On 2/26/2026 at 1:27 PM, Anton Rize said:

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

I decided to investigate this because of the mention of the fine-structure constant. I'm actually quite impressed with how close the derived Hubble constant matches the measured value. Unravelling the formula, what you appear to me to be saying is that:

[math]\dfrac{\rho_{\text{electromagnetic}}}{\rho_{\text{critical}}} = \alpha^2 \approx \dfrac{1}{18779}[/math]

However, I don't agree with your suggestion that you've figured out the fine-structure constant. The mystery of its particular numerical value remains, even though its relation to other physical notions is well known to physicists. In particular, it is known to be the value of [math]\beta[/math] of the electron in the lowest orbit of the Bohr model of the atom.

3 hours ago, KJW said:

I decided to investigate this because of the mention of the fine-structure constant. I'm actually quite impressed with how close the derived Hubble constant matches the measured value. Unravelling the formula, what you appear to me to be saying is that:

ρelectromagneticρcritical=α2≈118779

However, I don't agree with your suggestion that you've figured out the fine-structure constant. The mystery of its particular numerical value remains, even though its relation to other physical notions is well known to physicists. In particular, it is known to be the value of β of the electron in the lowest orbit of the Bohr model of the atom.

First, I must commend your algebraic reverse-engineering. You perfectly extracted the core relational consequence of the equation:

[math]\frac{\rho_{electromagnetic}}{\rho_{critical}} \propto \alpha^2[/math]

This is exactly the rigid geometric scaling law I am pointing to.

Regarding your skepticism about the fine-structure constant:

my methodology does not allow to introduce any standard QM postulates. So instead, the entire atomic structure is derived strictly from the intersection of three geometric closure principles. I do not import the Bohr radius or [math]\alpha[/math]; they emerge as necessary algebraic identities:
https://willrg.com/documents/WILL_RG_III.pdf#sec:alpha

I am not claiming to have discovered the numerical value of 1/137. What I am showing is its ontological origin. The fine-structure constant identified as the exact kinematic projection ([math]\beta_1[/math]) required to satisfy the geometric closure of a charged fermion on the [math]S^1[/math] and [math]S^2[/math] relational carriers.

What is truly striking is the predictions chain that Im doing. Its like all just locked in to placed... Statistically its pretty much impossible to achieve this randomly:

Unbroken Chain of 10 Derivations/Predictions. No Fitting Involved.

1. Hubble Constant ([math]H_0[/math])

Derived: 68.15 km/s/Mpc | Empirical: 67.4 [math]\pm[/math] 0.5 km/s/Mpc

System: Planck 2018 | Deviation: +1.0%

Formulation: Geometric saturation density derived from CMB temperature and [math]\alpha[/math]

2. CMB First Acoustic Peak ([math]l_1[/math])

Derived: 220.55 | Empirical: 220.60

System: Planck 2018 | Deviation: [math]\approx[/math] 0.02%

Formulation: Resonant harmonics of an [math]S^2[/math] topology loaded by 4.2% baryonic mass

3. CMB Quadrupole Power ([math]D_{l=2}[/math])

Derived: 0.156 - 0.320 (Corridor) | Empirical: [math]\approx[/math] 0.20

System: Planck 2018 | Deviation: Within predicted corridor

Formulation: Vacuum tension acting as a high-pass filter on a tensioned [math]S^2[/math] membrane

4. Galactic Rotation Curves Bias

Derived: [math]0.70 \times 10^{-10} m/s^2[/math] ([math]a_k[/math]) | Empirical: -2.26 km/s (Bias)

System: SPARC (175 galaxies) | Deviation: RMSE [math]\approx[/math] 0.065 dex

Formulation: Structural Projection Resonant Interference with Universal Fundamental Tone

5. Solar Orbital Velocity

Derived: 226.4 km/s | Empirical: 229 [math]\pm[/math] 6 km/s

System: Gaia DR3 / Milky Way | Deviation: Excellent agreement

Formulation: Geometric mean interference between local potential and global horizon

6. Wide Binary Gravity Boost ([math]\gamma[/math])

Derived: [math]\approx[/math] 1.47 | Empirical: [math]\approx[/math] 1.45 - 1.55

System: Gaia DR3 / Chae 2023 | Deviation: Exact Agreement

Formulation: Kinetic Resonance Scale ([math]S^1[/math] carrier coupling weight 1/3)

7. Type Ia Supernova Distance Modulus

Derived: Offset expected [math]\approx[/math] 0.150 mag | Empirical: Raw residual [math]\approx[/math] -0.151 mag

System: Pantheon+ | Deviation: Shape deviation [math]\le[/math] 0.02 mag

Formulation: Geometric Energy Budget Partitioning (2:1 ratio of [math]S^2[/math] tension to kinetic mass)

8. Strong Lensing Einstein Radius

Derived: 1.46'' | Empirical: 1.43 [math]\pm[/math] 0.01''

System: SDSSJ0946 +1006 (SLACS) | Deviation: [math]\approx[/math] 2%

Formulation: Phantom Inertia ([math]Q^2[/math]) acting as universal refractive medium

9. Recombination Epoch

Derived: [math]\approx[/math] 364,860 years | Empirical: [math]\approx[/math] 378,000 years

System: Standard Cosmological Dating | Deviation: [math]\approx[/math] 3.5%

Formulation: Unit Phase Condition ([math]\Omega_{crit} = 1[/math] radian) where arc length equals radius of curvature

10. Electron Mass ([math]m_e[/math])

Derived: [math]9.064 \times 10^{-31}[/math] kg | Empirical: [math]9.109 \times 10^{-31}[/math] kg

System: CODATA | Deviation: [math]\approx[/math] 0.49%

Formulation: Holographic Projection Principle / Geometric Capacity Resonance -> Mach Principle.

Edited March 3Mar 3 by Anton Rize

Just a friendly FYI the fine structure constant has been suggested in terms of the cosmological constant in the Machian universe as well as BSBM ( a form of TeVeS aka MOND) as well been looking into current constraints as it is interesting.

( I seem to recall a variation of LQC that also ties it in but I could be wrong on that).

BSBM paper relating to above

https://arxiv.org/pdf/1202.0069

Edited March 3Mar 3 by Mordred

10 hours ago, KJW said:

ρelectromagneticρcritical=α2≈118779

Just need to point out one important detail:
Because of my methodology I could not simply postulate the volumetric nature of mass and density so the terms that I derived are:
[math]\rho=\frac{\kappa^{2}\cdot c^{2}}{8\cdot\pi\cdot G\cdot r^{2}} [/math]
[math]\rho_{max}=\frac{c^{2}}{8\cdot\pi\cdot G\cdot a^{2}} [/math] when [math]\kappa=1 -> r=R_s [/math]
[math]m_0 = 4\pi r^3 \rho [/math]
You can find derivation here https://willrg.com/documents/WILL_RG_I.pdf#sec:density
So the relation between standard [math]\rho_{crit} \approx 9.5\times10^{-27}[/math] kg/m^3 and my
[math] \rho_{max}[/math] is [math] \frac{\rho_{crit}}{\rho_{max}} \approx 3 [/math]
so when written in my terms it looks like this
[math]\frac{\rho_{gamma}}{3\rho_{max}}=[/math]
[math] \frac{4\sigma_{SB}T_{CMB}^{4}}{c^{3}}\ \frac{8\pi GR_{H}^{2}}{3c^{2}}=\alpha^2[/math]
My interpretation is that its got to do with total relational shift [math]Q^2[/math].
When system is energetically closed no leakage (we not speculating anything apart of the universe so there's no outside the system is closed) we getting
[math] Q^2=\kappa^{2}+\beta^{2}=3\beta^2 -> \beta=\alpha -> 3\alpha^2 -> [/math]
[math] -> H_0=\sqrt{8\pi G\ \frac{\rho_{gamma}}{3\alpha^{2}}}[/math]

What do you think about it? Does it make sense?

Edited March 4Mar 4 by Anton Rize

I realize you likely have me blocked or simply refuse to respond to me but for others even if you do not respond ( which is fine) the above looks remarkably familiar with the cosmological coincidence problem.

If you do see this then my recommendation is that you pursue that direction of research.

Your approach on the last post is better than many I have read. So +1 for that.

Edit should be more specific the fine structure constant , gravitational constant as well as numerous other fundamental constants are also related. This includes DM and DE in some examinations. Granted it likely wasn't your goal but it would work well into that line of research..

Truth be told though it was @KJW earlier comment that brought the above to mind

Edited March 4Mar 4 by Mordred

19 hours ago, Mordred said:

I realize you likely have me blocked or simply refuse to respond to me

The reason I stopped responding to you is that our dialogue lost scientific honesty. You were asking for derivations and then dismissing them using standard definitions without engaging with the actual geometric equations I provided:

On 2/26/2026 at 2:32 PM, Mordred said:

Where is the pressure term

On 2/26/2026 at 2:46 PM, Anton Rize said:

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

On 2/27/2026 at 11:22 PM, Anton Rize said:

Since you want to talk about equation of state - yes please tell me how I deriving the pressure term and what w=? I end up and why?

On 2/27/2026 at 11:53 PM, Mordred said:

The answer would require use of three fields each describing a specified order of relations

Linear acceleration and stress

math]f(r) = 4\pi r^2[/math] can be plotted on a 2D graph does not make the expression itself "graphical coordinates

Thats a 2 dimensional graph using the double cover of the SO(3) group under SU(2)

I have no interest in semantic debates with someone who doesn't read the answers to questions the one asked. I am here for a scientifically honest, rigorous mathematical discussion. If you are willing to actually read the derivations and engage with them directly - whether to mathematically falsify them or explore their consequences - I will gladly continue dialog with you in respectful and friendly manner.


Regarding your specific point about the Cosmological Coincidence Problem: it is highly relevant and the full pattern might tell us something more...

The hypothesis that lead me to all those predictions:
1. The Geometric Budget The total relational shift: [math]Q^2 = \kappa^2 + \beta^2[/math]

2. The Invariant Partitioning Under closure condition ([math]\kappa^2 = 2\beta^2[/math]), this budget naturally partitions into exact, invariant ratios:
[math]\Omega_{potential} = \frac{\kappa^2}{Q^2} = \frac{2}{3}[/math] interpretation: (What standard cosmology labels "Dark Energy")
[math]\Omega_{kinetic} = \frac{\beta^2}{Q^2} = \frac{1}{3}[/math] interpretation: (What standard cosmology labels "Dark Matter")

3. Density Scaling via Alpha These macroscopic geometric partitions are rigidly locked to the microscopic radiation floor ([math]\rho_{gamma}[/math])
via EM kinetic limit ([math]\alpha[/math]): [math]\frac{\rho_{max}}{\rho_{gamma}} = \frac{1}{3\alpha^2}[/math]
[math]\frac{\rho}{\rho_{gamma}} = \frac{2}{3\alpha^2}[/math]
[math]\frac{\rho}{\rho_{max}} = \kappa^2[/math]

So it might be that here is no "coincidence epoch". The universe does not require fine-tuning because the 2:1 ratio of "Dark Energy" to "Dark Matter" is simply the native
[math]\Omega_{potential}[/math] to [math]\Omega_{kinetic}[/math] projection ratio required to geometrically close the system.

Regardless of mind-blowing numerical agreement - my interpretations could be wrong. Still more research is needed...

Edited March 4Mar 4 by Anton Rize

let me ask you a couple of questions it may help understand why I was pushing for equations beyond scalar relations. I will be 100% honest with you, whenever I look at your relations I see no involvement of spacetime curvature itself.

you can take every equation and relation you have posted here and if you were to parallel transport 2 or more separate laser beams your equations don't include how those beams will converge or diverge due to curvature. The simplest example is the center of mass. Take 2 objects and drop em. Yes they will fall at precisely the same rate however the distance between them decreases as they approach the center of mass. In regards to CCP ( Cosmological coincidence problem this directly relates as the Rayleigh equations are considered a solution).

Second example take your Mercury Sun system. Do you show how we can see Hades star system directly behind the sun at specific locations and times ?

in regards to pressure itself. Yes the Gamma law equations pertaining to Boltzmann are straightforward even when applying Van de Walls corrections if your spacetime is flat they work great but what if the area your examining is not homogeneous and isotropic where the thermodynamic state will not expand in an adiabatic and isentropic thermodynamic state ? The 3P relation specifically equates to 3 dimensional flux in a given area and will deviate if say the x axis component has a density difference from the Y and Z components. The critical density formula only works in situations where the cosmological principle applies accurately. As our universe is becoming more and more anistropic sometime in the future it will no longer be accurate.

Take Doppler type shifts

Doppler, gravitational, and cosmological redshift. they each naively look the same but when you get to higher examinations they become rather distinctive especially in causation. One excellent example is late time integrated Sache-Wolfe effect. Your signals are subjective to additional blueshift and red-shifting as the signal crosses underdense and overdense regions.

I could go on and on describing situations but you should see the point above. Fundamentally though its your work, your ontology, your philosophical approach. How far you choose to take it is your choice and not mine to make.

There isn't any FAULT with your mathematics they simply do not show the above. This is where your minimalization places restrictions on what dynamics you can describe. So the choice is simple ( ignore the situations where they can't ) or find ways to expand your treatments to encompass factors that equate to vector relations.

Your call it is your work not mine.

edit with some of the above one can apply a gradient much of your work also relates to the following. Principle of general Covariance. (all physics theories must account for this)

https://en.wikipedia.org/wiki/General_covariance

with CCP a fundamental question one can ask is " if the fundamental constants so easy to understand and develop relations to why is unification so hard ? "

Edited March 5Mar 5 by Mordred

1 hour ago, Mordred said:

let me ask you a couple of questions it may help understand why I was pushing for equations beyond scalar relations. I will be 100% honest with you, whenever I look at your relations I see no involvement of spacetime curvature itself.

you can take every equation and relation you have posted here and if you were to parallel transport 2 or more separate laser beams your equations don't include how those beams will converge or diverge due to curvature. The simplest example is the center of mass. Take 2 objects and drop em. Yes they will fall at precisely the same rate however the distance between them decreases as they approach the center of mass. In regards to CCP ( Cosmological coincidence problem this directly relates as the Rayleigh equations are considered a solution).

Second example take your Mercury Sun system. Do you show how we can see Hades star system directly behind the sun at specific locations and times ?

in regards to pressure itself. Yes the Gamma law equations pertaining to Boltzmann are straightforward even when applying Van de Walls corrections if your spacetime is flat they work great but what if the area your examining is not homogeneous and isotropic where the thermodynamic state will not expand in an adiabatic and isentropic thermodynamic state ? The 3P relation specifically equates to 3 dimensional flux in a given area and will deviate if say the x axis component has a density difference from the Y and Z components.

Take Doppler type shifts

Doppler, gravitational, and cosmological redshift. they each naively look the same but when you get to higher examinations they become rather distinctive especially in causation. One excellent example is late time integrated Sache-Wolfe effect. Your signals are subjective to additional blueshift and red-shifting as the signal crosses underdense and overdense regions.

I could go on and on describing situations but you should see the point above. Fundamentally though its your work, your ontology, your philosophical approach. How far you choose to take it is your choice and not mine to make.

There isn't any FAULT with your mathematics they simply do not show the above. This is where your minimalization places restrictions on what dynamics you can describe. So the choice is simple ( ignore the situations where they can't ) or find ways to expand your treatments to encompass factors that equate to vector relations.

Your call it is your work not mine.

My first reaction was to provide the explicit derivations for these scenarios.

However, you have consistently moved the goalposts and dismissed derivations by reverting to standard definitions instead of engaging with the math. Instead of just throwing more equations at you, let's analyze your reasoning.

Can you explain on what exact basis you are making the assertion that this framework cannot describe these dynamics?

Are your assertions actually falsifiable, or is the necessity of standard tensor formalism a dogmatic position you are willing to defend against empirical and mathematical facts?

look directly at many of your derivatives that uses critical density The formula itself is derived with a homegenous and isotropic fluid and applying Jeans instability. The FLRW equations and treatments you have shown I don't recall seeing the curvature equation of state nor have I seen a gradient operator. Did I miss some form of gradient operator ?

Do you have some form of parallel transport ? I don't recall seeing anything along those lines.

I do recall one of your lemmas on maximally symmetric which isn't curvature.

Its against your minimalization to employ tangent vectors as that is a form of curve fitting along with affine connections.

Edited March 5Mar 5 by Mordred

25 minutes ago, Mordred said:

look directly at many of your derivatives that uses critical density The formula itself is derived with a homegenous and isotropic fluid and applying Jeans instability. The FLRW equations and treatments you have shown I don't recall seeing the curvature equation of state nor have I seen a gradient operator. Did I miss some form of gradient operator ?

Do you have some form of parallel transport ? I don't recall seeing anything along those lines.

I do recall one of your lemmas on maximally symmetric which isn't curvature.

Its against your minimalization to employ tangent vectors as that is a form of curve fitting along with affine connections.

Look, you having major problems at the basic logic level:

If you can't see something does it mean it doesn't exist?

You assertion rests on a fallacy in formal logic known as argument of ignorance. Google it.

The second layer of your assertion is based on cycle reasoning:

GR uses formalism x and its true. there for everything that is not x is false.

Until you will learn the basis of logical reasoning - we have nothing to talk about.

Really lets look at this equation

On 3/3/2026 at 6:27 PM, Anton Rize said:

ρgamma3ρmax=
4σSBT4CMBc3 8πGR2H3c2=α2

quite frankly your describing an ideal gas. The critical density formula is an ideal gas solution.

Where is there any term in those equations that does not describe a homogenous and isotropic fluid ?

You stated were not to argue ontology or philosophy how about straight up mathematics ?

24 minutes ago, Anton Rize said:

Look, you having major problems at the basic logic level:

If you can't see something does it mean it doesn't exist?

You assertion rests on a fallacy in formal logic known as argument of ignorance. Google it.

The second layer of your assertion is based on cycle reasoning:

GR uses formalism x and its true. there for everything that is not x is false.

Until you will learn the basis of logical reasoning - we have nothing to talk about.

typical response perhaps You can understand my frustration when I receive responses such as this.

Why is it anytime I so much as mention anything outside your papers you respond negatively is beyond me when you also claim a desire to improve your applications of simplifying physics yet choose to ignore a huge bulk of those physics relations or react negatively when they are mentioned. That is what has been the real frustration in trying to discuss your papers.

What you choose to do with your work is your choice I'm simply describing what I see and feel should be included in better detail if they are included.

Edited March 5Mar 5 by Mordred

19 minutes ago, Mordred said:

You stated were not to argue ontology or philosophy how about straight up mathematics ?

Great idea! there's around 10 derivations I posted including the density derivation, and you ignored all of them. So yeh I agree lets get in to mathematics. You can start from any of my derivations.

22 minutes ago, Mordred said:

quite frankly your describing an ideal gas. The critical density formula is an ideal gas solution.

You have no idea what are you talking about and this is a major barrier in our communication. In logic this fallacy called The Straw Man. Look it up.

25 minutes ago, Mordred said:

typical response perhaps You can understand my frustration when I receive responses such as this.

Hold on! This is a big deal. Are you saying that you can't see the problem in your logic that I pointed out? Its just if we can't agree even at the most basic level it should be an alarm sing for both of us.

Look back at how often you shot me down for mentioning common physics treatments or would you rather I go through them and post them. That last post of yours was a good example. I specifically asked the question Do you have an equivalent to the gradient operator. Did you respond with the math ?

Edited March 5Mar 5 by Mordred

realized I had already asked the question I had posted, earlier so withdrawing this post

Edited March 5Mar 5 by Mordred

2 hours ago, Mordred said:

Look back at how often you shot me down for mentioning common physics treatments or would you rather I go through them and post them. That last post of yours was a good example. I specifically asked the question Do you have an equivalent to the gradient operator. Did you respond with the math ?

Look I don't want to shot you down. This is not the way I prefer to communicate. It's just the fact that no matter how many times Ill tell you: "I have a methodology I follow in my research," And I explicitly showed the principals. There's no container (x,y,z) - because it violates the core methodological principals that I follow. For once its shocking that you can't understand the difference between relationalism and substantivalism.

The Irony is that you unknowingly making way more philosophical claims than I do. And that's exactly the reason why I started this research. You think that you not doing philosofy - you doing physics but you wrong:

(x,y,z) - this is enormous philosophical claim that you not only just postulate silently but you also completely blissful about it!
Objects is a substance in empty (x,y,z) - another huge unproven speculation.
instead of the single SPACE-TIME-ENERGY relational structure you creating extra primitive by dividing it SPACETIME as empty box and ENERGY inside of this box. - not derived not empirically proven just postulated blindly
By thinking that you are not doing philosophy you end up making this wild assumptions blind.
Physics without philosophy = engineering (no offence to engineers out there, great respect)

And with all this you keep trying to fit my research in your made up empty box that you not even awear of - THAT is really frustrates me. And it would be ok if it would happened once or twice but its just non stop! You saying "You model dont have empty box there for it cannot predict a" - Im showing you mathematical prove that you are mistaken and explicitly showing the prediction of a, but you ignore it and demand prediction of b, I demonstrate, we moved to c. Are you planning to go through the hole "alphabet"? Do you ever learn anything when proven wrong?

I need a clear answer to this question:

3 hours ago, Anton Rize said:

Are you saying that you can't see the problem in your logic that I pointed out?

without the clear answer we can't move any further.

@Anton Rize, I've being looking at your first PDF, starting from the beginning, in order to gain insight as to how to solve a physics problem using your theory.

I believe I found an error with regards to your explanation of the deflection of light by a source of gravitation, specifically the factor of 2 which distinguishes Einstein's prediction from a Newtonian prediction. I would like you to derive a formula for the deflection of an object with non-zero mass by a source of gravitation. That formula should include all its dependencies. You may specify the gravitation as:

[math]\dfrac{r}{r_s}[/math]

and you may assume the mass of the deflected object, though non-zero, is sufficiently small that any gravitational radiation is negligible. It is up to you to determine what the deflection angle of the object depends upon.

Edited March 5Mar 5 by KJW

4 hours ago, Anton Rize said:

Look I don't want to shot you down. This is not the way I prefer to communicate. It's just the fact that no matter how many times Ill tell you: "I have a methodology I follow in my research," And I explicitly showed the principals. There's no container (x,y,z) - because it violates the core methodological principals that I follow. For once its shocking that you can't understand the difference between relationalism and substantivalism.

The Irony is that you unknowingly making way more philosophical claims than I do. And that's exactly the reason why I started this research. You think that you not doing philosofy - you doing physics but you wrong:

(x,y,z) - this is enormous philosophical claim that you not only just postulate silently but you also completely blissful about it!
Objects is a substance in empty (x,y,z) - another huge unproven speculation.
instead of the single SPACE-TIME-ENERGY relational structure you creating extra primitive by dividing it SPACETIME as empty box and ENERGY inside of this box. - not derived not empirically proven just postulated blindly
By thinking that you are not doing philosophy you end up making this wild assumptions blind.
Physics without philosophy = engineering (no offence to engineers out there, great respect)

And with all this you keep trying to fit my research in your made up empty box that you not even awear of - THAT is really frustrates me. And it would be ok if it would happened once or twice but its just non stop! You saying "You model dont have empty box there for it cannot predict a" - Im showing you mathematical prove that you are mistaken and explicitly showing the prediction of a, but you ignore it and demand prediction of b, I demonstrate, we moved to c. Are you planning to go through the hole "alphabet"? Do you ever learn anything when proven wrong?

I need a clear answer to this question:

without the clear answer we can't move any further.

I was never arguing logic to begin with. I was explaining where your articles lack detail as well as the examples used. If you look closely you would recognize most of my posts have not been theory dependent but rather that of mathematical relations that need to be presented in a more readily to relate to manner.

Math operations such as curl and divergence is a huge chunk of physics relations. So is ray castings when it comes to observational cosmology or non uniform mass density distributions and their effects.

These are major areas that any other theory such as GR, QFT etc deal with. Its also one of the reasons why the equations used get so complex.

In every theory above scalar field and scalar treatments is the easiest case scenarios. So if you really want to show the power behind your methodology try some of the examples I've provides.

Worse case scenario it will help others better relate to what you have

On 3/3/2026 at 5:17 PM, KJW said:

Unravelling the formula, what you appear to me to be saying is that:

ρelectromagneticρcritical=α2≈118779

Doesn't mass of dark matter taken into consideration when calculating critical density?

The author methodology refutes existence of dark matter and dark energy.

12 hours ago, KJW said:

@Anton Rize, I've being looking at your first PDF, starting from the beginning, in order to gain insight as to how to solve a physics problem using your theory.

I believe I found an error with regards to your explanation of the deflection of light by a source of gravitation, specifically the factor of 2 which distinguishes Einstein's prediction from a Newtonian prediction. I would like you to derive a formula for the deflection of an object with non-zero mass by a source of gravitation. That formula should include all its dependencies. You may specify the gravitation as:

rrs

and you may assume the mass of the deflected object, though non-zero, is sufficiently small that any gravitational radiation is negligible. It is up to you to determine what the deflection angle of the object depends upon.

@KJW, I'm so grateful that you decided to invest your time in my research. I highly appreciate it. Thank you!

So you got me all exited and I spend pretty much all night looking for error. No success... Maybe I just cant see it? Could you elaborate please?
But most important during this sleepless night I finally derived methodologically pure ontologically minimal and I think truly beautiful one input solution for light deflection. No mass, no G, no metric, no 4D spacetime curvature - pure relational geometry at it's best. Maybe its lack of sleep talking but Im feeling like its yet another triumph of Relational Geometry. Please have a look and let me know what you think:

---

Algebraic Derivation of Light Deflection
https://willrg.com/documents/WILL_RG_I.pdf#sec:light_deflection

In General Relativity, the deflection of light is obtained by integrating the null geodesic equations over a curved spacetime manifold, often relying on weak-field approximations and Taylor expansions. Within WILL Relational Geometry (RG), we reject both the background manifold and the use of mathematical approximations as non-operational ontological artifacts.

The system consists exclusively of its participants: the Source, the Lens (at periapsis [math]p[/math]), and the Receiver. The total deflection angle must be derived as a strict, exact algebraic difference between their measurable relational phase states, without resorting to series expansions.

Theorem: Algebraic Deflection of Light

Let [math]o[/math] be the orbital phase (true anomaly) representing the exact geometric angle between the lens periapsis and an observer. For an observer at local potential state [math]\kappa_o[/math], this angle is strictly determined by the algebraic identity:

[math]\cos(o) = \frac{\kappa_o^2 - \kappa_p^4}{\kappa_p^2 \kappa_{Xp}^2}[/math]

where [math]\kappa_p[/math] is the potential projection at periapsis, and [math]\kappa_{Xp}^2 = 1 - \kappa_p^2[/math] is the corresponding phase parameter.

Proof:

Step 1: The Photonic Closure Defect (Shape Parameter)

By the Single-Axis Transformation Principle (link: https://willrg.com/documents/WILL_RG_I.pdf#sec:nature_of_light) for light, the kinematic projection saturates the carrier ([math]\beta = 1[/math]), forcing the internal phase to vanish ([math]\beta_Y = 0[/math]). This eliminates the 1/2 partitioning factor inherent to massive bodies. Consequently, the closure defect at periapsis for a photon is defined exclusively by the total projections:

[math]\delta_\gamma = \frac{\kappa_p^2}{\beta_p^2} = \kappa_p^2[/math]

The geometric eccentricity (link: https://willrg.com/documents/WILL_RG_I.pdf#sec:rel_ecc) (shape parameter) of the light trajectory emerges directly from this closure defect:

[math]e_\gamma = \frac{1}{\delta_\gamma} - 1 = \frac{1}{\kappa_p^2} - 1[/math]

Step 2: Relational Phase State Equation

In Relational Orbital Mechanics (link: https://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom), the local potential [math]\kappa_o[/math] at any orbital phase [math]o[/math] is related to the periapsis potential [math]\kappa_p[/math] by the exact topological scaling:

[math]\kappa_o^2 = \kappa_p^2 \frac{1 + e_\gamma \cos(o)}{1 + e_\gamma}[/math]

Substituting the photonic shape parameter [math]1 + e_\gamma = \frac{1}{\kappa_p^2}[/math]:

[math]\kappa_o^2 = \kappa_p^2 \frac{1 + (\frac{1}{\kappa_p^2} - 1)\cos(o)}{1 / \kappa_p^2} = \kappa_p^4 (1 + (\frac{1-\kappa_p^2}{\kappa_p^2})\cos(o))[/math]

Expanding the bracket:

[math]\kappa_o^2 = \kappa_p^4 + \kappa_p^2(1-\kappa_p^2)\cos(o)[/math]

Recognizing that [math]1-\kappa_p^2 = \kappa_{Xp}^2[/math] (the phase component at periapsis), we solve for [math]\cos(o)[/math]:

[math]\cos(o) = \frac{\kappa_o^2 - \kappa_p^4}{\kappa_p^2 \kappa_{Xp}^2}[/math]

This completes the exact algebraic link between the measured potentials and the geometric angle.

Total Deflection (No Approximations)

In a purely flat geometry without gravitational phase ([math]\kappa_p = 0[/math]), this angle would be exactly [math]\frac{\pi}{2}[/math] (a straight line from a distant point to periapsis covers exactly a quarter of a circle). The presence of the gravitational gradient increases this angle by an exact one-sided deflection amount, which is half of the total deflection ([math]\frac{\Delta\varphi}{2}[/math]).

Let [math]o_\infty = \frac{\pi}{2} + \frac{\Delta\varphi}{2}[/math]. Applying the fundamental trigonometric identity [math]\cos(\frac{\pi}{2} + x) = -\sin(x)[/math]:

[math]-\sin(\frac{\Delta\varphi}{2}) = -\frac{\kappa_p^2}{\kappa_{Xp}^2} \rightarrow \sin(\frac{\Delta\varphi}{2}) = \frac{\kappa_p^2}{\kappa_{Xp}^2}[/math]

Solving directly for the total deflection angle [math]\Delta\varphi[/math] gives the absolute, exact, and non-linear equation for light deflection in Generative Physics:

[math]\Delta\varphi = 2 \arcsin(\frac{\kappa_p^2}{\kappa_{Xp}^2})[/math]

Desmos Project: Algebraic Light Deflection (one input derivation)

https://www.desmos.com/calculator/ldynwowqvi

Epistemological Triumph:

This result is achieved without a single differential equation, without background manifolds, and without Taylor series approximations.

The sine of the one-sided deflection angle is strictly equal to the ratio of the potential amplitude ([math]\kappa_p^2[/math]) to the potential phase ([math]\kappa_{Xp}^2[/math]) at periapsis.

For weak fields ([math]\kappa_p^2 \ll 1[/math]), the phase component [math]\kappa_{Xp}^2 \rightarrow 1[/math] and [math]\arcsin(x) \approx x[/math], recovering the empirical value [math]\Delta\varphi \approx 2\kappa_p^2[/math] (equivalent to legacy ontologically inflated form [math]4GM/rc^2[/math]). However, this equation remains exact and structurally unbroken across all interaction scales, demonstrating that spacetime curvature is simply the algebraic shadow of relational energy projections.

The WILL RG formula and GR exact agree to 2 × 10⁻⁷ arcseconds - six orders of magnitude below measurement uncertainty. Both sit comfortably within the observational error bars.

Second-order structure (honest disclosure): The series expansions in κ_p² differ at second order: WILL RG gives coefficient 2.0, GR gives (15π/16 − 1) ≈ 1.945. The discrepancy is ~2.8% of the second-order term, which translates to ~0.2 μas for the Sun - unmeasurable with any current or near-future technology. This could mean I missed something or in some future it could become another unique and falsifiable prediction from RG.

---

So what we got:
GR needs mass, G, differential formalism, tensors, 4D curvature of spacetime and lots of other questionable ontological baggage...
RG needs z_Sun. That's it!

@KJW , thank you again mate! You gave me motivation to finally derive it.

Edited March 6Mar 6 by Anton Rize

6 hours ago, MJ kihara said:

The author methodology refutes existence of dark matter and dark energy.

Corrections;not the author methodology,the author refutes existence of dark matter/energy.

13 hours ago, KJW said:

I would like you to derive a formula for the deflection of an object with non-zero mass

*Laughing hysterically*

Only just now I realized that you where asking for nonzero mass object.
Ok no worries it shouldn't hard at this point. Ill be back.

42 minutes ago, MJ kihara said:

Corrections;not the author methodology,the author refutes existence of dark matter/energy.

49 minutes ago, Anton Rize said:

@KJW, I'm so grateful that you decided to invest your time in my research. I highly appreciate it. Thank you!

So you got me all exited and I spend pretty much all night looking for error. No success... Maybe I just cant see it? Could you elaborate please?
But most important during this sleepless night I finally derived methodologically pure ontologically minimal and I think truly beautiful one input solution for light deflection. No mass, no G, no metric, no 4D spacetime curvature - pure relational geometry at it's best. Maybe its lack of sleep talking but Im feeling like its yet another triumph of Relational Geometry. Please have a look and let me know what you think:

---

Algebraic Derivation of Light Deflection
https://willrg.com/documents/WILL_RG_I.pdf#sec:light_deflection

In General Relativity, the deflection of light is obtained by integrating the null geodesic equations over a curved spacetime manifold, often relying on weak-field approximations and Taylor expansions. Within WILL Relational Geometry (RG), we reject both the background manifold and the use of mathematical approximations as non-operational ontological artifacts.

The system consists exclusively of its participants: the Source, the Lens (at periapsis p), and the Receiver. The total deflection angle must be derived as a strict, exact algebraic difference between their measurable relational phase states, without resorting to series expansions.

Theorem: Algebraic Deflection of Light

Let o be the orbital phase (true anomaly) representing the exact geometric angle between the lens periapsis and an observer. For an observer at local potential state κo, this angle is strictly determined by the algebraic identity:

cos(o)=κ2o−κ4pκ2pκ2Xp

where κp is the potential projection at periapsis, and κ2Xp=1−κ2p is the corresponding phase parameter.

Proof:

Step 1: The Photonic Closure Defect (Shape Parameter)

By the Single-Axis Transformation Principle (link: https://willrg.com/documents/WILL_RG_I.pdf#sec:nature_of_light) for light, the kinematic projection saturates the carrier (β=1), forcing the internal phase to vanish (βY=0). This eliminates the 1/2 partitioning factor inherent to massive bodies. Consequently, the closure defect at periapsis for a photon is defined exclusively by the total projections:

δγ=κ2pβ2p=κ2p

The geometric eccentricity (link: https://willrg.com/documents/WILL_RG_I.pdf#sec:rel_ecc) (shape parameter) of the light trajectory emerges directly from this closure defect:

eγ=1δγ−1=1κ2p−1

Step 2: Relational Phase State Equation

In Relational Orbital Mechanics (link: https://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom), the local potential κo at any orbital phase o is related to the periapsis potential κp by the exact topological scaling:

κ2o=κ2p1+eγcos(o)1+eγ

Substituting the photonic shape parameter 1+eγ=1κ2p:

κ2o=κ2p1+(1κ2p−1)cos(o)1/κ2p=κ4p(1+(1−κ2pκ2p)cos(o))

Expanding the bracket:

κ2o=κ4p+κ2p(1−κ2p)cos(o)

Recognizing that 1−κ2p=κ2Xp (the phase component at periapsis), we solve for cos(o):

cos(o)=κ2o−κ4pκ2pκ2Xp

This completes the exact algebraic link between the measured potentials and the geometric angle.

Total Deflection (No Approximations)

In a purely flat geometry without gravitational phase (κp=0), this angle would be exactly π2 (a straight line from a distant point to periapsis covers exactly a quarter of a circle). The presence of the gravitational gradient increases this angle by an exact one-sided deflection amount, which is half of the total deflection (Δφ2).

Let o∞=π2+Δφ2. Applying the fundamental trigonometric identity cos(π2+x)=−sin(x):

−sin(Δφ2)=−κ2pκ2Xp→sin(Δφ2)=κ2pκ2Xp

Solving directly for the total deflection angle Δφ gives the absolute, exact, and non-linear equation for light deflection in Generative Physics:

Δφ=2arcsin(κ2pκ2Xp)

Desmos Project: Algebraic Light Deflection (one input derivation)

https://www.desmos.com/calculator/ldynwowqvi

Epistemological Triumph:

This result is achieved without a single differential equation, without background manifolds, and without Taylor series approximations.

The sine of the one-sided deflection angle is strictly equal to the ratio of the potential amplitude (κ2p) to the potential phase (κ2Xp) at periapsis.

For weak fields (κ2p≪1), the phase component κ2Xp→1 and arcsin(x)≈x, recovering the empirical value Δφ≈2κ2p (equivalent to legacy ontologically inflated form 4GM/rc2). However, this equation remains exact and structurally unbroken across all interaction scales, demonstrating that spacetime curvature is simply the algebraic shadow of relational energy projections.

The WILL RG formula and GR exact agree to 2 × 10⁻⁷ arcseconds - six orders of magnitude below measurement uncertainty. Both sit comfortably within the observational error bars.

Second-order structure (honest disclosure): The series expansions in κ_p² differ at second order: WILL RG gives coefficient 2.0, GR gives (15π/16 − 1) ≈ 1.945. The discrepancy is ~2.8% of the second-order term, which translates to ~0.2 μas for the Sun - unmeasurable with any current or near-future technology. This could mean I missed something or in some future it could become another unique and falsifiable prediction from RG.

---

So what we got:
GR needs mass, G, differential formalism, tensors, 4D curvature of spacetime and lots of other questionable ontological baggage...
RG needs z_Sun. That's it!

@KJW , thank you again mate! You gave me motivation to finally derive it.

Future challenge

" what is the magnification"

How would you use the quoted relations to determine this.

As an assist to help you establish the above.

https://www.google.com/url?sa=i&source=web&rct=j&url=https://kamion.pha.jhu.edu/Ay127/week7.pdf&ved=2ahUKEwiu-evdloqTAxVGFzQIHciJJsUQ1fkOegQIDBAC&opi=89978449&cd&psig=AOvVaw2blqf2txIV8JzlX7EF6dwh&ust=1772848843742000

Note the use of angular diameter distance.

Just a little side note there is a lensing methodology to determine Hubble constant.

See equation 9.6

https://www.google.com/url?sa=i&source=web&rct=j&url=https://orbi.uliege.be/bitstream/2268/74098/7/Refsdal_Surdej_RepProgPhys_1994_56_117_185.pdf&ved=2ahUKEwiu-evdloqTAxVGFzQIHciJJsUQ1fkOegQIDBAU&opi=89978449&cd&psig=AOvVaw2blqf2txIV8JzlX7EF6dwh&ust=1772848843742000

14 hours ago, KJW said:

I would like you to derive a formula for the deflection of an object with non-zero mass by a source of gravitation. That formula should include all its dependencies. You may specify the gravitation as:

rrs

and you may assume the mass of the deflected object, though non-zero, is sufficiently small that any gravitational radiation is negligible. It is up to you to determine what the deflection angle of the object depends upon.

Ok here it is for mass body's:

---

The factor of 2 is exact geometric consequence of the Energy-Symmetry Law (https://willrg.com/documents/WILL_RG_I.pdf#sec:energy-symmetry) and the number of active relational axis on the kinematic carrier.

Here is the exact derivation:

1. Massive Bodies (Dual-Axis Partitioning)

For massive bodies, the transformation resource is partitioned equally between two orthogonal relational axis on the [math]S^1[/math] carrier: Amplitude ([math]\beta[/math]) and Phase ([math]\beta_Y[/math]). Because the energy budget is distributed across two axis, the invariant binding energy inherently carries a 1/2 partitioning factor:

[math]W_{mass} = \frac{1}{2}(\kappa^2 - \beta^2)[/math]

This gives the effective potential: [math]\Phi_{mass} = \frac{1}{2}\kappa^2 c^2[/math].

Applying this conserved, partitioned energy invariant between periapsis and apoapsis yields the exact shape parameter (eccentricity) for a massive body:

[math]e_m = \frac{2\beta_p^2}{\kappa_p^2} - 1[/math]
https://willrg.com/documents/WILL_RG_I.pdf#sec:rel_ecc

2. Light (Single-Axis Collapse)

By the Single-Axis Transformation Principle, a photon's kinematic projection completely saturates the carrier ([math]\beta = 1[/math]). This forces the internal phase component to vanish entirely ([math]\beta_Y = 0[/math]).

Because the Y-axis is absent, the entire relational resource is concentrated on the single X-axis. The 1/2 partitioning factor is strictly eliminated:

[math]W_{\gamma} = \kappa^2 - \beta^2 = \kappa^2 - 1[/math]

This gives the unpartitioned effective potential for light: [math]\Phi_\gamma = \kappa^2 c^2[/math].

The gravitational effect on light is exactly twice that on massive particles at the geometric level. This yields the photon shape parameter:

[math]e_\gamma = \frac{1}{\kappa_p^2} - 1[/math]
https://willrg.com/documents/WILL_RG_I.pdf#sec:nature_of_light

3. The Exact Deflection Equations

To find the deflection, we use the exact relational phase state equation: [math]\kappa_o^2 = \kappa_p^2 \frac{1 + e\cos(o)}{1 + e}[/math].

Distant observer ([math]\kappa_o \to 0[/math]) yields the angle [math]\cos(o_{obs}) = -1/e[/math].

Using the total angular phase [math]o_{obs} = \frac{\pi}{2} + \frac{\Delta\varphi}{2}[/math], we get the geometric relation:

[math]\sin(\frac{\Delta\varphi}{2}) = \frac{1}{e}[/math]

Substituting the respective shape parameters gives the exact, non-linear deflection angles without a single approximation:

For a massive body (partitioned budget):

[math]\Delta\varphi_m = 2 \arcsin(\frac{\kappa_p^2}{2\beta_p^2 - \kappa_p^2})[/math]

For light (unpartitioned budget, [math]\beta=1[/math]):

[math]\Delta\varphi_\gamma = 2 \arcsin(\frac{\kappa_p^2}{1 - \kappa_p^2})[/math]

The historical "factor of 2" discrepancy does not require to speculate a curved 4D spacetime manifold. It is the direct algebraic signature of the axis count in relational space: massive bodies distribute energy across two axes (requiring the 1/2 factor), while light collapses the geometry to a single axis, experiencing the full unpartitioned geometric effect.

---

As you can see everything is absolutely transparent and intuitive. Let me know what you think.

Edited March 6Mar 6 by Anton Rize

2 hours ago, Anton Rize said:

I'm so grateful that you decided to invest your time in my research. I highly appreciate it. Thank you!

That's because I see value in it. Nevertheless, I'm still scrutinising you and your work. And the nature of your work has made it very difficult to scrutinise in the way I feel is necessary. I'm not sure that what value I see is what you intend me to see as I'm not in full agreement with your philosophy, although some of it does align with relativity. What I would like to see is some form of mathematical proof that your theory fully agrees with general relativity. What you have provided so far is not such a proof, and I'm not sure what such a proof would look like.

2 hours ago, Anton Rize said:

Algebraic Derivation of Light Deflection

Thanks. This looks more detailed than what I saw earlier. I want to look through it to see if you have addressed the problem I see in your explanation. Bear in mind that I know why the defection of light under general relativity is twice that of Newtonian theory. It's not straightforward and I believe you have chosen the wrong explanation. I don't wish to reveal what I believe to be your mistake because I think it is important for your theory that it be able to derive things without the guidance that exists when deriving preexisting results.

2 hours ago, Anton Rize said:

Only just now I realized that you where asking for nonzero mass object.

I feel that what I'm asking for may be too difficult for the piece of information I'm looking for from you. While I asked for a complete formula, in fact I want to know how you handle a specific scenario that I don't wish to reveal, but maybe I should. I apologise for that. Anyway, if you can preempt why I am asking you about non-zero mass, perhaps you can address my concern without deriving the formula.

Nice post follows some of my concerns but far better worded +1.

One I've hinted at numerous times is how does one apply field treatment distributions which also happens to be one of my favorite studies in different formal methodologies.

Another being how would it handle multiple unknowns if say you had to determine all the Cosmological parameters given just volume and change in volume over time. Nothing else.

Edited March 6Mar 6 by Mordred