8 hours ago, KJW said:

However, because the time of an orbit of the earth around the sun is one year, the length of the helical trajectory in spacetime is about one light-year, and at this length, no longer seems significantly curved.

Nice way to visualise this +1

19 hours ago, Anton Rize said:

Light is simply the rightward topological limit of matter, where the internal phase βY reaches zero.

On 3/1/2026 at 8:29 AM, MJ kihara said:

...however I also realized I was doing a lot of new things in physics...things that have fundamental consequences like the way people view electromagnetic spectrum to me it's no longer the same,to cut the story short..

Am very surprised by this let me hope there is substance in what @ Anton Rize is doing...am pegging my hope also on @KJW ability to cross check the authors math.

@ Anton Rize,What specifically do you mean by saying that "where internal phase βY reaches zero."?

On 3/6/2026 at 12:02 PM, Mordred said:

Future challenge

" what is the magnification"

How would you use the quoted relations to determine this.

Dark lensing:

[math] \theta_E = 2 \frac{D_{LS}}{D_S} \arcsin( \frac{(\kappa_{bar}^2(\theta_E) + \kappa_{phantom}^2(\theta_E)) (1 + \beta_p^2)}{2\beta_p^2 - (\kappa_{bar}^2(\theta_E) + \kappa_{phantom}^2(\theta_E)) (1 + \beta_p^2)} )[/math]

Full derivation: https://willrg.com/documents/WILL_RG_II.pdf#sec:lensing
Colab notebook: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/Dark_Lensing.ipynb
OUTPUT:

WILL RG AB INITIO PREDICTIONS (Zero Free Parameters)
    LensID  sigma_obs  sigma_pred  theta_E_obs  theta_E_pred
J0037-0942        279      282.11         1.53          1.49
J0216-0813        333      286.12         1.16          0.78
J0737+3216        323      285.34         1.00          0.95
J0946+1006        287      282.96         1.43          1.37
J0956+5100        334      296.55         1.33          1.15
J1250+0523        252      268.53         1.13          1.37
J1430+4105        322      292.07         1.52          1.14
J1627-0053        290      298.59         1.23          1.46
-----------------------------------------------------------------
Kinematics Mean Absolute Error: 23.02 km/s
Lensing Mean Absolute Error:    0.195 arcsec
-----------------------------------------------------------------

I guess one could call it a mic drop 😁

Edited March 7Mar 7 by Anton Rize

Thank you that is all I've been trying to do is provide challenges and other considerations not to compete with you but to provide methodological comparisons.

I will assume you can use the above to determine mirror images and determine the angular diameter distance of the original object.

The alternative method I wanted to see how you handle was flux vectors which also involved specific and mean intensity.

example Einstein ring or different distortions as per the images in article below

https://web.pa.msu.edu/people/abdo/GravitationalLensing.pdf

Edited March 7Mar 7 by Mordred

20 hours ago, MJ kihara said:

  On 3/7/2026 at 5:45 AM, KJW said:

Firstly, there are three space dimensions and one time dimension. The spacetime metric itself separates space and time into separate notions by their opposite signs in the signature.

Don't know if am wrong but the opposite signs in spacetime metric... signature -1,+1,+1,+1 or +1,-1,-1,-1 makes sure the spacetime is unified as it's also stated by
□=0. ( To whoever get concerned...Sorry I couldn't not comment in another thread...when experts discuss, sometimes it's reasonable to keep quite).

The three sides of de'Alembert operator magnitude(space axis x,y,z) being equivalent to magnitude of one side of the de'Alembert operator ( square,four sided, x,y,z,t) opposite sign ensure the total magnitude is equal to Zero...each degree of freedom is equivalent...t equivalent to -x,t equivalent to -y,t equivalent to -z and t equivalent to -(x,y,z).

Sorry corrections, each degree of freedom is equivalent...t = x,t = y,t = z and t = (x,y,z).

□=0.

It seems there is no need of correction,it depends on which signature you are using signature -1,+1,+1,+1 or +1,-1,-1,-1.

It is the metric being expressed in terms of all four dimensions that ensures spacetime is a unified notion. The signature of the metric provides finer detail about the geometry of spacetime, giving rise to the distinction between space and time, even though spacetime is a unified notion. In four dimensions, there are three distinct signatures: (+,+,+,+)/(–,–,–,–), (+,–,–,–)/(–,+,+,+), and (+,+,–,–). In the case of (+,+,+,+)/(–,–,–,–), there is no distinction at all between the dimensions, and there is no notion of a speed of light. Space is four-dimensional and time does not exist. In the case of (+,+,–,–), there are two space dimensions and two time dimensions, and the notion of a speed of light. However, because changing all the signs of a signature doesn't change the geometry, there is nothing to indicate which dimensions are space and which are time. But it is the case of (+,–,–,–)/(–,+,+,+) that is our spacetime in which there is the notion of a speed of light, and there are unequivocally three dimensions of space and one dimension of time. Solving the equation

[math]\pm\ x^2 \pm y^2 \pm z^2 \pm t^2 = 0[/math]

for the various sign combinations shows how the partition between space and time depends on the signature.

On 3/6/2026 at 1:45 PM, KJW said:

I'm not sure that what value I see is what you intend me to see as I'm not in full agreement with your philosophy, although some of it does align with relativity. What I would like to see is some form of mathematical proof that your theory fully agrees with general relativity. What you have provided so far is not such a proof, and I'm not sure what such a proof would look like.

Ok lets think together what would you consider a proof? You can give me a list of predictions and Ill show you derivations. You can ask WILL-AI he will show as well. All major GR predictions I already derived as far as I know. Gravitational waves I discard my derivation because math wasn't pretty enough😅. Im looking for the solution that would be as simple as the rest.
Discrepancy's are far to small for detection tools accuracy so far... I diverge with GR at cosmic and quantum levels but in a good way.
Do you have something specific in mind?

In terms of mathematical proofs the following properties should be followed

Concise (not unnecessarily long)

Clear (not ambiguous)

Complete (no missing intermediate steps)

Logical (every statement logically follows)

Rigorous (uses mathematical expressions)

Convincing (does not raise questions)

The way a proof is presented might be different from the way the proof is discovered. Proofs will include defining the limits of a set, step by step derivatives to answer questions such as

-How did you arrive at this equation ?

- What limits does the equation have ?

-does the equation follow other related and well established equations that do the same operation.

If I was developing a new theory, methodology or set of equations the above is the steps I would follow.

On 3/7/2026 at 9:57 AM, Anton Rize said:

In standard General Relativity, to explain the the observed phenomena, one must postulate several heavy ontological primitives:

* Mass (as an intrinsic substance).

* A background 4D spacetime manifold.

* The metric tensor.

* Spatial curvature as a distinct geometric entity.

I don't agree with this assessment of what general relativity must postulate.

Mass is not a postulate. It is a physical quantity that is part of the connection between the pure mathematics of Ricci calculus and physical reality. Obtaining the Schwarzschild solution involves solving a first-order ordinary differential equation, which produces a single arbitrary constant. Physically, it is directly proportional to the mass of the black hole, but to obtain the proportionality constant, one needs to compare the behaviour of the Schwarzschild solution under weak-field conditions, with the Newtonian formula. You have chosen to use the Schwarzschild radius to represent mass. That's ok until you need to specify mass in terms of mass units. Note that using the Schwarzschild radius as the arbitrary constant of integration is purely mathematical, and therefore requires some form of physical measurement to connect your Schwarzschild solution to physical reality.

What do you mean by a background 4D spacetime manifold? Actually, the fundamental fields of general relativity describe a spacetime manifold rather than sit upon a background. Fields that sit upon a general relativistic manifold are probably not themselves part of general relativity. For example, the metric tensor field is a set of coefficients of the metric that describe the distance between infinitesimally separated points of the spacetime manifold. And the various spacetime curvature fields are mathematically derived from the metric tensor field. The existence of the curvature fields is guaranteed given the existence of the metric tensor field. It is not a postulate that the curvature fields exist. And because it is not assumed that the metric tensor field is special, it is not assumed that the curvature fields are zero.

Although it is desirable to minimise the number of postulates, the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact. And from this fact, I can derive guaranteed theoretical results by deliberately avoiding assumptions about physical reality. I've already discussed this earlier in this thread, so I won't repeat myself here.

The method I have used to derive general relativity, I have so far been unable to derive the metric tensor field from first principles. The problem isn't the invertible matrix field itself. That emerges naturally from the connection. The problem is that given an arbitrary connection, it is not guaranteed that a covariantly constant invertible matrix field exists. And it is covariant constancy that distinguishes the metric tensor field from an arbitrary invertible matrix field. However, the postulate of the existence of the metric tensor field can be justified by the notion of magnitude. Nevertheless, I have a keen eye on what can and can't be derived without the metric tensor field.

As for "spatial curvature as a distinct geometric entity", that is not a postulate of general relativity. That is a notion that emerges from the Schwarzschild solution as well as an understanding of the nature of familiar gravity.

Although the theory of general relativity doesn't use coordinates, specific solutions of the Einstein field equations do use coordinates, and often those coordinates are space and time coordinates. Often, interesting solutions possess symmetries that render splitting the spacetime into space and time quite natural (because symmetry is manifestly covariant). And with that natural splitting, the notion of spatial curvature is also natural.

@KJW , thank you for detailed and deep respond. I understand and respect your point. But I have to disagree with it.
Let's examine the logical structure of this ontological defense:

You outlined a linear dependency in General Relativity:
The spacetime manifold exists $\to$ we postulate a metric tensor field upon it $\to$ spatial curvature mathematically emerges from this tensor. I understand this chain perfectly.

However, the critical epistemological flaw in your methodology is revealed in this exact sentence:

"the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact."

This is a textbook logical leap known as reification (treating an abstract mathematical model as a physical entity), and it is the exact anthropocentrism my methodology filters out. In the scientific method, a successful postulate remains a model; it does not magically transform into a physical "fact" just because it aligns with human intuition of "empty space."

As for your comment on covariant symmetries naturally splitting spacetime: the reason you never see me struggling to enforce covariance is because Relational Geometry is natively background-independent. The problem of non-covariant relations you face is an artifact of your initial ontological choice to use coordinates. When you discard coordinates and deal strictly with energetic capacities, covariance is absolute by definition. (You can see the precise ontological difference on my logic map: https://willrg.com/logos_map/).

But philosophy will not give us concrete results, physics and math - will.

My claim is not merely that RG can reproduce GR's post-Newtonian results algebraically. My claim is that RG delivers empirically testable, closed-form analytical results where standard mechanics cannot. Let's look at a categorical application:

---

Title: Analytical Resolution of the M sin(i) Degeneracy via Transverse Baseline Asymmetry

Over the past months, we have debated the empirical results of Relational Geometry and Relational Orbital Mechanics (R.O.M.). Since the discussion has reached a consensus on the predictive accuracy of the method for deflection, I am presenting the formal algebraic proof of how R.O.M. breaks the classical [math]M \sin i[/math] degeneracy in closed form, using only 1D spectroscopic extrema.

In classical radial velocity analysis, the semi-amplitude [math]K \propto \beta \sin i[/math] leaves the true kinematic projection [math]\beta[/math] and inclination [math]i[/math] degenerate. R.O.M. resolves this by restoring the systemic transverse baseline [math]Z_{sys}[/math], which depends strictly on [math]\beta[/math] and is completely independent of [math]i[/math].

1. The Observer Equations at Extrema

[math]\kappa^2 = 1-\frac{1}{(1+z_{\kappa})^2}[/math] ([math]z_{\kappa}[/math] = gravitational redshift)

[math]\beta^2 = 1-\frac{1}{(1+z_{\beta})^2}[/math] ([math]z_{\beta}[/math] = transverse Doppler shift)

Observational Z Inputs

[math]Z_{sys}(o) = (1+z_{\kappa o}(o))(1+z_{\beta o}(o)) = \tau_{Wo}(o)^{-1}[/math] (product of gravitational redshift and transverse Doppler shift)

[math]\tau_{Wo}(o) = \kappa_{Xo}(o)\beta_{Yo}(o) = (Z_{sys}(o))^{-1}[/math] (product of projectional phase factors on [math]S^1[/math] and [math]S^2[/math] carriers)

[math]z_{\kappa} = \frac{1}{\kappa_{X}}-1[/math] (gravitational redshift)

[math]z_{\beta} = \frac{1}{\beta_{Y}}-1[/math] (transverse Doppler shift)

The raw spectroscopic shifts at maximum and minimum radial velocity (phases [math]o = -\omega_i[/math] and [math]o = \pi - \omega_i[/math]) are products of the line-of-sight Doppler projection and the transverse baseline [math]Z_{sys}[/math] (denoted here as [math]D_{max}[/math] and [math]D_{min}[/math]):

[math]D_{max}(\beta, e, \omega_i) = \sqrt{1 - 2\beta^2\frac{1+e\cos\omega_i}{1-e^2}} \sqrt{1 - \beta^2\frac{1+e^2+2e\cos\omega_i}{1-e^2}}[/math]

[math]D_{min}(\beta, e, \omega_i) = \sqrt{1 - 2\beta^2\frac{1-e\cos\omega_i}{1-e^2}} \sqrt{1 - \beta^2\frac{1+e^2-2e\cos\omega_i}{1-e^2}}[/math]

The observed extrema are:

[math]Z_{rawmax} \cdot D_{max} = 1 + K_i(1+e\cos\omega_i)[/math]

[math]Z_{rawmin} \cdot D_{min} = 1 - K_i(1-e\cos\omega_i)[/math]

2. Algebraic Decoupling (The Decryption Invariant)

Subtracting the equations isolates the observed semi-amplitude:

[math]2K_i = Z_{rawmax} D_{max} - Z_{rawmin} D_{min}[/math]

Adding the equations and substituting [math]2K_i[/math] back into the sum yields a strict algebraic invariant where the inclination angle [math]i[/math] is completely eliminated:

[math]Z_{rawmax} D_{max} (1 - e\cos\omega_i) + Z_{rawmin} D_{min} (1 + e\cos\omega_i) = 2[/math]

This equation proves that the true kinematic projection [math]\beta[/math] and the argument of periapsis [math]\omega_i[/math] are locked strictly by the asymmetry of the observed extrema [math]Z_{rawmax}[/math] and [math]Z_{rawmin}[/math], regardless of the viewing angle.

3. Analytical Extraction of sin(i)

Once [math]\beta[/math] is constrained by the invariant above, the true inclination is trivially extracted without invoking standard metric priors, G, or M:

[math]\sin i = \frac{\sqrt{1-e^2}}{2\beta} \left[ Z_{rawmax} D_{max}(\beta, e, \omega_i) - Z_{rawmin} D_{min}(\beta, e, \omega_i) \right][/math]

4. Empirical Challenge

I have verified this algebraic closure against the S0-2 (GRAVITY) dataset and synthetic 1PN data. The math holds absolutely. I invite anyone to find an algebraic flaw in the derivation from Step 1 to Step 3, or to explain how a supposedly "fundamental" physical degeneracy can be resolved through pure relational geometry if the classical 1D linear projection framework is complete.

* Full derivation: https://willrg.com/documents/WILL_RG_I.pdf#sec:analytical_sini

* ROM parameters and equations: https://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom

* Detailed results analysis: https://willrg.com/msini_test

* Colab notebook Synthetic: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/ROM_Validation_via_Synthetic_1PN_Data.ipynb

Colab notebook real data: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/ROM_S2_SOLVER.ipynb
Colab deep analyses: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/beta_i_comparison.ipynb

---

So far it seems to me that RG delivers more and clearer results while requiring less.

Edited March 8Mar 8 by Anton Rize

13 hours ago, Anton Rize said:

the critical epistemological flaw in your methodology is revealed in this exact sentence:

"the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact."

This is a textbook logical leap known as reification (treating an abstract mathematical model as a physical entity)

Bear in mind that we had already discussed in greater detail the logical basis I used to derive general relativity, and that I didn't want to repeat that discussion here, choosing only to give a fairly brief statement relevant to the current discussion. So, in effect, you attacked a strawman.

I actually make a point of distinguishing the mathematics from the physics. Yet you chose to interpret my statement:

the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact.

as:

treating an abstract mathematical model as a physical entity

No, my statement does not lead to your interpretation. I said the mathematical model was reasonable on the basis of the physical entity. I had already outlined earlier in this thread the reason that connects the physical entity to the mathematical model. That is, we measure the physical entity to create a description that is the theoretical basis of a mathematical model. It is a fact that the physical spacetime is amenable to being described in terms of a mathematical spacetime manifold. Although one can challenge the spacetime manifold in terms of how close it is to first principles, one can't challenge its validity.

It should be noted that accuracy of the description is a separate issue. I assume that the descriptions are perfectly accurate. I know that isn't true in reality but consider inaccuracies to be outside the scope of my interest. My interest is actually what physical reality must be, based on what mathematical descriptions of it must be. The key requirement is that all constraints are logical rather than ad hoc. Thus, I am interested in why spacetime is four-dimensional, and why the signature of the metric is (+,–,–,–)/(–,+,+,+). Much of the mathematics of Ricci calculus is non-specific in terms of the number of dimensions or the signature of the metric. Some formulae explicitly mention n dimensions. Formulae that specify four dimensions and the signature of the metric are specifically about general relativity. The question of how the metric tensor field arises is a challenge to the key requirement that all constraints are logical rather than ad hoc, the specific constraint in this case being to the connection. Thus, although my interest is in the physics, my focus goes beyond the physics.

13 hours ago, Anton Rize said:

The problem of non-covariant relations you face is an artifact of your initial ontological choice to use coordinates.

I don't see the non-covariance that arises due to coordinates as a problem, but as an opportunity. It literally creates the fields of reality, not just in general relativity, but in gauge theory in general.

Edited March 8Mar 8 by KJW

2 hours ago, KJW said:

Thus, I am interested in why spacetime is four-dimensional, and why the signature of the metric is (+,–,–,–)/(–,+,+,+).

In what sense? Is it not from the fact that space and time form spacetime... Space equivalency to time.

Sometimes I feel explanations afterwards provided for GR is complicated than how the thought process of Einstein was as he was deriving it.

1 hour ago, MJ kihara said:

In what sense? Is it not from the fact that space and time form spacetime... Space equivalency to time.

Sometimes I feel explanations afterwards provided for GR is complicated than how the thought process of Einstein was as he was deriving it.

Take a close look at the Lorentz transforms in terms of distance with time as an interval (ct) as you increase velocity you have an inverse relation with proper time also you get length contraction. Consider further velocity itself requires a start and end point with a rate for the object to transverse that distance. That should help you understand why time has the opposite metric signature to space components.

4 hours ago, KJW said:

I actually make a point of distinguishing the mathematics from the physics. Yet you chose to interpret my statement:

the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact.

as:

treating an abstract mathematical model as a physical entity

I apologize if I misinterpreted your intent. I actually make a genuine effort to separate the mathematical models from the physical reality as well, so that was definitely not my intention.

To help improve our communication, I want to point out exactly what type of wording I read that way. When you write something like:

4 hours ago, KJW said:

I don't see the non-covariance that arises due to coordinates as a problem, but as an opportunity. It literally creates the fields of reality, not just in general relativity, but in gauge theory in general.

This specific wording is what would lead me to interpretation of your stance as treating the mathematical tool (coordinates) as the creator of physical reality. If that wasn’t your intent, then it is simply a semantic misunderstanding between us, and I am glad we cleared it up.

4 hours ago, KJW said:

Thus, I am interested in why spacetime is four-dimensional, and why the signature of the metric is (+,–,–,–)/(–,+,+,+)

I am looking for the fundamental answers to those exact same questions! However, I don't currently have enough evidence to rule out other descriptions in favor of strict priors like "spacetime must be 4D" or "the metric must have this specific signature." My fear is that if I adopt such restrictive assumptions too early, I might inadvertently rule out the unknown Truth I am seeking, simply because the field of search was artificially narrowed by the form of the question itself.

---

Anyway, this is philosophy, and we could spend years debating it with zero tangible results. I have a much more solid question, or rather, a direct request for your help:

These recent results of mine regarding the M sin(i) degeneracy might be a huge deal. This degeneracy is considered mathematically unsolvable using standard 1D linear projection methods. I remain inherently skeptical of my own work and the idea that I might have found a solution. However, I cannot find any mistakes in the derivation, and the empirical data aligns perfectly with the predictions.

I would be incredibly grateful if you could bring your rigorous analytical skills to this specific derivation. Could you help me figure it out? Either by finding the mathematical mistake I might be making, or confirming that you couldn't find any.

Solving this definitively will affect astrophysics substantially, allowing us to extract much more true kinematic data from the same radial velocity signals. And potentially give us tools for solving the Hubble tension (maybe).

"The key requirement is that all constraints are logical rather than ad hoc."

At this fundamental level, we are in absolute, 100% agreement.

On 3/6/2026 at 10:45 PM, KJW said:

The spacetime metric itself separates space and time into separate notions by their opposite signs in the signature.

I tend to think opposites sign unify them... ensuring □=0

On 3/6/2026 at 10:45 PM, KJW said:

However, because the time of an orbit of the earth around the sun is one year, the length of the helical trajectory in spacetime is about one light-year

Just a question,if you make light keep orbiting the sun using whichever means for one year...what will be the length of helical trajectory?

18 minutes ago, Mordred said:

Take a close look at the Lorentz transforms in terms of distance with time as an interval (ct) as you increase velocity you have an inverse relation with proper time also you get length contraction. Consider further velocity itself requires a start and end point with a rate for the object to transverse that distance. That should help you understand why time has the opposite metric signature to space components.

should have added to this one could also alternately use celerity (proper velocity) which is distinct from coordinate velocity.

There is no real hard and fast methodology all math methods used to describe a system or state if they can produce the same results and fully describe that system or state should never have any restrictions based on philosophy. After all why should it ? Mathematics itself is a collection of bookkeeping tools it merely describes what we observe or measure and offers means of predicting cause and effect. The mathematics itself never defines reality.

After all the universe couldn't care less how we measure it nor describe it.

3 hours ago, MJ kihara said:

  5 hours ago, KJW said:

Thus, I am interested in why spacetime is four-dimensional, and why the signature of the metric is (+,–,–,–)/(–,+,+,+).

In what sense? Is it not from the fact that space and time form spacetime... Space equivalency to time.

Sometimes I feel explanations afterwards provided for GR is complicated than how the thought process of Einstein was as he was deriving it.

It seems to me to be a common view that the laws of physics are ad hoc, as if given to us from above. I reject this view. For example, according to Noether's theorem, each conservation law corresponds to a particular symmetry. Also, in the mathematics of general relativity, every scalar functional of the metric tensor and its partial derivatives of any order corresponds to a covariantly-conserved second-order tensor field (the symmetry in this case is the invariance of the scalar functional with respect to a change in coordinates, noting the severe restriction on what can actually be a scalar functional of the metric tensor and its partial derivatives). Such laws of physics arise for purely mathematical reasons.

But, that the number of dimensions of physical reality appears to be four rather than some other value is something that also needs to be explained logically. It turns out that in the mathematics of general relativity, four dimensions is rather special compared to other numbers of dimensions. For example:

• Four is the smallest number of dimensions for which pure gravitation can exist.

• Four is the only number of dimensions for which pure gravitation has the same algebraic freedom as energy-momentum.

• Four is the only number of dimensions for which the dual of the Riemann tensor is the same order as the Riemann tensor, and for which the dual of the Weyl tensor is the same as the Weyl tensor.

• Four is the only number of dimensions for which the integrand of the generalised Gauss-Bonnet topological invariant is quadratic in the curvature tensor.

As significant as these properties of four-dimensional spaces are, it is not clear to me how they actually constrain the number of dimensions.

Edited March 9Mar 9 by KJW

3 hours ago, MJ kihara said:

Sometimes I feel explanations afterwards provided for GR is complicated than how the thought process of Einstein was as he was deriving it.

Im having the same feeling.
One of my absolute favorite results RG research led me to is: "Mathematical complexity is the symptom of philosophical negligence." it is the logical conclusion that come up naturally after proving 2 theorems https://willrg.com/documents/WILL_RG_Substantialism_vs._Relationalism.pdf
Or if you prefer more visual approach its in the end of this Logos Map https://willrg.com/logos_map/

And regarding your question about "there's no massless particles" - its just poor choice of words on my side. I was trying to say that the lensing results once again suggesting that mass as a concept is not a necessary primitive. This result come up multiple times in my research. When Ill get a bit more time Ill put them all together for a comprehensive review for all of us.
And I know it sounds crazy at first but then when you start to think about it, it becomes so clear...

7 minutes ago, KJW said:

It seems to me to be a common view that the laws of physics are ad hoc, as if given to us from above. I reject this view. For example, according to Noether's theorem, each conservation law corresponds to a particular symmetry. Also, in the mathematics of general relativity, every scalar functional of the metric tensor and its partial derivatives of any order corresponds to a covariantly-conserved second-order tensor field (the symmetry in this case is the invariance of the scalar functional with respect to a change in coordinates, noting the severe restriction on what can actually be a scalar functional of the metric tensor and its partial derivatives). Such laws of physics arise for purely mathematical reasons.

But, that the number of dimensions of physical reality appears to be four rather than some other value is something that also needs to be explained logically. It turns out that in the mathematics of general relativity, four dimensions is rather special compared to other numbers of dimensions. For example:

• Four is the smallest number of dimensions for which pure gravitation can exist.

• Four is the only number of dimensions for which pure gravitation has the same algebraic freedom as energy-momentum.

• Four is the only number of dimensions for which the dual of the Riemann tensor is the same order as the Riemann tensor, and for which the dual of the Weyl tensor is the same as the Weyl tensor.

• Four is the only number of dimensions for which the kernel of the generalised Gauss-Bonnet topological invariant is quadratic in the curvature tensor.

As significant as these properties of four-dimensional spaces are, it is not clear to me how they actually constrain the number of dimensions.

Its remarkable how we are giving absolute different answers to the same question at the same exact time! 😄

By the way you might find this interesting. Its a little hint on 3D stricture of reality I got in my derivations:

Geometric Signature of Spatial Dimension

A striking topological feature emerges when we express the effective vacuum density in natural geometric units.

Substituting [math]\rho_{\Lambda} = \frac{2}{3}\rho_{\max}[/math] into the explicit definition of [math]\rho_{\max}[/math]:

[math]\rho_{\Lambda}(r) = \frac{2}{3} \frac{c^2}{8\pi G r^2} = \frac{c^2}{12\pi G r^2}[/math]

Stripping away dimensional scaling factors ([math]c, G, r[/math]) reveals a purely dimensionless geometric coefficient:

[math]\hat{\rho}_{\Lambda} = \frac{1}{12\pi} = \frac{1}{3 \times 4\pi}[/math]

This factorization suggests a profound geometric origin for 3D space:

* The factor [math]4\pi[/math] represents the intrinsic capacity of the relational carrier [math]S^2[/math].

* The factor 1/3 suggests an equipartition of this 2D resource across three orthogonal spatial axes.

This hints that the dimensionality of observable space is not arbitrary but is a structural consequence of distributing the [math]S^2[/math] energy budget into a volume.

The full section you can find in here: https://willrg.com/documents/WILL_RG_II.pdf#sec:dark_energy

Edited March 9Mar 9 by Anton Rize

3 hours ago, Anton Rize said:

  7 hours ago, KJW said:

I don't see the non-covariance that arises due to coordinates as a problem, but as an opportunity. It literally creates the fields of reality, not just in general relativity, but in gauge theory in general.

This specific wording is what would lead me to interpretation of your stance as treating the mathematical tool (coordinates) as the creator of physical reality. If that wasn’t your intent, then it is simply a semantic misunderstanding between us, and I am glad we cleared it up.

Perhaps what I said had a bit of hyperbole. In fact, I should've said:

I don't see the non-covariance that arises due to coordinates as a problem, but as an opportunity. It literally creates the fields of the mathematical descriptions of reality, not just in general relativity, but in gauge theory in general.

But then again, pretty much everything I am saying is about mathematical descriptions of reality rather than reality itself, so it is natural that I abbreviate. On the other hand, because of the correspondence between the reality and mathematical descriptions of the reality, the notion that non-covariance creates a field in the mathematical description would imply that non-covariance in a way creates a field in the reality itself also. For example, spacetime curvature exists in physical reality for the same reason it does in mathematical descriptions of reality. And yes, spacetime curvature is real and not just an abstract notion in a mathematical model.

On 3/6/2026 at 2:26 PM, Anton Rize said:

Light is simply the rightward topological limit of matter

What do you mean with topological limit of matter?

10 hours ago, Anton Rize said:

I was trying to say that the lensing results once again suggesting that mass as a concept is not a necessary primitive.

Am not getting it " not a necessary primitive " meaning mass is emergent or what?

40 minutes ago, MJ kihara said:

Am not getting it " not a necessary primitive " meaning mass is emergent or what?

Another question one can ask is if mass is a primitive concept " why do different particles or other objects require greater force to accelerate the same distance than others "?

15 hours ago, MJ kihara said:

  On 3/7/2026 at 5:45 AM, KJW said:

The spacetime metric itself separates space and time into separate notions by their opposite signs in the signature.

I tend to think opposites sign unify them

Why would opposite signs unify them?

15 hours ago, MJ kihara said:

... ensuring □=0

□ψ=0 is a second-order partial differential equation to be solved for ψ. The sign of the individual terms of the □ operator correspond to the signature of the metric. Solutions exist regardless of the signature, though the solutions themselves do depend on the signature. For the signature (+,+,+,+), □ψ=0 is called a "potential equation", though such an equation is best known in two and three dimensions.

15 hours ago, MJ kihara said:

  On 3/7/2026 at 5:45 AM, KJW said:

However, because the time of an orbit of the earth around the sun is one year, the length of the helical trajectory in spacetime is about one light-year

Just a question,if you make light keep orbiting the sun using whichever means for one year...what will be the length of helical trajectory?

When I wrote this, it was my intention that it be visualised in an ordinary three-dimensional Euclidean space where the vertical dimension represents time that is scaled correctly but ignoring such things as time dilation or other relativistic effects. That is why I said "about one light-year" instead of "exactly one light-year" or even just "one light-year". In the case of light, relativity can't be ignored. The corresponding trajectory of light will have a completely different length in spacetime than the Euclidean space in which it is intended to be visualised. In spacetime, lightlike trajectories have zero length. And it is the opposite signs in the signature that enable such trajectories to exist.

On 3/8/2026 at 9:14 AM, Anton Rize said:

Ok lets think together what would you consider a proof? You can give me a list of predictions and Ill show you derivations.

No, that will not do. That would be like trying to prove the Riemann hypothesis by brute force.

18 hours ago, Anton Rize said:

I have a much more solid question, or rather, a direct request for your help:

These recent results of mine regarding the M sin(i) degeneracy might be a huge deal. This degeneracy is considered mathematically unsolvable using standard 1D linear projection methods. I remain inherently skeptical of my own work and the idea that I might have found a solution. However, I cannot find any mistakes in the derivation, and the empirical data aligns perfectly with the predictions.

I would be incredibly grateful if you could bring your rigorous analytical skills to this specific derivation. Could you help me figure it out? Either by finding the mathematical mistake I might be making, or confirming that you couldn't find any.

Solving this definitively will affect astrophysics substantially, allowing us to extract much more true kinematic data from the same radial velocity signals. And potentially give us tools for solving the Hubble tension (maybe).

Unfortunately, what you are asking me about is galaxy rotation curves, of which I have little knowledge or interest. I don't think I can help you. I should remark that I am firmly in the dark matter camp for two reasons: 1, the alternative is a modified law of gravitation, which I reject on the basis of my wholehearted acceptance of general relativity; and 2, the Bullet Cluster and other galaxy cluster collisions, exhibit gravitational lensing that strongly support dark matter over a modified law of gravitation. However, I don't have any particularly strong opinions about dark matter candidates.

16 hours ago, Anton Rize said:

https://willrg.com/documents/WILL_RG_II.pdf#sec:dark_energy

In the beginning, I rejected a cosmological constant on the basis that any such constant could be absorbed into the energy-momentum source term of the Einstein field equation. But sometime later, I had an epiphany. The cosmological constant has a property that is not possessed by energy-momentum in general: It is invariant to Lorentz transformations. This means it is not possible to establish one's velocity relative to the cosmological constant. The reason why one can determine the velocity of an ordinary object (relative to oneself) is because that object breaks Lorentz symmetry. That is, the object appears different at different velocity, and that difference is used to determine the velocity. But the inability to determine the velocity of the cosmological constant means that the cosmological constant looks like empty space. However, that empty space still appears curved though that curvature can't establish its velocity.

It's worth noting that although the speed of light is invariant to Lorentz transformations, light itself is not. A Lorentz transformation changes both the frequency and wavelength of a given light. Empty space does not behave this way, so light does distinguish itself from empty space.

Thus, the cosmological constant will naturally distinguish itself from the energy-momentum in the Einstein field equation.

Edited March 9Mar 9 by KJW

7 hours ago, MJ kihara said:

What do you mean with topological limit of matter?

Am not getting it " not a necessary primitive " meaning mass is emergent or what?

16 hours ago, KJW said:

And yes, spacetime curvature is real and not just an abstract notion in a mathematical model.

These are excellent, fundamental questions and one very important statement.

When I say that "mass is not a necessary primitive," I am not just speaking philosophically; I mean it in a strict, operational, mathematical sense. In standard physics, mass ([math]M[/math]) is treated as a fundamental building block of reality. You need to plug [math]M[/math] (along with the gravitational constant [math]G[/math]) into equations to figure out the gravitational scale of a system, such as the Schwarzschild radius [math]R_s = \frac{2GM}{c^2}[/math] and other gravitational phenomenon.

But what if we could derive the exact same absolute scale of a stellar system without ever knowing its "Mass" and without ever using [math]G[/math]? If the geometry of the system can be completely solved using only clocks and light, then Mass is not a fundamental pillar of reality. So mass is a bookkeeping label we paste on later for convenience. The phenomena themselves are governed by the single observable length R_s.

Here is the formal proof of this concept from my research. I call it the Chrono-Spectroscopic Theorem. It proves that the absolute system scale is generated exclusively from pure chronometry (time) and spectroscopy (light shifts):

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The Epistemological Bottleneck of Spatial Distance

In classical orbital mechanics, the absolute gravitational scale [math]R_s[/math] requires [math]G[/math] and mass [math]M[/math], which in turn require measuring distances in meters (tethering physics to the cosmic distance ladder). In Relational Orbital Mechanics (R.O.M.), physical distance is not an a priori container; distance emerges macroscopically as the geometric "tension" between two relational energetic potentials.

The Theorem

By integrating the local relational spacetime factor over a closed phase interval, the absolute scale of the system [math]R_s[/math] decouples entirely from spatial coordinates and mass. For a complete orbital cycle, it algebraically collapses to:

[math]R_s = T c \frac{\kappa^2 \beta}{2\pi}[/math]

Here, [math]\kappa^2[/math] and [math]\beta[/math] are the global potential and kinematic projections. The absolute scale is derived exclusively from Chronometry (orbital period [math]T[/math]) and Spectroscopy (light shifts), eliminating any dependency on [math]G[/math], [math]M[/math], or spatial parallax.

Empirical Validation (The Solar System)

We validate this using Mercury's state at perihelion. We extract the necessary parameters entirely from Earth-based dimensionless observables:

Direct Optical Radius ([math]\theta_{\odot}[/math]): The angular radius of the Sun ([math]\approx 0.004652[/math] rad).

Chronometric Scaling ([math]T_M / T_{\oplus}[/math]): The ratio of orbital periods ([math]\approx 0.3871[/math]).

Kinematic Eccentricity ([math]e[/math]): Derived from angular velocity extrema ([math]e \approx 0.2056[/math]).

The pure relational scale factor at perihelion is computed without any reliance on [math]G[/math], [math]M[/math], or the Astronomical Unit (meters):

[math]R_{ratio} = \frac{\theta_{\odot}}{(T_M / T_{\oplus})^{2/3} (1-e)} \approx 0.01512[/math]

Using the solar gravitational redshift ([math]z_{sun} = 2.1224 \times 10^{-6}[/math]) and Mercury's transverse kinematic shift ([math]\beta_p = 1.967 \times 10^{-4}[/math]), the global invariants are recovered algebraically.

Substituting these strictly relational observables into the R.O.M. absolute scale equation yields:

[math]R_s \approx 2953.3[/math] m

This result perfectly matches the classical derivation [math]\frac{2GM_{sun}}{c^2}[/math], yet it is achieved strictly through internal system clocks ([math]T[/math]) and spectroscopic shifts ([math]\beta_p, z_{sun}[/math]).

Conclusion:

WILL RG is operationally independent from mass and [math]G[/math]. The physical scale of a closed orbital system is algebraically equivalent to the ratio of geometric tension ([math]\kappa^2\beta[/math]) to local clock ticks ([math]T[/math]). This is what I mean when I say mass is not a necessary primitive. The geometry works without it.

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To make it absolutely transparent and easy to verify here's locked and loaded desmos project: https://www.desmos.com/calculator/iymnd3tw3z
Here's a full section: https://willrg.com/documents/WILL_RG_I.pdf#sec:absolute_scale
Here's more evidence of mass and G independence: https://willrg.com/documents/WILL_RG_R.O.M..pdf#sec:operational

Under SR/GR all particles that are following a geodesic are in freefall. So mass is irrelevant so is proper acceleration.

Where mass/energy density distribution becomes relevant is determining the geodesic itself.

I'll let you discover by yourself where you are wrong in this context.