Is anybody else here struck by the similarities between Anton's ideas and Category Theory? recall that this theory, which is 80 yers old and highly respectable, seeks to unify all mathematics in a sort of meta-mathematiccs and where the different "objects" in the different categories (of sets, of rings, of groups etc) are of no real interest, the relations between them ("morphisms") of passing interest only, the main focus being on the relations between categories and thus the relations between maps between different categorical object, ("maps of maps")

I can can see how, in outline, it might be applied to both the special and general theories, but having no real knowledge of physics, cannot see the details. Maybe that's where the devil lies?

I beleive that John Baez (Riverside, U. Cal??) has an interest in applying category theory to physics problems. Might be worth Anton's while to look up is work

Edited October 29Oct 29 by Xerxes

Thanks for reminding me of the terminology +1

That is exactly what I was getting at when I was discussion 'relational geometry'.

On 10/26/2025 at 2:45 PM, Markus Hanke said:

Can you show us exactly how you calculate the trajectory of this test particle?

On 10/27/2025 at 1:45 PM, Anton Rize said:

Hit-or-miss in WILL (short, coordinate-free)

On 10/27/2025 at 3:14 PM, Markus Hanke said:

Some of these things are also observer-dependent, which you’re not taking into consideration at all.

Showing complete misunderstanding.

On 10/27/2025 at 3:14 PM, Markus Hanke said:

nowhere do you actually take into consideration the nature and strength of the energy-momentum distribution

Proven Wrong.

On 10/27/2025 at 3:14 PM, Markus Hanke said:

Also, real-world gravity isn’t linear, but nowhere do you account for that non-linearity

Proven Wrong.

On 10/27/2025 at 3:14 PM, Markus Hanke said:

If I were to model this situation in standard GR, the result for the trajectory of the test particle is very different - not surprisingly, because it accounts for all relevant relativistic effects.

Due to my results allying with empirical data, your different results would be just plain wrong. So again proven wrong.

On 10/27/2025 at 11:21 PM, Anton Rize said:

I made 2 desmos projects for you:
https://www.desmos.com/geometry/nrtnjramrl - calculates aphelion of Mercury using the set of algebraic equations I listed above r_{a}=\frac{-R_{s}-\sqrt{R_{s}^{2}-8E_{d}\left(-h^{2}\right)}}{4E_{d}}=6.9762118617\times10^{10} m. empirical value r_a = 6.982×10^10 m (discrepancy due to estimated input values, but you got the point)
https://www.desmos.com/geometry/hkxjqfkchp - calculates perihelion precession of Mercury \Delta_{WILL}=\frac{2\pi Q_{Merc}^{2}}{\left(1-e_{Merc}^{2}\right)}\ = 5.0208724126\times10^{-7} radians/orbit. empirical value \Delta_{Merc}=5.02 \times10^{-7} radians/orbit.

On 10/27/2025 at 4:19 PM, Anton Rize said:

You are assuming these concepts imply a metric. I am deriving them from an algebraic foundation that precedes it. The burden is now on you to show which definition is faulty, not which word sounds like GR.

@Markus Hanke stop building argument on personal (often wrong) opinion and show which definition is faulty not which word sounds like GR.

6 hours ago, studiot said:

Are you suggesting that

c = -(e^πi) ? or any of your natural constants, set to unity ?

Sorry I have no idea what you mean by this. Cant see any connections at all. There must be some misunderstanding between us.

6 hours ago, studiot said:

Since you are listed as living in Brisbane I am suprised you need AI to write English for you. Do you not speak English ?

I moved to AU some time ago in my 30's. English is my third languedge. I can talk I can read but I cant spell at all 🙃. Also for topics like this I prefer to think and wright in my native language due to my English vocabulary isn't big enough sometimes.

23 minutes ago, Anton Rize said:

7 hours ago, studiot said:

Are you suggesting that

c = -(e^πi) ? or any of your natural constants, set to unity ?

Sorry I have no idea what you mean by this. Cant see any connections at all. There must be some misunderstanding between us.

Have you heard of Euler's identity ?

https://en.wikipedia.org/wiki/Euler%27s_identity

e, i and π are all pure numbers, 1. That is they have no units at all.

So if you are saying that c, etc = 1 as a pure number you are saying that it satisfies what I wrote above.

That is why these constants cannot do without units and why you are incorrect to say.

23 hours ago, Anton Rize said:

The blog post you quoted is correct that c=1 is a formal change of unit system, but it is philosophically incorrect to call it "WRONG".

That is why marcus at the university of waikato is correct in his statement of the units of c that cannot be left out.

My further apologies I did say by mistake it was the university of Otago (where my contact is a professor of geology, not physics).

My esxcuse is that it was late here. The link I gave is correct.

My concern is that mathematically you have not started at the beginning, but have invoked mathematics which have nothing to do with algebra, whilst claiming this as you 'proof'.

You want to talk about a manifold, OK, but you need to start with a basic set.

You can turn that set into a manifold by specifying suitable additional structure.

Conventionally we go even further by specifying a metric etc but that is not necessary.

If you do not specify a metric you cannot use the properties of the disk as is done in complex analysis (since you have the whole plane to play with).

However you require a circle, not a disk.

This is also why it is taking me so long to unravel what you are doing. It was as though Euclid had started halfway through book 1 instead of at his 5 postulates.

Edited October 29Oct 29 by studiot

22 hours ago, Anton Rize said:

From my perspective, metrics, distances, and angles are the bookkeeping (the 'how'), not the physics (the 'why'). They are the anthropocentric representation, the "grammar of measurement" we use to describe relations.

You’ve offered your framework up as ontology. i.e. the why is philosophy. Physics tells you how the universe behaves.

Physics equations have no inherent units attached to them. The only requirement is that you use a consistent set, once you start solving them.

7 hours ago, Xerxes said:

Is anybody else here struck by the similarities between Anton's ideas and Category Theory? recall that this theory, which is 80 yers old and highly respectable, seeks to unify all mathematics in a sort of meta-mathematiccs and where the different "objects" in the different categories (of sets, of rings, of groups etc) are of no real interest, the relations between them ("morphisms") of passing interest only, the main focus being on the relations between categories and thus the relations between maps between different categorical object, ("maps of maps")

I can can see how, in outline, it might be applied to both the special and general theories, but having no real knowledge of physics, cannot see the details. Maybe that's where the devil lies?

I beleive that John Baez (Riverside, U. Cal??) has an interest in applying category theory to physics problems. Might be worth Anton's while to look up is work

Welcome @Xerxes
This is a geat observation. Im not familuuer with Category Theory as much as I should yet... But I can see the resemblance. Thank you for pointing it out.

1 hour ago, studiot said:

Have you heard of Euler's identity ?

Yes but I cant follow your logic from this c=1 to this c = -(e^πi).

1 hour ago, studiot said:

This is also why it is taking me so long to unravel what you are doing. It was as though Euclid had started halfway through book 1 instead of at his 5 postulates.

This is a misunderstanding. Everything derived from essentially just methodology. Have a look in the beginning of this document https://github.com/AntonRize/WILL/raw/main/documents/WILL_PART_I_SR_GR.pdf
from where im starting and how derivation progress. If you don't want to look at .pdf let me know and ill post this part here but its a bit lengthy.

Thank you for joining us @swansont

37 minutes ago, swansont said:

You’ve offered your framework up as ontology. i.e. the why is philosophy. Physics tells you how the universe behaves.

Yes but why do we have to limit ourselves? Why not both the "HOW?" and the "WHY?"? This is exactly what im trying to do.

39 minutes ago, swansont said:

Physics equations have no inherent units attached to them. The only requirement is that you use a consistent set, once you start solving them.

Yes! 100% agree.

stop building argument on personal (often wrong) opinion and show which definition is faulty not which word sounds like GR

You’re still not getting this - the point I was making was that the structures you are using already presuppose and require some combination of the things you have initially rejected as unnecessary, and I have highlighted some specific examples. Others here have attempted to point out the same.

What you do with these criticisms we offered is up to yourself, it’s your hypothesis after all.

1 hour ago, Markus Hanke said:

What you do with these criticisms we offered is up to yourself, it’s your hypothesis after all.

@Markus Hanke

Your last statement is an assertion, not a criticism. For it to become a criticism, it has to obtain an objective form like an equation or a well defined logical construct.

But let's explore your assertion:

1. Let's assume you are correct, and I am "tacitly using" concepts that require a metric.

2. However, as you've seen, I can't find this "hidden metric" in my algebra. You can't find it. It doesn't appear in the equations.

3. And most importantly, it is not needed to derive the correct precession of Mercury to [math]10^{-7}[/math] precision, as shown by my algebraic formula: [math]\Delta\phi = (2\pi Q^2)/(1-e^2)[/math].

So, if this "hidden metric" is mathematically invisible, algebraically unnecessary, and operationally redundant... what is its physical meaning? It becomes a useless entity.

Sorry, but this is why your assertion not only is not a valid criticism, but also just not valid at all.

Edited October 30Oct 30 by Anton Rize

9 hours ago, Anton Rize said:

Yes but why do we have to limit ourselves? Why not both the "HOW?" and the "WHY?"? This is exactly what im trying to do.

No reason, but I think you shouldn’t try and sell philosophy as physics. You can solve physics problems without the why; we’ve been doing it for hundreds of years now, quite successfully. And as ontology, I don’t see it.

1 hour ago, swansont said:

No reason, but I think you shouldn’t try and sell philosophy as physics. And as ontology, I don’t see it.

Its funny that among physicists I'm seen as philosopher and among philosophers I'm seen as physicist. Im like a nomad 😩.
I don't know where exactly the line between physics and philosophy lies but im quiet sure that if "philosophy" makes unique numerical testable predictions + recovering standard results it becomes ""physics" is it not?

10 hours ago, Anton Rize said:

Yes but I cant follow your logic from this c=1 to this c = -(e^πi).

You have made four propositions . viz

that you only need simple arithmetic

that of the three independent variables required for mechanics (normally called mass length and time) two may be fixed (leaving you only one degree of freedom in your equations) by the expedient of defining

c= 1 and hbar =1.

Here are some interesting consequences of doing this

Propositions

[math]\hbar = 1[/math]

[math]c = 1[/math]

Mathematical Identities

[math]{e^{\pi i}} = - 1[/math]

[math]{\sin ^2}x + {\cos ^2}x - 1 = 0[/math]

Consequences

[math]c = - ( - 1) = - {e^{\pi i}}[/math]

[math]\hbar - c = 1 - 1 = 0[/math]

[math]{\sin ^2}x + {\cos ^2}x - 1 = 0 = \hbar - c[/math]

[math]{\sin ^2}x + {\cos ^2}x = \hbar = - {e^{\pi i}}[/math]

Edited October 30Oct 30 by studiot

On 10/28/2025 at 1:19 PM, Anton Rize said:

On 10/28/2025 at 10:38 AM, KJW said:

Please explain how you obtained this formula:

[math]\Delta_{WILL}=\dfrac{2\pi Q_{Merc}^{2}}{\left(1-e_{Merc}^{2}\right)}[/math]

𝐃𝐢𝐦𝐞𝐧𝐬𝐢𝐨𝐧𝐥𝐞𝐬𝐬 𝐩𝐫𝐨𝐣𝐞𝐜𝐭𝐢𝐨𝐧 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫𝐬 𝐟𝐨𝐫 𝐌𝐞𝐫𝐜𝐮𝐫𝐲:
κₘₑᵣ꜀ = √(Rₛₛᵤₙ / aₘₑᵣ꜀)
βₘₑᵣ꜀ = √(Rₛₛᵤₙ / 2aₘₑᵣ꜀)

𝐂𝐨𝐦𝐛𝐢𝐧𝐞𝐝 𝐞𝐧𝐞𝐫𝐠𝐲 𝐩𝐫𝐨𝐣𝐞𝐜𝐭𝐢𝐨𝐧 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫:
Qₘₑᵣ꜀ = √(κₘₑᵣ꜀² + βₘₑᵣ꜀²)

𝐂𝐨𝐫𝐫𝐞𝐜𝐭𝐢𝐨𝐧 𝐟𝐚𝐜𝐭𝐨𝐫 𝐟𝐨𝐫 𝐭𝐡𝐞 𝐞𝐥𝐥𝐢𝐩𝐭𝐢𝐜 𝐨𝐫𝐛𝐢𝐭 𝐝𝐢𝐯𝐢𝐝𝐞𝐝 𝐛𝐲 𝐨𝐧𝐞 𝐨𝐫𝐛𝐢𝐭𝐚𝐥 𝐩𝐞𝐫𝐢𝐨𝐝:
(1 − eₘₑᵣ꜀²) / 2π

𝐅𝐢𝐧𝐚𝐥 𝐖𝐈𝐋𝐋 RG 𝐩𝐫𝐞𝐜𝐞𝐬𝐬𝐢𝐨𝐧 𝐫𝐞𝐬𝐮𝐥𝐭:
Δφ₍WILL₎ = (Qₘₑᵣ꜀²) / [(1 − eₘₑᵣ꜀²) / 2π] = (2πQₘₑᵣ꜀²) / (1 − eₘₑᵣ꜀²)

I didn't look at the link because I assume that (as required by the forum rules) you've posted all that I requested. I don't see any of the physics that your formula is describing. Nevertheless, I am quite intrigued by the maths. I checked your maths to see if your claim is valid:

[math]\text{From above:}\ \ \ Q^2_{\text{merc}} = \dfrac{3r_{\text{s sun}}}{2a_{\text{merc}}} = \dfrac{3GM_{\text{sun}}}{c^2 a_{\text{merc}}}[/math]

[math]\text{From Kepler's third law:}\ \ \ GM_{\text{sun}} = \dfrac{4\pi^2 a^3_{\text{merc}}}{T^2_{\text{merc}}}[/math]

[math]Q^2_{\text{merc}} = \dfrac{12\pi^2 a^2_{\text{merc}}}{c^2 T^2_{\text{merc}}}[/math]

[math]\text{Perihelion shift:}\ \ \ \sigma = \dfrac{2\pi Q^2_{\text{merc}}}{1 - e^2_{\text{merc}}} = \dfrac{24\pi^3 a^2_{\text{merc}}}{c^2 T^2_{\text{merc}}(1 - e^2_{\text{merc}})}[/math]

which agrees with the known formula. However, how do you justify the "correction factor for the elliptic orbit" [math](1 - e^2_{\text{merc}})/2\pi[/math]?

[If the above LaTeX doesn't render, please refresh the webpage.]

2 hours ago, swansont said:

No reason, but I think you shouldn’t try and sell philosophy as physics. You can solve physics problems without the why; we’ve been doing it for hundreds of years now, quite successfully. And as ontology, I don’t see it.

And also I don't think we should separate philosophy from physics ever. Allow me the attempt to prove it:



Derivation of statement:
"Complex mathematics is the consequence of bad philosophy"

Classical Keplerian Energy as a WILL-Minkowski Projection

For a test body of mass m on a circular orbit of radius a about a central mass [math]M_\oplus[/math], classical Newtonian mechanics gives:

[math]\Delta U = -\frac{GM_\oplus m}{a} + \frac{GM_\oplus m}{R_\oplus}[/math]

[math]K = \frac{1}{2} m\frac{GM_\oplus}{a}[/math]

Adding these and dividing by the rest-energy [math]E_0 = m c^2[/math] yields the dimensionless total:

[math]\frac{E_{\text{tot}}}{E_0} = \frac{GM_\oplus}{R_\oplus c^2} - \frac{1}{2}\frac{GM_\oplus}{a c^2}[/math]

---

Projection Parameters and Minkowski-like Form

Define the WILL projection parameters for the surface and the orbit:

[math]\kappa_\oplus^2 \equiv \frac{2GM_\oplus}{R_\oplus c^2}[/math]

[math]\beta_{\text{orbit}}^2 \equiv \frac{GM_\oplus}{a c^2}[/math]

Substituting into the classical equation gives the exact identity:

[math]\frac{E_{\text{tot}}}{E_0} = \frac{1}{2}\bigl(\kappa_\oplus^2 - \beta_{\text{orbit}}^2\bigr)[/math]

This is already in the form of a hyperbolic difference of squares. If we set [math]x \equiv \kappa_\oplus[/math] and [math]y \equiv \beta_{\text{orbit}}[/math], then:

[math]\frac{E_{\text{tot}}}{E_0} = \frac{1}{2}\,(x^2 - y^2)[/math]

This is structurally identical to a Minkowski interval in [math](1+1)[/math] dimensions.

---

Physical Interpretation

In classical derivations, this is just the sum [math]\Delta U + K[/math] with a particular choice of potential zero. In WILL Relational Geometry, this form emerges directly from the Energy-Symmetry Law:

[math]\Delta E_{A\to B} = \frac{1}{2}\bigl((\kappa_A^2 - \kappa_B^2) + \beta_B^2\bigr)[/math]

(with [math](A,B) = (\text{surface},\ \text{orbit})[/math])

This shows that the Keplerian total energy is not an isolated Newtonian artifact but a special case of a deeper geometric structure. The emergence of this Minkowski-like structure from purely energetic principles is a powerful indicator of the deep identity between the geometry of spacetime and the geometry of energy transformation.

---

Lagrangian and Hamiltonian as Ontologically Corrupted Approximations

The familiar Lagrangian and Hamiltonian formalisms are not fundamental principles but approximations that arise from collapsing the two-point relational structure of the Energy-Symmetry Law into a single-point description.

The "Ontological Collapse"

If one commits the ontological violation by identifying the two distinct points, [math]r_A = r_B = r[/math], the relational structure degenerates into a local, single-point function:

[math]L(r,\dot r,\dot\phi)=\tfrac{1}{2} m(\dot r^2+r^2\dot\phi^2)+\frac{GMm}{r}[/math]

This is precisely the standard Newtonian Lagrangian.

Defining the Hamiltonian via the Legendre transformation [math]H = p_r \dot r + p_\phi \dot\phi - L[/math] evaluates to the total energy of the collapsed system:

[math]H = T+U = \tfrac{1}{2} m\left(\dot r^2 + r^2 \dot\phi^2\right) - \frac{GMm}{r}[/math]

Interpretation

In terms of the collapsed WILL projections [math]\beta^2 = v^2/c^2[/math] and [math]\kappa^2 = 2GM/(rc^2)[/math], the match becomes explicit:

[math]L = \tfrac{1}{2} m v^2 + \frac{GMm}{r} \;\;\;\longleftrightarrow\;\;\; \tfrac{1}{2} m c^2\bigl(\beta^2 + \kappa^2\bigr)[/math]

[math]H = \tfrac{1}{2} m v^2 - \frac{GMm}{r} \;\;\;\longleftrightarrow\;\;\; \tfrac{1}{2} m c^2\bigl(\beta^2 - \kappa^2\bigr)[/math]

Here the + or - signs do not come from [math]\kappa^2[/math] itself (which is always positive), but from the ontological collapse of the two-point relational energy law into a single-point formalism.

> Key Message: The Lagrangian and Hamiltonian are not fundamental principles. They are degenerate shadows of a deeper relational Energy-Symmetry Law. Classical mechanics, Special Relativity, and General Relativity all operate within this corrupted approximation. WILL restores the underlying two-point relational clarity.

---

Third Ontological Collapse: Derivation of Newton's Third Law

Newton's Third Law is another "degenerate shadow" that arises as a necessary mathematical consequence of this same ontological collapse.

Theorem: Newton's Third Law as a Degenerate Consequence

The Energy-Symmetry Law ([math]\Delta E_{A \to B} + \Delta E_{B \to A} = 0[/math]) mathematically necessitates Newton's Third Law ([math]\vec{F}_{AB} = - \vec{F}_{BA}[/math]) in the classical limit where the two-point relational energy budget is collapsed into a single-point potential function [math]U(\vec{r})[/math].

Proof:

Begin with the foundational Energy-Symmetry Law:

[math]\Delta E_{A \to B} + \Delta E_{B \to A} = 0[/math]

In the classical limit, this is "corrupted" into a single-point potential function U that depends only on the relative positions:

[math]U = U(\vec{r}) \quad \text{where} \quad \vec{r} = \vec{r}_B - \vec{r}_A[/math]

The force [math]\vec{F}[/math] is defined as the negative gradient of this potential.

(1) Force on B by A:

[math]\vec{F}_{AB} = - \nabla_B U(\vec{r}_B - \vec{r}_A) = - \left( \frac{d U}{d \vec{r}} \right) \cdot \left( \frac{\partial \vec{r}}{\partial \vec{r}_B} \right) = - \nabla U(\vec{r})[/math]

(2) Force on A by B:

[math]\vec{F}_{BA} = - \nabla_A U(\vec{r}_B - \vec{r}_A) = - \left( \frac{d U}{d \vec{r}} \right) \cdot \left( \frac{\partial \vec{r}}{\partial \vec{r}_A} \right) = + \nabla U(\vec{r})[/math]

(3) Conclusion:

By direct comparison:

[math]\vec{F}_{AB} = - \nabla U(\vec{r}) \quad \text{and} \quad \vec{F}_{BA} = + \nabla U(\vec{r})[/math]

Therefore, it is a mathematical tautology of the collapsed formalism that:

[math]\boxed{ \vec{F}_{AB} = - \vec{F}_{BA} }[/math]

This completes the proof. The law of "equal and opposite forces" is revealed to be a degenerate approximation of the more fundamental, generative law of Relational Geometry.

---

General Consequence

Bad philosophy (the ontological separation of Structure and Dynamics) has measurable effects:

1. Inflated Formalism: Equations multiply to compensate for the error.

2. Loss of Transparency: Physical meaning is hidden behind coordinate dependencies.

By contrast, good philosophy (**epistemic hygiene**) enforces relational closure and yields simplicity through necessity.

> Daring Remark: The historical escalation of mathematical complexity in physics did not reveal deeper reality—it compensated for a philosophical mistake. Once the ontological symmetry is restored, Nature’s laws reduce to algebraic self-consistency.

>

> [math]\boxed{\text{Bad Philosophy}} \Rightarrow \boxed{\text{Ontological Duplication}} \Rightarrow \boxed{\text{Mathematical Inflation}}[/math]

> [math]\boxed{\text{Complex mathematics is the consequence of bad philosophy.}}[/math]



Yes I know its to daring and to radical. But this is exactly why I love it so much. It makes you think it makes you feel and maybe it will help someone to see the bigger picture...

Edited October 30Oct 30 by Anton Rize

@Anton Rize , the "math" tags and everything between them needs to be on the same line, though the line is allowed to be long enough for the text to wrap.

44 minutes ago, KJW said:

I checked your maths to see if your claim is valid

💥💥💥💯🦀💃 It's meant to be fireworks and we celebrating hahahaha!!! WE GOT A WINER!!!!!

seriously @KJW , thank you so much mate because I was already losing hope completely. Mathematical validation and testing was the soul purpose of this whole post and you are the first and only one who actually did it. thank you so much!

Thank you. This is the correct analysis. You've correctly shown my algebraic formula
[math]\Delta_{WILL} = \frac{2\pi Q_{merc}^2}{1-e_{merc}^2}[/math] agrees with the known GR result.

To answer your excellent question:

My formula is composed of two parts:

1. The Numerator ([math]2\pi Q_{merc}^2[/math]): This is the new physics from WILL. The term
   [math]Q^2 = \kappa^2 + \beta^2[/math] is the instantaneous relational energy budget derived from my framework. The [math]2\pi[/math] converts it to one full orbit.

2. The Denominator ([math]1-e_{merc}^2[/math]): This is standard geometry, not new physics. It's the classical orbital mechanics factor needed to average the instantaneous shift (at radius a) over the full eccentric ellipse, relating the semi-major axis (a) to the semi-latus rectum [math]p = a(1-e^2)[/math].

In short, I am applying the new WILL relational physics (the numerator) to the known, standard geometry of an ellipse (the denominator) to get the final correct result. But I prefer to think about it this way: a circular orbit (e=0) gives a baseline precession of
[math]2\pi Q^2[/math]. An elliptical orbit (e>0) simply introduces a standard geometric "deviation from circularity" factor, which is [math]\frac{1}{1-e^2}[/math].

40 minutes ago, KJW said:

the "math" tags and everything between them needs to be on the same line, though the line is allowed to be long enough for the text to wrap.

Thank you for spotting it out mate.

2 hours ago, studiot said:

Here are some interesting consequences of doing this

@studiot I honestly tried to understand what are you saying but - no clue. I just cant see any physics behind this numbers and symbols manipulation.
If it make you feel better: "I promise not to use c=1 in my derivation's"🙂

Edited October 30Oct 30 by Anton Rize

1 hour ago, Anton Rize said:

@studiot I honestly tried to understand what are you saying but - no clue. I just cant see any physics behind this numbers and symbols manipulation.
If it make you feel better: "I promise not to use c=1 in my derivation's"🙂

So you can't follow the fact that if I replace a 1 in any of your equations (for example the one you are having a love-in with KJW over) by c, or c² or c³ or... or hbar or e^πi or (sin²x + cos²x), where x is any number, you have changes the physics of that eqaution ?

Edited October 30Oct 30 by studiot

6 hours ago, studiot said:

So you can't follow the fact that if I replace a 1 in any of your equations (for example the one you are having a love-in with KJW over) by c, or c² or c³ or... or hbar or e^πi or (sin²x + cos²x), where x is any number, you have changes the physics of that eqaution ?

@studiot

Ah, I see! This finally clears up our misunderstanding. Your question is about confusing different types of "1".
Let's use an analogy:

Imagine we want to create an abstract "Feynman Scale" to measure how funny someone is as a physicist.
Now, we know a mathematical identity:
[math]\sin^2(x) + \cos^2(x) = 1[/math]
This '1' is an abstract, dimensionless number. It's the result of a calculation.

Separately, we create our convention (our "language"): We set the benchmark for maximum funniness as "1 Full Feynman" (or [math]R_F = 1[/math]).
This '1' is the axiom of our system. It's a definition, not a result.

Here is the question that shows the category error:
Can we say that [math]\sin^2(x) + \cos^2(x)[/math] is equal to "1 Full Feynman"?"

Of course not! 😄
One is an abstract mathematical identity. The other is a foundational definition of a system.

You are doing the exact same thing by confusing the mathematical identity [math]-e^{\pi i} = 1[/math] (an abstract result) with the physical unit convention [math]c=1[/math] (the foundational axiom of the Natural Units language).

That's why I couldn't follow your logic - you are brilliantly mixing two completely different categories!

So no, replacing '1' in my equations with [math]c[/math] is not the same as replacing it with [math]\sin^2(x) + \cos^2(x)[/math]. One is a change of language (SI vs. Natural Units), the other is just... mathematical salad. 😉
And Im aware that "1 Full Feynman" is a unit so my analogy is not ideal.
But I hope this will clear out this misunderstanding between us so we can concentrate on the substance of the equations instead of notation.

23 minutes ago, Anton Rize said:

Here is the question that shows the category error:
Can we say that sin2(x)+cos2(x) is equal to "1 Full Feynman"?"

Of course not! 😄
One is an abstract mathematical identity. The other is a foundational definition of a system.

You are doing the exact same thing by confusing the mathematical identity −eπi=1 (an abstract result) with the physical unit convention c=1 (the foundational axiom of the Natural Units language).

Actually I am not doing it at all.

I agree that not only is Can we say that sin2(x)+cos2(x) is NOT equal to "1 Full Feynman"?" , I have been saying that all along !

If you go back you will find I have been conditionals like ' if 'could' and so on to try to demonstrate the folly of doing so.

I claim, as do others such as in my reference and you apparantly now agree with that the 1 in natural units is different from the 1 in say 1+ 1 = 2.

The nature of that difference is all importance.

As a matter of interest we now teach up to 4 different meaning for the symbols 1, 2, 3 ... etc in primary school.

Have you come across these ?

It is actually a very difficult subject at the foundations of mathematics.

As a matter of interest as a new member, my respect for your mathematical knowledge and ability has grown immensely during this thread.

Very few new members come as well equipped.

But I had to tease out a lot of it with seemingly trivial questions.

This is why is has been so disappointing to only ever have have my comments addressed at a time.

So in the case of natural units c is 1 natural velocity unit.

And in Physics, equations must agree in both units and numbers.

I don't know if you are aware that 'number' as in pure number or count has recently been added to the primary list of SI quantities ?

Some equations in physics or chemistry do not make sense without it.

You haven't respomded to my construction of the natural unit system where you can indeed choose energy as your one free variable by constraining the other two of the 3 required for mechanics.

This does then lead your equation E² = p² + m² but in suitable units.

And it comes with a very high price in lack of convenience in the everyday world.

Nor have you explained what you mean by emergent, I think of it as a very special term, not applicable to common or garden situations as already explained in an earlier post.

But progress has been made.

Finally do you need any further help with Latex or MathMl ?

I often post various alternative for scientific notation for members.

Edited October 31Oct 31 by studiot

@studiot

Thank you for your last post and your very kind words about my mathematical preparation (though it would be better if you would not make any assumptions just based on account's age). I'm glad we've made progress on the c=1 issue and can now move forward.

You are correct that I haven't properly responded to your construction:

12 hours ago, studiot said:

Here are some interesting consequences of doing this

Propositions

ℏ=1

c=1

Mathematical Identities

eπi=−1

sin2x+cos2x−1=0

Consequences

c=−(−1)=−eπi

ℏ−c=1−1=0

sin2x+cos2x−1=0=ℏ−c

sin2x+cos2x=ℏ=−eπi

I must be honest: at first glance, it looked like an intentionally absurd "mathematical salad". Your final equation

[math]{\sin ^2}x + {\cos ^2}x = \hbar = - {e^{\pi i}}[/math]

...seemed designed to be nonsensical.

However, I'm starting to wonder if I misunderstood you.

By deliberately introducing [math]\hbar=1[/math] (witch I didn't brought up) into a discussion about Relativity, were you trying to make a deep point about Quantum Mechanics?

If so, that's a fascinating and important topic and I have plenty of results to share and discuss. But it's also a separate one, as it brings QM into a thread focused on GR.

Before I can answer your other points, I need to know: was that your intention? Are you asking how RG model can possibly unify with Quantum Mechanics?

1 hour ago, studiot said:

Finally do you need any further help with Latex or MathMl ?

I often post various alternative for scientific notation for members.

It seems like your firs help about [math]...[/math] solved my problems. Thank you for offer I appreciate it.

1 hour ago, studiot said:

As a matter of interest we now teach up to 4 different meaning for the symbols 1, 2, 3 ... etc in primary school.

Have you come across these ?

It is actually a very difficult subject at the foundations of mathematics.

No I didn't but it sounds interesting.

Edited October 31Oct 31 by Anton Rize

11 hours ago, Anton Rize said:

I must be honest: at first glance, it looked like an intentionally absurd "mathematical salad". Your final equation

sin2x+cos2x=ℏ=

...seemed designed to be nonsensical.

However, I'm starting to wonder if I misunderstood you.

Yes indeed it was designed to show that you have to be careful.

It is not actually nonsense, since all the terms are pure numbers with magnitudes.

I can repalce the one in the identity with -e^πi since they are both the same type of object - a magnitude.

But I can't replace it by c since 1 is a magnitude and c is a velocity.

But it is very strange.

Similarly c and hbar are different since one is a velocity and the other an action.

So I can't just replace one with the other and say (ℏ−c)=1−1=0

Those consequences were designed to show that it is nonsensical

11 hours ago, Anton Rize said:

By deliberately introducing ℏ=1 (witch I didn't brought up) into a discussion about Relativity, were you trying to make a deep point about Quantum Mechanics?

If so, that's a fascinating and important topic and I have plenty of results to share and discuss. But it's also a separate one, as it brings QM into a thread focused on GR.

Before I can answer your other points, I need to know: was that your intention? Are you asking how RG model can possibly unify with Quantum Mechanics?

No I was not trying to introduce QM or Unification.

I have always been happy with the separation.

I think it is a very good idea to work from the known to the unknown or to try out ideas on simple, well defined examples before plunging into the complicated.

This is why I mentioned number and SI.

For various calculations in chemistry it is necessary to work simultaneous equations balancing the number of atoms, the number of charges, and perhaps the number of phases or the energy input.

For other calculations in the Gas Laws you need the number of molecules.

This discussion of number and dimensions (and circles) is worth reviewing

https://scienceforums.net/topic/80023-the-dimensions-of-numbers/

11 hours ago, Anton Rize said:

No I didn't but it sounds interesting.

Please note I do not mean different types of numbers that you will find if you ask the question of Google - Natural, integer, rational, fractional, decimal, real, imaginary and so on.

What I was referring to is a different viewpoint.

I will use the conventional # symbol to denote numbers in this context.

Consider a horse race.

Horse #52 wins with #19 second and #5 third.

Here we have two other uses of numbers.

As labels horse # 52 etc

As an ordering first, second and third. - The ordinal numbers

Neither of these are magnitudes and do not obey the ordinary rules of arithmetic - adding first and third makes no sense any more than adding #52 + #5

Nor does the set of them need to be complete.

As symbols for example on car number plates.

Again ordinary arithmetic is not followed but some information may be conveyed, nevertheless

As magnitudes - the cardinal numbers

Which do obey both ordering and the normal rules of arithmetic.

These numbers are often used as coefficients to be combined with some quality or property, which may well imply a scale, units and magnitudes.
Or they may be specific coefficients of variables in expressions where the variables are also pure numbers (but obviously a range of pure numbers).

Finally you can generate the natural numbers from nothing using copies of the empty set.

On 10/31/2025 at 12:17 AM, Anton Rize said:

To answer your excellent question:

My formula is composed of two parts:

1. The Numerator ([math]2\pi Q_{merc}^2[/math]): This is the new physics from WILL. The term
   [math]Q^2 = \kappa^2 + \beta^2[/math] is the instantaneous relational energy budget derived from my framework. The [math]2\pi[/math] converts it to one full orbit.

2. The Denominator ([math]1-e_{merc}^2[/math]): This is standard geometry, not new physics. It's the classical orbital mechanics factor needed to average the instantaneous shift (at radius a) over the full eccentric ellipse, relating the semi-major axis (a) to the semi-latus rectum [math]p = a(1-e^2)[/math].

In short, I am applying the new WILL relational physics (the numerator) to the known, standard geometry of an ellipse (the denominator) to get the final correct result. But I prefer to think about it this way: a circular orbit (e=0) gives a baseline precession of
[math]2\pi Q^2[/math]. An elliptical orbit (e>0) simply introduces a standard geometric "deviation from circularity" factor, which is [math]\frac{1}{1-e^2}[/math].

Thanks. However, you didn't mention the physics that is described by the formula:

[math]\text{Perihelion shift:}\ \ \ \sigma = \dfrac{2\pi Q^2_{\text{merc}}}{1 - e^2_{\text{merc}}}[/math]

In particular, in what way does the derivation of this formula indicate that it is describing a perihelion shift?

I'm still not convinced about your overall theory. I would like to explore how it is that these two quantities:

[math]\beta^2 = \dfrac{v^2}{c^2}\\\kappa^2 = \dfrac{2GM}{c^2 r}[/math]

lead to the known formula for the perihelion shift. One thing to note is that:

[math]\sqrt{1 - \beta^2}[/math]

is the time dilation associated with speed, and that:

[math]\sqrt{1 - \kappa^2}[/math]

is the time dilation associated with gravitation.

The total time dilation for both speed and gravitation:

[math]\sqrt{1 - \beta^2} \sqrt{1 - \kappa^2}[/math]

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

[math]= \sqrt{1 - (\beta^2 + \kappa^2) + \beta^2 \kappa^2}[/math]

[math]\approx \sqrt{1 - (\beta^2 + \kappa^2)}\ \ \ \text{ for } \beta \approx 0 \text{ and } \kappa \approx 0[/math]

[math]= \sqrt{1 - Q^2}[/math]

If [math]v[/math] is the orbital speed, then:

[math]\beta^2 \approx \dfrac{GM}{c^2 r} = \dfrac{1}{2} \kappa^2[/math]

and therefore:

[math]Q^2 = \beta^2 + \kappa^2 = \dfrac{3}{2} \kappa^2 = \dfrac{3GM}{c^2 r}[/math]

I note that [math]\beta^2 + \kappa^2[/math] is a weak field approximation. However, it is not clear to me the extent to which the standard formula for the perihelion shift is an approximation. I find it interesting that the standard formula for the perihelion shift gives a non-zero value for circular orbits. Although I expect this value to be a limit value for infinitesimal eccentricity, it is not clear to me if it is the same value as given by the deviation from Newtonian theory.

Edited November 2Nov 2 by KJW

However, it is not clear to me the extent to which the standard formula for the perihelion shift is an approximation

Well, the standard formula is derived from the Schwarzschild metric, so it is an approximation to the extent that the SS metric itself is an approximation to the actual physical situation in the solar system. Actually quantifying this error would be not so easy - has anyone ever produced a numerical GR simulation of the solar system, taking into account the other sources of gravity, including the angular momentum of the sun itself? I wasn’t immediately able to find such a solution.

It should be noted though that the standard formula matches the observed precession value to ~0.1%, so the approximation error is quite small when compared to actual observation.

> @studiot

Ok, thanks for explaining - now I see your point much more clearly.

And I genuinely appreciate that you raise these foundational concerns; it shows mathematical care, which is precisely what I value too.

What you said about different meanings of numbers makes perfect sense. We can’t just throw objects of completely different categories into the same “bucket” and pretend it’s fine - I actually wish I had been taught this way back in school.

You may recall that you mentioned earlier that you’re a mathematician. I’d like to take that seriously and ask for your help in exactly the way I originally came here for - independent verification.

The thing is, I’m just a self-educating enthusiast. That’s why I don’t ask anyone to believe my results - in fact, I don’t fully believe them myself until they’re independently checked. That’s why I’m sharing all derivations openly.

I’ve done everything I can to uncover potential mistakes, but at this point I can’t find any - and neither could any AI system I’ve tested it with. Those are the only resources I have.

At the same time, it’s hard for me to accept the results as correct - they seem too grandiose, and I’m too sceptical for that. So I’m stuck in the middle: can’t find where I’m wrong, and can’t quite accept that I might be right.

If you have time and interest, I’d be truly grateful if you could look through one or two of the key relations purely algebraically and tell me whether you can spot any hidden assumptions or inconsistencies in the reasoning itself.

That kind of objective audit would mean more to me than any praise or criticism. Thanks for engaging so thoughtfully - it’s rare, and I genuinely appreciate it.

---

@KJW

Thank you for this very sharp question - it goes straight to the physics. Lets try to figure it out together.

5 hours ago, KJW said:

In particular, in what way does the derivation of this formula indicate that it is describing a perihelion shift?

1) Methodological frame (RG vs GR)

GR treats geometry as a pre-existing stage and derives the shift from a small difference between the azimuthal and radial frequencies in the Schwarzschild field. RG is ontology-first: there is no background stage. The “geometry” of motion emerges from two conjugate energetic projections that form a closed cycle. In symbols I keep using β (kinetic projection) and κ (potential projection). Their sum

Q² = β² + κ²

is a dimensionless measure of the cycle’s energetic content at the characteristic orbital scale. Its like total projectional scale.

2) Why the formula really is a perihelion shift

In RG the orbit is not integrated in the classical sense - there is no metric-based trajectory.

Instead, we follow one full κ–β energy-exchange cycle until the system returns to its initial total energy.

At that point, the angular phase of the cycle has advanced slightly beyond 2π.

That excess phase, when mapped onto the geometric ellipse defined by (a, e), corresponds exactly to the observed perihelion shift:

Δφ = 2π Q² / (1 − e²).

3) Checks and limits

• Circular limit: for e → 0 the expression gives Δφ = 2π Q². In the weak-field regime β² ≈ GM/(rc²) and κ² = 2GM/(rc²), hence Q² = 3GM/(rc²). This reproduces the familiar 6π GM/(rc²) circular value. There is no paradox here: a perfect circle has no distinguished perihelion point, so the quantity is a formal phase surplus per cycle. For small e it coincides with the limit from the elliptic family.

• Weak field vs not-only: the algebra itself is not tied to a perturbative metric expansion; it is a relational closure. Numerically, in weak fields it agrees with the standard GR value. In stronger fields the same expression remains predictive.

Example: star S2 around Sgr A*

Using the observational parameters

M = 4.30×10⁶ M☉, a ≈ 1000 AU, e ≈ 0.884,

we obtain

R_s = 1.27×10⁷ km, r_p ≃ 116 AU, r_a ≃ 1884 AU, v_p ≃ 7.9×10³ km/s,

and a precession per orbit

Δφ ≃ 12.6 arcmin,

matching the observed general-relativistic shift without invoking the Schwarzschild metric. In other words, the same measurable angle emerges from the relational closure.

A brief aside: the same Q² also correctly flags the photon-sphere separatrix. At r = 1.5 R_s one has κ² = 2/3, β² = 1/3, hence Q² = 1, which marks the null circular configuration - consistent with Q² carrying information about orbit families. It also plays a main role in predicting rotation curves of the 175 galaxys from SPARC database. I got RMSE=20.23 km/s with no DM involved no speculations or free parameters and fixed Y*=0.66 for every galaxy which is quite remarkable I think. You can find the details here: https://antonrize.github.io/WILL/calculator/

But most important is to remember that ontologically GR and RG describing very different Universes. One is "interactions in the box" the other is more of a “there’s-no-spoon” situation if you know what I mean.

Your question is exactly the right one to ask here; thanks again for pressing on the physics.

P. S. Also lets not forget that my interpretation could be completely wrong.

@Markus Hanke

Thank you very much for this thoughtful comment - and actually, for your earlier questions as well.

It was your request for an explicit orbital trajectory of a test particle that pushed me to go back to the algebra and work out a fully relational version of orbital dynamics. As a result, Part I of the main document now includes a new section called

“Relational Orbital Dynamics”.

So, in a very direct sense, that development exists thanks to your challenge.

I fully agree with your point here: the standard perihelion formula is only as approximate as the Schwarzschild metric itself is to the real solar system. Quantifying that deviation would indeed require a full numerical GR simulation including the solar angular momentum and perturbations from other masses - which, to my knowledge, hasn’t been done in complete generality either.

That’s exactly why I find it valuable to approach orbital motion from a strictly relational side: no background metric, just the internal balance of the energy projections themselves. It gives us a complementary way to look at the same physics, while staying anchored to measurable quantities.

So, thank you again - your question about the trajectory turned out to be the seed for an entire new section of the framework.

P. S. If you interested your expertise in GR would be of grate help to falsify or confirm this relational orbital mechanics. I can't relay on my own judgment from within.

Edited November 2Nov 2 by Anton Rize

1 hour ago, Anton Rize said:

The “geometry” of motion emerges from two conjugate energetic projections that form a closed cycle. In symbols I keep using β (kinetic projection) and κ (potential projection). Their sum

Q² = β² + κ²

is a dimensionless measure of the cycle’s energetic content at the characteristic orbital scale. Its like total projectional scale.

1 hour ago, Anton Rize said:

That’s exactly why I find it valuable to approach orbital motion from a strictly relational side: no background metric, just the internal balance of the energy projections themselves. It gives us a complementary way to look at the same physics, while staying anchored to measurable quantities.

This is exactly what I mean when I say you have not started in the beginning.

Please define mathematically what you mean by both energy and metric.

Note you definition/derivations should be clean of depracatory comparisons with something else.

Otherwise you are not starting with an agreed basis.