Anton Rize

@Markus Hanke Ok. Lets just make sure that this time you after asking question DOES NOT IGNORE THE ANSWER ok? You claim [math]R_s[/math] explicitly depends on M and G. That's incorrect. In RG, G and M are output calibration tools, not physical inputs. Check the Two-Point Method. We derive [math]R_s[/math] strictly from kinematics: [math]R_s = \frac{r_1 r_2}{r_2 - r_1}(\beta_1^2 - \beta_2^2)[/math] Look at the right side. It's just geometry [math]r[/math] and velocity [math]\beta[/math]. Where are G and M? They are absent. The system scale is determined by the orbit itself, not a "Mass" label. I only use G if we need to convert [math]R_s[/math] to legacy units (kg): [math]M \equiv \frac{R_s c^2}{2G}[/math] Full prove of operational independency from M, G you can find here https://willrg.com/documents/WILL_RG_I.pdf#sec:operational For this derivation we dont need to. We using formula without a: [math]\frac{3\cdot\pi}{2}\frac{\kappa_{p}^{4}}{\beta_{p}^{2}}[/math] You are mistaken. z_sun is a raw observable we take from spectroscopy. Neither G or M needed for its measurement. Source: {IAU_2015_ResB3} Mamajek, E.E. et al. (2015). IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants. \textit{arXiv:1510.07674}. Observation, radioastronomy, spectroscopy etc... This exact value I took from {NASA_Eclipse_Mercury} Espenak, F. (2014). Seven Century Catalog of Mercury Transits: 2001 to 2700. \textit{NASA Eclipse Web Site}. From astronomical observations. We dont need absolute values to find the ratio between lines in the sky. Here's the source: {UniverseToday_Mercury} Williams, M. (2016). How Far is Mercury From the Sun? \textit{Universe Today} (citing NASA data). Exactly. You fix boundary conditions with Mass. I fix them with observables ([math]z_{sun}[/math], [math]\beta_p[/math]). Both are valid boundary conditions. But mine do not require the assumption of a "Mass" entity they only require the measurement of a signal. You see I dont need to invent unmeasurable nonphysical parameters to solve the problem - you do. So, when you say [math]z_{sun}[/math] depends on M, you are mistaking the map for the territory. [math]z_{sun}[/math] is a raw observable. M is just one way to model it. I am showing you a way to model it without M, using only the relations between potentials and kinematics. I hope you learn something new today.

Yes I see how it could seem that way, but no. It has nothing to do with MOND. MOND is an embodiment of the exact phenomenological approach that I critic. Good call that's exactly what I thought when I first derived it. But later it become clear that alpha is a geometrical invariant. Almost like pi. You can find the details here: https://willrg.com/documents/WILL_RG_II.pdf#sec:invariant_alpha In short it independently predicts H_0 and the CMB first peak \ell_1. So [math]\frac{\partial H_0}{\partial \ell_{peak}} = 0[/math]. And dynamically temperature changing [math]H(t) \propto T(t)^2 / \alpha[/math] but [math] \alpha=[/math]constant. This is actually a very useful tool. Im constantly comparing my results with LCDM and this calculator will help me learn less python witch is great! thank you.

I appreciate your clarification regarding the vacuum equations. But in this case you are mistaking. My derivation is not a reverse-engineering of GR. It is a direct result of applying a specific set of methodological principles that strictly forbid the use of a spacetime manifold: 1. Epistemic hygiene: We rely only on directly measurable signal relations (spectroscopy and astrometry), discarding unobservable theoretical entities (like mass). 2. Ontological minimalism: We do not multiply entities beyond necessity. If [math]G[/math], [math]M[/math], and the metric tensor are not required for the calculation, they are excluded from the ontology. 3. Relational origin: Everything originate from direct relations between potentials and kinetic states, not objects residing in a container (the unspoken assumption in modern physics). 4. Mathematical transparency: The connection between observables must be algebraic and direct. Each mathematical object must correspond to explicitly identifiable relation between observers with transparent ontological origin. 5. Simplicity: Everything must be expressed in the simplest form possible. Any unjustified complexity risks reintroducing metaphysical artefacts and contradicts the foundational insight of Epistemic Hygiene. Here is the proof that RG is a superset of the Schwarzschild solution, not a copy of it. 1. The General Definition of the Horizon (The Limit of Causality) In standard GR (Schwarzschild), the horizon is defined strictly spatially via mass ([math]r_s = 2GM/c^2[/math]). This describes a static boundary where escape velocity equals [math]c[/math]. In Relational Geometry, the horizon is defined by the saturation of the Total Relational Shift [math]Q[/math]. An observer defines the state of any external system via two orthogonal projections: 1. Gravitational Potential ([math]\kappa[/math]) 2. Kinematic State ([math]\beta[/math]) The total state difference is the norm of these projections: [math]Q^2 = \beta^2 + \kappa^2[/math] ( it is NOT Pythagorizes identity) The natural causal horizon occurs when this relational difference saturates to unity ([math]Q=1[/math]). This leads to a distinction: * WILL RG Horizon: [math]\beta^2 + \kappa^2 = 1[/math] (The limit depends on both motion and potential). * Schwarzschild Horizon: This is merely the static slice of the generalized horizon where [math]\beta \to 0[/math]. In this degenerate case, [math]\kappa^2 = 1[/math], which we label [math]R_s[/math]. Thus, the Schwarzschild radius is just the "potential-axis intercept" of the true relational horizon. 2. Deriving Precession from the General State [math]Q[/math] The precession is intrinsic to the accumulation of this state difference [math]Q[/math]. The system accumulates a state mismatch over every closed cycle. The total angular shift is the full phase ([math]2\pi[/math]) scaled by the intensity of the shift ([math]Q^2[/math]), normalized by geometry: [math]\Delta\varphi = \underbrace{2\pi}_{\text{Cycle}} \cdot \underbrace{Q^2}_{\text{Intensity}} \cdot \underbrace{\frac{1}{1-e^2}}_{\text{Shape Factor}}[/math] To solve this for a stable orbit, we apply the Closure Condition (proved in the full paper). At the reference scale [math]a[/math], the relation between kinetic and potential states stabilizes as: [math]Q^2(a) = \frac{3}{2}\kappa^2(a)[/math] Recognizing that in the static limit [math]\kappa^2[/math] corresponds to [math]R_s/r[/math], we can substitute: [math]Q^2 = \frac{3R_s}{2a}[/math] Mapping this to the periapsis ([math]p[/math]) to use direct observables (Doppler [math]\beta_p[/math] and Redshift [math]z_{sun}[/math]), we arrive at the operational equation: [math]\Delta\varphi = \frac{3}{2}\pi \frac{\kappa_p^4}{\beta_p^2}[/math] Conclusion 1. Genealogy: The formula is derived from the accumulation of [math]Q[/math], a quantity that generalizes the horizon concept beyond the static Schwarzschild definition. 2. Independence: We define [math]R_s[/math] not as a mass parameter, but as the geometric saturation point ([math]\kappa=1[/math]) of the potential axis. 3. Epistemology: The fact that a model defining the horizon as [math]\beta^2+\kappa^2=1[/math] perfectly predicts a phenomenon traditionally explained by curvature tensors suggests that the tensor formalism is an emergent, albeit limited, map of a deeper relational terrain. I am not hiding the metric. I am showing that the metric is a static approximation of a broader kinematic-potential relation obtain only from observables: Inputs: 1. Kinetic projection (Doppler): [math]\beta_p[/math] 2. Gravitational potential (Redshift): [math]z_{sun}[/math] 3. Geometric ratio (Astrometry): [math]R_{ratio}[/math] The Crucial Question: If philosophy that categorically denies the existence of a metric tensor naturally yields the exact predictions attributed to the metric tensor, does this not imply that the metric is an epistemological artifact (a map) rather than a fundamental entity (the terrain)? What do you think? P. S. We already touched this topic with you earlier in this post. It went badly. Maybe this time we will be able to reach some consensus? P.P.S. I forgot to say thank you to you. Your first question in this post lead me in to developing a full Relational Orbital Mechanics: https://willrg.com/documents/WILL_RG_I.pdf#eq:rom Thank you for the question.

Yes thats the explanation that I encounter most often. But when Im asking what observations exactly im getting bunch of model dependant non physical assumptions. So committed to the cause of scraping the bottom of this mystery well. Here's model independent raw observations that "Cosmological Dark Sector" boils down to according to my research: 1. Orbital Speeds in Galaxies Light Path Deflections Around Structures Microwave Background Temperature Variations Distant Supernova Flux Levels Wide Binary Star Motions So as far as I understood accurate prediction and ontological explanation of all this phenomena, without introducing new entity's or speculations would solve the "Dark Sector Problem". I didn't include structure formation, its not directly observable, and here I just wanted to boil it down to the solid measurements. Do you guys agree with the list and criteria or am I missing something? @Markus Hanke @Mordred We often blinded by the formalisms forgetting how mesmerizing Cosmos is. Thank you for reminding.

Ahhhh... now it makes more sense... Sorry I forgot that I put a permalink on to opening page. Its a heavily shortened and significantly outdated. The core idea is the same and math too. Its just there's no dynamic systems explicitly described in this version. So now I understand why you were saying: So you thought Im postulating circular maximally symmetrical spacetime geometry? No Im not. Its just so upsetting that Im putting so much effort in to making the website easy to read and methodically transparent and you wasting your time reading outdated document. But its my fault I shouldn't put a permalink there in a first place. Can you elaborate on this please? I genuinely want to understand if its a well known method (as I understood from this comment) why I couldn't find it anywhere? And if we can predict observations that in GR are caused by mass and G without involving mass and G at all what does it tells us about the nature of gravity and spacetime geometry?

@Mordred That is a very specific point regarding the Equation of State ([math]w=0[/math]) and the necessity of a pressureless component to recover the correct expansion history and [math]H_0[/math]. You argued that without this particle component, the Universe would transition to [math]\Lambda[/math]-dominance incorrectly, making it impossible to match the observed Hubble constant and structure formation timing. This is exactly the constraint I addressed in my open research. I found that if you strictly couple the vacuum geometry to the fine-structure constant ([math]\alpha[/math]) and Thermodynamics ([math]T_{CMB}[/math]), you derive [math]H_0[/math] analytically without any Dark Matter parameter: [math]H_0 = \sqrt{8\pi G \frac{\rho_{\gamma}}{3\alpha^2}} \approx 68.15 \text{ km/s/Mpc}[/math] This matches the Planck 2018 result ([math]67.4 \pm 0.5[/math]) within 1% purely from first principles. Since your main argument for the "particle nature" is the necessity of matching these expansion constraints, does this exact geometric derivation of [math]H_0[/math] (which bypasses the need for a fitted [math]w=0[/math] component) count as counter-evidence in your view? Or do you see a physical flaw in linking the horizon scale to [math]\alpha[/math]?

@Mordred thank you for detailed response So this seems to me like circular logic. Isn't it?: 1. We basically have 2 (rotation curves and structure formation) model-observation inconsistency's. 2. We patching them with phenomenology driven speculations 3. We trying to justify first speculation by using second speculation as a core of the argument. I must have been missing something because this is clearly a methodological no no. And also Im having major problems coherently structuring this physical process in my head: Its just 1. we have to assume the existence of unmeasured, very model dependent particle (not happy about it but ok...) 2. Then we have to assume some truly bizarre interaction protocols for this particle 3. Then we have to assume almost magical halo formation evolution that happened to have exact the same invariant relation with baryonic matter that would have to form exactly same way in Blue Compact Dwarfs (BDC) and in Intermediate Spiral (Sb) And after all this assumptions and free parameters introduced we still cant explain all observations like Wide Binary's, some week lensing systems, structure formation, H_0, weak amplitude ($\ell=2$) in the CMB quadrupole moment, etc... So my thinking is: "From my pov its all falling apart, but yet particle origin DM model remains dominant in modern cosmology. Therefor Im just missing something" And problem is that the more I dig in to it I keep finding only new inconsistency's but not the explanation of why particle DM remains dominant theory. It can't be that Im the first one who formulating this question like this right? Yes this is exactly what im trying to find. Can we pinpoint exactly why do we consider it a strong possibility?

Did you even open my Desmos project? Out of 3 inputs precession derived: 3 IINPUTS NO MASS NO G NO c PRECESSION DERIVATION 1. Mercury's kinetic projection beta_p at perihelium unitless ( transverse Doppler shift, obtained via spectroscopy or radio signal.) \beta_{p}=0.000196736103348 2. Gravitational Redshift at the surface of the sun (R_sun) (obtain via spectroscopy) z_{sun}=2.1224\cdot10^{-6} 3. Ratio between R_sun and r_p (Mercury radius at perihelium) R_ratio=R_sun/r_p from astrometric data. No absolute scales needed. R_{ratio}=0.0151235185169 THATS ALL OUR INPUTS if gravitation is the curvature of 4D spacetime induce by mass, then mass and G has to be primary parameters. But in my calculation they are unphysical unmeasurable redundant values: \kappa_{p}=\sqrt{\left(1-\left(1+z_{sun}\right)^{-2}\right)\cdot R_{ratio}}=0.000253369506895 \Delta_{precession}=\frac{3\cdot\pi}{2}\cdot\frac{\kappa_{p}^{4}}{\beta_{p}^{2}}=5.0175347157\times10^{-7} after units conversion we getting 42.9710621566 arcsec/100years DATA SOURCES: \cite{NASA_Eclipse_Mercury}, \cite{UniverseToday_Mercury}, \cite{IAU_2015_ResB3}. Dare to interpret this result? P. S. I don't know what article you reading probably not mine. im talking about WILL_GR_I.pdf on my website. You still dont get it. When you think about physical process you assuming that there's a flexible "box" (4D manifold) where mass/energy/fields are interacting. Im NOT MAKING THIS ASSUMPTION. I argue that the hole concept of the "box" thinking is an anthropocentric speculation inbuild silently in foundations of modern physics. If you would red anything that I wrote this would be the first thing you would understand. Its a very old debate going back to Newton vs Leibniz. I don't get it... Why are you engaging in criticism without a clue about the subject of your critic? Whats the point? To show your arrogance? Thats not something you should be proud about.

@Mordred , buddy, hold on. You still having major problems in understanding the most basic things. If you did read my paper as you claim than your reading skills rising concerns: 1. Spacetime is about meters and seconds. Relations are unitless. Its NOT "Maximally symmetric spacetime". Its a relational carrier of conservation. 2. The relational geometry of Sun-Mercury system will not change regardless of your point of view. 3. Lying is bad. You haven't read my paper and by continue lying about it you only make it worse. 4. Im not going to engage in philosophy with you because you have no clue what you talking about. 5. I gave you VERY SIMPLE solution for Mercury's precession with no mass no G and asked you "How would you explain this?" And the unswear you gave me is: "Do you honestly believe that simply because you can plot a 2d orbital that this encompasses all possible observer from other angles ?" - Your answer sounds like you saying that precession will change when viewed from different frame. Im just hoping that this some kind of misunderstanding. And in general try to remember that before you want to critic anything make sure that you at least can repeat your opponents point of view without twisting it. Without it everything we say is meaningless.

Maybe you guys can clear this to me?: How particle origin of DM can explain Tully-Fisher Relation and Wide Binary's phenomena without shameless curve fitting? Isn't it require exactly the same evolution history of halos in vastly different galaxies? And where the halo is hiding in Wide Binary systems?

@Mordred , we achieved incredible results in physics so far but there's fundamental challenges still remain. By blindly repeating textbooks we cant achieve any progress. That's why challenging anthropocentric beliefs is the main engine in scientific progress. Unfortunately you still have no clue what are you talking about. Spacetime interval is a mathematical tool we invented, Its just one of many possible interpretations. Same as fundamentality of mass and G. But look, when we ignore interpretations and dealing with directly measured phenomena we can see that neither mass or G needed to predict Mercury's presession. Here's Desmos project explicitly showing this https://www.desmos.com/calculator/wimnrykbvy How would you explain this?

This is getting ridicules! All my results are publicly open but instead of reading you accuse me in not showing. We have nothing to discuss because you never engaged with the arguments provided.

So for nearly 3 months this post was in Relativity section without a word about moving it and then after your public humiliation you suddenly remembered the the rules. I see... @Mordred , buddy, you have no clue what you are talking about. Memorizing and understanding are not the same thing. Google the words I suggested. If you would actually read what you trying to critic you would know that there's several sections dedicated to this precise question. Its Part I, you can start from \section{Operational Independence and the Role of Constants} and finish with \section{Relational Orbital Mechanics (R.O.M.) Without Mass or $G$} And if you will provide criticism please make sure that its not just your opinion. Support it with mathematical proves. For example prediction of the Mercury, S2, and S4716 stars precession. You can try to disprove it. Good luck.

Yes you are right thank you for pointing this out. This argument + contradiction that comes from the fact that the Sun orbiting the SgtrA BH at around [math]\approx 229 \pm 6 km/s[/math] instead of classically predicted [math]\approx 170 \pm 5 \, \mathrm{km/s}[/math] completely falsify my "Carousel" hypothesis. But my latest's results are just mind blowing. Have a look at https://willrg.com/results/ its in Part II .pdf. Here's just a glimpse:

I'm sorry in advance but ill be blunt with you on this. To say something like this one has to be either stupid or dishonest. We all know that GR is incomplete and we have mountains of empirical data undeniably confirming it. It is probably the best that we got at the moment arguably yes... But to put GR in rank of absolute Truth is either sing of stupidity or dishonesty. In both outcomes you lost all remaining respect that I still had towards you . You don't have the right to call yourself scientist. You see that's the main difference between us. Im running publicly open research out of pure interest and love of science. All my methods and results are open for public. And don't claim to be right, im just sharing results and open for corrections and critic. But with you it is completely another level or the complete lack of it. All you do is just throwing empty insults and Appeal to tradition/popularity (argumentum ad populum). In stead of engaging with the actual evidence and arguments presented (this is science) you keep repeating same logical fallacy's over and over again. You call it speculation? Prove it as scientist! All codes and data and results are open at willrg.com and if you will I only will be grateful for helping me be less wrong. But instead you silently moving the thread and when confronted retreat to insults. You don't have the right to call yourself scientist! Shame! Another example of Appeal to authority logical fallacy presented to us by yet another "moderator". Good start mate! @MJ kihara Im not enjoying this too. I came here with hope to meet likeminded people who with united front engaging in the scientific attempt to be a bit less wrong, but instead Im facing.. this... Glad you found it entertaining though @Mordred , buddy well done you memorized some words from a text book. I suggest you to engage with basics of philosophy of science. Start from googling the words "ontology" and "epistemology". Than try to apply this methods to classical mechanics. There's a fascinating world of physics awaits you as soon as you learn to separate anthropocentric interpretations from empirical data. Good luck!

@swansont On what basis this thread was moved to speculations? What did you find so speculative? Was it the remarkable agreement with empirical data across 10^20 orders of magnitude range? Or was it the lack of adjustable parameters and refusal of curve fitting? Or was it full open source methodology derivations calculations and results publicly available at willrg.com ? At least have some dignity and scientific honesty for open scientific dispute! Moving this thread without a word while I was away is just pathetic! Ahhhh I know! You moved it here because you ashamed of how your glairing incompetence was publicly reviled on the last few pages. This is worse than pathetic! You have no right to call yourself a scientist! Shame!

@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: 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!

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. 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. 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: 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. 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:

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. --- 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. --- Here is another earlier answer you “don’t see”: 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”. --- Ill help you recover the full context. Here it is: Now when we have full context above lets have a look how you cherry-picked my quote: “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.

I have not. 1st time you explicitly wrote: 2nd time you reinforced the accusation by treating cherry-picking as a methodological issue.: You cannot simultaneously say: “I pointed out an instance of cherry-picking” and “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: 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: 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) —: 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: 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: 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: 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: 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.

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: Definition of Cherry-Picking: Selecting only favorable data while suppressing unfavorable data. My Action: In the post you criticised, I explicitly presented NGC0801 as a case where the model deviates (overshoots at the bulge): 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: 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: 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.

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 formal accusation of scientific dishonesty. It is a serious charge. I will not take such false accusation lightly. I demand that you substantiate it or retract it. The Data: I presented a statistical analysis of 175 galaxies . This is the entirety of the SPARC database. The Method: I used the Global Median RMSE. By definition, a median over the full dataset cannot be "cherry-picked." The Code: The analysis scripts are open-source and linked . "Cherry-picking" means selecting only data that fits. Using every single data point available is the exact opposite. Making baseless accusations of data manipulation against a transparent, full-dataset analysis reflects poorly on the accuser, not the accused. If you cannot point to a specific galaxy I excluded or a specific line of code that filters the data, then your accusation is factually false and scientifically unethical.

@swansont Thank you for the structured critique. You raise specific, testable objections regarding the statistical standard and the physical shape of the rotation curves. 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. Open-source: You welcome to test it yourself. All my google colab notebooks you can find here: https://antonrize.github.io/WILL/predictions/ I didn't had to speculate any "dark" entities. 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. This premise relies on a misunderstanding of my claim. I do not argue that General Relativity is "wrong" in its predictions; I argue that it is ontologically redundant. Therefore, a mass [math]M[/math] inferred via standard Keplerian/GR dynamics is algebraically consistent with the mass inferred via WILL. Since the predictive equations converge, the "inferred values" do not need to be recalculated - they are valid inputs for both frameworks. Furthermore, even if there were a higher-order divergence between the theories in the strong-field regime, it would be irrelevant for this specific test: * SMBHs dominate kinematics only within a few parsecs (sphere of influence). * SPARC rotation curves measure dynamics at kiloparsecs ([math]R > 1[/math] kpc). * At these radii, the potential is dominated by the Stellar Disk and Gas ($10^{9}-10^{11} M_\odot$). The contribution of the central SMBH is vanishingly small. The masses are safe. The geometric factor [math]\sqrt{3}[/math] is tested on scales where the specific model of the central black hole acts merely as a point-source correction, negligible compared to the galaxy's bulk mass. Since dropping a link to a PDF is often where discussion dies, I will write the explicit short algebraic derivation right here. It requires no metric tensors, only the conservation of the relational energy budget. The Geometric Derivation of [math]V_Q = \sqrt{3} V_{bary}[/math] 1. Inputs: In WILL, the state of any system is defined by projections: [list] [*] Kinetic projection (Motion on [math]S^1[/math]): [math]\beta = v/c[/math] [*] Potential projection (Gravity on [math]S^2[/math]): [math]\kappa = \sqrt{R_s/r}[/math] [/list] 2. The Closure Condition (Geometric "Virial-like"): For a self-contained, gravitationally bound system in equilibrium, the energy capacity of the potential field ([math]S^2[/math], 2 degrees of freedom) must balance the kinetic capacity ([math]S^1[/math], 1 degree of freedom). This enforces the exchange rate of 2:1. [math]\boxed{\kappa^2 = 2\beta^2}[/math] 3. The Observable (Total Projection): An external observer (inter-galactic) measures the total energy budget [math]Q^2[/math] required to maintain this structure against the vacuum. [math]Q^2 = \kappa^2 + \beta^2[/math] 4. Substitution: Substitute the closure condition (2) into the total budget (3): [math]Q^2 = (2\beta^2) + \beta^2 = 3\beta^2[/math] 5. Velocity Translation: Convert back to velocities ([math]V = c \cdot \text{projection}[/math]): [list] [*] Baryonic velocity (visible matter): [math]V_{bary} = c \cdot \beta[/math] [*] Total Observed velocity (dynamic mass): [math]V_{Q} = c \cdot Q[/math] [/list] [math]V_{Q}^2 = c^2 Q^2 = c^2 (3\beta^2) = 3 (c\beta)^2 = 3 V_{bary}^2[/math] Final Result: [math]\boxed{V_{Q} = \sqrt{3} \cdot V_{bary}}[/math] --- Why this matters: This factor [math]\sqrt{3} \approx 1.73[/math] is not a fitted parameter. It is a geometric constant arising from the topology of a closed system ([math]S^1 + S^2[/math]). Standard Dark Matter models must add an invisible halo with 2-3 free parameters to bridge the gap between [math]V_{bary}[/math] and [math]V_{obs}[/math]. WILL derives the gap as a necessary geometric consequence of the system's unity. You can download .pdf with all the details here: https://antonrize.github.io/WILL/results/ P.S. Forgot to highlight: Its the same Q parameter that predicts orbital perihelium. Isn't it fascinating!?

So can we still interpret gravity as bending of spacetime due to mass or not? What do you think?

You are absolutely right about the factor of 3 difference. I suspect this isn't an error, but the precise "fingerprint" of the ontological difference we are discussing. In standard theory, we assume vacuum energy fills the Volume. Mathematically, getting the total energy requires integrating the surface area over the radius: [math]\int r^2 dr = r^3/3[/math] That’s where the factor of 3 (or 1/3) comes from, right? In RG, energy is defined by the Surface projection on [math]S^2[/math]. Since I don't assume a "bulk container" that needs to be filled, I don't integrate over [math]r[/math], so the 3 never appears. What do you think? Could the factor of 3 be just a mathematical artifact of the "volume-filling" assumption? This derivation is impressive. Thank you, @KJW, for taking the time to write this out explicitly. This allows for a precise, line-by-line comparison between the two frameworks. Let's analyze your final result for the velocity parameter [math]\beta^2[/math] for a timelike circular orbit: [math]\beta_{GR}^2 = \frac{GM}{c^2 R} \left(1 - \frac{2GM}{c^2 R}\right)^{-1}[/math] This result is fascinating because it allows us to translate directly into WILL RG terms. 1. The term [math]\frac{GM}{c^2 R}[/math] is exactly the RG local invariant [math]\beta_{local}^2[/math], derived immediately from the closure condition [math]\kappa^2=2\beta^2[/math]. 2. The term [math]\left(1 - \frac{2GM}{c^2 R}\right)[/math] is exactly the RG potential projection [math]\kappa_X^2=1-\kappa^2=cos^2(\theta_2)[/math] (which is the gravitational time dilation factor). So, your complex GR formula collapses into a remarkably simple ratio of projections: [math]\beta_{GR}^2 = \frac{\beta_{local}^2}{\kappa_X^2}[/math] Interpretation: Your derivation mathematically proves that the "coordinate velocity" measured from infinity ([math]\beta_{GR}[/math]) is simply the local invariant velocity ([math]\beta_{local}[/math]) scaled by the gravitational dilation ([math]\kappa_X[/math]). This confirms that RG generates the core dynamic invariant ([math]\beta_{local}[/math]) directly in two lines of algebra. GR requires a full metric derivation to obtain the same value, wrapped in the necessary coordinate transformations to relate it to a distant observer. This effectively demonstrates "Ontological Minimalism": RG yields the naked invariant, while GR dresses it in coordinate effects. Oh I think you'll find it interesting. I derived it just today after thinking "Is mass fundamental or maybe its just our human-centric artifact?..." I want to share this result with you that I suspect might be significant: Massless Orbital Reconstruction (S2 Star Test) Using the relational framework, it is possible to reconstruct the orbital dynamics (specifically precession) using only dimensionless observables, without ever knowing the Mass ([math]M[/math]), Gravitational Constant ([math]G[/math]), or even the physical size of the orbit ([math]a[/math]). Here is the step-by-step logic: 1. The Operational Inputs: We rely on two dimensionless ratios obtained from kinematics and astrometry: * Eccentricity ([math]e[/math]**):** The shape of the orbit. * Periapsis Velocity Projection ([math]\beta_p = v_p/c[/math]**):** The maximum redshift/Doppler shift at the closest approach. (Note: The absolute radius [math]a[/math] is not needed to find the precession angle, only to find the physical value of [math]R_s[/math] in meters). 2. The Derivation: In standard dynamics, velocity is governed by the Vis-Viva equation. In RG, we express this purely through relational projections. The kinetic projection [math]\beta^2[/math] at periapsis relates to the semi-major geometric potential [math]\kappa^2(a)[/math] as: [math]\beta_p^2 = \frac{\kappa^2(a)}{2} \left( \frac{2a}{r_p} - 1 \right)[/math] Since [math]r_p = a(1-e)[/math], the scale [math]a[/math] cancels out of the bracket, leaving only the shape [math]e[/math]: [math]\beta_p^2 = \frac{\kappa^2(a)}{2} \left( \frac{2}{1-e} - 1 \right)[/math] Solving for [math]\kappa^2(a)[/math] gives the defining relation purely in terms of [math]\beta_p[/math] and [math]e[/math]: [math]\boxed{\kappa^2(a) = \frac{\beta_p^2}{\frac{1}{1-e} - \frac{1}{2}}}[/math] **3. The Test (Star S2 around Sgr A*):** Let's plug in the observational data for S2: * [math]e \approx 0.88466[/math] * [math]v_p \approx 7.7 \times 10^3 \text{ km/s} \implies \beta_p \approx 0.02568[/math] First, we compute the geometric scale [math]\kappa^2(a)[/math] (which represents the ratio [math]R_s/a[/math]): [math]\kappa^2(a) = \frac{(0.02568)^2}{\frac{1}{0.11534} - 0.5} \approx \frac{6.59 \times 10^{-4}}{8.167} \approx 8.07 \times 10^{-5}[/math] Then, we compute the orbit-level displacement norm [math]Q_{orbit}^2 = \frac{3}{2}\kappa^2(a)[/math]: [math]Q_{orbit}^2 \approx 1.21 \times 10^{-4}[/math] Finally, the precession follows from the geometric closure: [math]\Delta\varphi = \frac{2\pi Q_{orbit}^2}{1-e^2} \approx \frac{2\pi (1.21 \times 10^{-4})}{1 - 0.88466^2}[/math] Result: [math]\Delta\varphi \approx 12.0 \text{ arcmin/orbit}[/math] Conclusion: This matches the GR prediction exactly. However, at no point did I use the Mass of the Black Hole ($4.3 \times 10^6 M_\odot$) or [math]G[/math]. The precession emerges directly from the dimensionless relationship between velocity ([math]\beta[/math]) and shape ([math]e[/math]). Mass is not a primary cause here; it is a secondary description of this geometric closure.