15 hours ago, Anton Rize said:

You claim Rs explicitly depends on M and G.

That's incorrect.

No it is not. Rs of the sun is not the same as Rs of earth, for example. It depends on the mass of the body in question, as well as the relative strength of gravity.

15 hours ago, Anton Rize said:

You are mistaken. z_sun is a raw observable we take from spectroscopy. Neither G or M needed for its measurement.

Again, the gravitational potential depends on both the mass of the body as well as the relative strength of gravity. The potential function of the sun isn’t the same as that of earth. I don’t know what “measurement” has to do with the basic fact that not all bodies share the same potential.

15 hours ago, Anton Rize said:

The system scale is determined by the orbit itself, not a "Mass" label.

The orbit itself depends on M and G.

15 hours ago, Anton Rize said:

Both are valid boundary conditions. But mine do not require the assumption of a "Mass" entity

So why do you observe those quantities, orbits etc to be different for different bodies? What is it about those bodies that makes them different?

You asked me before what I think of all this. I’m sorry to say that the only fitting word that comes to mind regarding your reasoning here is “bizarre”, especially after this last reply of yours. For my part, I’m not interested in investing more time in this, but I wish you all the best.

@Mordred Great! Now I can see that you actually red something.
Let me answer your questions:

2 hours ago, Mordred said:

You have parameters that do vary by definition you have effective degrees of freedom. They may or may be independent degrees of freedom.

The system is algebraically locked. So the change in 1 parameter inevitably triggering the cascade of changes in all codependent parameters in order to conserve invariants.

2 hours ago, Mordred said:

You also have trigonometric relations between your effective degrees of freedom within your article. So direction is inherent in your S^2 manifold regardless of what parameters you use to determine each unique point on said 2d manifold.

Not quiet. This is a very common misreading. That's exactly why I'm emphasising in the document: https://willrg.com/documents/WILL_RG_I.pdf#rem:nonspatial
"Throughout this paper, S^1 and S^2 are not to be interpreted as spacetime geometries. They are relational carriers that encode the closure, conservation, and isotropy of the transformational resource. Ordinary spatial and temporal notions are emergent descriptors of patterns within WILL.".

Here's the ontology clarification:

---

Ontological Status of the Relational Carriers [math]S^1[/math] and [math]S^2[/math]

A natural question arises regarding the ontological status of the circle [math]S^1[/math] and the sphere [math]S^2[/math]: What are they, and where do they "exist"?

The answer requires a shift in perspective. In WILL Relational Geometry, [math]S^1[/math] and [math]S^2[/math] are not spatial entities existing within a pre-defined container. They are the necessary relational architectures that implement the core identity [math]\text{SPACETIME} \equiv \text{ENERGY}[/math].

Energy as Relational Transformation Capacity

Recall that energy is defined as the relational measure of difference between possible states. It is not an intrinsic property but a relational potential for change. It is never observed directly, only through transformations.

The Carriers as Protocols of Interaction

The Carriers [math]S^1[/math] and [math]S^2[/math] are the minimal, unique mathematical structures capable of hosting this relational "bookkeeping" for directional and omnidirectional transformations, respectively. They enforce closure, conservation, and symmetry by their very topology.

Imagine two observers, [math]A[/math] and [math]B[/math]:

* Observer [math]A[/math] is the center of their own relational framework. Observer [math]B[/math] is a point on [math]A[/math]'s [math]S^1[/math] (for kinematic relations) and [math]S^2[/math] (for gravitational relations).

* Simultaneously, observer [math]B[/math] is the center of their own framework. Observer [math]A[/math] is a point on [math]B[/math]'s [math]S^1[/math] and [math]S^2[/math].

There is no privileged "master" carrier. Each observable interaction is structured by these mutually-centered relational protocols. The parameters [math]\beta[/math] and [math]\kappa[/math] are the coordinates within these relational dimensions, and the conservation laws (e.g., [math]\beta^2 + \beta_Y^2 = 1; \quad \kappa_X^2 + \kappa^2 = 1[/math]) are the innate accounting rules of these protocols.


So its like Im looking at you and asking how is your state is different from mine? Because my velocity and potential in my rest frame are always [math](\beta, \kappa)= (0, 0) [/math] - Im always the origin of my relational frame. Therefor I can map your state reletive to mine as a point on [math](\beta, \kappa) [/math] plane. And because the same rules applies to you, you can map me as a point on [math](\beta, \kappa) [/math] plane of your relational frame. So the total relational shift [math]Q=\sqrt{\beta^2+\kappa^2}[/math] stays invariant between frames. And that's what Principal of Relational Reciprocity is all about. Its way simpler then boosts, spacetime intervals, geodesics, Christoffel symbols, tensors etc...

---

3 hours ago, Mordred said:

Now why am I pointing this out. Well fundamentally your going to eventually want to take these relations and eventually apply them to some coordinate system otherwise how do you determine the force the sun exerts on mercury when you have a force with a 1/r^2 relation.

Im not really using the concept of force but we can translate in to force if you want:
[math]\frac{F}{E_{0}} = \frac{R_{s}}{2r_{o}(o)^{2}} \equiv \frac{\kappa_{o}(o)^{2}}{2r_{o}(o)}[/math]

Where:

[math]\frac{F}{E_{0}} = \frac{R_{s}}{2r_{o}(o)^{2}} \equiv \frac{\kappa_{o}(o)^{2}}{2r_{o}(o)}[/math]

Where:

[math]E_{0}[/math] = rest energy

[math]o[/math] = orbital phase in radians

[math]\kappa_{o} = \sqrt{\frac{R_{s}}{r}} = 1-(1+z_{ko}(o))^{-2}[/math] (local potential projection at phase [math]o[/math])

[math]r = r_o(o)= a\frac{1-e^{2}}{1+e\cos o} = \frac{R_s}{\kappa_o^2}[/math] (radial distance at phase [math]o[/math])

[math]R_{s} = \kappa^{2}a = \frac{2Gm_{0}}{c^{2}} = \frac{2}{3}Q_{o}(O_{o})^{2}a = \frac{r_{1} r_{2}}{r_{2} - r_{1}}(\beta_{1}^{2} - \beta_{2}^{2}) =[/math]
[math] \frac{a}{2}(3-\sqrt{1+8\tau_{Wo}(O_{o})^{2}}) = \frac{r_{o}(o)}{2(2a-r_{o}(o))}(4a-r_{o}(o)-\sqrt{(4a-r_{o}(o))^{2}-8a(2a-r_{o}(o))(1-\tau_{Wo}(o)^{2})})[/math] (Schwarzschild radius - system scale)
[math]z_{ko}(o) = \frac{1}{\kappa_{Xo}(o)}-1[/math] (gravitational redshift at phase o)

4 hours ago, Mordred said:

The title of your thread specifically states "Simplifying SR and GR" yet I don't anything relating to observer effects and what different observers will see or measure. Relative motion from one frame of reference to another etc. I find that curious as well evidently its not in the scope of the work

I guess I already answered this one above with relational reciprocity. In your case you jumping to conclusions about the scope without any proper reasoning. Look I understand its a lot to take in and you dont have to if you dont want. The only thing I demand is scientific honesty. So jumping in to conclusion after misreading few pages is scientifically dishonest.

54 minutes ago, Markus Hanke said:

You asked me before what I think of all this. I’m sorry to say that the only fitting word that comes to mind regarding your reasoning here is “bizarre”, especially after this last reply of yours. For my part, I’m not interested in investing more time in this, but I wish you all the best.

Thats ok we tried second time and result is the same. Im just fascinated how can't you see that all your arguments are based on just an interpretation. Regardless...
this is mathematical and empirical fact [math]\Delta_{precession} = \frac{3\pi}{2} (\frac{R_{ratio}}{\beta_{p}} (1-(1+z_{sun})^{-2}))^2=5.0175347157\times10^{-7}[/math] if you instead of facing it and openly discuss it prefer to hide behind dogmatic thinking - its your choice.

4 hours ago, Mordred said:

I don't anything relating to observer effects and what different observers will see or measure.

I dont get it. What's your reasoning here? Dont you think that if you dont see it doesnt mean that it doesnt exist? Dont you think that look again or ask me would be more reasenoble then jump in to conclusion based on you glassing through the paper? If thats how you do science Id better doublecheck the accuracy of your cosmic calculator...

Edited Wednesday at 04:51 AM2 days by Anton Rize

There's no jumping to conclusion I read your documents you dont have anything relating to a reference frame. That requires geometry. How else do you describe kinematic motion relative to an observer or to multiple observers.

How do you relate the angles of one observer motion relative to another. Try for example an emitter in transverse motion to the observer. Unless Im mistaken every single equation you have outputs a scalar value. You don't have any vector addition rules with regards to distance and angle of travel. Relativity involves more than just scalar ratios.

So tell me without any geometry how do you apply a Galilean or even a Lorentz transformation between multiple events ?

You and I also have difference of opinion of a conserved system. Freefall is a conserved state. There is no external influence such as force acting upon the object in motion. Yet planetary orbits is not a conserved system you have change in direction aka acceleration. In GR this requires the transformation matrix.

You dont have one so how do you translate the freefall state to one of acceleration and stay conserved ? A boost ( change in velocity under the Minkowskii metric is just a type of rotation ).

How do you relate an observer measuring kinematic motion of that orbiting body without geometry to equate an angle of view ?

Aside from the statement closure whats your mathematical proof of closure ?

You describe orthogonal projections but in the same breath state there is no geometry yet an orthogonal projection is 90 degrees relative to the axis its projecting from classical example x axis is orthogonal to the y axis.

I dont care if your manifolds involve spacetime. Thats not a requirement of a manifold it doesn't even require spatial coordinates if a manifold only requires one parameter to uniquely identify each point that's a 1 d manifold. If the manifold requires 2 or more parameters to uniquely identify each point. The number of parameter required is the dimensionality of that manifold. It doesn't require any coordinate basis the number of required parameters or dimension is the number of effective degrees of freedom.

With regards to boosts in Lorentz for the benefit of other readers here's a listing

Lorentz group

Lorentz transformations list spherical coordinates (rotation along the z axis through an angle ) \[\theta\]

\[(x^0,x^1,x^2,x^3)=(ct,r,\theta\phi)\]

\[(x_0,x_1,x_2,x_3)=(-ct,r,r^2,\theta,[r^2\sin^2\theta]\phi)\]

 

\[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\]

\[\Lambda^\mu_\nu=\begin{pmatrix}1&0&0&0\\0&\cos\theta&\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}\]

generator along z axis

\[k_z=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}\]

generator of boost along x axis::

\[k_x=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}=-i\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix}\]

boost along y axis\

\[k_y=-i\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{pmatrix}\]

generator of boost along z direction

\[k_z=-i\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{pmatrix}\]

the above is the generator of boosts below is the generator of rotations.

\[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\]

\[J_x=-i\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&-1&0 \end{pmatrix}\]

\[J_y=-i\begin{pmatrix}0&0&0&0\\0&0&0&-1\\0&0&1&0\\0&0&0&0 \end{pmatrix}\]

\[J_z=-i\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0 \end{pmatrix}\]

they obey commutations

\[[A,B]=AB-BA\]

Does your work do anything to replace the above ?

The above applies for the Minkowskii metric essentially SR.

@Markus Hanke, I believe what @Anton Rize is saying is that Schwarzschild radius can act as a substitute for mass. In other words, mass and Schwarzschild radius are equivalent, and this equivalence can be applied even if the mass is not of a spherical object.

[math]r_s = \dfrac{2GM}{c^2}\ \ \ \ \ ;\ \ \ \ \ M = \dfrac{c^2 r_s}{2G}\ \ \ \text{or}\ \ \ GM = \dfrac{c^2}{2}\ r_s[/math]

Thus, any formula with [math]M[/math] or [math]GM[/math] can be replaced with the corresponding formula with [math]r_s[/math].

It is interesting to note that the radius of a Newtonian "black hole" (where the escape velocity at the surface is [math]c[/math]) is the Schwarzschild radius of its mass. I think this and other formulae for which the Newtonian version is the same as general relativity is due to the special nature of the [math]r[/math] coordinate of the Schwarzschild metric (the surface area at [math]r[/math] is equal to the Euclidean surface area). In this discussion, this appears to create an uncertainty as to whether @Anton Rize is in the realm of general relativity.

@Mordred
Thank you for this detailed and rigorous comment. It is refreshing to see someone who clearly understands the Group Theory and the mathematical foundations of SR/GR. I appreciate the time you took to write out the Lorentz generators.

I have a detailed derivation ready that addresses your questions regarding kinematic motion, vectors, and boosts solely through the algebraic relations of WILL RG.

However, I suspect we are operating on different ontological premises. You seem to view the mathematical artifact (the transformation matrix/geometry) as the physics itself, whereas I view it as a descriptive tool for the physics.

Before I provide my answer, I want to establish a clear criterion for its acceptance to avoid moving the goalposts later. I need to know if an answer that satisfies you is even possible within your current view.

The Question: If I can demonstrate - using strictly algebraic relations of the S1/S2 projections - that I can derive the exact same empirical results as the Lorentz transformation (e.g., the exact numerical values for time dilation, aberration of light, and Doppler shift observed by a moving observer), but without invoking a 4D manifold or a transformation matrix...

...will you accept this as a valid physical solution?

Or is your position that unless the specific mathematical formalism of the Lorentz Group (matrices/tensors) is used, the result is invalid regardless of its agreement with experiment?

Put simply: Are we judging the theory by its ability to predict observable data (Empiricism), or by its adherence to a specific geometric tradition (Formalism)?

Once you clarify your standard of evidence, I will post the derivation.

All mathematics is nothing more than a useful tool. To me the mathematical methodology isn't an issue. I don't have strict adherence to any formalism. My comments above relate to versatility of a methodology or ontology.

The question is " does your methodology have the same predictive and descriptive ability as that of the entirety of SR or GR"

If you can demonstrate that it does and still follow mathematic rules then great. If its lacking then that's something to improve upon.

19 minutes ago, Mordred said:

The question is " does your methodology have the same predictive and descriptive ability as that of the entirety of SR or GR"

Agreed. please provide the list of empirically measured phenomena post diction of witch will be a sufficient evidence of predictive power, for you. Don't be shy. Any SR, GR but no grav waves please I don't want to sit here till morning.
For now here's a quick derivation of Constant of Aberration:

---

In standard SR, this requires calculating the angle transformation between reference frames using vectors. In WILL RG, this is treated as a projection of the observer's kinetic intensity onto the transverse signal carrier.

Here is the derivation using strictly the algebraic relations of the [math]S^1[/math] carrier:

1. The Physical Setup

Consider a star (e.g., Gamma Draconis) positioned at the true geometric zenith relative to Earth's orbital plane.

Input Data (Empirical): Earth's mean orbital velocity [math]v \approx 29.78[/math] km/s.

Relativistic Kinetic Projection ([math]\beta[/math]):

[math]\beta = \frac{v}{c} = \frac{29.7827}{299792.458} \approx 0.0000993444[/math]

2. The Generative Logic (WILL RG)

In my framework, "angles" are not primary entities. They are derived ratios of projections.

When an observer possesses a kinetic projection [math]\beta[/math] (along the line of motion), it "contaminates" the purely transverse signal from the star. The observer perceives their own kinetic intensity as a longitudinal component in the incoming light.

On the unitary circle [math]S^1[/math] (where the signal radius is always 1), the relationship between the Transverse Projection (Line of Sight) and the Kinetic Projection (Motion) is fixed by the Pythagorean theorem.

The "Aberration Angle" [math]\alpha[/math] is simply the angle whose sine equals the kinetic projection:

[math]\sin(\alpha) = \beta[/math]

Note: There are no vectors here. This is a direct conversion of Energy ([math]\beta[/math]) into Geometry ([math]\alpha[/math]).

3. The Calculation

[math]\alpha_{rad} = \arcsin(0.0000993444)[/math]

[math]\alpha_{rad} \approx 0.0000993445 \text{ radians}[/math]

Converting to arcseconds ([math]1 \text{ rad} \approx 206264.8"[/math]):

[math]\alpha \approx 20.4913"[/math]

4. Comparison

WILL RG Prediction: [math]20.491"[/math]

Standard IAU Constant of Aberration: [math]20.495"[/math]

The result is effectively identical (variance is due to averaging Earth's orbital eccentricity).

Conclusion

I have derived the exact observational consequence of Special Relativity (Stellar Aberration) using zero vectors, zero tensors, and zero coordinate transformations.

I simply treated the observer's velocity as a spectral intensity ([math]\beta[/math]) and calculated its projection on the [math]S^1[/math] carrier.

This answers your question: Yes, the methodology has the same predictive ability, but it achieves it with significantly fewer mathematical entities (Occam's Razor).

---

Does this derivation satisfy your requirement for 'predictive ability'? If not - provide me the list of phenomena and I'll provide the derivations.

I noted you have the beta function from the beginning. The part Ive been trying to fathom is when do you correlate time dilation. With SR the Interval ( ct) is what allows dimensionality of length.

How do you handle distance? How do you handle location of each reference frame.

I assume for acceleration specifically change in velocity your using instantaneous velocity. Your math above would certainly work with it.

However a change in direction is also a form of acceleration and this is where I do not understand your spacetime = geometry correlation. The energy momentum equation doesn't contain the terms for change in direction. The total energy of the object is unaffected by change in direction. If you have 2 observers and 1 emitter say a spaceship flying by both observers they should be getting different results each for relative velocity etc.

Lol if you look back through my posts its not certainly not the first Ive mentioned distance including seperation distance pertaining to the ds^2 line element. Specifically the null geodesic worldline

Edit should mention the total energy measured will vary between different observers (variant mass and variant energy as opposed to invariant mass and energy).

For other readers assuming x axis is direction of motion the transformations between reference frames is \{\acute{x}= x-vt\} other axis are unchanged with time absolute.

3 hours ago, Mordred said:

I noted you have the beta function from the beginning. The part Ive been trying to fathom is when do you correlate time dilation. With SR the Interval ( ct) is what allows dimensionality of length.

How do you handle distance? How do you handle location of each reference frame.

I assume for acceleration specifically change in velocity your using instantaneous velocity. Your math above would certainly work with it.

However a change in direction is also a form of acceleration and this is where I do not understand your spacetime = geometry correlation. The energy momentum equation doesn't contain the terms for change in direction. The total energy of the object is unaffected by change in direction. If you have 2 observers and 1 emitter say a spaceship flying by both observers they should be getting different results each for relative velocity etc.

Lol if you look back through my posts its not certainly not the first Ive mentioned distance including seperation distance pertaining to the ds^2 line element. Specifically the null geodesic worldline

Edit should mention the total energy measured will vary between different observers (variant mass and variant energy as opposed to invariant mass and energy).

For other readers assuming x axis is direction of motion the transformations between reference frames is \{\acute{x}= x-vt\} other axis are unchanged with time absolute.

@Mordred , these are excellent questions. You are getting to the core of the Generative Ontology.

DIFFERENT ONTOLOGY AND PHILOSOPHY WARNING - all your life you was thinking about space as a container and time as a flow. Im suggesting a radical departure from this philosophy. Its not going to be easy to rearrange your thinking. I suggest starting from the question: "If there would be only one elementary particle in the hole Universe, would it have energy, velocity, position etc...? And if this particle would despiser what would left of the Universe?" We going back to debate Leibniz vs Newton.

You are asking how I handle the 'fabric' of reality (distance, location, direction) without a pre-existing metric container ([math]ds^2[/math]).

Here is the step-by-step mechanism of WILL RG:

---

1. Time Dilation Correlation (The Mechanism)

In WILL RG, Time is not a dimension you travel through; it is the rate of process execution derived from available energy.

Standard View: You move fast --> Your worldline tilts --> Time dilates.

WILL RG View: You allocate energy to motion ([math]\beta[/math]) --> You have less energy for internal oscillation ([math]\beta_Y = \sqrt{1-\beta^2}[/math]) --> Your clock slows down.

The correlation is instantaneous and intrinsic.

[math]dt_{observer} = dt_{source} \sqrt{1-\beta^2}[/math]

Distance didn't cause this. Energy allocation caused this.

---

2. Handling Distance and Location (No A Priori Space)

You asked: "How do you handle distance?"

In my model, Space is a consequence of Time, not a container for it.

Distance is strictly defined as Signal Delay.

[math]Distance \equiv c (t_{reception} - t_{emission})[/math]

I do not assign coordinate [math](x, y, z)[/math] as a primary property. I assign a Relational Lag.

If I say "Star A is at distance [math]L[/math]", I physically mean "The information from Star A is [math]L/c[/math] seconds old".

The metric [math]ds^2[/math] in SR is simply a way to keep this causality consistent. In RG, I enforce causality directly via signal exchanges.

---

3. Acceleration and Direction (The [math]S^1[/math] Vector Rotation)

You noted correctly: "The energy momentum equation doesn't contain the terms for change in direction."

Exactly! And that is a feature, not a bug.

On the kinematic carrier [math]S^1[/math] (Velocity Circle):

Total Energy ([math]E[/math]) is the Radius.

Momentum ([math]p[/math]) is the vector projection.

If a spaceship turns at constant speed:

Total Energy ([math]E[/math]) stays constant.

Magnitude of Momentum ([math]|p|[/math]) stays constant.

The Vector Components rotate on the [math]S^1[/math] circle.

Changes in direction are treated as Phase Shifts on the carrier.

The observer sees the projection of the spaceship change (e.g., transverse velocity becomes longitudinal), but the spaceship's internal state (Energy) remains invariant.

---

4. Multiple Observers (The "Private Reality" Principle)

You asked: "If you have 2 observers and 1 emitter... they should be getting different results."

Absolutely. And this is handled natively without a global coordinate system.

Emitter (Spaceship): Has a single, invariant Total Energy State [math]E_{source}[/math].

Observer A: Receives the signal and projects it onto their Relational Axis (based on relative [math]\beta_A[/math]). They measure Energy [math]E_A'[/math] and Frequency [math]f_A'[/math].

Observer B: Receives the same signal but projects it onto their Relational Axis (relative [math]\beta_B[/math]). They measure [math]E_B'[/math] and [math]f_B'[/math].

There is no "God's Eye View" coordinate system where the spaceship "really is". There is only the Source State and the Projection of that state onto each observer's local reality.

Lorentz Transformations are simply the translation rules between Observer A's projection and Observer B's projection.

Summary:

I replace the Coordinate Grid with a Network of Relations.

Location --> Signal Phase/Delay.

Distance --> Time Interval.

Direction --> Projection Ratio on [math]S^1[/math].

The core principle that SPACETIME ≡ ENERGY means that the geometry of the universe is the manifestation of energy relations.

Does this clarify how the "Spacetime = Energy" concept handles geometry?

P. S. WILL-AI gave a lengthy and deep answer to your questions. Im not sure witch answer is better mine or he's: https://github.com/AntonRize/WILL/blob/63bc924f61891a434a976d303eca773cc21cf07e/assistant/logs/2026-02-19T02-48-41-070Z.md

P.P.S. The fastest and easiest way to grasp the hole shebang is through this chart: https://willrg.com/LOGOS_MAP/

34 minutes ago, Anton Rize said:

1. Time Dilation Correlation (The Mechanism)

In WILL RG, Time is not a dimension you travel through; it is the rate of process execution derived from available energy.

Standard View: You move fast --> Your worldline tilts --> Time dilates.

WILL RG View: You allocate energy to motion ([math]\beta[/math]) --> You have less energy for internal oscillation ([math]\beta_Y = \sqrt{1-\beta^2}[/math]) --> Your clock slows down.

The correlation is instantaneous and intrinsic.

[math]dt_{observer} = dt_{source} \sqrt{1-\beta^2}[/math]

Distance didn't cause this. Energy allocation caused this.

How do you handle the twin paradox? If observers A and B are moving away from each other, then A sees B's clock tick slower while B also sees A's clock tick slower. And if observers A and B are moving towards each other, then A sees B's clock tick faster while B also sees A's clock tick faster. This is because of the symmetry of the relationship between A and B. But in the twin paradox, one of the twins is definitely younger than the other twin when they eventually meet. Thus, the symmetry of the relationship between A and B does not exist between the two twins. Why?

18 minutes ago, KJW said:

How do you handle the twin paradox? If observers A and B are moving away from each other, then A sees B's clock tick slower while B also sees A's clock tick slower. And if observers A and B are moving towards each other, then A sees B's clock tick faster while B also sees A's clock tick faster. This is because of the symmetry of the relationship between A and B. But in the twin paradox, one of the twins is definitely younger than the other twin when they eventually meet. Thus, the symmetry of the relationship between A and B does not exist between the two twins. Why?

@KJW, Great question! you are correctly describing the Optical Symmetry (Doppler Effect). Lets get on to it:

The Twin Paradox: Optical Symmetry vs. Energetic Asymmetry

While moving apart, both see the other redshifted. While moving together, both see the other blueshifted.

So, where does the physical age difference come from?

In WILL RG, the asymmetry is Energetic. Motion is not just a change of coordinates; it is a change of State allocation on the [math]S^1[/math] carrier.

Here is the breakdown:

1. The Energy Audit (Breaking the Symmetry)

The relationship is not symmetric because only one Twin changed their energy state relative to the system's origin.

Twin A (Earth): Remains in the Low-Kinetic State ([math]\beta \approx 0[/math]). Their Total Energy is almost entirely projected onto the Time Axis ([math]\beta_Y \approx 1[/math]). They maximize their aging rate.

Twin B (Traveler): Expends energy to accelerate. On the [math]S^1[/math] diagram, they rotate their state vector. They convert a portion of their 'Time Flow' ([math]\beta_Y[/math]) into 'Spatial Motion' ([math]\beta_X[/math]).

2. The Cost of Motion

The 'Paradox' disappears when you realize that Velocity is not free.

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

Twin B 'bought' spatial velocity ([math]\beta_X[/math]) by spending temporal velocity ([math]\beta_Y[/math]). Twin A did not spend anything.

3. The Turnaround (The Reality Check)

The moment Twin B turns around, they prove they are the Active Agent. They must expend energy to reverse their [math]\beta_X[/math]. This confirms that their deviation from the 'Time Axis' was real, not just a coordinate artifact.

Conclusion:

Twin B is younger not because of 'geometry' or 'paths', but because they spent a significant portion of their existence being motion rather than experiencing time.

Symmetry exists in perception (light signals), but asymmetry exists in state (energy allocation).

In the future please do no place my questions to you into your AI especially as a direct copy paste. I am having this discussion with you and not AI. However thank you for your reply I will always prefer your reply over any AI generated response.

That being said thank you for providing the details relating to your manifold structures with regards to S1 and S2. You do have a distance relation via signal ct. Fine its an interval you have direction so your structure does indeed have vector quantities. Given the above can you mathematically describe Observer A's reference frame and Bobs reference frame ? Are they assigned to coordinate space or are they assigned to your manifold ?

2 hours ago, Mordred said:

In the future please do no place my questions to you into your AI especially as a direct copy paste. I am having this discussion with you and not AI.

Ok. Im not going to use your questions to test WILL-AI.
Im finding it fascinating that AI trained on my papers can extend the core ontology in the documents to generate meaningful answers that the source material does not cover directly. Isn't it amazing that well tuned and weighted "massive equation" (AI) can mimic human comprehension to the point that its hard to tell sometimes who's answers are better mine or AI? We live in remarkable time...

3 hours ago, Mordred said:

can you mathematically describe Observer A's reference frame and Bobs reference frame ? Are they assigned to coordinate space or are they assigned to your manifold ?

Using coordinate space would be direct violation of the core methodological principals of my research:

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Epistemic Hygiene (as Refusal to Import Unjustified Assumptions)
This line of reasoning derive physics by removing hidden assumptions, rather than introducing new postulates. This construction is deliberate and contains zero free parameters. This is not a simplification - it is a deliberate epistemic constraint. No assumptions are introduced and no constructs are retained unless they are geometrically or energetically necessary.

Ontological Minimalism
Any fundamental theory must proceed from the minimum possible number of ontological assumptions. The burden of proof lies with any assertion that introduces additional complexity or new entities. This principle is not a statement about the nature of reality, but a rule of logical hygiene for constructing a theory.

Relational Origin
All physical quantities must be defined by their relations.
Any introduction of absolute properties risks reintroducing metaphysical artefacts and contradicts the foundational insight of relationalism.

Simplicity
Everything} must be expressed in the simplest form possible.
Any unjustified complexity risks reintroducing metaphysical artefacts and contradicts the foundational insight of Epistemic Hygiene.

Mathematical Transparency
Every mathematical phrase, operational choice, or identity caries its own ontological statement.
Each mathematical object must correspond to explicitly identifiable relation between observers with transparent ontological origin.
Every symbol must be anchored to unique physical idea.
Introducing symbols without explicit necessity constitutes semantic inflation: the proliferation of symbols without corresponding physical meaning.
Number of symbols = Number of independent physical ideas.


Every prediction or result can be directly backtrack to this 5 core methodological principals.

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So as you can see coordinate space violates all 5 core principals.

Regarding Alice and Bob perspectives: I already posted it here so Ill just quote myself:

On 2/18/2026 at 2:45 PM, Anton Rize said:

Energy as Relational Transformation Capacity

Recall that energy is defined as the relational measure of difference between possible states. It is not an intrinsic property but a relational potential for change. It is never observed directly, only through transformations.

The Carriers as Protocols of Interaction

The Carriers S1 and S2 are the minimal, unique mathematical structures capable of hosting this relational "bookkeeping" for directional and omnidirectional transformations, respectively. They enforce closure, conservation, and symmetry by their very topology.

Imagine two observers, A and B:

* Observer A is the center of their own relational frame. Observer B is a point on A's S1 (for kinematic relations) and S2 (for gravitational relations).

* Simultaneously, observer B is the center of their own framework. Observer A is a point on B's S1 and S2.

There is no privileged "master" carrier. Each observable interaction is structured by these mutually-centered relational protocols. The parameters β and κ are the coordinates within these relational dimensions, and the conservation laws (e.g., β^2+β_Y^2=1; κ_X^2+κ^2=1) are the innate accounting rules of these protocols.


So its like Im looking at you and asking how is your state is different from mine? Because my velocity and potential in my rest frame are always (β,κ)=(0,0) - Im always the origin of my relational frame. Therefor I can map your state relative to mine as a point on (β,κ) plane. And because the same rules applies to you, you can map me as a point on (β,κ) plane of your relational frame. So the total relational shift [math]Q=\sqrt{\kappa^2+\beta^2}[/math] stays invariant between frames. And that's what Principal of Relational Reciprocity is all about. Its way simpler then boosts, spacetime intervals, geodesics, Christoffel symbols, tensors etc...

So everyone get's to be the center of its own "Universe" and the relational reciprocity [math]Q_A=Q_B[/math] same as the Energy Symmetry Law:
[math]\Delta E_{A \to B} + \Delta E_{B \to A} = 0. [/math] https://willrg.com/documents/WILL_RG_I.pdf#sec:energy-symmetry
are keeping this framework from falling down in to solipsism.
Here's a simple Desmos project that mapping this relations: https://www.desmos.com/geometry/2nkjtezjpi

Edited yesterday at 11:44 AM1 day by Anton Rize

The reason I asked concerning a mathematical defined reference frame has more to do with tracking.

Take all the signals from 3 dimensional space and try tracking a large quantity of individual objects say several thousand.

This is where I further question your methodology with regards to reference frames.

Specifically a large number of events / observers lying on axis x, y and Z axis. Particularly when no total energy from what I see above will at any point in motion have constant total energy.

Particularly when it comes to multiparticle field energy conservation laws.

Take for example the statement " an EM field has two fields E and B where any work performed is by the E field and the work is not performed by the B field."

Then use your methodology to show why this is the case..

This other problem I can foresee is linearizing any graph of E

Assuming your methodology was developed prior to SR and GR being realized by Einstein...and assuming SR and GR developed much later..could your methodology lead to practical technological innovations e.g assuming it was like that for like 50 years prior to 1945 could atomic bomb been developed within that time frame?

Ok this is something I can't shake.

If you have a static event at x=10, y=10 and z=10. How do you intuitively identify their orientation from (0,0,0) including angles between principle axis ?

In essence Occams razor

Recognizing you also want to avoid any mappings pertaining to geometry

If you were to hand me an E value how would I identify and seperate the components that make up the total E. ( in essence working backwards from total E in reverse and correctly identifying each contributor including quantity of each contribution.

When you get a chance here's another test for you. According to Newtons Shell theorem if you energy mass distribution is uniform throughout your system/ state. G=0 Regardless of what the energy mass density is.

@KJW , @Mordred , @Markus Hanke , Guys something incredible is happening. Please help me conduct the blind test. I want to fully isolate myself from the data source. That's the last explanation I need rule out before considering the possibility of success. The critic in me is ruthless. So I call out to you guy's. Lets do science! Please participate in this blind testing:

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Call for Blind Test Data: Resolving Inclination Degeneracy Exclusively via 1D Spectroscopy (R.O.M. Framework)

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In classical Newtonian and Keplerian orbital mechanics, the amplitude of a radial velocity (RV) curve is governed by a single inseparable parameter: [math]K \propto \beta \sin(i)[/math]. Consequently, it is mathematically impossible to independently extract the true orbital velocity [math]\beta[/math] and the inclination angle [math]i[/math] exclusively from a 1D spectroscopic curve. Resolving this degeneracy traditionally requires independent 3D spatial data (astrometry) or transit observations.

Within the WILL Relational Orbital Mechanics (R.O.M.), this geometric limitation does not exist. The framework isolates a second-order systemic scalar invariant, [math]Z_{sys}[/math], which is strictly proportional to the absolute kinetic projection [math]\beta^2[/math] and the potential projection, but is fundamentally independent of the observer's line of sight [math]i[/math].

By applying a dynamic 5-parameter inversion (Differential Evolution + MCMC) based strictly on R.O.M. invariants, I recently succeeded in blindly extracting the complete 3D spatial geometry of the S0-2 star ([math]e, \omega_0, i[/math]), its internal precessional shift, and the background drift ([math]v_{z0}[/math]) using nothing but 1D Keck radial velocity data. The extracted inclination matched the independent GRAVITY 3D-interferometer consensus ([math]\sim 134^\circ[/math]) to within the instrumental noise limits.

I am now calling for a strictly blind test to prove that im not going crazy.

I request that anyone provide me with an anonymized, raw 1D RV/redshift dataset. I do not want to know the system's identity, assumed masses, distance, or established geometric parameters. I will process the raw numbers through the R.O.M. equations and return the full 3D orbital geometry that we will compare with anonymized data source.

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CRITICAL DATA REQUIREMENTS:

For the [math]Z_{sys}[/math] invariant shift to mathematically exceed the noise floor of modern spectrographs, the system must be highly relativistic.

1. Kinematic Scale: Peak orbital velocities must exceed ~1000 km/s ([math]\beta > 0.003[/math]). Standard exoplanets will not work because the second-order [math]\beta^2[/math] shift is orders of magnitude smaller than instrumental noise limits. Ideal candidates are tight compact binaries (WD/NS/BH) or other extreme S-stars.

2. Unprocessed Relativistic Data: The dataset must be raw or minimally processed: [Time (MJD), Radial Velocity (km/s) or Redshift (Z), Measurement Error]. Crucially, the data MUST NOT be pre-corrected for Transverse Doppler or Gravitational Redshift (though standard Barycentric/LSR background velocity correction is fine).

3. Optional (for computational efficiency): Providing the Period ([math]P[/math]) and Epoch of Periapsis ([math]T_{peri}[/math]) is helpful to bound the MCMC sampler, but entirely optional if the data covers at least one full orbit.

Please drop the raw CSV data or a link below. Do not provide the system name or accepted parameters. Let the pure relational geometry speak for itself.

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DATASET EXAMPLE:

MJD,RV_km_s,sigma_km_s,Instrument

51718.50000,1192,100,NIRSPEC

52427.50000,-491,39,NIRC2

52428.50000,-494,39,NIRC2

52739.23275,-1571,59,VLT

52769.18325,-1512,40,VLT

52798.50000,-1608,34,NIRC2

52799.50000,-1536,36,NIRC2

52803.15150,-1428,51,VLT

53179.00000,-1157,47,NIRC2

53200.90875,-1055,46,VLT

53201.63925,-1056,37,VLT

53236.33800,-1039,39,VLT

53428.45950,-1001,77,VLT

53448.18300,-960,37,VLT

53449.27875,-910,54,VLT

53520.50000,-983,37,NIRC2

53554.50000,-847,18,OSIRIS

53904.50000,-721,25,OSIRIS

53916.50000,-671,25,OSIRIS

53917.50000,-692,26,OSIRIS

54300.29167,-485,22,OSIRI
...

Here's my results for S2 star just from 2 inputs stream MJD, RV_km_s:

=== R.O.M. DYNAMIC PRECESSION RECOVERY ===

Eccentricity (e): 0.88498 (GRAVITY Ref: 0.88466)

Base Arg of Periapsis (ω0):66.26° (GRAVITY Ref: 66.13°)

Internal R.O.M. Precession:0.207° / orbit

---------------------------------------------------

Global Kin. Proj. (β): 0.006448

Extracted Inclination (i): 135.68° (GRAVITY Ref: ~134°)

Background Drift (v_z0): -20.56 km/s

Fit Quality (χ²): 166.87

As far as I understand according to Kepler and GR it shouldn't work right?

19 hours ago, Mordred said:

The reason I asked concerning a mathematical defined reference frame has more to do with tracking.

Take all the signals from 3 dimensional space and try tracking a large quantity of individual objects say several thousand.

This is where I further question your methodology with regards to reference frames.

Specifically a large number of events / observers lying on axis x, y and Z axis. Particularly when no total energy from what I see above will at any point in motion have constant total energy.

Im afraid I can't understand the question... Is there a question?

19 hours ago, Mordred said:

Particularly when it comes to multiparticle field energy conservation laws.

Take for example the statement " an EM field has two fields E and B where any work performed is by the E field and the work is not performed by the B field."

Then use your methodology to show why this is the case..

This other problem I can foresee is linearizing any graph of E

This is spot on! EM theory was developed inherently as substantival theory and to rederive it in purely relational terms remains a challenge for now. Mainly due to my knowledge gaps in this domain. But I'm working on it.

19 hours ago, Mordred said:

This other problem I can foresee is linearizing any graph of E

Im not sure what you mean but as long as its real (not a formalism artifact) Im up for a good challenge. That's how we progress isn't it?

18 hours ago, MJ kihara said:

Assuming your methodology was developed prior to SR and GR being realized by Einstein...and assuming SR and GR developed much later..could your methodology lead to practical technological innovations e.g assuming it was like that for like 50 years prior to 1945 could atomic bomb been developed within that time frame?

This is extremely speculative question my friend. But due to the post was moved to Speculations forum I guess why not? Let's speculate!

What do we have to speculate from? We have one individual (me) without specific education and experience in the field, out personal curiosity allegedly (I still hoping that someone finally verify my math and logic, that's actually suppose to be the main purpose of this post. So far out of 150+ comments I got 1 verification from @KJW . Thank you mate.) derives SR, GR, Cosmology, QM in about one year time. I see 2 possible explanation:
1. This individual is a genius
2. The methodology is covering some fundamental truths about the Universe.

We can rule out the first option straight away. I have almost 40 years of empirical data that undeniably proves that I'm far from genius. So it must be the method. How ironic is that? The philosophy becomes the key factor of scientific progress in our speculative alternative history thought experiment.

Lets say the method was discovered some wearer in late 18 hundreds. The peak of ultraviolet catastrophe in physics. Be cues quantisation, de Broglie Relation, Geometric Quantization of Angular Momentum, Atomic Structure, alpha, Dirac equation are all inevitable results of relational geometry: https://willrg.com/documents/WILL_RG_III.pdf#calibration
and in the same time it works wanders in GR realm. So we wouldn't have this incompatibility problem. The development of QM would stream line leading to earlier development of nuclear weapons. Considering that back then Germany was world capital of science we looking in to horrific scenario where Fascist Germany due to overwhelmingly superior technology's developing a nuclear weapon first. German nuclear U-boats bombing the Manhattan project in Los Alamos, New Mexico killing the one and only bongo playing Richard Feynman. The world in tears and Hitler ruling with the iron fist riding the T-Rex. Terror and Horror everywhere. The end.

Ha-ha-ha-ha! Yeh it was fun!

Edited 4 hours ago4 hr by Anton Rize

13 hours ago, Mordred said:

Ok this is something I can't shake.

If you have a static event at x=10, y=10 and z=10. How do you intuitively identify their orientation from (0,0,0) including angles between principle axis ?

You see this doesn't make any sense to me sorry. How do you determine x=10, y=10 and z=10? In relation to what? How do you choose your (0,0,0) ? On what basis you assume that spacetime can be forced in to a grid like an empty lunchbox? You are smuggling a background manifold into the premise of your question.

If there is only me (the observer) and a single static event, there are no "x, y, z" axes. There is no grid. There is only a direct relational link (line of sight).

To define x=10, y=10, z=10, you must arbitrarily invent a background coordinate system (an empty lunchbox) with arbitrarily chosen orthogonal axes pointing into empty space. This violates the principle of relational origin.

In RG, I self-center at my relational origin [math](\beta=0, \kappa=0)[/math].

If I observe an event, I measure only two things:

1. The relational shift (redshift/Doppler) which gives me [math]\kappa[/math] and [math]\beta[/math].

2. The geometric scale [math]r[/math] (via luminosity or parallax or signal delay etc...).

Angles do not exist a priori between an object and an imaginary spatial axis. Angles only exist between two distinct observable objects (e.g., measuring the angular separation between Event A and Event B).

So the system under study is me at the origin and the event. That's all our system is. there's no need to speculate "the empty lunchbox" scenario with arbitrarily chosen values in arbitrarily chosen metric.

Geometry is constructed from actual relations between bodies, not by pinning bodies onto an imaginary Cartesian grid.

14 hours ago, Mordred said:

If you were to hand me an E value how would I identify and seperate the components that make up the total E. ( in essence working backwards from total E in reverse and correctly identifying each contributor including quantity of each contribution.

This is precisely what Im doing in this blind test I called out. Send me the data lets test not by just agreeing with GR but by predicting accurate results where GR can not! I know it sounds crazy. I don't believe my results as well. There's only one way to find out. Blind empirical test. Lets do it!

10 hours ago, Mordred said:

When you get a chance here's another test for you. According to Newtons Shell theorem if you energy mass distribution is uniform throughout your system/ state. G=0 Regardless of what the energy mass density is.

Isn't it's the net gravitational force (or acceleration, [math]g[/math]) that equals zero? not the gravitational constant [math]G[/math]? [math]G[/math] remains constant in the Newtonian framework as far as I know...

But let's address physics of uniform distribution using RG. This is actually a perfect example of why RG is ontologically cleaner (Occams razor)

Standard mechanics explains the zero net force inside a uniform shell via vector cancellation: an infinite number of force vectors pulling in all directions through a background space perfectly cancel each other out. Don't you think its a bit of an overkill ontologically? Do we really need to speculate infinite forces from infinite points just to explain the rest state? Its violates all 5 methodological principals we building on isn't it? Actually Im not sure... having methodological principals in the foundation of the theory is it a common practice? The only thing that comes to mind is equivalence and least action principals...

I'm offering a relational definition of Energy, as the relational measure of difference between possible states. https://willrg.com/documents/WILL_RG_I.pdf#def:energy

If a distribution is perfectly uniform, there is absolutely zero geometric or physical difference between State A and State B.

If there is no difference in state, the relational shift is zero ([math]\Delta Q = 0[/math]).

Therefore, the potential projection gradient [math]\Delta\kappa = 0[/math].

Gravity in RG is not a "force" pulling you through a grid; it is the geometric gradient of the potential projection [math]\kappa[/math]. If the density is uniform everywhere, there is no gradient. No gradient means no relational difference, which operationally means zero gravity.

We don't need infinite vectors cancelling each other out in a background container. We just have a uniform state with zero relational difference. The math perfectly matches the Shell Theorem outcome, but completely eliminates the unnecessary ontological complexity.

@Anton Rize, you are correct in saying that only one of the twins actually turns around, but you are incorrect in how that turnaround manifests in that twin ending up younger than the stay-at-home twin. I actually provided a little bit of a hint when I mentioned the symmetry between observers who see each other either redshifted or blueshifted depending on whether they are moving away from or towards each other. Instead of a pair of twins, consider three observers A, B, and C. Let A be the same as the stay-at-home twin, B be the same as the travelling twin on the outbound leg of the journey, and C be the same as the travelling twin on the inbound leg of the journey. Whereas the travelling twin accelerates at the turnaround, observers B and C are two different observers, neither of whom accelerate when they meet at the turnaround location of the travelling twin. Observers A and B are moving away from each other and see each other redshifted by the same amount. Observers A and C are moving towards each other and see each other blueshifted by the same amount. Thus, there is symmetry between observers A and B as well as between observers A and C. Yet in spite of this symmetry, the total time of B and C is less than the time of A. It should be noted that A and B, as well as A and C, are observing each other's clocks (comparing with their own clocks), the comparison being according to the Doppler effect for light. That is, the observed Doppler effect can be used to establish that the travelling twin returns younger than the stay-at-home twin.

Remarkably, it also explains precisely how the turnaround manifests this asymmetry. At the turnaround, the redshift of A's clock observed by B immediately becomes the blueshift of A's clock observed by C. By contrast, the redshift of B's clock observed by A doesn't become the blueshift of C's clock observed by A until light from the turnaround has reached A. For the travelling twin, half the time the stay-at-home twin's clock is observed as redshifted, and half the time the stay-at-home twin's clock is observed as blueshifted. For the stay-at-home twin, more than half the time the travelling twin's clock is observed as redshifted and less than half the time the travelling twin's clock is observed as blueshifted. And the faster the travelling twin travelled, the further away from home the turnaround location will be, and therefore the longer the time it takes light to travel the distance, and the greater the proportion of time the travelling twin's clock is observed to be redshifted.

When one actually calculates the Doppler effect observed by each twin of the other twin's clock, in both cases, the time of the travelling twin is less than the time of the stay-at-home twin by the expected time dilation factor (consistent and not symmetric).

Edited 39 minutes ago39 min by KJW

4 hours ago, Anton Rize said:

Ha-ha-ha-ha! Yeh it was fun!

The reason I ask that question is that...so far it seems to me your method require prior knowledge that's why maybe you require to cross check your results from people who are aquainted with GR and SR...also how many people so far can use your arguments to get results the way you are doing..

On 2/19/2026 at 6:37 AM, Anton Rize said:

You asked: "How do you handle distance?"

In my model, Space is a consequence of Time, not a container for it.

Distance is strictly defined as Signal Delay.

Distance≡c(treception−temission)

I do not assign coordinate (x,y,z) as a primary property. I assign a Relational Lag.

You use signal delay to determine distance..how do you determine where the signal is coming from and how do you determine the delay?