Anton Rize

@KJW, Great question! you are correctly describing the Optical Symmetry (Doppler Effect). Lets get on to it: The Twin Paradox: Optical Symmetry vs. Energetic Asymmetry While moving apart, both see the other redshifted. While moving together, both see the other blueshifted. So, where does the physical age difference come from? In WILL RG, the asymmetry is Energetic. Motion is not just a change of coordinates; it is a change of State allocation on the [math]S^1[/math] carrier. Here is the breakdown: 1. The Energy Audit (Breaking the Symmetry) The relationship is not symmetric because only one Twin changed their energy state relative to the system's origin. Twin A (Earth): Remains in the Low-Kinetic State ([math]\beta \approx 0[/math]). Their Total Energy is almost entirely projected onto the Time Axis ([math]\beta_Y \approx 1[/math]). They maximize their aging rate. Twin B (Traveler): Expends energy to accelerate. On the [math]S^1[/math] diagram, they rotate their state vector. They convert a portion of their 'Time Flow' ([math]\beta_Y[/math]) into 'Spatial Motion' ([math]\beta_X[/math]). 2. The Cost of Motion The 'Paradox' disappears when you realize that Velocity is not free. [math]\beta_Y = \sqrt{1 - \beta_X^2}[/math] Twin B 'bought' spatial velocity ([math]\beta_X[/math]) by spending temporal velocity ([math]\beta_Y[/math]). Twin A did not spend anything. 3. The Turnaround (The Reality Check) The moment Twin B turns around, they prove they are the Active Agent. They must expend energy to reverse their [math]\beta_X[/math]. This confirms that their deviation from the 'Time Axis' was real, not just a coordinate artifact. Conclusion: Twin B is younger not because of 'geometry' or 'paths', but because they spent a significant portion of their existence being motion rather than experiencing time. Symmetry exists in perception (light signals), but asymmetry exists in state (energy allocation).

@Mordred , these are excellent questions. You are getting to the core of the Generative Ontology. DIFFERENT ONTOLOGY AND PHILOSOPHY WARNING - all your life you was thinking about space as a container and time as a flow. Im suggesting a radical departure from this philosophy. Its not going to be easy to rearrange your thinking. I suggest starting from the question: "If there would be only one elementary particle in the hole Universe, would it have energy, velocity, position etc...? And if this particle would despiser what would left of the Universe?" We going back to debate Leibniz vs Newton. You are asking how I handle the 'fabric' of reality (distance, location, direction) without a pre-existing metric container ([math]ds^2[/math]). Here is the step-by-step mechanism of WILL RG: 1. Time Dilation Correlation (The Mechanism) In WILL RG, Time is not a dimension you travel through; it is the rate of process execution derived from available energy. Standard View: You move fast --> Your worldline tilts --> Time dilates. WILL RG View: You allocate energy to motion ([math]\beta[/math]) --> You have less energy for internal oscillation ([math]\beta_Y = \sqrt{1-\beta^2}[/math]) --> Your clock slows down. The correlation is instantaneous and intrinsic. [math]dt_{observer} = dt_{source} \sqrt{1-\beta^2}[/math] Distance didn't cause this. Energy allocation caused this. 2. Handling Distance and Location (No A Priori Space) You asked: "How do you handle distance?" In my model, Space is a consequence of Time, not a container for it. Distance is strictly defined as Signal Delay. [math]Distance \equiv c (t_{reception} - t_{emission})[/math] I do not assign coordinate [math](x, y, z)[/math] as a primary property. I assign a Relational Lag. If I say "Star A is at distance [math]L[/math]", I physically mean "The information from Star A is [math]L/c[/math] seconds old". The metric [math]ds^2[/math] in SR is simply a way to keep this causality consistent. In RG, I enforce causality directly via signal exchanges. 3. Acceleration and Direction (The [math]S^1[/math] Vector Rotation) You noted correctly: "The energy momentum equation doesn't contain the terms for change in direction." Exactly! And that is a feature, not a bug. On the kinematic carrier [math]S^1[/math] (Velocity Circle): Total Energy ([math]E[/math]) is the Radius. Momentum ([math]p[/math]) is the vector projection. If a spaceship turns at constant speed: Total Energy ([math]E[/math]) stays constant. Magnitude of Momentum ([math]|p|[/math]) stays constant. The Vector Components rotate on the [math]S^1[/math] circle. Changes in direction are treated as Phase Shifts on the carrier. The observer sees the projection of the spaceship change (e.g., transverse velocity becomes longitudinal), but the spaceship's internal state (Energy) remains invariant. 4. Multiple Observers (The "Private Reality" Principle) You asked: "If you have 2 observers and 1 emitter... they should be getting different results." Absolutely. And this is handled natively without a global coordinate system. Emitter (Spaceship): Has a single, invariant Total Energy State [math]E_{source}[/math]. Observer A: Receives the signal and projects it onto their Relational Axis (based on relative [math]\beta_A[/math]). They measure Energy [math]E_A'[/math] and Frequency [math]f_A'[/math]. Observer B: Receives the same signal but projects it onto their Relational Axis (relative [math]\beta_B[/math]). They measure [math]E_B'[/math] and [math]f_B'[/math]. There is no "God's Eye View" coordinate system where the spaceship "really is". There is only the Source State and the Projection of that state onto each observer's local reality. Lorentz Transformations are simply the translation rules between Observer A's projection and Observer B's projection. Summary: I replace the Coordinate Grid with a Network of Relations. Location --> Signal Phase/Delay. Distance --> Time Interval. Direction --> Projection Ratio on [math]S^1[/math]. The core principle that SPACETIME ≡ ENERGY means that the geometry of the universe is the manifestation of energy relations. Does this clarify how the "Spacetime = Energy" concept handles geometry? P. S. WILL-AI gave a lengthy and deep answer to your questions. Im not sure witch answer is better mine or he's: https://github.com/AntonRize/WILL/blob/63bc924f61891a434a976d303eca773cc21cf07e/assistant/logs/2026-02-19T02-48-41-070Z.md P.P.S. The fastest and easiest way to grasp the hole shebang is through this chart: https://willrg.com/LOGOS_MAP/

Agreed. please provide the list of empirically measured phenomena post diction of witch will be a sufficient evidence of predictive power, for you. Don't be shy. Any SR, GR but no grav waves please I don't want to sit here till morning. For now here's a quick derivation of Constant of Aberration: In standard SR, this requires calculating the angle transformation between reference frames using vectors. In WILL RG, this is treated as a projection of the observer's kinetic intensity onto the transverse signal carrier. Here is the derivation using strictly the algebraic relations of the [math]S^1[/math] carrier: 1. The Physical Setup Consider a star (e.g., Gamma Draconis) positioned at the true geometric zenith relative to Earth's orbital plane. Input Data (Empirical): Earth's mean orbital velocity [math]v \approx 29.78[/math] km/s. Relativistic Kinetic Projection ([math]\beta[/math]): [math]\beta = \frac{v}{c} = \frac{29.7827}{299792.458} \approx 0.0000993444[/math] 2. The Generative Logic (WILL RG) In my framework, "angles" are not primary entities. They are derived ratios of projections. When an observer possesses a kinetic projection [math]\beta[/math] (along the line of motion), it "contaminates" the purely transverse signal from the star. The observer perceives their own kinetic intensity as a longitudinal component in the incoming light. On the unitary circle [math]S^1[/math] (where the signal radius is always 1), the relationship between the Transverse Projection (Line of Sight) and the Kinetic Projection (Motion) is fixed by the Pythagorean theorem. The "Aberration Angle" [math]\alpha[/math] is simply the angle whose sine equals the kinetic projection: [math]\sin(\alpha) = \beta[/math] Note: There are no vectors here. This is a direct conversion of Energy ([math]\beta[/math]) into Geometry ([math]\alpha[/math]). 3. The Calculation [math]\alpha_{rad} = \arcsin(0.0000993444)[/math] [math]\alpha_{rad} \approx 0.0000993445 \text{ radians}[/math] Converting to arcseconds ([math]1 \text{ rad} \approx 206264.8"[/math]): [math]\alpha \approx 20.4913"[/math] 4. Comparison WILL RG Prediction: [math]20.491"[/math] Standard IAU Constant of Aberration: [math]20.495"[/math] The result is effectively identical (variance is due to averaging Earth's orbital eccentricity). Conclusion I have derived the exact observational consequence of Special Relativity (Stellar Aberration) using zero vectors, zero tensors, and zero coordinate transformations. I simply treated the observer's velocity as a spectral intensity ([math]\beta[/math]) and calculated its projection on the [math]S^1[/math] carrier. This answers your question: Yes, the methodology has the same predictive ability, but it achieves it with significantly fewer mathematical entities (Occam's Razor). Does this derivation satisfy your requirement for 'predictive ability'? If not - provide me the list of phenomena and I'll provide the derivations.

@Mordred Thank you for this detailed and rigorous comment. It is refreshing to see someone who clearly understands the Group Theory and the mathematical foundations of SR/GR. I appreciate the time you took to write out the Lorentz generators. I have a detailed derivation ready that addresses your questions regarding kinematic motion, vectors, and boosts solely through the algebraic relations of WILL RG. However, I suspect we are operating on different ontological premises. You seem to view the mathematical artifact (the transformation matrix/geometry) as the physics itself, whereas I view it as a descriptive tool for the physics. Before I provide my answer, I want to establish a clear criterion for its acceptance to avoid moving the goalposts later. I need to know if an answer that satisfies you is even possible within your current view. The Question: If I can demonstrate - using strictly algebraic relations of the S1/S2 projections - that I can derive the exact same empirical results as the Lorentz transformation (e.g., the exact numerical values for time dilation, aberration of light, and Doppler shift observed by a moving observer), but without invoking a 4D manifold or a transformation matrix... ...will you accept this as a valid physical solution? Or is your position that unless the specific mathematical formalism of the Lorentz Group (matrices/tensors) is used, the result is invalid regardless of its agreement with experiment? Put simply: Are we judging the theory by its ability to predict observable data (Empiricism), or by its adherence to a specific geometric tradition (Formalism)? Once you clarify your standard of evidence, I will post the derivation.

@Mordred Great! Now I can see that you actually red something. Let me answer your questions: The system is algebraically locked. So the change in 1 parameter inevitably triggering the cascade of changes in all codependent parameters in order to conserve invariants. Not quiet. This is a very common misreading. That's exactly why I'm emphasising in the document: https://willrg.com/documents/WILL_RG_I.pdf#rem:nonspatial "Throughout this paper, S^1 and S^2 are not to be interpreted as spacetime geometries. They are relational carriers that encode the closure, conservation, and isotropy of the transformational resource. Ordinary spatial and temporal notions are emergent descriptors of patterns within WILL.". Here's the ontology clarification: Ontological Status of the Relational Carriers [math]S^1[/math] and [math]S^2[/math] A natural question arises regarding the ontological status of the circle [math]S^1[/math] and the sphere [math]S^2[/math]: What are they, and where do they "exist"? The answer requires a shift in perspective. In WILL Relational Geometry, [math]S^1[/math] and [math]S^2[/math] are not spatial entities existing within a pre-defined container. They are the necessary relational architectures that implement the core identity [math]\text{SPACETIME} \equiv \text{ENERGY}[/math]. Energy as Relational Transformation Capacity Recall that energy is defined as the relational measure of difference between possible states. It is not an intrinsic property but a relational potential for change. It is never observed directly, only through transformations. The Carriers as Protocols of Interaction The Carriers [math]S^1[/math] and [math]S^2[/math] are the minimal, unique mathematical structures capable of hosting this relational "bookkeeping" for directional and omnidirectional transformations, respectively. They enforce closure, conservation, and symmetry by their very topology. Imagine two observers, [math]A[/math] and [math]B[/math]: * Observer [math]A[/math] is the center of their own relational framework. Observer [math]B[/math] is a point on [math]A[/math]'s [math]S^1[/math] (for kinematic relations) and [math]S^2[/math] (for gravitational relations). * Simultaneously, observer [math]B[/math] is the center of their own framework. Observer [math]A[/math] is a point on [math]B[/math]'s [math]S^1[/math] and [math]S^2[/math]. There is no privileged "master" carrier. Each observable interaction is structured by these mutually-centered relational protocols. The parameters [math]\beta[/math] and [math]\kappa[/math] are the coordinates within these relational dimensions, and the conservation laws (e.g., [math]\beta^2 + \beta_Y^2 = 1; \quad \kappa_X^2 + \kappa^2 = 1[/math]) are the innate accounting rules of these protocols. So its like Im looking at you and asking how is your state is different from mine? Because my velocity and potential in my rest frame are always [math](\beta, \kappa)= (0, 0) [/math] - Im always the origin of my relational frame. Therefor I can map your state reletive to mine as a point on [math](\beta, \kappa) [/math] plane. And because the same rules applies to you, you can map me as a point on [math](\beta, \kappa) [/math] plane of your relational frame. So the total relational shift [math]Q=\sqrt{\beta^2+\kappa^2}[/math] stays invariant between frames. And that's what Principal of Relational Reciprocity is all about. Its way simpler then boosts, spacetime intervals, geodesics, Christoffel symbols, tensors etc... Im not really using the concept of force but we can translate in to force if you want: [math]\frac{F}{E_{0}} = \frac{R_{s}}{2r_{o}(o)^{2}} \equiv \frac{\kappa_{o}(o)^{2}}{2r_{o}(o)}[/math] Where: [math]\frac{F}{E_{0}} = \frac{R_{s}}{2r_{o}(o)^{2}} \equiv \frac{\kappa_{o}(o)^{2}}{2r_{o}(o)}[/math] Where: [math]E_{0}[/math] = rest energy [math]o[/math] = orbital phase in radians [math]\kappa_{o} = \sqrt{\frac{R_{s}}{r}} = 1-(1+z_{ko}(o))^{-2}[/math] (local potential projection at phase [math]o[/math]) [math]r = r_o(o)= a\frac{1-e^{2}}{1+e\cos o} = \frac{R_s}{\kappa_o^2}[/math] (radial distance at phase [math]o[/math]) [math]R_{s} = \kappa^{2}a = \frac{2Gm_{0}}{c^{2}} = \frac{2}{3}Q_{o}(O_{o})^{2}a = \frac{r_{1} r_{2}}{r_{2} - r_{1}}(\beta_{1}^{2} - \beta_{2}^{2}) =[/math] [math] \frac{a}{2}(3-\sqrt{1+8\tau_{Wo}(O_{o})^{2}}) = \frac{r_{o}(o)}{2(2a-r_{o}(o))}(4a-r_{o}(o)-\sqrt{(4a-r_{o}(o))^{2}-8a(2a-r_{o}(o))(1-\tau_{Wo}(o)^{2})})[/math] (Schwarzschild radius - system scale) [math]z_{ko}(o) = \frac{1}{\kappa_{Xo}(o)}-1[/math] (gravitational redshift at phase o) I guess I already answered this one above with relational reciprocity. In your case you jumping to conclusions about the scope without any proper reasoning. Look I understand its a lot to take in and you dont have to if you dont want. The only thing I demand is scientific honesty. So jumping in to conclusion after misreading few pages is scientifically dishonest. Thats ok we tried second time and result is the same. Im just fascinated how can't you see that all your arguments are based on just an interpretation. Regardless... this is mathematical and empirical fact [math]\Delta_{precession} = \frac{3\pi}{2} (\frac{R_{ratio}}{\beta_{p}} (1-(1+z_{sun})^{-2}))^2=5.0175347157\times10^{-7}[/math] if you instead of facing it and openly discuss it prefer to hide behind dogmatic thinking - its your choice. I dont get it. What's your reasoning here? Dont you think that if you dont see it doesnt mean that it doesnt exist? Dont you think that look again or ask me would be more reasenoble then jump in to conclusion based on you glassing through the paper? If thats how you do science Id better doublecheck the accuracy of your cosmic calculator...

@KJW Im so exited!! Check this out: So on my website im testing WILL-AI trained on my papers. And I asked him your question. And here's what he answered: The derivation of the Schwarzschild radius [math]R_s[/math] without invoking the gravitational constant [math]G[/math] or mass [math]M[/math] in WILL Relational Geometry (RG) comes directly from the principle of the [b]Conservation of the Relational Invariant[/b] ([math]W[/math]). This principle states that for any closed system, the specific energy difference between the potential ([math]\kappa^2[/math]) and kinematic ([math]\beta^2[/math]) projections remains constant throughout the orbit (doc. Part I - Section Energy-Symmetry Law). Here's the step-by-step derivation, as presented in WILL Part I (Section 15.2): 1. [b]The Relational Invariant:[/b] The core of this derivation is the conserved energy invariant [math]W[/math]. For a bound system, the specific energy invariant (per unit rest energy) is given by: [math]W = \frac{1}{2}(\kappa^2 - \beta^2)[/math] This invariant is conserved at every point along an orbit. 2. [b]Applying the Invariant to Two Points:[/b] Consider any two arbitrary points, 1 and 2, along the trajectory of an orbiting object. Since [math]W[/math] is conserved, the invariant at point 1 must equal the invariant at point 2: [math]\frac{1}{2}(\kappa_1^2 - \beta_1^2) = \frac{1}{2}(\kappa_2^2 - \beta_2^2)[/math] We can simplify this by canceling the factor of [math]1/2[/math]: [math]\kappa_1^2 - \beta_1^2 = \kappa_2^2 - \beta_2^2[/math] Rearranging the terms to group potential and kinematic projections: [math]\kappa_1^2 - \kappa_2^2 = \beta_1^2 - \beta_2^2[/math] 3. [b]The Potential Projection ([math]\kappa^2[/math]) Definition:[/b] In WILL RG, the potential projection [math]\kappa^2[/math] is defined as the dimensionless ratio of the system's Schwarzschild radius ([math]R_s[/math]) to the current radial distance ([math]r[/math]) (doc. Part I - Section Potential Energy Projection on [math]S^2[/math]): [math]\kappa^2 = \frac{R_s}{r}[/math] It's crucial to understand that [math]R_s[/math] here is treated as a geometric scale factor, not as a parameter directly dependent on [math]G[/math] or [math]M[/math]. Its value is inferred from relational observations. 4. [b]Substitution and Algebraic Solution:[/b] Now, substitute the definition of [math]\kappa^2[/math] into the invariant equation: [math]\frac{R_s}{r_1} - \frac{R_s}{r_2} = \beta_1^2 - \beta_2^2[/math] Factor out [math]R_s[/math] from the left-hand side: [math]R_s \left( \frac{1}{r_1} - \frac{1}{r_2} \right) = \beta_1^2 - \beta_2^2[/math] Combine the terms within the parentheses: [math]R_s \left( \frac{r_2 - r_1}{r_1 r_2} \right) = \beta_1^2 - \beta_2^2[/math] Finally, solve for [math]R_s[/math]: [math]R_s = \frac{\beta_1^2 - \beta_2^2}{\frac{r_2-r_1}{r_1 r_2}} = \frac{r_1 r_2}{r_2 - r_1}(\beta_1^2 - \beta_2^2)[/math] This derivation (doc. Part I - Section 15.2, Theorem 15.2) shows that [math]R_s[/math] can be determined purely from two measurements of radial position ([math]r_1, r_2[/math]) and their corresponding kinematic projections ([math]\beta_1, \beta_2[/math]). These kinematic projections ([math]\beta = v/c[/math]) are directly observable through Doppler shifts, and radii ([math]r[/math]) are observable through astrometry. No information about the central mass [math]M[/math] or the gravitational constant [math]G[/math] is required as input for this calculation. [math]G[/math] and [math]M[/math] become output calibration tools if one wishes to translate [math]R_s[/math] into conventional units, rather than fundamental inputs (doc. Part I - Section The Role of G as Translation Constant). You can explore this method and test it yourself using the interactive Desmos calculators available on the [Relativistic Foundations](https://willrg.com/relativistic-foundations/) page of the WILL RG website. I don't think I could of explain better myself... Isn't it cool!!?? Im so proud of my little guy...

Hey welcome back! I derived it like this: https://willrg.com/documents/WILL_RG_I.pdf#thm:two_point There's also two other methods you will find Method B: Geometric Resonance (Balance Point Method): [math] R_s = \frac{3a \pm a\sqrt{1 + 8\tau_W^2}}{2} = \frac{a}{2}(3 \pm \sqrt{1 + 8\tau_W^2}) [/math] and Method C: Instantaneous (Arbitrary Phase Method): [math] R_s = \frac{r}{2(2a-r)} \left( (4a-r) \pm \sqrt{(4a-r)^2 - 8a(2a-r)(1-\tau_{Wo}^2)} \right) [/math] I tested all of them numerically on real data - works like a charm. I couldn't find anything similar anywhere. Do you think its new? So this one is interesting... Somehow im not using any space vectors in my system at all. Im not using any coordinate grid its all intrinsic parametrization. Its pure relationalism just like Leibnitz, Mach and Einstein always wanted. Im dealing with relational states of the objects in the system. And the question is not where the object is moving but where the object is allowed to be. And it turns out that there's only one legit solution for any given moment/state that satisfy all the constrains. Thats why I calling it WILL because there's 0 freedom whatsoever. Its an ironic joke... Take a look at my Closed Algebraic System of Relational Orbital Mechanics (R.O.M.): https://willrg.com/documents/WILL_RG_I.pdf#eq:rom There's no space vectors involved.

@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: