3 hours ago, Anton Rize said:

Thus, the factor [math]\sqrt{3}[/math] is the signature of a holistic observation of a closed system from the outside (Galactic Scale), while Newtonian dynamics represents the differential observation from the inside (Local Scale).

One isn't going to have a [math]\sqrt{3}[/math] factor difference between observations made from outside and observations made from inside of a non-relativistic system.

It is my understanding that it is the virial theorem that indicates the existence of dark matter in galaxies.

6 hours ago, Anton Rize said:

"Cherry-picking" means selecting only data that fits. Using every single data point available is the exact opposite.

But you did the former, and not the latter, when you posted “Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.” which is what I was referring to.

“many like it” ≠ all

IOW, you basically said your model is a great fit for a subset of the data, which is cherry-picking.

“Yes, for some dense galaxies (like NGC0801), there is indeed an overshoot at the bulge. This suggests that the transition from "Newtonian" (center) to "Relational/Dark" (outskirts) dynamics might depend on the local potential depth (the "internal observer" effect I mentioned in the paper that Ill link below”

You don’t explain why this “internal observer effect” only happens for some observations. Aren’t all of our observations subject to this? Seems to me you are just substituting a fudge factor so you can say there’s no dark matter.

“Thus, the factor –√ 3 is the signature of a holistic observation of a closed system from the outside (Galactic Scale), while Newtonian dynamics represents the differential observation from the inside (Local Scale).”

How can a constant factor arise in galaxies that could vary widely in their gravitational potentials? If it’s because of our own potential, where does that come from? I don’t see where our orbital parameters show up in your model. Can you show a detailed calculation of this?

***CORRECTION***

This post is a correction of the following text in the post about circular orbits of the Schwarzschild metric:

On 11/17/2025 at 3:42 PM, KJW said:

[math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math]

...

[math]\text{For:}\ \ \ r = R\ \ \ ;\ \ \ dr = 0\ \ \ ;\ \ \ \theta = \dfrac{\pi}{2}\ \ \ ;\ \ \ d\theta = 0\ \ \ ;\ \ \ \varphi = 0\ \ \ ;\ \ \ d\varphi = 0[/math]

[math](ds)^2 = \Big(c^2 - \dfrac{2GM}{R} - \omega^2 R^2\Big) (dt)^2[/math]

[math]-a_r = \dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = \dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r}[/math]

[math]= \dfrac{\dfrac{GM}{R^2} - \omega^2 R}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}[/math]

The above contains an error of sorts. I mishandled the [math]r[/math] and [math]\theta[/math] coordinate variables. Specifically, I substituted the location of the object in the rotating coordinate system before obtaining the partial derivatives of [math]g_{tt}[/math]. Anticipating the correct result for [math]a_r[/math] (and [math]a_\theta[/math]) of the object, the error was confined to the above and did not affect the remainder of the post.

Below is obtained the acceleration components [math]a_r[/math] and [math]a_\theta[/math] for an arbitrarily located stationary object in the rotating coordinate system. Only then is the particular location of the object substituted.

[math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math]

[math]g_{tt} = c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta[/math]

[math]a_r = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r} = \dfrac{\omega^2 r \sin^2\theta - \dfrac{GM}{r^2}}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math]

[math]= \dfrac{\omega^2 R - \dfrac{GM}{R^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math]

[math]a_\theta = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial \theta} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial \theta} = \dfrac{\omega^2 r^2 \sin\theta \cos\theta}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math]

[math]= 0\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math]

Edited November 20Nov 20 by KJW

3 hours ago, swansont said:

But you did the former, and not the latter, when you posted “Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.” which is what I was referring to.

“many like it” ≠ all

You have now twice accused me of "cherry-picking" data. This is a direct accusation of scientific misconduct. Since you represent the administration of this forum, I expect you to adhere to the standards of evidence you demand from others.

Your accusation is factually false, and here is the proof:

1. Definition of Cherry-Picking: Selecting only favorable data while suppressing unfavorable data.

2. My Action: In the post you criticised, I explicitly presented NGC0801 as a case where the model deviates (overshoots at the bulge):
   

   22 hours ago, Anton Rize said:

   This is a physically sound intuition based on the "Maximum Disk" hypothesis, assuming galaxy centers are always baryon-dominated ([math]V_{obs} \approx V_{bary}[/math]). If this were universally true, a uniform scaling of [math]\sqrt{3} \approx 1.73[/math] would systematically overshoot the centers.

   However, the data shows something unexpected.

   I invite you to look at the actual SPARC profiles using the open visualizer I built for this verification: https://antonrize.github.io/WILL/calculator/

   Case 1: The Counter-Example (Low Surface Brightness)
   
   

   Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.

   My parameter-free prediction [math]V = \sqrt{3} V_{bary}[/math] tracks the observed data perfectly from [math]r \to 0[/math] outwards.

   If your intuition were universally correct, I should see a massive overshoot here. I do not.

   Case 2: The Mixed Bag (High Surface Brightness)
   

   Yes, for some dense galaxies (like NGC0801), there is indeed an overshoot at the bulge. This suggests that the transition from "Newtonian" (center) to "Relational/Dark" (outskirts) dynamics might depend on the local potential depth (the "internal observer" effect I mentioned in the paper that Ill link below).

   The Verdict:

   If my formula were systematically wrong at small [math]r[/math], the global Median RMSE would be inflated by these "center errors" across the board. The fact that the Global Median RMSE is only 20.23 km/s proves that for a significant portion of the dataset, the geometric relation [math]V = \sqrt{3} V_{bary}[/math] holds surprisingly well even at small radii.

   I am not "guessing" the shape. I am reporting that the geometric factor [math]\sqrt{3}[/math] fits the data of diverse galactic morphologies better than the standard assumption that "baryons must dominate the center" or that magic invisible "dark matter" is a real thing.
   
   
   

   
   
   
   3. Your Action: You ignored my inclusion of the "bad" result (NGC0801), quoted only the "good" result (IC2574), and then accused me of selecting only data that fits:
   

   4 hours ago, swansont said:

   “Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.” which is what I was referring to.

Do you realize the irony? To construct your accusation that I am cherry-picking data, you had to cherry-pick my quote, deliberately cutting out the paragraph where I highlighted the model's limitations.

I presented a Global Median RMSE for 175 galaxies - the entire database.
I presented links to all my python scripts and datasets showing complete transparency:

23 hours ago, Anton Rize said:

You welcome to test it yourself. All my google colab notebooks you can find here: https://antonrize.github.io/WILL/predictions/

23 hours ago, Anton Rize said:

You can download .pdf with all the details here: https://antonrize.github.io/WILL/results/

23 hours ago, Anton Rize said:

I invite you to look at the actual SPARC profiles using the open visualizer I built for this verification: https://antonrize.github.io/WILL/calculator/

I presented specific counter-examples where the model struggles.


To accuse an author of "hiding bad data" immediately after they explicitly presented that bad data is not just wrong; it is a gross misrepresentation of the discussion.

I am here to defend the model's mathematics, but I will also defend my integrity. I expect you to either substantiate where I hid data or retract the accusation.

8 hours ago, Anton Rize said:

You have now twice accused me of "cherry-picking" data

I have not.

I pointed out an instance of cherry-picking and then added discussion. You can’t legitimately count these as unique events.

I presented a Global Median RMSE for 175 galaxies - the entire database.

The fact that you admit your model doesn’t work well with a certain subset of the data should be enough to show that using RMSE is not a good measure of its quality. It’s why there’s a joke about the statistician with their feet in a tub of ice water and head in an oven says, “On average, I feel great!”

I presented links to all my python scripts and datasets showing complete transparency:

Nobody is required to go to external links, per rule 2.7

Do you realize the irony? To construct your accusation that I am cherry-picking data, you had to cherry-pick my quote, deliberately cutting out the paragraph where I highlighted the model's limitations.

The conclusion from this is that the model is flawed, but you haven’t admitted that.

You’re also neglecting to respond to the other issues raised.

On 11/21/2025 at 12:26 AM, swansont said:

  On 11/20/2025 at 3:52 PM, Anton Rize said:

You have now twice accused me of "cherry-picking" data

I have not.

1st time you explicitly wrote:

On 11/19/2025 at 9:23 PM, swansont said:

That tends to happen when you cherry-pick data.

2nd time you reinforced the accusation by treating cherry-picking as a methodological issue.:

On 11/20/2025 at 11:22 AM, swansont said:

IOW, you basically said your model is a great fit for a subset of the data, which is cherry-picking.

You cannot simultaneously say:

1. “I pointed out an instance of cherry-picking”
   and

2. “I have not accused you of cherry-picking.”

These positions contradict each other.
The evidence above shows that your statements contradict your own earlier claims.

---

Misrepresentation of my RMSE analysis

You wrote:

On 11/21/2025 at 12:26 AM, swansont said:

The fact that you admit your model doesn’t work well with a certain subset of the data should be enough to show that using RMSE is not a good measure

This is an invented premise.
I did not “admit” that the model “doesn’t work well”.
I explicitly showed:

• the global median RMSE over 175 galaxies,

• the distribution,

• identification of outliers,

• and the physical reason for the residuals (internal vs external observer geometry).

You ignored this and replaced it with an interpretation I never wrote.

That is not a rebuttal - it is quote-mining.

---

You stated RMSE might mask shape mismatches:

On 11/18/2025 at 7:21 AM, swansont said:

I’d like to see the derivation, and also the justification that RMSE is the appropriate standard to apply.

I addressed this directly with an alternative statistical test.

Since you raised the issue of RMSE potentially masking profile differences, I performed an independent Reduced Chi-Squared analysis (χ²_ν) on the full SPARC dataset.

This was done specifically to answer your concern.

Here are the results you are ignoring — FULL χ² ANALYSIS (175 galaxies) —:

On 11/19/2025 at 4:41 PM, Anton Rize said:

I have completed the full statistical analysis to address your concerns about the metric and the fit quality.

You rightly pointed out that RMSE can mask shape mismatches, so I ran the test using Reduced Chi-Squared ([math]\chi_\nu^2[/math]).

The results reveal exactly what is happening physically. I compared my strict Zero-Parameter geometric model against a minimal One-Parameter variation (standard astrophysical practice).

--- COMPARISON OF PREDICTIVE POWER (SPARC, 175 Galaxies) ---

1. FIXED QWILL (0 Free Parameters):

Constraint: Fixed Global [math]\Upsilon_* = 0.66[/math]

Law: [math]V = \sqrt{3} V_{bary}[/math]

Median [math]\chi_\nu^2[/math]: 34.47

Median RMSE: 20.23 km/s

2. TUNED QWILL (1 Free Parameter per galaxy):

Constraint: [math]\Upsilon_*[/math] allowed to vary (representing stellar population differences)

Law: [math]V = \sqrt{3} V_{bary}[/math]

Median [math]\chi_\nu^2[/math]: 6.52 <-- THE SIGNAL

Median RMSE: 11.62 km/s

The Smoking Gun:

The massive drop in [math]\chi_\nu^2[/math] (from ~34 down to ~6.5) when allowing just one degree of freedom (mass-to-light ratio) proves that the "shape problem" you suspected is not intrinsic to the geometric law [math]\sqrt{3}[/math].

If the geometric law were wrong (e.g., wrong shape at small [math]r[/math]), adjusting the amplitude [math]\Upsilon_*[/math] would NOT fix the [math]\chi^2[/math] so dramatically. The fact that it [i]does[/i] drop to near-acceptable levels implies that the geometric profile is correct, and the residuals in the Fixed Model are dominated purely by astrophysical scatter (old vs. young stellar populations).

Context on Complexity:

Standard Dark Matter halo models typically employ 3 free parameters per galaxy (halo scale, density, plus [math]\Upsilon_*[/math]) to achieve [math]\chi_\nu^2 \approx 1[/math].

WILL RG achieves [math]\chi_\nu^2 \approx 6.5[/math] and RMSE [math]\approx 11[/math] km/s with only 1 parameter.

Conclusion:

The fact that a parameter-free geometric law performs comparably to tuned Dark Matter models suggests that the $\sqrt{3}$ factor captures the fundamental driver of galactic dynamics, while astrophysical variations account for the residuals. The [math]\sqrt{3}[/math] factor potentially might replace the multi-parameter Dark Matter halo. The remaining deviation is just standard astrophysics.

You raised a concern.
I ran the more rigorous test.
I reported the results.
You ignored them.
This is inappropriate and scientifically non-compliant behaviour.

---

You wrote:

On 11/21/2025 at 12:26 AM, swansont said:

Nobody is required to go to external links, per rule 2.7

This does not justify ignoring the evidence.

If you choose not to inspect the scripts and datasets, that is fine - but then you cannot make methodological accusations whose truth depends precisely on the content you refuse to examine.

You cannot simultaneously:

• critique the analysis,

• and declare that you are not required to look at the analysis.

That is not a scientific position.

---

Deliberate removal of the paragraph where I described the limitations

In your earlier reply, you quoted a fragment of my message and removed the paragraph where I explicitly described the model’s limitations:

On 11/20/2025 at 11:22 AM, swansont said:

  On 11/20/2025 at 4:59 AM, Anton Rize said:

"Cherry-picking" means selecting only data that fits. Using every single data point available is the exact opposite.

But you did the former, and not the latter, when you posted “Look at galaxy IC2574 (and many LSB galaxies like it).

You used that truncated quote as the basis of your accusation.

This is exactly the behavior you attributed to me. I asked you to provide the evidence - you failed to do so. Your accusation is false but you haven’t admitted that.

---

"The conclusion is that the model is flawed." You wrote:

On 11/21/2025 at 12:26 AM, swansont said:

The conclusion from this is that the model is flawed, but you haven’t admitted that.

You haven’t shown where.

You have not:

• identified one incorrect equation,

• provided a contradicting datapoint,

• pointed out a mathematical error,

• or referenced any inconsistency with observations.

A conclusion without argumentation is not a scientific conclusion, simply an empty statement.

If the model is flawed, show the step.

---

You wrote:

On 11/21/2025 at 12:26 AM, swansont said:

You’re also neglecting to respond to the other issues raised.

This is false.

I replied to every concrete point you raised:

• IC2574

• NGC0801

• global RMSE distribution

• χ² analysis

• internal vs external observational setup

• full dataset transparency

• astrophysical scatter

• mass-to-light variation

• physical interpretation of the deviations

If you believe there is an unresolved issue, quote it directly.
General accusations are not actionable, they do not constitute actionable criticism.

---

Summary

Your reply contains:

• denial of your own prior accusation,

• misrepresentation of my statements,

• ignoring of the χ² analysis you requested,

• shifting the burden of proof,

• an unsupported claim that the model is “flawed”,

• and the contradictory stance that you can critique the analysis while refusing to examine the analysis.
  
  Instead of analysing presented results and be a part of productive scientific discussion you resort to personal attacks and false accusations. This directly contradicts the standards of scientific discussion.

If you wish to continue the discussion, the next steps are simple:

1. Retract your false accusations.
2. Present a specific empirical or mathematical inconsistency.

A claim of “flaw” without a demonstration of the flaw is not a scientific argument.


P.S. If the thread is closed without addressing the specific scientific points listed above, it will be objectively unclear which of them - if any - were considered incorrect.
For clarity and fairness, please identify the specific issue before taking any administrative action.

Edited November 25Nov 25 by Anton Rize

2 hours ago, Anton Rize said:

I replied to every concrete point you raised:

• IC2574

• NGC0801

• global RMSE distribution

• χ² analysis

• internal vs external observational setup

• full dataset transparency

• astrophysical scatter

• mass-to-light variation

• physical interpretation of the deviations

I don’t think I made these points. There are your responses. Chi-squared, for example, is still a curve-fitting test, but I don’t see where you answered my question about whether dark matter was included.

I don’t see where you addressed the issue of your internal observer effect only being applied to some observations.

2 hours ago, Anton Rize said:

You ignored this and replaced it with an interpretation I never wrote.

I tried to quote directly from your posts.

I did not “admit” that the model “doesn’t work well”.

“Yes, for some dense galaxies (like NGC0801), there is indeed an overshoot at the bulge.”

2 hours ago, swansont said:

Chi-squared, for example, is still a curve-fitting test

You didn’t accept RMSE; I then provided chi-squared. Now you dismiss χ² as well, without proposing any alternative statistical criterion. This shows that you are prepared to disregard any quantitative measure that does not support your expectations. That is exactly the evidence of methodological cherry-picking you falsely accused me of, and for which you have provided no evidence.

---

2 hours ago, swansont said:

but I don’t see where you answered my question about whether dark matter was included.

On 11/19/2025 at 4:41 PM, Anton Rize said:

  On 11/18/2025 at 7:21 AM, swansont said:

Also, if this calculation is including dark matter.

I didn't had to speculate any "dark" entities.

I already answered this explicitly. The fact that you now claim you “don’t see” this answer shows that you are not engaging with the responses you requested.

---

3 hours ago, swansont said:

I don’t see where you addressed the issue of your internal observer effect only being applied to some observations.

Here is another earlier answer you “don’t see”:

On 11/20/2025 at 4:59 AM, Anton Rize said:

  On 11/19/2025 at 9:23 PM, swansont said:

But for galaxies who have little DM at small r, the orbits are just Newtonian, which suggests that your theory doesn’t predict such orbits properly. What happens when you calculate the orbit of the earth about the sun using the same method?

Hypothesis: Internal vs. External Observation (The "Carousel" Effect)

• Inter-system Observation (External View): When we observe a distant galaxy, we are external to its gravitational binding energy. We are not part of its "system." Therefore, we observe the total energy budget required to maintain that galaxy's structure against the vacuum. We see both the kinetic motion ([math]\beta^2[/math]) and the structural tension ([math]\kappa^2[/math]) required for closure. [math]Q^2_{\mathrm{ext}} = \beta^2 + \kappa^2 = 3\beta^2 \quad \Longrightarrow \quad V = \sqrt{3} V_{\mathrm{bary}}[/math]

• Intra-system Observation (Internal View): When we observe the Solar System, we are embedded within the same gravitational potential well ([math]\kappa_{\mathrm{local}}[/math]) as the planets. We are, effectively, "riding the same carousel." The background potential [math]\kappa^2[/math] is a shared baseline for both the observer (Earth) and the target (Jupiter).

Potential Screening Principle: For an observer embedded within the system, the binding potential [math]\kappa^2[/math] acts as a common background frame, not as an observable kinematic difference. The relative measurement cancels out the structural tension, leaving only the kinetic differential:
[math]Q^2_{\mathrm{int}} \approx \beta^2 \quad \Longrightarrow \quad V \approx V_{\mathrm{bary}}[/math]

Thus, the factor [math]\sqrt{3}[/math] is the signature of a holistic observation of a closed system from the outside (Galactic Scale), while Newtonian dynamics represents the differential observation from the inside (Local Scale).

This is a direct answer to your question about internal vs external observations.
Given this, it is difficult to understand how you can still claim that the issue was “not addressed”.

---

3 hours ago, swansont said:

I tried to quote directly from your posts.

Ill help you recover the full context. Here it is:

On 11/19/2025 at 4:41 PM, Anton Rize said:

Case 1: The Counter-Example (Low Surface Brightness)

Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.

My parameter-free prediction [math]V = \sqrt{3} V_{bary}[/math] tracks the observed data perfectly from [math]r \to 0[/math] outwards.

If your intuition were universally correct, I should see a massive overshoot here. I do not.

Case 2: The Mixed Bag (High Surface Brightness)

Yes, for some dense galaxies (like NGC0801), there is indeed an overshoot at the bulge. This suggests that the transition from "Newtonian" (center) to "Relational/Dark" (outskirts) dynamics might depend on the local potential depth

Now when we have full context above lets have a look how you cherry-picked my quote:

3 hours ago, swansont said:

  Quote

I did not “admit” that the model “doesn’t work well”.

“Yes, for some dense galaxies (like NGC0801), there is indeed an overshoot at the bulge.”

And the very next sentence that you choose not to include states: "This suggests that the transition from "Newtonian" (center) to "Relational/Dark" (outskirts) dynamics might depend on the local potential depth". This is a textbook example of cherry-picking and the second evidence of your scientific misconduct.

---

Conclusion

Taken together, this shows that:

-you repeatedly claim “I don’t see where you answered X”,
-while the answers to X are already present in the thread in direct response to your own questions.
-you once again fail to provide any evidence supporting your false accusations or statements, yet you still refuse to retract them.

You are not engaging with the answers you requested.
Instead of analysing the provided results and statistics, you fall back to repeated accusations which you do not substantiate and which contradict the actual record of the discussion.

Under these conditions I do not see any realistic way to have a constructive scientific exchange.

To resolve the issue:

1. Retract your false accusation. Acknowledge that your accusation of cherry-picking and your later denials are mutually inconsistent.

2. If you wish to continue this discussion in good faith, correct the record and engage with the answers you requested.


Until these basic problems are addressed, there is no possibility of progress in this discussion.

44 minutes ago, Anton Rize said:

I already answered this explicitly. The fact that you now claim you “don’t see” this answer shows that you are not engaging with the responses you requested.

I wasn’t asking about DM in your hypothesis. I will rephrase: You gave a result for “Newtonian Baryonic” with a RMSE of 43 km/s. I wanted to confirm that this did not include DM.

Because the obvious followup is what do you get with DM? Comparing your result to DM-free data is meaningless. Who cares what the RMSE is for an incomplete, biased data set?

You didn’t accept RMSE; I then provided chi-squared. Now you dismiss χ² as well, without proposing any alternative statistical criterion. This shows that you are prepared to disregard any quantitative measure that does not support your expectations. That is exactly the evidence of methodological cherry-picking you falsely accused me of, and for which you have provided no evidence.

Both suffer from the same issue of being the wrong tool, IMO. It’s not very informative.

This is a direct answer to your question about internal vs external observations.
Given this, it is difficult to understand how you can still claim that the issue was “not addressed”.

It’s AN answer, to be sure, but it doesn’t address the objection.

For HSB/dense galaxies you invoke the internal observer effect to explain why there is a poor match, but for LSB galaxies you do not. But later you define internal observer as looking at a system in their own potential well. All galaxies (other than the Milky Way) should therefore be external observers, not internal.

There was also an issue of physically justifying the sqrt(3) value, which I think KJW asked.

And the very next sentence that you choose not to include states: "This suggests that the transition from "Newtonian" (center) to "Relational/Dark" (outskirts) dynamics might depend on the local potential depth". This is a textbook example of cherry-picking and the second evidence of your scientific misconduct

Yes, you invoke this factor that doesn’t align with its definition, and even though you invoke a hand wave, it doesn’t change the fact that the equation is a poor match, which you admitted to.

2 hours ago, swansont said:

You gave a result for “Newtonian Baryonic” with a RMSE of 43 km/s. I wanted to confirm that this did not include DM.

Because the obvious followup is what do you get with DM? Comparing your result to DM-free data is meaningless. Who cares what the RMSE is for an incomplete, biased data set?

This question alone would be enough to fail an undergraduate exam on galaxy dynamics.

“Newtonian baryonic, RMSE ≈ 43 km/s” is not “DM-free data”. It is a DM-free model: the rotation curve predicted from the observed baryons only, using Newtonian gravity.
The SPARC dataset is the same in every row of the table. What changes is the theoretical model applied to that dataset.

The “obvious follow-up” you demand - “what do you get with DM?” - is already in the table:

• MOND is precisely “Newtonian + modified dynamics”

• CDM / Burkert / NFW is precisely “Newtonian + dark halo”

Those rows are the “with DM” cases, with their own median RMSE values. You are asking for a result that is already explicitly listed, and then declaring the comparison “meaningless” on the basis of a distinction (DM-free data) that simply does not exist here.

If, as a moderator on a physics forum, you cannot tell the difference between:

• a dataset (SPARC velocities and baryonic components), and

• a model applied to that dataset (Newtonian baryons, MOND, CDM, WILL, …),

and you call the baseline Newtonian model an “incomplete, biased data set”, then the problem is not with my analysis. It is with your grasp of the very framework you are trying to criticise.
This alone exposes a striking level of incompetence in the very subject you are trying to present yourself as an expert in. It is frankly embarrassing.

2 hours ago, swansont said:

Both suffer from the same issue of being the wrong tool, IMO. It’s not very informative.

It is logically flawed to reject standard error metrics like RMSE or [math]\chi_\nu^2[/math] without giving a clear rationale.. The burden is on the critic to specify why the standard is insufficient and what replaces it. Otherwise, the objection is arbitrary, not scientific.

2 hours ago, swansont said:

It’s AN answer, to be sure, but it doesn’t address the objection.

For HSB/dense galaxies you invoke the internal observer effect to explain why there is a poor match, but for LSB galaxies you do not. But later you define internal observer as looking at a system in their own potential well. All galaxies (other than the Milky Way) should therefore be external observers, not internal.

This again highlights a serious misunderstanding of the basic mass–model structure you are trying to criticise.
The dense–bulge mismatch comes from an intentionally crude mass–to–light approximation, which I explicitly stated:

V_bary² = V_gas² + Υ* · (V_disk² + V_bul²),

with a single fixed Υ* = 0.66 for all galaxies.
Anyone with a basic familiarity with stellar populations knows that a universal Υ* is a rough first–order approximation. The term it multiplies,
Υ*(V_disk² + V_bul²),
is exactly the contribution of the central disk + bulge – the region where you are complaining about overshoot.
When I relax this crude assumption and allow Υ* to vary per galaxy (one free parameter), the χ² collapses and the “poor match” in HSB systems largely disappears. This is already shown in the numbers I posted:

On 11/19/2025 at 4:41 PM, Anton Rize said:

TUNED QWILL (1 Free Parameter per galaxy):

Constraint: [math]\Upsilon_*[/math] allowed to vary (representing stellar population differences)

Law: [math]V = \sqrt{3} V_{bary}[/math]

Median [math]\chi_\nu^2[/math]: 6.52 <-- THE SIGNAL

Median RMSE: 11.62 km/s

The Smoking Gun:

The massive drop in [math]\chi_\nu^2[/math] (from ~34 down to ~6.5) when allowing just one degree of freedom (mass-to-light ratio) proves that the "shape problem" you suspected is not intrinsic to the geometric law [math]\sqrt{3}[/math].

If the geometric law were wrong (e.g., wrong shape at small [math]r[/math]), adjusting the amplitude [math]\Upsilon_*[/math] would NOT fix the [math]\chi^2[/math] so dramatically. The fact that it [i]does[/i] drop to near-acceptable levels implies that the geometric profile is correct, and the residuals in the Fixed Model are dominated purely by astrophysical scatter (old vs. young stellar populations).

Your objection is built on your own misreading.
So your statement

“for HSB galaxies you invoke the internal observer effect to explain the poor match”

is simply false: the HSB issue is an astrophysical M/L modelling problem, not a geometric “observer” effect, and I have already demonstrated how it behaves when Υ* is treated properly.

At this point your criticism is based either on not reading what I actually wrote, or on not understanding the very mass model you are trying to criticise. In both cases, it has nothing to do with the real content of the analysis.

2 hours ago, swansont said:

There was also an issue of physically justifying the sqrt(3) value, which I think KJW asked.

I was having a productive technical dialogue with KJW on this point before it was derailed by your false accusations. Instead of continuing that discussion, I now have to respond to your misreadings, false accusations and unsubstantiated claims. You are directly preventing scientific discussion in this thread.
At this stage, this should be genuinely embarrassing for you...



As I have already stated:

4 hours ago, Anton Rize said:

Under these conditions I do not see any realistic way to have a constructive scientific exchange.

To resolve the issue:

1. Retract your false accusation. Acknowledge that your accusation of cherry-picking and your later denials are mutually inconsistent.

2. If you wish to continue this discussion in good faith, correct the record and engage with the answers you requested.

@KJW

I'm sorry that it took so long to reply to your questions. Hopefully now we can continue our dialog without interventions.
I want to thank you for the correction on the acceleration derivation . And thank you for correcting my blunder with orbital velocity written backwards. Your rigor in re-checking the partial derivatives is exactly the kind of scrutiny I am looking for.

I also want to address your objection regarding the distinction between Internal and External observations. You wrote:

On 11/20/2025 at 9:19 AM, KJW said:

One isn't going to have a [math]\sqrt{3}[/math] factor difference between observations made from outside and observations made from inside of a non-relativistic system.

It is my understanding that it is the virial theorem that indicates the existence of dark matter in galaxies.

This is an important physical objection you raised. Previously, I described this as a "Carousel effect," which might have sounded like a heuristic analogy.

However, if we treat the relational displacements strictly as vectors in the ([math]\beta, \kappa[/math]) projection space, this effect becomes a rigorous consequence of Potential Screening. It works exactly like voltage difference in electrostatics.

Here is the formal derivation that resolves your objection:

Hypothesis: Internal vs. External Observation (The "Carousel" Effect)

A fundamental question arises: if the universal rotation law is
[math]V = \sqrt{3} V_{\mathrm{bary}}[/math], why does the Solar System follow pure Newtonian dynamics ([math]V = V_{\mathrm{bary}}[/math])?

The answer lies in the relational nature of observation. We must distinguish between two modes of measurement:

- Inter-system Observation (External View):

When we observe a distant galaxy, we are external to its gravitational binding energy. We are not part of its "system." Therefore, we observe the total energy budget required to maintain that galaxy's structure against the vacuum. We see both the kinetic motion ([math]\beta^2[/math]) and the structural tension ([math]\kappa^2[/math]) required for closure.

[math]Q^2_{\mathrm{ext}} = \beta^2 + \kappa^2 = 3\beta^2 \quad \Longrightarrow \quad V = \sqrt{3} V_{\mathrm{bary}}[/math]

- Intra-system Observation (Internal View):

When we observe the Solar System or Milky Way, we are embedded within the same gravitational potential well ([math]\kappa_{\mathrm{local}}[/math]) as the planets or stars. We are, effectively, "riding the same carousel." The background potential [math]\kappa^2[/math] is a shared baseline for both the observer (Earth) and the target (Jupiter).

Potential Screening Principle - Local Potential Screening:

For an observer embedded within the system, the binding potential [math]\kappa^2[/math] acts as a common background frame, not as an observable kinematic difference. The relative measurement cancels out the structural tension, leaving only the kinetic differential:

[math]Q^2_{\mathrm{int}} \approx \beta^2 \quad \Longrightarrow \quad V \approx V_{\mathrm{bary}}[/math]

Thus, the factor [math]\sqrt{3}[/math] is the signature of a holistic observation of a closed system from the outside (Galactic Scale), while Newtonian dynamics represents the differential observation from the inside (Local Scale).

________________________________________

Vector Analysis of Observation Modes

To resolve the apparent discrepancy between galactic dynamics (where [math]V \approx \sqrt{3} V_{\text{bary}}[/math]) and local solar system dynamics
(where [math]V \approx V_{\text{bary}}[/math]), we must treat the relational displacement [math]Q[/math] strictly as a vector quantity in the [math](\beta, \kappa)[/math] plane.

1. Definition of Relational Vector

Any physical state is characterized by a relational displacement vector [math]\mathbf{Q}[/math] relative to the observer's origin:

[math]\mathbf{Q} = \begin{pmatrix} \beta \\ \kappa \end{pmatrix}[/math]

The magnitude of this vector determines the total observable energy budget (and thus the effective orbital velocity):

[math]V_{\text{obs}}^2 = c^2 |\mathbf{Q}|^2 = c^2 (\beta^2 + \kappa^2)[/math]

2. Case 1: Inter-system Observation (External View)

Consider an observer located far outside the target system (e.g., measuring a distant galaxy). The observer resides in the asymptotic vacuum relative to the target's potential well.

- Observer State: The observer defines the relational zero: [math]\mathbf{Q}_{\text{obs}} = (0, 0)[/math].

- Target State: The target system (galaxy) exhibits both kinematic motion and structural potential binding: [math]\mathbf{Q}_{\text{sys}} = (\beta, \kappa)[/math].

The measured displacement is the absolute vector:

[math]\mathbf{Q}_{\text{ext}} = \mathbf{Q}_{\text{sys}} - \mathbf{Q}_{\text{obs}} =[/math]
[math]= \begin{pmatrix} \beta \\ \kappa \end{pmatrix} - \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} \beta \\ \kappa \end{pmatrix}[/math]

Applying the closure condition for stable systems ([math]\kappa^2 = 2\beta^2[/math]):

[math]|\mathbf{Q}_{\text{ext}}|^2 = \beta^2 + 2\beta^2 = 3\beta^2 \quad \Longrightarrow \quad V_{\text{ext}} = \sqrt{3} V_{\text{bary}}[/math]

This explains the "Dark Matter" effect as the observation of the full vector magnitude, including the orthogonal potential component [math]\kappa[/math].

3. Case 2: Intra-system Observation (Internal View)

Consider an observer embedded within the same system as the target (e.g., Earth observing Jupiter). Both the observer and the target share the same background gravitational potential scale defined by the central mass (Sun/Galaxy).

- Common Potential: [math]\kappa_{\text{background}} \approx \text{const}[/math] locally.

- Observer State: [math]\mathbf{Q}_{\text{obs}} = (\beta_{\text{obs}}, \kappa_{\text{background}})[/math].

- Target State: [math]\mathbf{Q}_{\text{target}} = (\beta_{\text{target}}, \kappa_{\text{background}})[/math].

The observable is the relative displacement vector between the two bodies:

[math]\mathbf{Q}_{\text{int}} = \mathbf{Q}_{\text{target}} - \mathbf{Q}_{\text{obs}} = [/math]
[math]=\begin{pmatrix} \beta_{\text{target}} - \beta_{\text{obs}} \\ \kappa_{\text{background}} - \kappa_{\text{background}} \end{pmatrix} = \begin{pmatrix} \Delta\beta \\ 0 \end{pmatrix}[/math]

The common structural potential component [math]\kappa[/math] subtracts out. The observer perceives only the differential kinetic projection:

[math]|\mathbf{Q}_{\text{int}}|^2 = (\Delta\beta)^2 \quad \Longrightarrow \quad V_{\text{int}} \approx V_{\text{bary}}[/math]

Thus, internal observation naturally recovers Newtonian dynamics without requiring screening mechanisms or adjustable parameters. The "Dark" component ([math]\kappa[/math]) exists but is geometrically invisible to an internal observer, just as voltage difference is zero between two points at the same high potential.

Remark:

The remaining scatter (RMSE 20.23 km/s) is expected due to the assumption of a universal [math]\Upsilon^*[/math] and perfect geometric virial equilibrium. The fact that a parameter-free geometric law performs comparably to tuned Dark Matter models suggests that the [math]\sqrt{3}[/math] factor captures the fundamental driver of galactic dynamics, while astrophysical variations account for the residuals.


---

Regarding your point on the Circular Orbit Formula:

You noted that I might have the interpretation "backwards" regarding local vs. infinity observations in GR. I think I got it backwards but Im not sure now... Its so easy to get lost in different frames. Lets think together.

Even if we accept that [math]\beta_{\infty}^2 \approx GM/R[/math] (Newtonian) in standard GR for a point source, the key insight of the Vector Analysis above is that for a distributed system (Galaxy), the binding energy itself ([math]\kappa[/math]) contributes to the global energy budget measured by an external observer ([math]Q_{ext}[/math]).

The "Dark Matter" phenomenon is simply the observation of the magnitude of the full vector [math]|\mathbf{Q}_{ext}|[/math], whereas local (internal) dynamics only measure the differential [math]|\mathbf{Q}_{int}|[/math].

Does this vector formulation make the distinction physically clearer to you? What do you think?

P. S. If I skipped any of your questions - please repeat them so we can get back on our thought train. Lets go!

Edited November 26Nov 26 by Anton Rize

12 hours ago, Anton Rize said:

This question alone would be enough to fail an undergraduate exam on galaxy dynamics.

“Newtonian baryonic, RMSE ≈ 43 km/s” is not “DM-free data”. It is a DM-free model: the rotation curve predicted from the observed baryons only, using Newtonian gravity.
The SPARC dataset is the same in every row of the table. What changes is the theoretical model applied to that dataset.

The “obvious follow-up” you demand - “what do you get with DM?” - is already in the table:

• MOND is precisely “Newtonian + modified dynamics”

• CDM / Burkert / NFW is precisely “Newtonian + dark halo”

Thank you. Getting a response to a pretty straightforward question was like pulling teeth.

12 hours ago, Anton Rize said:

If, as a moderator on a physics forum, you cannot tell the difference between:

• a dataset (SPARC velocities and baryonic components), and

• a model applied to that dataset (Newtonian baryons, MOND, CDM, WILL, …),

and you call the baseline Newtonian model an “incomplete, biased data set”, then the problem is not with my analysis. It is with your grasp of the very framework you are trying to criticise.
This alone exposes a striking level of incompetence in the very subject you are trying to present yourself as an expert in. It is frankly embarrassing.

This isn’t a physics forum, it’s a science forum, and a moderator enforces rules (my role as moderator is mostly irrelevant here, as there has been no need for moderator action). I never claimed expertise in cosmology.

12 hours ago, Anton Rize said:

This again highlights a serious misunderstanding of the basic mass–model structure you are trying to criticise.
…

At this point your criticism is based either on not reading what I actually wrote, or on not understanding the very mass model you are trying to criticise. In both cases, it has nothing to do with the real content of the analysis.

You might consider that misunderstanding can also be based on inadequacies in the presentation, and that asking for clarification is an effort to gain understanding. Also that berating people for asking for clarification is perhaps not the best approach. (this observation is why my role as moderator is not completely irrelevant)

12 hours ago, Anton Rize said:

I was having a productive technical dialogue with KJW on this point before it was derailed by your false accusations. Instead of continuing that discussion, I now have to respond to your misreadings, false accusations and unsubstantiated claims. You are directly preventing scientific discussion in this thread.

Thank you for the laugh.

@Anton Rize

The point I was making about the distinction between observing from outside and observing from inside is that there isn't going to be a [math]\sqrt{3}[/math] factor difference in any observed quantity because in any non-relativistic system [math]\kappa^2 = \dfrac{r_s}{r} \ll 1[/math] and [math]\beta^2 = \dfrac{r_s}{2r} \ll 1[/math], and therefore [math]1 - (\kappa^2 + \beta^2) \approx 1[/math]. This is the factor that governs any difference between observations made from inside and observations made from outside. For example, at the surface of the sun, [math]\dfrac{r_s}{r} = 4.24 \times 10^{-6}[/math], requiring very accurate measurments to even detect the gravitational redshift of spectral lines.