13 hours ago, Mordred said:

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

Thank you for proving my earlier comment last night.

That disregard of the mathematical proof of the critical density formula that you consider invalid despite all its success and clear match to observational evidence simply because of your refusal to apply vectors in understanding that proof is a clear example.

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

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

I have to admit that this is a completely new experience to me - you are the first person in this 250+ comments thread who is not dismissing my ontology... I didn't realize how horribly sad it is until I typed it out...

I also would like to emphasize the redefinition of energy wasn't my philosophical choice. The standard "ability to do work" - violates the principle of relational origin. So my definition, same as everything else in my research is forced by the core method that Im committed to.
If my research ever finds wider acceptance it is this generative method of mine I consider the biggest contribution. It shows a different way to do science where ontology comes first. No mindless phenomenology no curve fitting no unfalsifiable models.

P.S. I would be very curious to hear your "controversial" definition - it seems our intuitions are vibrating on the same frequency.

Edited yesterday at 05:57 PM1 day by Anton Rize

On 2/25/2026 at 5:34 AM, MJ kihara said:

I tend to think eventually it generates the mainstream definition, therefore, by default am not against mainstream definitions,however, I don't regard them as fundamental...

I'm surprised by this, as ever since 1918, we know energy to be a derived quantity, not a defined one, and it's based on the Lagrangian.

If the dynamical system is amenable to a Lagrangian formulation, and if that Lagrangian does not explicitly depend on time, then there is an energy, and the different expressions can be obtained with a precise recipe.

So it's been a while since we know all these "mainstream definitions" are nothing but the corollaries of a master theorem.

1 hour ago, joigus said:

I'm surprised by this, as ever since 1918, we know energy to be a derived quantity, not a defined one, and it's based on the Lagrangian.

If the dynamical system is amenable to a Lagrangian formulation, and if that Lagrangian does not explicitly depend on time, then there is an energy, and the different expressions can be obtained with a precise recipe.

So it's been a while since we know all these "mainstream definitions" are nothing but the corollaries of a master theorem.

Well said, for that matter that's even explained in my 1920 publication textbook on physics rather interesting reading as the atom is only described by protons and electrons and no other particle was known.

The textbook also specifically went into classical examples on refrigeration, power generators etc.

The amusing part is the term Superposition is included even though it doesn't even refer to QM...not surprised of course but still amusing

Just a side note Newtons gravitational law had a different form then.

\[F=k\frac{m\acute{m}}{r^2}\]

with units given in dynes. K was simply some constant.

15 hours ago, Anton Rize said:

It is the observable signature of a star’s resonant coupling to the cosmic horizon inside a closed relational geometry.

That's is the definition of dark matter according to you,therefore, it ceases being dark...by explaining it's nature, confirming it's existence, it no longer become dark...I think the issue not by it being called dark...it's just an indication of physics at that particular era not being able to explain the source of ' missing mass'

The fact that 'dark matter and dark energy ' makes up almost 95% of the universe is not a small issue.

According to your formulation how does matter or rather how does things acquire mass?

Also from your definition above,how do you explain the discrepancy between rotation velocity in the outer of the Galaxy and the inner region? Does gravitational wave/force move at/transmitted at speed of light.?

Without referring to any of your links or articles ( post the math here)

Answer the following.

Why do you use the \[8\pi r^2\] relation

If you are not using the volume comparision between 2 spheres.

Follow up question What single geometric object has the relation \{ 8\pi r^2\}.

On 2/25/2026 at 2:34 PM, MJ kihara said:

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

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

You have raised a crucial philosophical point, but I must respectfully disagree with your underlying premise. You are conflating mathematical flexibility with physical constraint.

Discoveries like Neptune were not made because the model was "flexible." They were made precisely because classical mechanics was rigid. When the orbit of Uranus violated predictions, Le Verrier did not have the "flexibility" to alter Newton's gravitational constant or invent a new force distance parameter. Because the model's constraints were absolute, the anomaly strictly demanded a localized physical explanation. That rigidity is exactly what pointed the telescope at Neptune.

If a model is "flexible" enough that you can just add a mathematical parameter every time an observation deviates from theory (like adding epicycles, or adding an invisible "Dark Matter" halo to fix a galactic rotation curve), you stop looking for physical reality. You just fit the curve. Flexibility is often the enemy of discovery; it hides structural failures under mathematical patches.

This brings me to your comment about my math. You suggested that my model "incorporates Hamiltonian, Klein Gordon equations... and manipulation of physics units/SI units in a smart way" to give short answers.

I need to be absolutely clear: it does not.

There are no hidden quantum mechanical operators, no Hamiltonians, and no dimensional juggling under the hood. I'm keeping everything dimensionless at all times and units used only when its absolutely necessary. The reason the derivations are so short and direct is that the framework relies entirely on generative geometric constraints, not phenomenological equations. Hamiltonian same as Lagrangian are derived: Lagrangian and Hamiltonian as Single-Point Limits of the Relational Energy-Symmetry Law

A rigid, zero-parameter model like WILL RG does not prevent the prediction of the unknown. On the contrary: because it has zero flexibility, any future observation that deviates from its predictions cannot simply be "patched" with a new parameter. It will immediately expose the exact location of actual new physics or exact limitation of the current model. There's no place for dark sector in RG.

2 hours ago, Mordred said:

Without referring to any of your links or articles ( post the math here)

Answer the following.

Why do you use the \[8\pi r^2\] relation

If you are not using the volume comparision between 2 spheres.

Ok I will copy and paste and convert the formatting of math for you, but you have to explain what difficulties are you facing when clicking on to direct link like this?: https://willrg.com/documents/WILL_RG_I.pdf#sec:density the same derivation from this link Ill provide below:

Translating RG (2D) to Conventional Density (3D).

In RG [math]\kappa^2[/math] is the 2D parameter defined in the relational carrier [math]S^2[/math]. In conventional physics, the source term is volumetric density [math]\rho[/math], a 3D concept defined by the "cultural artifact" (a Newtonian "cannonball" model) of mass-per-volume.

To bridge our 2D theory with 3D empirical data, we must create a "translation interface". We do this by explicitly adopting the conventional (Newtonian) definition of density, [math]\rho \propto m_0/r^3[/math], as our "translation target".

From the projective analysis established in the previous sections:

[math]\kappa^2 = \frac{R_s}{r}[/math]

where [math]\kappa[/math] emerges from the energy projection on the area of unit sphere [math]S^2[/math], and [math]R_s = \frac{2Gm_0}{c^2}[/math] links to the mass scale factor [math]m_0 = \frac{E_0}{c^2}[/math].

This leads to mass definition:

[math]m_0 = \frac{\kappa^2 c^2 r}{2G}[/math]

To translate this into a volumetric density, we first adopt the conventional 3D (volumetric) proxy, [math]r^3[/math]. This is not a postulate of RG, but the first step in applying the legacy (3D) definition of density:

[math]\frac{m_0}{r^3} = \frac{\kappa^2 c^2}{2G r^2}[/math]

This expression, however, is incomplete. Our [math]\kappa^2[/math] "lives" on the 2D surface [math]S^2[/math] (which corresponds to [math]4\pi[/math]), while the [math]r^3[/math] proxy implicitly assumes a 3D volume. To correctly normalize the 2D parameter [math]\kappa^2[/math] against the 3D volume, we must apply the geometric normalization factor of the [math]S^2[/math] carrier by dividing by the area of the sphere, which introduces the factor [math]\frac{1}{4\pi}[/math].

This normalization is the necessary geometric step to interface the 2D relational carrier ([math]S^2[/math]) with the 3D legacy definition of density:

[math]\rho = \frac{1}{4\pi}( \frac{\kappa^2 c^2}{2G r^2} )[/math]

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

[math]\text{Local Density} \equiv \text{Relational Projection}[/math]

Maximal Density.

At [math]\kappa^2 = 1[/math] (the horizon condition (for non rotating systems), [math]r=R_s[/math]), this density reaches its natural bound, [math]\rho_{\max}[/math], which is derived purely from geometry:

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

Normalized Relation.

Thus, our "translation" reveals an identity: the geometric projection [math]\kappa^2[/math] is simply the ratio of density to the maximal density:

[math]\kappa^2 = \frac{\rho}{\rho_{\max}} \rightarrow \kappa^2 \equiv \Omega[/math]

Self-Consistency Requirement

The mass scale factor can be expressed in two equivalent ways.

From the geometric definition:

[math]m_0 = \frac{\kappa^2 c^2 r}{2G}[/math]

From the energy density:

[math]m_0 = \alpha r^n \rho[/math]

Substituting [math]\rho = \frac{\kappa^2 c^2}{8\pi G r^2}[/math] into [math]m_0 = \alpha r^n \rho[/math] gives:

[math]m_0 = \frac{\alpha \kappa^2 c^2 r^{n-2}}{8\pi G}[/math]

Equating the two forms:

[math]\frac{\alpha r^{n-2}}{8\pi} = \frac{r}{2}[/math]

For the mass [math]m_0[/math] to remain a constant independent of the measurement scale [math]r[/math], the exponent must be [math]n=3[/math], yielding [math]\alpha=4\pi[/math]. Hence:

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

which closes the consistency loop between the geometric and density-based formulations.

You have switched from 4\pi r^2 to 8 \ pi r^2 in the above one term to the next without explaining why.

\[ A= 4 \pi r^2\] is the area of a sphere.

The question I asked why are you using that term if your not using spheres.

Edited 19 hours ago19 hr by Mordred

6 hours ago, joigus said:

I'm surprised by this, as ever since 1918, we know energy to be a derived quantity, not a defined one, and it's based on the Lagrangian.

If the dynamical system is amenable to a Lagrangian formulation, and if that Lagrangian does not explicitly depend on time, then there is an energy, and the different expressions can be obtained with a precise recipe.

So it's been a while since we know all these "mainstream definitions" are nothing but the corollaries of a master theorem.

Welcome back @joigus !
I will assume that you came not just to drop a comment but also to engage in conversation. So Ill gladly provide you with transparent derivations of Lagrangian and Hamiltonian. Next in line is Newtons third law you can find here https://willrg.com/documents/WILL_RG_I.pdf#thm:third_law

We now demonstrate that the familiar Lagrangian and Hamiltonian formalisms arise as limiting cases of the two-point relational Energy-Symmetry Law. Specifically, they emerge when the relational structure between two distinct observers A and B is collapsed into a single-point local description. This collapse preserves computational utility but reduces the ontological transparency of the underlying relational structure.

Definitions of Parameters

We consider a central mass [math]M[/math] and a test mass [math]m[/math]. The state of the test mass is described in polar coordinates

[math](r,\phi)[/math] relative to the central mass.

* [math]r_A[/math] --- reference radius associated with observer A (e.g., planetary surface).

* [math]r_B[/math] --- orbital radius of the test mass [math]m[/math] (position of observer B).

* [math]v_B^2 = \dot r_B^2 + r_B^2 \dot\phi^2[/math] --- total squared orbital speed at B.

* [math]\beta_B^2 = v_B^2/c^2[/math] --- dimensionless kinematic projection at B.

* [math]\kappa_A^2 = 2GM/(r_A c^2)[/math] --- dimensionless potential projection defined at A.

The Relational Lagrangian

Instead of a relational energy, we define the clean relational Lagrangian [math]L_{rel}[/math], which represents the kinetic budget at point B relative to the potential budget at point A:

[math]L_{rel} = T(B) + U(A) = \frac{1}{2} m(\dot r_B^2 + r_B^2 \dot\phi^2) + \frac{GMm}{r_A}[/math]

In dimensionless form, using the rest energy [math]E_0 = mc^2[/math], this is:

[math]\frac{L_{rel}}{E_0} = \frac{1}{2}(\beta_B^2 + \kappa_A^2)[/math]

This two-point, relational form is the clean geometric statement.

First Ontological Collapse: The Newtonian Lagrangian

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

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

This is precisely the standard Newtonian Lagrangian. Its origin is not fundamental but arises from the collapse of the two-point relational Energy Symmetry law into a one-point formalism.

Second Ontological Collapse: The Hamiltonian

Introducing canonical momenta,

[math]p_r = \frac{\partial L}{\partial \dot r} = m\dot r[/math]

[math]p_\phi = \frac{\partial L}{\partial \dot\phi} = mr^2\dot\phi[/math]

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

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

Interpretation

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

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

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

Here the "+" or "-" signs do not come from [math]\kappa^2[/math] itself, which is always positive, but from the ontological collapse of the two-point relational energy law into a single-point formalism. In WILL, both projections are clean and positive; in standard mechanics, the apparent sign difference arises only after this collapse.

Both the Lagrangian and Hamiltonian thus emerge from the same relational Energy-Symmetry Law under the identification

[math]r_A = r_B = r[/math].

The apparent sign difference between [math]L[/math] and [math]H[/math] is not a fundamental feature but an artifact of this single-point collapse: in the full two-point relational law, both projections [math]\beta^2[/math] and [math]\kappa^2[/math] are strictly positive.

The Lagrangian and Hamiltonian arise as single-point limiting cases of the two-point relational Energy-Symmetry Law

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

They remain computationally valid within their domain of applicability.

RG provides the more general two-point relational structure from which these formalisms can be systematically derived.

Edited 18 hours ago18 hr by Anton Rize

Explain in detail how you get Just and only just the \[4\ pi r^2\] nothing else.

Then once you do that maybe just maybe it might help others including myself what your doing to correlate an observable 3d universe without any time coordinate.

10 minutes ago, Mordred said:

You have switched from 4\pi r^2 to 8 \ pi r^2 in the above one term to the next without explaining why.

A=4πr2

is the area of a sphere.

The question I asked why are you using that term if your not using spheres.

What do you mean " without explaining why" ?

"To translate this into a volumetric density, we first adopt the conventional 3D (volumetric) proxy, [math]r^3[/math]. This is not a postulate of RG, but the first step in applying the legacy (3D) definition of density:

[math]\frac{m_0}{r^3} = \frac{\kappa^2 c^2}{2G r^2}[/math]

This expression, however, is incomplete. Our [math]\kappa^2[/math] "lives" on the 2D surface [math]S^2[/math] (which corresponds to [math]4\pi[/math]), while the [math]r^3[/math] proxy implicitly assumes a 3D volume.

To correctly normalize the 2D parameter [math]\kappa^2[/math] against the 3D volume, we must apply the geometric normalization factor of the [math]S^2[/math] carrier by dividing by the area of the sphere, which introduces the factor [math]\frac{1}{4\pi}[/math].

This normalization is the necessary geometric step to interface the 2D relational carrier ([math]S^2[/math]) with the 3D legacy definition of density:

[math]\rho = \frac{1}{4\pi}( \frac{\kappa^2 c^2}{2G r^2} )[/math]

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

[math]\text{Local Density} \equiv \text{Relational Projection}[/math]"

Please specify what exactly seems unclear to you?

Its a simple geometry question nothing more

\[ 4\ pi r^2\]

Mathematically describes the area of a sphere.

Edited 18 hours ago18 hr by Mordred

6 minutes ago, Mordred said:

Its a simple geometry question nothing more

4 pir2

Mathematically describes the area of a sphere.

Sorry Im not following where you leading with this. Can you elaborate please?

Oh my please tell me you know basic geometry relations.

One sphere

\[A=4\pi r^2\]

2 spheres

\[A=8\pi r^2\]

One singular geometric object with \{8\pi r^2\] is a cylinder with 4 times the radius in height.

Now look at the mathematical proof I posted again.

You have the above relations throughout all your documents apparently without realizing those terms automatically describe geometric objects that are not a circle.

16 minutes ago, Mordred said:

Oh my please tell me you know basic geometry relations.

One sphere

A=4πr2

2 spheres

A=8πr2

One singular geometric object with \{8\pi r^2\] is a cylinder with 4 times the radius in height.

Now look at the mathematical proof I posted again.

You have the above relations throughout all your documents apparently without realizing those terms automatically describe geometric objects that are not a circle.

There is no single geometric object with an area of [math]8\pi r^2[/math]. The number 8 is simply the result of multiplying 4 by 2.

Please look closely at the two lines of the derivation I posted:

Line 1: [math]\rho = \frac{1}{4\pi} \left( \frac{\kappa^2 c^2}{2 G r^2} \right)[/math]

Line 2: [math]\rho = \frac{\kappa^2 c^2}{8\pi G r^2}[/math]

The [math]4\pi[/math] in the denominator is the normalization factor from the [math]S^2[/math] carrier area.

The [math]2G[/math] in the denominator comes from the mass scale factor ([math]R_s = 2GM/c^2[/math]).

When you multiply the denominators, [math]4\pi \times 2G = 8\pi G[/math].

I am not "switching" terms. I am executing basic algebraic multiplication. Does this clarify the geometry for you?

I know it's 2 spheres I stated that above.

In application to my proof area then compared to area now to determine the Hubble constant.

The surface area of both objects are 3 dimensional spheres that is the area being described by whichever radius its at.

That is not a circle

( you need 3 dimensions to define area.)

Would you like to know the math relations the very term ( orthogonal ) tells me.

One projection in a 90 degree relation to the other

\[ A \cdot B\]

I have no identity of what A or B has.

But I can automatically apply the Kronecker Delta to that inner product ( Hilbert Space).

Regardless if it's coordinates or not...

( if both A and B are finite and continous) with regards to Hilbert

Edited 18 hours ago18 hr by Mordred

3 hours ago, MJ kihara said:

That's is the definition of dark matter according to you,therefore, it ceases being dark...by explaining it's nature, confirming it's existence, it no longer become dark...I think the issue not by it being called dark...it's just an indication of physics at that particular era not being able to explain the source of ' missing mass'

The fact that 'dark matter and dark energy ' makes up almost 95% of the universe is not a small issue.

According to your formulation how does matter or rather how does things acquire mass?

Also from your definition above,how do you explain the discrepancy between rotation velocity in the outer of the Galaxy and the inner region? Does gravitational wave/force move at/transmitted at speed of light.?

You asked exactly the right question: "how do you explain the discrepancy between rotation velocity in the outer of the Galaxy and the inner region?"

In standard cosmology, because the Newtonian velocity drops off as
[math]1/\sqrt{r}[/math], they have to invent a halo of invisible mass (Dark Matter) that grows with [math]r[/math] to keep the rotation curve flat.

In WILL Relational Geometry, the explanation requires zero invisible particles and no adjustable parameters.
It is a direct consequence of SPACETIME ≡ ENERGY.

I will show it in 4 steps:

Step 1 The Necessity of Global Resonance

Step 2 Deriving [math]H_0[/math] from CMB Temperature and [math]\alpha[/math]

Step 3 The Resonant Horizon Interference

Step 4 Derivation of the Baryonic Tully-Fisher Relation

All this leads us to conclusion:
The discrepancy in the outer regions of the galaxy is not caused by "missing mass." It is the physical manifestation of the star hitting the energetic floor (the Fundamental Tone) supported by the global resonance of the Universe. It cannot slow down any further because it is coupled to the horizon.

To make sure that everyone could test it against empirical data I developed this page https://willrg.com/Galactic_Dynamics/
Its an interactive LAB where you can calculate rotation speed of any galaxy in SPARC database (175 total). This page development was a peace of work I have to admit. Hope you will find it useful.


Also here's RAR graph:

Radial Acceleration Relation (RAR) for 175 SPARC galaxies. The green dots shows the density of [math]>3000[/math] individual data points. The cyan line represents the WILL Resonance Interference prediction
([math]g_{obs} = g_{bar} + \sqrt{g_{bar}a_{\kappa}}[/math]) using the [math]H_0[/math] value derived from CMB thermodynamics. The remarkable agreement ([math]RMSE \approx 0.065[/math] dex) without free parameters strongly suggests that galactic dynamics are regulated by the global horizon.

Colab notebook link: [RAR test.ipynb](https://github.com/AntonRize/WILL/blob/8967a5fac8a7a42433e9cc66fffbbbbc4e18b6a1/Colab_Notebooks/RAR_test.ipynb)

---

And here's the wide binary stars graph based on Gaia DR3 catalog where we comparing WILL and MOND:

The Kinetic Resonance Test. The plot compares the gravity boost factor as a function of Newtonian acceleration.

Blue dashed line: Standard MOND prediction ([math]a_0 = 1.2 \times 10^{-10} m/s^2[/math]), which systematically overestimates the anomaly.

Red solid line: WILL RG Kinetic Resonance prediction
([math]a_{\beta} = cH_0/6\pi \approx 0.35 \times 10^{-10} m/s^2[/math]) passes precisely through the observational data points, matches the reported trend of Wide Binary observations without any parameter fitting.

Colab notebook link: [Wide binary Chae 2023.ipynb](https://github.com/AntonRize/WILL/blob/8967a5fac8a7a42433e9cc66fffbbbbc4e18b6a1/Colab_Notebooks/Wide_binary_Chae_2023.ipynb)

---

You can find all my derivations and all my python scripts on this page: https://willrg.com/results/

Edited 17 hours ago17 hr by Anton Rize

So can't stick to simply the geometry relations.

Graph \[4\ pi r^2\] pick any radius.

What shape do to you get.

Does or does not your S^2 group produce the same graph.

Plain and simple

1 hour ago, Anton Rize said:

Lagrangian Lrel, which represents the kinetic budget at point B relative to the potential budget at point A:

Lrel=T(B)+U(A)

What is,if your are not paraphrasing Hamiltonian.

1 hour ago, Anton Rize said:

Discoveries like Neptune were not made because the model was "flexible." They were made precisely because classical mechanics was rigid. When the orbit of Uranus violated predictions, Le Verrier did not have the "flexibility" to alter Newton's gravitational constant or invent a new force distance parameter. Because the model's constraints were absolute, the anomaly strictly demanded a localized physical explanation. That rigidity is exactly what pointed the telescope at Neptune

There is a place in the thread you used the term 'crystallization'..... rigidity after those ideas crystallize..am sorry for this shallow explanation....remember in physics they are two major fields classical mechanics and quantum mechanics.

1 hour ago, Anton Rize said:

I'm keeping everything dimensionless at all times and units used only when its absolutely necessary.

The one you deem necessary were derived from somewhere.

1 hour ago, Anton Rize said:

because it has zero flexibility, any future observation that deviates from its predictions cannot simply be "patched" with a new parameter

''Any future observation" that is the main reason why I said your model seems to give answers to already established facts...it can't predict what is made up of dark matter but it will end up giving answers as if dark matter doesn't exist....

9 minutes ago, Anton Rize said:

The discrepancy in the outer regions of the galaxy is not caused by "missing mass." It is the physical manifestation of the star hitting the energetic floor (the Fundamental Tone) supported by the global resonance of the Universe. It cannot slow down any further because it is coupled to the horizon.

Haaaaa I like this explanation🙏....the reason I say answers without because of, where from?

Again how do things acquire mass?

Important note :we have been trying to follow your language...please try to answer mainstream questions as presented by @ Mordred in a precise way so that we can follow your understanding,I salute him for being the residential expert being patient with all this.

Take your time to answer in a format close to mainstream way... don't panic.

12 minutes ago, Mordred said:

So can't stick to simply the geometry relations.

Graph

4 pir2

pick any radius.

What shape do to you get.

Does or does not your S^2 group produce the same graph.

Plain and simple

I notice you completely ignored the empirical data, the SPARC RAR graph, and the Wide Binary tests demonstrating 0-parameter accuracy, choosing instead to retreat to elementary geometry. But I will humor your question. Also you still ignoring my goal pole question. You have to admit it not to me but to yourself: If your believes unfalsifiable then they are NOT scientific.

---

Yes, plain and simple: if you graph [math]y = 4\pi r^2[/math] against [math]r[/math], you get a parabola. Yes, the topological [math]S^2[/math] carrier possesses the exact metric area of
[math]4\pi r^2[/math].

If you leading me to: "Since it is a 3D sphere, it must enclose a volume of [math]\frac{4}{3}\pi r^3[/math], therefore your density equation is missing the [math]1/3[/math] factor!"

Let me stop you right there and refer you back to the ontology you refuse to read.

In classical Euclidean space, a sphere is a boundary enclosing a physical volumetric void that gets filled with a fluid.

In WILL Relational Geometry, the [math]S^2[/math] carrier is not a container; it is the geometric capacity of the potential projection itself. The energy state is the surface projection ([math]\kappa^2[/math]). There is no independent "inside volume" to integrate over.

That is exactly why the algebraic closure in my model is strictly [math]m_0 = 4\pi r^3 \rho[/math], without Newtonian [math]1/3[/math] fluid coefficient.

Now that I have answered your geometry question, are you going to address my goal pole question and the physical rotation curves + the Wide Binary data I just provided, or are we going to continue plotting parabolas?

Im applying strictly mathematical rules in my line of questioning that has nothing to do with physics.

You do realize the very fact that

\[4\pi r^2\] can be graphed it is in effect graphical coordinates?

Edited 14 hours ago14 hr by Mordred

2 hours ago, MJ kihara said:

What is,if your are not paraphrasing Hamiltonian.

There is a place in the thread you used the term 'crystallization'..... rigidity after those ideas crystallize..am sorry for this shallow explanation....remember in physics they are two major fields classical mechanics and quantum mechanics.

The one you deem necessary were derived from somewhere.

''Any future observation" that is the main reason why I said your model seems to give answers to already established facts...it can't predict what is made up of dark matter but it will end up giving answers as if dark matter doesn't exist....

Haaaaa I like this explanation🙏....the reason I say answers without because of, where from?

Again how do things acquire mass?

Important note :we have been trying to follow your language...please try to answer mainstream questions as presented by @ Mordred in a precise way so that we can follow your understanding,I salute him for being the residential expert being patient with all this.

Take your time to answer in a format close to mainstream way... don't panic.

I appreciate your honesty, and I understand why the dynamic of this conversation might look confusing from the outside. Let me address your points calmly and systematically. There is no panic here - only the rigorous execution of a geometric proof.

1. On the Hamiltonian:

The difference is fundamental. A standard Hamiltonian describes the total energy of a single, isolated state at one point in space. My equation is a two-point relational difference. It describes the kinetic budget at an orbital point (B) relative to the potential budget at a central point (A). The standard Hamiltonian only appears when you collapse those two distinct points into one
([math]r_A = r_B[/math]). I am not paraphrasing the Hamiltonian; I am showing the deeper geometric architecture it is derived from.

2. On "Predicting the Unknown":

Please look closely at the Wide Binary Stars graph I posted and the section https://willrg.com/documents/WILL_RG_II.pdf#sec:wide-binary. The standard "flexible" model (MOND) failed to predict that trajectory. WILL RG predicted that exact curve with zero parameter fitting. Eliminating a 40-year-old hallucination (Dark Matter) by mathematically proving it is just the tension of the macroscopic horizon is a novel, testable prediction. Have a closer look at this web page https://willrg.com/Galactic_Dynamics/ this is 175 unique testable predictions. Also take a look at this section
https://willrg.com/documents/WILL_RG_I.pdf#sec:M_sin(i) this is a completely novel method achieving predictions that standard approach consider mathematically impossible. Right now Im going through astronomical databases in order to make predictions about 3D parametrisation (orbital shape and angle relative to us) using only spectroscopy data before the release of astrometric data. This line of unique predictions we will be able to test in the near future with new astronomical data releases. The list can go on...

3. "How do things acquire mass?"

Short answer: I don't know. But I'm working on it.

4. On answering "Mainstream Questions" from Mordred:

I salute your desire for clarity. However, when Mordred asks me to explain the volume of a 3D sphere, he is asking me to defend a Newtonian container model that my theory explicitly mathematically rejects. It is like asking a quantum physicist to explain electron orbits using gears and springs. I am answering his questions directly, but I will not adopt a broken ontological framework just to make the answers sound "mainstream."

I am glad the "energetic floor" concept resonated with you. That is exactly what happens when you let the geometry guide the physics, rather than forcing the geometry to fit the mainstream.

1 hour ago, Mordred said:

Im applying strictly mathematical rules in my line of questioning that has nothing to do with physics.

You do realize the very fact that

4πr2

can be graphed it is in effect graphical coordinates?

Thank you for explicitly confirming that your line of questioning "has nothing to do with physics."

This perfectly explains why you are completely ignoring the empirical galactic rotation curves, the Wide Binary data, and the Baryonic Tully-Fisher derivation I provided. You have abandoned the physical territory because you cannot contest the results.

Regarding your mathematical claim: the fact that a function [math]f(r) = 4\pi r^2[/math] can be plotted on a 2D graph does not make the expression itself "graphical coordinates." You can graph temperature versus time on a piece of paper; that does not turn temperature into a spatial dimension. Conflating a mathematical plot with an ontological coordinate space is a severe category error.

Since you have openly admitted your questions have nothing to do with physics, and you continue to refuse to state your falsifiability criteria...

Look... it's just... I'm so disappointed. I was sincerely hoping that I could talk with you about these results. And you have to admit they are far from trivial. I wanted to discuss weak and strong points, potential implications, agreement with other physics domains, ways to empirically test it etc... But instead you are completely ignoring my questions and my derivations and trying to quiz me on high school geometry. Really? Look, if high school homework problems are what you are after - there are subforums on this website dedicated to them specifically.

Edited 13 hours ago13 hr by Anton Rize

Its too bad you do not understand your mathematics do not match what you verbally describe as your ontology.

S^2 You show as a 2 dimensional plane.

Which does not produce a 3 dimensional sphere.

You lectured me on my proof of critical density because it used spheres yet your very own mathematics also produces spheres.

However in case you never noticed not once did that 3 show up in the geometry mapping generated by

\[ 4\pi r^2\].

I also asked previous " where is the pressure"

Would you like to know why pressure to energy density has a 1 to 3 ratio ?

\[ \rho=\frac{3p}{c^2\]

Via stress energy momentum tensor

\[T_{00}, T_{i,j}\]

Pressure has 3 specific mathematical relations to account for. Engineers will love this part being described.

Edited 10 hours ago10 hr by Mordred

27 minutes ago, Mordred said:

I also asked previous " where is the pressure"

Yes you did and I provided you with an unswear that you once again completely ignored. You asking questions and then ignoring the answers. Is that the type of communication you want to be a part of?

33 minutes ago, Mordred said:

Would you like to know why pressure to energy density has a 1 to 3 ratio ?

Since you want to talk about equation of state - yes please tell me how I deriving the pressure term and what w=? I end up and why?
And this time please try not to make your typical category error where you assuming the Universe is a container filled with fluid. Are you capable?