post how you incorperated pressure into your calculation using the equation above.

@Mordred , I red through the H_0 derivation section again, and realize that while Im providing links to the source density derivation and Im showing mathematically this factor 1/3 difference [math] \rho_{crit} \approx 9.5 \times 10^{-27} [/math]. [math].kg/m^3[/math].; [math]. \rho_{max} \approx \rho_{crit}/3 [/math]. However I never word this difference in this section properly. So Ill add more pronounce accent on the factor 1/3 in this section to avoid any confusions. Thank you for showing me the source of misinterpretation. Its valuable.

5 minutes ago, Mordred said:

No offense buts that's another article your asking to go through this one being 77 pages.

Please post how you incorperated pressure into your calculation using the equation above.

Oh no I wouldn't expect you to go through the hole thing. That's why I always sharing hyperlinks like this https://willrg.com/documents/WILL_RG_I.pdf#sec:pressure

Just click left mouse button on it and it will redirect you to the exact derivation. Just remember if you will use open new tab command - the hyper part will not work and you will see the 1st page of .pdf instead of specific section. It has to be left mouse button only.

Edited Thursday at 05:34 AM1 day by Anton Rize

Ok lets compare by looking at just the kinetic and potential energy terms in just a sphere with gravitational or otherwise pull in terms of the critical density. Its mathematical proof in more detail.

kinetic energy \(\frac{1}{2}mv^2=\frac{1}{2}m\dot{R}^2\)

potential energy \(U -\frac{GMm}{R}\)

mass \(\rho \times volume=\rho \times \frac{4}{3}R^3\)

substituting M into U

\[U=-\frac{G(\rho \cdot \frac{4}{3}\pi R^3m}{R}=-\frac{4\pi G\rho R^2}{3}\]

total energy

\[E_t=K+U=\frac{1}{2}m\cdot {R}^2-\frac{4\pi G\rho R^2}{3}\]

set curvature to zero energy balance for critically dense universe E=0 the point of when expansion halts or it can be described where the escape velocity is identical to the expansion speed.

solve for density \(\rho\rightarrow \rho_c\)

\[\frac{1}{2}\cdot{R}^2=\frac{4\pi G\rho R^2}{3}\]

incorperate Hubble parameter

\[\frac{v}{R}=\frac{ \dot{R}^2}{R^2}\]

\[H^2=\frac{\dot{R}^2}{R^2}\]

\[\rho_c=\frac{3H^2}{8\pi G}\]

that is what you should have

Edited Thursday at 06:43 AM1 day by Mordred

39 minutes ago, Mordred said:

mass ρ×volume=ρ×43R3

Thank you for posting this exact derivation. This is the perfect pedagogical moment, because it highlights the precise ontological divergence between the Standard Model and WILL RG in a single line.

Look carefully at the third step of your proof:

[math] \text{mass} \quad \rho \times \text{volume} = \rho \times \frac{4}{3}\pi R^3 [/math]

Where does the factor of 3 come from? It comes exclusively from the geometric formula for the volume of a 3D sphere.

Your derivation explicitly assumes that the Universe is an expanding Newtonian 3D container filled with a uniform fluid/dust. This is the classical derivation of the Friedmann equation.

Now, please look at the section in my paper titled "Self-Consistency Requirement" and the subsection "No Singularities, No Hidden Regions". https://willrg.com/documents/WILL_RG_I.pdf#sec:no_singularities

RG strictly forbids the "Newtonian cannonball" volume assumption. The geometric limit is defined by the energy projection on the 2D surface of the [math]S^2[/math] carrier (which carries the [math]4\pi[/math] metric).

As derived in the text, the algebraic closure in WILL RG is:

[math]m_0 = 4\pi r^3 \rho[/math]

There is no [math]1/3[/math] volume coefficient because the geometry is a 2D surface-scaled projection, not a 3D volume integration.

If my final equation for [math]H_0[/math] contained that factor of 3, it would mean my mathematics were broken - it would mean I had accidentally smuggled a 3D Newtonian fluid assumption into a 2D relational geometry.

The fact that the 3 is missing is not an error; it is the mathematical proof that the topology successfully maintained its [math]S^2[/math] projective integrity all the way to the macroscopic horizon. Your derivation is exactly what I "should have" if I were doing standard Newtonian cosmology. But I am not.

Edited Thursday at 07:16 AM1 day by Anton Rize

Correct our universe is a 3d sphere for the observable universe. However note that the critical density formula itself only applies 2 dimensions. Or did you miss the R^2 term ?

Here is another relation you missed look at dimensionality of the cross product and the dimensionality of the dot product space. ( inner product space) with above

\[\mu \cdot \nu= \nu \cdot \mu\]

For relevant vector addition rules regarding inner and cross products. The above defines this as an inner product space and the dot product of 2 vectors potential energy (collapse) vs kinetic energy outward ( exoansion).

You only require 2 dimensions to describe that relation as shown above.

In stress energy tensor terms the critical density enerrgy density and pressure relations are

\[\rho =T_{00} ]

\[P= T_{i,j}\]

Edited Thursday at 08:27 AM1 day by Mordred

34 minutes ago, Mordred said:

Correct our universe is a 3d sphere for the observable universe. However note that the critical density formula itself only applies 2 dimensions. Or did you miss the R^2 term ?

Its just an misunderstanding.

The [math]1/R^2[/math] dependence in the critical density formula is not a sign of "2-dimensionality." It is the signature system at its saturation limit.

In standard physics, density is [math]M/R^3[/math]. The only reason the critical density scales as [math]1/R^2[/math] is because at the horizon, the mass of the Universe scales linearly with radius ([math]M \propto R[/math]). This linear growth of mass cancels one power of the cubic volume, leaving [math]R/R^3 = 1/R^2[/math].

In RG, this isn't a "coincidence" - it is derived. The density still has 3DOF:

[math]\rho = \frac{R_s c^2}{8\pi G r^3} = \frac{\kappa^{2}c^{2}}{8\pi Gr^{2}} [/math]

[math] \rho_{max}=\frac{c^{2}}{8\cdot\pi\cdot G\cdot r^{2}} [/math]

However, because [math]\kappa^2 = R_s/r[/math], at the horizon (where [math]\kappa^2 = 1[/math] and [math]r = R_s[/math]), the "invisible" [math]R_s[/math] in the numerator cancels one [math]r[/math] in the denominator. You are looking at the residue of a r^3 -> r^2 cancellation.

But let's step back from moving goalposts. I want to ask you directly:

What are your specific, objective criteria for a model to be taken seriously?

1. Derive [math]H_0[/math] from [math]T_0[/math] and [math]\alpha[/math] with zero free parameters? (Done).

2. Solve the parameter degeneracy in orbital mechanics? (Done).

3. Derive the recombination epoch without Dark Matter? (Done).

Is there a threshold of mathematical evidence that would shift your stance from "dismissing the derivation" to "engaging with the ontology," or are the goalposts intended to stay in motion indefinitely?

Edited Thursday at 08:27 AM1 day by Anton Rize

Your wrong look at the inner and cross product vector addition rules.

An inner product space is 2 dimensional reread the above relations.

Edited Thursday at 08:31 AM1 day by Mordred

6 minutes ago, Mordred said:

Your wrong look at the inner and cross product vector addition rules

That is a very vague redirection.

1. On the Math: Could you be specific? Which exact RG equation violates which rule of inner or cross-product addition? [math]\kappa^2[/math] is explicitly derived as the square of a relational projection (inner product geometry) on an [math]S^2[/math] carrier (cross-product area geometry). Dropping terms like "vector addition rules" without showing a specific algebraic contradiction is not a critique; it is a distraction.

2. On the Logic: You completely avoided my primary question. I will ask it again: What are your specific, objective criteria for a model to be taken seriously?

I have derived [math]H_0[/math] with zero parameters, solved the orbital degeneracy problem, and computed the recombination epoch. If these results are not enough, what is? Do you have a defined threshold for evidence, or are you simply going to throw random jargon from a physics textbook every time the derivation holds up?

You are attempting to judge a fundamental generative geometry using the secondary mathematical artifacts of a 100-year-old descriptive model.

1. On "Inner Products" and "Vector Rules": These are not physical laws; they are bookkeeping tools designed for 3D Euclidean containers. An inner product requires a pre-defined metric space to exist. RG does not assume a metric space - it generates one from the relational energy projections (alpha, light wavelength) things we can physically measure. You trying to legitimize unphysical unmeasurable model specific mathematical tools. Using vector addition rules to "correct" RG is like trying to use a thermometer to measure the volume of a shadow. It is a category error.

2. Observed Reality vs. Tensors: The Stress-Energy Tensor (Tμν) is an ad-hoc mathematical construct used to balance equations in a fluid-dynamic model that requires Dark Matter to function. RG operates on physical observables: absolute temperatures, coupling constants, and geometric closure limits.

3. The Goalpost Question: You have repeatedly retreated into textbook jargon to avoid a simple fact: RG produces accurate physical results (H0, recombination, orbital tests and many more) with zero free parameters - results your "3 methodologies" can only achieve by "throwing in" invisible dark substances.

I will not waste more time debating the commutative properties of your mathematical "toys." If you have a specific physical observation that contradicts the output of my equations, present it. If not, then your critique is nothing more than a defense of a map that is no longer the territory.

What is your objective threshold for a model to be valid? Does it have to be true, or does it just have to use your favorite tensors? Is your physical believes based on dogmas or scientific principals?

On 2/25/2026 at 1:33 PM, MJ kihara said:

Meaning dark matter doesn't exist?

According to my results in https://willrg.com/documents/WILL_RG_II.pdf
The vast majority (only week lensing left unsolved; just haven't touched it yet) of observational phenomena that we interpreting as DE, DM are inevitable consequents of Relational Geometry. We just don't have to speculate any "dark" entities to accurately predict and explain observables. The “missing mass” is not missing mass. It is the observable signature of a star’s resonant coupling to the cosmic horizon inside a closed relational geometry.

Dark Matter becomes redundant.

CRITICAL NOTE: Regardless of complete publicity of all data and tools and methodological discipline of my Open Research - My results are still remain unverified independently.

Until independent verification is confirmed we should avoid such categorical formulations: "dark matter doesn't exist".

There is nothing indirect on my last statement and the mathematics I posted has nothing to do with any particular model.

Its literally geometry and vectors under a geometry.relating kinetic and potential energy terms without any specified particle contributor.

You do know what the dot product or cross product means with regards to vectors do you not ?

Those math operations and symbols are included in the above proof for critical density. The mass term has zero model dependency neither does the gravitational terms. It is any arbitrary inward generation of force of attraction to any arbitrary force of expansion in mathematical displacement terms using vectors.

I only related those mathematics with regards to the FLRW metric with regards to Hubble constant.

Edited Thursday at 01:08 PM1 day by Mordred

Vectors are not a bookkeeping device beyond mathematic relations directly related to displacement. Your ontology arguments do not matter in the slightest.

This is strictly mathematics.