Anton Rize

Sorry guys for late reply. Im back now and I decided to commit to the open research concept fully. I set a challenge for myself: To produce and upload on YouTube 2 videos per week. So Im happy to share with you the first video from this challenge: https://www.youtube.com/watch?v=6YkDZGLLxnY Also this is probably my best desmos project so far https://www.desmos.com/calculator/mjen4ms452 Would you agree with my Corollary?: Corollary (Epistemic Mandate and Ontological Redundancy): In information theory and formal logic, if a parameter is strictly absent from the complete algebraic generative chain of a system, its reintroduction constitutes an epistemic violation. Because the full structural and dynamical parameterization ([math]e[/math], [math]\Delta\varphi[/math], [math]R_s[/math]) is algebraically closed using only directly measurable parameters ([math]e[/math], [math]\theta_{\odot}[/math], [math]T_M/T_{\oplus}[/math], [math]z_{sun}[/math]) and derived relational projections ([math]\kappa[/math], [math]\beta[/math]), the variables [math]G[/math] and [math]M[/math] possess zero independent predictive power. They are not fundamental primitives. Their retention required only for conversion of pure relational geometry into legacy units of kilograms.

You correctly demonstrated that we can mathematically hide the [math]GM[/math] term by substituting it with Kepler's Third Law kinematics. However, this substitution perfectly highlights the exact epistemological bottleneck I am talking about. Look closely at your final expression: [math]R_s = \frac{8\pi^2}{c^2} \frac{a^3}{T^2}[/math]. 1. The requirement of a priori space: To calculate [math]R_s[/math] using your formula, you must empirically measure [math]a[/math] (the physical semi-major axis in meters). How do you measure [math]a[/math]? You must rely on the cosmic distance ladder (parallax, radar ranging). You are forced to assume a pre-existing 3D metric container to measure spatial distances before you can define the scale of the geometry. 2. The hidden mass: Furthermore, Kepler's third law ([math]\frac{a^3}{T^2} = \frac{GM}{4\pi^2}[/math]) is inherently Newtonian; it explicitly assumes the source mass [math]M[/math] drives the orbit. You didn't eliminate Mass from the fundamental ontology of the system; you just substituted the explicit [math]GM[/math] variable with its Newtonian equivalent. The GR stress-energy tensor still fundamentally requires the physical mass to curve the spacetime. In stark contrast, look at the WILL RG Chrono-Spectroscopic equation: [math]R_s = T c \frac{\kappa^2\beta}{2\pi}[/math] There is no spatial distance [math]a[/math] here. There are no meters. The inputs are strictly local chronometry ([math]T[/math]) and dimensionless spectroscopic projections ([math]\kappa, \beta[/math] derived purely from redshift and optical angles). WILL RG doesn't need to measure spatial distances to find the scale. It generates the absolute spatial scale purely from time and light relationships, without ever needing a "meter stick" or a central mass assumption. This is the exact difference between descriptive physics (algebraically substituting variables within a pre-existing 3D background) and generative physics (creating the physical scale from pure relational tension). Here's the desmos project: https://www.desmos.com/calculator/iymnd3tw3z (P.S. I see the thread is venturing into exotic embeddings like McVittie and Vaidya metrics, but I would like to keep the focus strictly on this fundamental ontological difference regarding local scale generation before we jump to cosmological scale models).

Yes I see where you pointing at. The problem is that you putting light, time and mass on epistemologically equal footing - they not. Light, time - directly measurable. mass - model output. They are on vastly different epistemological levels. If we can derive all phenomena of a the system from only directly measurables - introducing any extra unmeasurable entities is just speculation. What was my derivation about and how do you understand it? In the end its about this derivation your comments supposed to be. But with you its hard to tell...

@Mordred , So you just incapable of admitting your own mistakes? I see... Non of this has anything to do with my derivation or your clear misunderstanding of it: At this point you only making it worse. You basically forcing me to ignore you. Is that what you trying to achieve?

I think there is a slight, but very important, misunderstanding here. The [math]M \sin(i)[/math] degeneracy has absolutely nothing to do with galaxy rotation curves, Modified Gravity (MOND), or Dark Matter. It is a strictly local, classical problem in orbital mechanics and stellar kinematics (specifically, radial velocity measurements of binary stars and exoplanets). When we observe a star orbiting a companion (like the S0-2 star orbiting the supermassive compact object at the center of our galaxy, which I used in my data), we don't know its incantation (the orbital tilt), we can only measure its 1D line-of-sight velocity via the Doppler shift. Standard methods state that we cannot disentangle the true orbital velocity from the inclination angle ([math]i[/math]) of the orbital plane. They are locked in the formula [math]K \propto v \sin(i)[/math] where [math]K=\frac{\beta}{\sqrt{1-e^{2}}}\sin(i) [/math]. So currently in order to fully determent the orbital system we have to make assumptions about its distance and relay on less accurate optical data sources. Its a concrete limitation of our current observational methods and theoretical models. My algebraic derivation resolves this exact orbital problem strictly through the geometric asymmetry of the transverse baseline at the apsides, isolating the true velocity and the inclination angle using only 1D spectroscopic extrema. I understand your stance on Dark Matter and the Bullet Cluster, and I am not asking you to abandon your wholehearted faith in GR or to evaluate cosmology. I am asking you, as someone who understands pure math and standard orbital geometry, to look at the solution I provided in the previous post. It is pure kinematics and trigonometry. Since you have a keen eye for mathematical consistency, your audit of this specific orbital derivation would be highly valued. Read the exact Wikipedia text you just posted: "...where the source of curvature is the stress–energy tensor (representing matter...)". You are confusing Test Mass ([math]m[/math]) with Source Mass ([math]M[/math]). 1. Test Mass ([math]m[/math]): Drops out of the geodesic equation in freefall. We agree. 2. Source Mass ([math]M_{sun}[/math]): Is absolutely required in GR to generate the stress-energy tensor that curves the spacetime in the first place. In standard GR, you cannot calculate the absolute scale of the geometry ([math]R_s[/math]) without knowing the central mass [math]M[/math] and [math]G[/math]. My derivation calculated the exact absolute scale of the Solar System ([math]R_s \approx 2953.3[/math] m) using strictly local clock ticks and light shifts - **completely bypassing the stress-energy tensor, [math]M_{sun}[/math], and [math]G[/math]**. You argued that the falling object's mass is irrelevant. I never said it was. I showed that the Central Star's mass is not a necessary primitive to generate the geometry. I don't know how to communicate with you and I can't see any reasons why I should. Since you are arguing against the very text you are quoting, I will leave it at that.

Since you explicitly invited me to apply GR to prove you wrong, I will do so. Your statement: "Under SR/GR all particles that are following a geodesic are in freefall. So mass is irrelevant" demonstrates a fundamental confusion between the test mass m and the source mass M. In GR, the trajectory of a free-falling test particle is governed by the geodesic equation. You are correct that the mass of the falling particle m does not appear here. However, the Christoffel symbols are strictly defined by the metric tensor. In standard General Relativity, the geometry of spacetime is dependent on M (the mass of the central body). Without M, the metric reduces to flat Minkowski space. You cannot determine the absolute scale of the system (R_s) in GR without knowing M and G. My Chrono-Spectroscopic Theorem demonstrated exactly the opposite: the absolute system scale (R_s = 2953.3 m for the Sun) can be derived strictly from chronometry and spectroscopy, without ever invoking the central mass M_{sun} or the constant G. You confused the mass of the observer with the mass of the system. I suggest before you commenting read what you actually attempt to critic. Throughout this long dialog you keep showing your absolute disrespect to me by ignoring the materials presented. This remains the main barrier in communication between us.

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

These are excellent, fundamental questions and one very important statement. When I say that "mass is not a necessary primitive," I am not just speaking philosophically; I mean it in a strict, operational, mathematical sense. In standard physics, mass ([math]M[/math]) is treated as a fundamental building block of reality. You need to plug [math]M[/math] (along with the gravitational constant [math]G[/math]) into equations to figure out the gravitational scale of a system, such as the Schwarzschild radius [math]R_s = \frac{2GM}{c^2}[/math] and other gravitational phenomenon. But what if we could derive the exact same absolute scale of a stellar system without ever knowing its "Mass" and without ever using [math]G[/math]? If the geometry of the system can be completely solved using only clocks and light, then Mass is not a fundamental pillar of reality. So mass is a bookkeeping label we paste on later for convenience. The phenomena themselves are governed by the single observable length R_s. Here is the formal proof of this concept from my research. I call it the Chrono-Spectroscopic Theorem. It proves that the absolute system scale is generated exclusively from pure chronometry (time) and spectroscopy (light shifts): The Epistemological Bottleneck of Spatial Distance In classical orbital mechanics, the absolute gravitational scale [math]R_s[/math] requires [math]G[/math] and mass [math]M[/math], which in turn require measuring distances in meters (tethering physics to the cosmic distance ladder). In Relational Orbital Mechanics (R.O.M.), physical distance is not an a priori container; distance emerges macroscopically as the geometric "tension" between two relational energetic potentials. The Theorem By integrating the local relational spacetime factor over a closed phase interval, the absolute scale of the system [math]R_s[/math] decouples entirely from spatial coordinates and mass. For a complete orbital cycle, it algebraically collapses to: [math]R_s = T c \frac{\kappa^2 \beta}{2\pi}[/math] Here, [math]\kappa^2[/math] and [math]\beta[/math] are the global potential and kinematic projections. The absolute scale is derived exclusively from Chronometry (orbital period [math]T[/math]) and Spectroscopy (light shifts), eliminating any dependency on [math]G[/math], [math]M[/math], or spatial parallax. Empirical Validation (The Solar System) We validate this using Mercury's state at perihelion. We extract the necessary parameters entirely from Earth-based dimensionless observables: Direct Optical Radius ([math]\theta_{\odot}[/math]): The angular radius of the Sun ([math]\approx 0.004652[/math] rad). Chronometric Scaling ([math]T_M / T_{\oplus}[/math]): The ratio of orbital periods ([math]\approx 0.3871[/math]). Kinematic Eccentricity ([math]e[/math]): Derived from angular velocity extrema ([math]e \approx 0.2056[/math]). The pure relational scale factor at perihelion is computed without any reliance on [math]G[/math], [math]M[/math], or the Astronomical Unit (meters): [math]R_{ratio} = \frac{\theta_{\odot}}{(T_M / T_{\oplus})^{2/3} (1-e)} \approx 0.01512[/math] Using the solar gravitational redshift ([math]z_{sun} = 2.1224 \times 10^{-6}[/math]) and Mercury's transverse kinematic shift ([math]\beta_p = 1.967 \times 10^{-4}[/math]), the global invariants are recovered algebraically. Substituting these strictly relational observables into the R.O.M. absolute scale equation yields: [math]R_s \approx 2953.3[/math] m This result perfectly matches the classical derivation [math]\frac{2GM_{sun}}{c^2}[/math], yet it is achieved strictly through internal system clocks ([math]T[/math]) and spectroscopic shifts ([math]\beta_p, z_{sun}[/math]). Conclusion: WILL RG is operationally independent from mass and [math]G[/math]. The physical scale of a closed orbital system is algebraically equivalent to the ratio of geometric tension ([math]\kappa^2\beta[/math]) to local clock ticks ([math]T[/math]). This is what I mean when I say mass is not a necessary primitive. The geometry works without it. To make it absolutely transparent and easy to verify here's locked and loaded desmos project: https://www.desmos.com/calculator/iymnd3tw3z Here's a full section: https://willrg.com/documents/WILL_RG_I.pdf#sec:absolute_scale Here's more evidence of mass and G independence: https://willrg.com/documents/WILL_RG_R.O.M..pdf#sec:operational

Im having the same feeling. One of my absolute favorite results RG research led me to is: "Mathematical complexity is the symptom of philosophical negligence." it is the logical conclusion that come up naturally after proving 2 theorems https://willrg.com/documents/WILL_RG_Substantialism_vs._Relationalism.pdf Or if you prefer more visual approach its in the end of this Logos Map https://willrg.com/logos_map/ And regarding your question about "there's no massless particles" - its just poor choice of words on my side. I was trying to say that the lensing results once again suggesting that mass as a concept is not a necessary primitive. This result come up multiple times in my research. When Ill get a bit more time Ill put them all together for a comprehensive review for all of us. And I know it sounds crazy at first but then when you start to think about it, it becomes so clear... Its remarkable how we are giving absolute different answers to the same question at the same exact time! 😄 By the way you might find this interesting. Its a little hint on 3D stricture of reality I got in my derivations: Geometric Signature of Spatial Dimension A striking topological feature emerges when we express the effective vacuum density in natural geometric units. Substituting [math]\rho_{\Lambda} = \frac{2}{3}\rho_{\max}[/math] into the explicit definition of [math]\rho_{\max}[/math]: [math]\rho_{\Lambda}(r) = \frac{2}{3} \frac{c^2}{8\pi G r^2} = \frac{c^2}{12\pi G r^2}[/math] Stripping away dimensional scaling factors ([math]c, G, r[/math]) reveals a purely dimensionless geometric coefficient: [math]\hat{\rho}_{\Lambda} = \frac{1}{12\pi} = \frac{1}{3 \times 4\pi}[/math] This factorization suggests a profound geometric origin for 3D space: * The factor [math]4\pi[/math] represents the intrinsic capacity of the relational carrier [math]S^2[/math]. * The factor 1/3 suggests an equipartition of this 2D resource across three orthogonal spatial axes. This hints that the dimensionality of observable space is not arbitrary but is a structural consequence of distributing the [math]S^2[/math] energy budget into a volume. The full section you can find in here: https://willrg.com/documents/WILL_RG_II.pdf#sec:dark_energy

I apologize if I misinterpreted your intent. I actually make a genuine effort to separate the mathematical models from the physical reality as well, so that was definitely not my intention. To help improve our communication, I want to point out exactly what type of wording I read that way. When you write something like: This specific wording is what would lead me to interpretation of your stance as treating the mathematical tool (coordinates) as the creator of physical reality. If that wasn’t your intent, then it is simply a semantic misunderstanding between us, and I am glad we cleared it up. I am looking for the fundamental answers to those exact same questions! However, I don't currently have enough evidence to rule out other descriptions in favor of strict priors like "spacetime must be 4D" or "the metric must have this specific signature." My fear is that if I adopt such restrictive assumptions too early, I might inadvertently rule out the unknown Truth I am seeking, simply because the field of search was artificially narrowed by the form of the question itself. Anyway, this is philosophy, and we could spend years debating it with zero tangible results. I have a much more solid question, or rather, a direct request for your help: These recent results of mine regarding the M sin(i) degeneracy might be a huge deal. This degeneracy is considered mathematically unsolvable using standard 1D linear projection methods. I remain inherently skeptical of my own work and the idea that I might have found a solution. However, I cannot find any mistakes in the derivation, and the empirical data aligns perfectly with the predictions. I would be incredibly grateful if you could bring your rigorous analytical skills to this specific derivation. Could you help me figure it out? Either by finding the mathematical mistake I might be making, or confirming that you couldn't find any. Solving this definitively will affect astrophysics substantially, allowing us to extract much more true kinematic data from the same radial velocity signals. And potentially give us tools for solving the Hubble tension (maybe). At this fundamental level, we are in absolute, 100% agreement.

We can boil down the idea to a single question: If the Universe consists only from one single object - would this object "poses" parameters like mass, energy, position, velocity, etc... and if then this object would vanish what would left of this Universe?

@chron44 I believe the quote that you looking for is this one: Also I have to worn you, my experience on this forum I can describe as "Fundamental Church of container space and its holy trinity (x,y,z)) ". Einstein was deeply inspired by Mach's relational ideas and couldn't come to terms with postulated manifold in GR as form of background that preexists interactions and relations of mass and energy.

@KJW , thank you for detailed and deep respond. I understand and respect your point. But I have to disagree with it. Let's examine the logical structure of this ontological defense: You outlined a linear dependency in General Relativity: The spacetime manifold exists $\to$ we postulate a metric tensor field upon it $\to$ spatial curvature mathematically emerges from this tensor. I understand this chain perfectly. However, the critical epistemological flaw in your methodology is revealed in this exact sentence: "the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact." This is a textbook logical leap known as reification (treating an abstract mathematical model as a physical entity), and it is the exact anthropocentrism my methodology filters out. In the scientific method, a successful postulate remains a model; it does not magically transform into a physical "fact" just because it aligns with human intuition of "empty space." As for your comment on covariant symmetries naturally splitting spacetime: the reason you never see me struggling to enforce covariance is because Relational Geometry is natively background-independent. The problem of non-covariant relations you face is an artifact of your initial ontological choice to use coordinates. When you discard coordinates and deal strictly with energetic capacities, covariance is absolute by definition. (You can see the precise ontological difference on my logic map: https://willrg.com/logos_map/). But philosophy will not give us concrete results, physics and math - will. My claim is not merely that RG can reproduce GR's post-Newtonian results algebraically. My claim is that RG delivers empirically testable, closed-form analytical results where standard mechanics cannot. Let's look at a categorical application: Title: Analytical Resolution of the M sin(i) Degeneracy via Transverse Baseline Asymmetry Over the past months, we have debated the empirical results of Relational Geometry and Relational Orbital Mechanics (R.O.M.). Since the discussion has reached a consensus on the predictive accuracy of the method for deflection, I am presenting the formal algebraic proof of how R.O.M. breaks the classical [math]M \sin i[/math] degeneracy in closed form, using only 1D spectroscopic extrema. In classical radial velocity analysis, the semi-amplitude [math]K \propto \beta \sin i[/math] leaves the true kinematic projection [math]\beta[/math] and inclination [math]i[/math] degenerate. R.O.M. resolves this by restoring the systemic transverse baseline [math]Z_{sys}[/math], which depends strictly on [math]\beta[/math] and is completely independent of [math]i[/math]. 1. The Observer Equations at Extrema [math]\kappa^2 = 1-\frac{1}{(1+z_{\kappa})^2}[/math] ([math]z_{\kappa}[/math] = gravitational redshift) [math]\beta^2 = 1-\frac{1}{(1+z_{\beta})^2}[/math] ([math]z_{\beta}[/math] = transverse Doppler shift) Observational Z Inputs [math]Z_{sys}(o) = (1+z_{\kappa o}(o))(1+z_{\beta o}(o)) = \tau_{Wo}(o)^{-1}[/math] (product of gravitational redshift and transverse Doppler shift) [math]\tau_{Wo}(o) = \kappa_{Xo}(o)\beta_{Yo}(o) = (Z_{sys}(o))^{-1}[/math] (product of projectional phase factors on [math]S^1[/math] and [math]S^2[/math] carriers) [math]z_{\kappa} = \frac{1}{\kappa_{X}}-1[/math] (gravitational redshift) [math]z_{\beta} = \frac{1}{\beta_{Y}}-1[/math] (transverse Doppler shift) The raw spectroscopic shifts at maximum and minimum radial velocity (phases [math]o = -\omega_i[/math] and [math]o = \pi - \omega_i[/math]) are products of the line-of-sight Doppler projection and the transverse baseline [math]Z_{sys}[/math] (denoted here as [math]D_{max}[/math] and [math]D_{min}[/math]): [math]D_{max}(\beta, e, \omega_i) = \sqrt{1 - 2\beta^2\frac{1+e\cos\omega_i}{1-e^2}} \sqrt{1 - \beta^2\frac{1+e^2+2e\cos\omega_i}{1-e^2}}[/math] [math]D_{min}(\beta, e, \omega_i) = \sqrt{1 - 2\beta^2\frac{1-e\cos\omega_i}{1-e^2}} \sqrt{1 - \beta^2\frac{1+e^2-2e\cos\omega_i}{1-e^2}}[/math] The observed extrema are: [math]Z_{rawmax} \cdot D_{max} = 1 + K_i(1+e\cos\omega_i)[/math] [math]Z_{rawmin} \cdot D_{min} = 1 - K_i(1-e\cos\omega_i)[/math] 2. Algebraic Decoupling (The Decryption Invariant) Subtracting the equations isolates the observed semi-amplitude: [math]2K_i = Z_{rawmax} D_{max} - Z_{rawmin} D_{min}[/math] Adding the equations and substituting [math]2K_i[/math] back into the sum yields a strict algebraic invariant where the inclination angle [math]i[/math] is completely eliminated: [math]Z_{rawmax} D_{max} (1 - e\cos\omega_i) + Z_{rawmin} D_{min} (1 + e\cos\omega_i) = 2[/math] This equation proves that the true kinematic projection [math]\beta[/math] and the argument of periapsis [math]\omega_i[/math] are locked strictly by the asymmetry of the observed extrema [math]Z_{rawmax}[/math] and [math]Z_{rawmin}[/math], regardless of the viewing angle. 3. Analytical Extraction of sin(i) Once [math]\beta[/math] is constrained by the invariant above, the true inclination is trivially extracted without invoking standard metric priors, G, or M: [math]\sin i = \frac{\sqrt{1-e^2}}{2\beta} \left[ Z_{rawmax} D_{max}(\beta, e, \omega_i) - Z_{rawmin} D_{min}(\beta, e, \omega_i) \right][/math] 4. Empirical Challenge I have verified this algebraic closure against the S0-2 (GRAVITY) dataset and synthetic 1PN data. The math holds absolutely. I invite anyone to find an algebraic flaw in the derivation from Step 1 to Step 3, or to explain how a supposedly "fundamental" physical degeneracy can be resolved through pure relational geometry if the classical 1D linear projection framework is complete. * Full derivation: https://willrg.com/documents/WILL_RG_I.pdf#sec:analytical_sini * ROM parameters and equations: https://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom * Detailed results analysis: https://willrg.com/msini_test * Colab notebook Synthetic: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/ROM_Validation_via_Synthetic_1PN_Data.ipynb Colab notebook real data: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/ROM_S2_SOLVER.ipynb Colab deep analyses: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/beta_i_comparison.ipynb So far it seems to me that RG delivers more and clearer results while requiring less.

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

Dark lensing: [math] \theta_E = 2 \frac{D_{LS}}{D_S} \arcsin( \frac{(\kappa_{bar}^2(\theta_E) + \kappa_{phantom}^2(\theta_E)) (1 + \beta_p^2)}{2\beta_p^2 - (\kappa_{bar}^2(\theta_E) + \kappa_{phantom}^2(\theta_E)) (1 + \beta_p^2)} )[/math] Full derivation: https://willrg.com/documents/WILL_RG_II.pdf#sec:lensing Colab notebook: https://colab.research.google.com/github/AntonRize/WILL/blob/main/Colab_Notebooks/Dark_Lensing.ipynb OUTPUT: WILL RG AB INITIO PREDICTIONS (Zero Free Parameters) LensID sigma_obs sigma_pred theta_E_obs theta_E_pred J0037-0942 279 282.11 1.53 1.49 J0216-0813 333 286.12 1.16 0.78 J0737+3216 323 285.34 1.00 0.95 J0946+1006 287 282.96 1.43 1.37 J0956+5100 334 296.55 1.33 1.15 J1250+0523 252 268.53 1.13 1.37 J1430+4105 322 292.07 1.52 1.14 J1627-0053 290 298.59 1.23 1.46 ----------------------------------------------------------------- Kinematics Mean Absolute Error: 23.02 km/s Lensing Mean Absolute Error: 0.195 arcsec -----------------------------------------------------------------I guess one could call it a mic drop 😁

This once again shows your complete lack of engagement with presented materials. For the last 4 pages on this forum I was presenting evidence that mass as a concept is not necessarily primitive. But how would you know if you only glance at my posts and then going back to make the same logically false arguments over and over again never admitting your mistakes and never clearly stating your own falsification conditions. As I told you before: without the clear answer we can't move any further. Sorry but I will have to ignore you until you will provide a clear unswear. This once again shows your complete lack of engagement with presented materials. For the last 4 pages on this forum I was presenting evidence that mass as a concept is not necessarily primitive. But how would you know if you only glance at my posts and then going back to make the same logically false arguments over and over again never admitting your mistakes and never clearly stating your own falsification conditions. As I told you before: without the clear answer we can't move any further. Sorry but I will have to ignore you until you will provide a clear unswear.

@KJW @MJ kihara (and everyone following the mathematical results) Guys lets put aside our personal preferences and beliefs and engage with the results that this research provides us so far. The guiding principle of this research is Ontological Minimalism: any physical phenomenon must be interpreted using the absolute minimum number of primitives required for its complete description and prediction. In standard General Relativity, to explain the the observed phenomena, one must postulate several heavy ontological primitives: * Mass (as an intrinsic substance). * A background 4D spacetime manifold. * The metric tensor. * Spatial curvature as a distinct geometric entity. However, the consistent quantitative convergence of this research (including the Python script I just presented) demonstrate a hard, undeniable fact: these are not necessary primitives. The same empirical predictions are generated purely from the algebraic distribution of relational projections ([math]\kappa[/math] and [math]\beta[/math]) on closed carriers, without ever invoking a spatial metric, a manifold, or spatial curvature. When a theoretical model relies on constructs beyond the minimal necessary primitives, it inevitably introduces non-physical, artifactual structures. The concept of an "empty 3D space," a "background time axis," or a "curved fabric" are legacy concepts rooted deeply in anthropocentrism. They are projections of how the human brain biologically parses its environment, not fundamental physical entities. My methodology acts as a strict epistemic filter against this anthropocentrism. If WILL Relational Geometry can algebraically generate the same empirical results strictly from the depletion of the internal phase buffer ([math]\beta_Y \to 0[/math] directly measurable transvers Doppler shift [math]z_b=\frac{1}{\beta_Y}-1[/math]) - without using a single metric tensor or spatial curve - then by the fundamental rules of the scientific method (Occam's Razor), spatial curvature is an ontological ghost. It is a mathematical over-parameterization of a purely energetic relation. We do not need to imagine Earth moving in a "helical trajectory in 4D spacetime." We only need to measure the total relational shift ([math]Q[/math]) between energetic states. The math works. The extra ontology is just cultural baggage.

First, thank you for your honesty. Hearing "I see value in it" from someone who is actively and rigorously scrutinizing the math means a great deal to me. I am not looking for easy agreement; ruthless mathematical scrutiny is exactly what RG needs. This is exactly what I have been looking for, and it is the highest form of respect one researcher can show another. Thank you! You are 100% correct. My assumption that the factor of 2 arose simply because light is "massless" (and hardcoding the 1/2 partitioning factor for massive bodies) was a critical error. As you rightly pointed out, a massive object traveling at 0.999999*c must asymptotically approach the deflection of light, which my previous discrete geometric separation failed to do. Further more without the graduate rebalancing of the weight of itch axis the hole geometric structure goes haywire and energy comes from nothing go nowhere. So I spend pretty much all day deriving it. It was way harder than I expected but in the end something incredible happened. I don't know how to prove it but guys I swear I did not try to copy GR results. You will see my ontology and derivation logic has nothing to do with GR. And when I was going to send the results here, I saw your message @Markus Hanke and decided to test my result against this GR equation. Any way you'll see the results below. I don't know how did it happened... Here's the new section on gravitational diflection: The Unified Interaction Gradient and Relativistic Deflection Energy projections distribution among axis of relational carriers has to obey the conservation law (distribution between axis does not create nor destroy energy). There must exist a strict, algebraically closed gradient connecting all states, governed entirely by the kinematic projection [math]\beta[/math]. Theorem (Unified Interaction Gradient): The gravitational interaction capacity of any entity is determined by its available phase buffer [math]\beta_Y[/math]. The geometric scaling factor [math]\Gamma[/math] that dictates the distribution of the potential projection [math]\kappa^2[/math] onto the spatial trajectory is strictly defined by the arithmetic mean of the saturated carrier [math]S^1[/math]: [math]\Gamma(\beta) = \frac{1 + \beta^2}{2} = 1 - \frac{\beta_Y^2}{2}[/math] Proof: By the [math]S^1[/math] closure invariant ([math]\beta^2 + \beta_Y^2 = 1[/math]), an object at rest ([math]\beta = 0[/math]) possesses maximum internal phase ([math]\beta_Y = 1[/math]). This phase acts as a geometric buffer, absorbing half of the relational gradient, yielding the classical partitioning [math]\Gamma = 1/2[/math]. As the spatial projection [math]\beta[/math] increases, the internal clock slows, and the phase buffer [math]\beta_Y[/math] depletes. At the topological limit [math]\beta \to 1[/math] (light), the internal phase collapses ([math]\beta_Y \to 0[/math]). The buffer is exhausted, forcing the entity to absorb the full, unpartitioned gravitational gradient, yielding [math]\Gamma = 1[/math]. Using this gradient, we define the Unified Closure Defect for any trajectory—from a slow asteroid to a photon—as: [math]\delta_{unified} = \frac{\kappa_p^2}{\beta_p^2} \Gamma(\beta_p) = \frac{\kappa_p^2 (1 + \beta_p^2)}{2\beta_p^2}[/math] The geometric shape parameter (eccentricity) of the trajectory is derived directly from this defect: [math]e_{unified} = \frac{1}{\delta_{unified}} - 1 = \frac{2\beta_p^2}{\kappa_p^2(1+\beta_p^2)} - 1[/math] Applying the exact algebraic transit equation for a distant observer ([math]\kappa_o \to 0[/math]), we have [math]\cos(o_\infty) = -1/e_{unified}[/math]. Using the same trigonometric extraction as established for light ([math]\sin(\frac{\Delta\varphi}{2}) = \frac{1}{e_{unified}}[/math]), we arrive at the absolute, unified equation for gravitational deflection: [math]\Delta\varphi_{unified} = 2 \arcsin(\frac{\kappa_p^2 (1 + \beta_p^2)}{2\beta_p^2 - \kappa_p^2 (1 + \beta_p^2)})[/math] Verification of Topological Limits: * Newtonian Limit ([math]\beta_p \ll 1[/math]): The phase buffer is full ([math]\Gamma \to 0.5[/math]). The eccentricity is dominated by [math]2\beta_p^2 / \kappa_p^2[/math]. The deflection reduces to the classical Rutherford/Newton scattering: [math]\Delta\varphi \approx \frac{\kappa_p^2}{\beta_p^2}[/math]. * Relativistic Limit ([math]\beta_p \to 0.99[/math]): The phase buffer is nearly depleted ([math]\Gamma \to 0.99[/math]). The trajectory stiffens, and the deflection angle approaches the photonic maximum, smoothly capturing the post-Newtonian factor without Taylor expansions. * Photonic Limit ([math]\beta_p = 1[/math]): The phase buffer is completely exhausted ([math]\Gamma = 1[/math]). Substituting [math]\beta_p = 1[/math] yields [math]e = \frac{2}{\kappa_p^2(2)} - 1 = \frac{1}{\kappa_p^2} - 1 = \frac{\kappa_{Xp}^2}{\kappa_p^2}[/math]. The equation resolves perfectly into the exact light deflection identity: [math]\Delta\varphi = 2 \arcsin(\frac{\kappa_p^2}{\kappa_{Xp}^2})[/math]. Elimination of the "Massless" Myth: This formulation proves that light is not a distinct ontological category governed by separate physical laws. There are no "massless particles" in RG. Light is simply the rightward topological limit of matter, where the internal phase [math]\beta_Y[/math] reaches zero. The historical factor of 2 in gravitational lensing is not an anomaly of curved spacetime, but the inevitable geometric consequence of an exhausted phase buffer. Here's my python script comparing the results of my derived arcsin against the scary equation that @Markus Hanke posted: import math from scipy.integrate import quad z_sun = 2.1224e-6 kappa_p_sq = 1 - 1 / ((1 + z_sun)**2) def gr_deflection(beta_p, kp2): def S(u): term1 = (1 - kp2) / (beta_p**2) term2 = (1 - kp2 * u) * ((1 - beta_p**2) / (beta_p**2) + u**2) return term1 - term2 def integrand(u): s_val = S(u) if s_val <= 1e-15: return 0.0 return 1.0 / math.sqrt(s_val) phi, _ = quad(integrand, 0, 1) return (2 * phi - math.pi) * (180.0 / math.pi) * 3600.0 def will_deflection(beta_p, kp2): b2 = beta_p**2 num = kp2 * (1 + b2) den = 2 * b2 - num ratio = num / den if ratio > 1.0 or ratio < -1.0: return float('nan') return 2 * math.asin(ratio) * (180.0 / math.pi) * 3600.0 betas = [0.01, 0.5, 0.9, 0.99, 0.9999, 1.0] print("Sun kappa_p^2: {:.6e}".format(kappa_p_sq)) print("-" * 75) print("{:<10} | {:<20} | {:<20} | {:<15}".format('beta_p', 'GR Integral (arcsec)', 'WILL RG (arcsec)', 'Diff (%)')) print("-" * 75) for b in betas: gr_val = gr_deflection(b, kappa_p_sq) will_val = will_deflection(b, kappa_p_sq) if math.isnan(will_val): print("{:<10.4f} | {:<20.6f} | {:<20} | {:<15}".format(b, gr_val, 'Captured', '-')) else: diff = abs(gr_val - will_val) / gr_val * 100 if gr_val != 0 else 0 print("{:<10.4f} | {:<20.6f} | {:<20.6f} | {:<15.4e}".format(b, gr_val, will_val, diff)) print("-" * 75)Output: Sun kappa_p^2: 4.244786e-06 --------------------------------------------------------------------------- beta_p | GR Integral (arcsec) | WILL RG (arcsec) | Diff (%) --------------------------------------------------------------------------- 0.0100 | 8946.966119 | 8946.971578 | 6.1021e-05 0.5000 | 4.377794 | 4.377797 | 5.9810e-05 0.9000 | 1.956485 | 1.956485 | 1.6577e-05 0.9900 | 1.768885 | 1.768885 | 2.0381e-05 0.9999 | 1.751282 | 1.751283 | 3.3894e-05 1.0000 | 1.751107 | 1.751108 | 1.3785e-05 Output with z= z_sun = 0.0901Sun kappa_p^2: 1.584744e-01 --------------------------------------------------------------------------- beta_p | GR Integral (arcsec) | WILL RG (arcsec) | Diff (%) --------------------------------------------------------------------------- 0.0100 | -648000.000000 | Captured | - 0.5000 | 275119.856251 | 295243.875599 | 7.3146e+00 0.9000 | 88523.699944 | 89458.026859 | 1.0555e+00 0.9900 | 78536.915426 | 79109.733546 | 7.2936e-01 0.9999 | 77621.268245 | 78162.819773 | 6.9768e-01 1.0000 | 77612.175741 | 78153.418327 | 6.9737e-01 --------------------------------------------------------------------------- I don't know about you guys but Im shocked.

Ok here it is for mass body's: The factor of 2 is exact geometric consequence of the Energy-Symmetry Law (https://willrg.com/documents/WILL_RG_I.pdf#sec:energy-symmetry) and the number of active relational axis on the kinematic carrier. Here is the exact derivation: 1. Massive Bodies (Dual-Axis Partitioning) For massive bodies, the transformation resource is partitioned equally between two orthogonal relational axis on the [math]S^1[/math] carrier: Amplitude ([math]\beta[/math]) and Phase ([math]\beta_Y[/math]). Because the energy budget is distributed across two axis, the invariant binding energy inherently carries a 1/2 partitioning factor: [math]W_{mass} = \frac{1}{2}(\kappa^2 - \beta^2)[/math] This gives the effective potential: [math]\Phi_{mass} = \frac{1}{2}\kappa^2 c^2[/math]. Applying this conserved, partitioned energy invariant between periapsis and apoapsis yields the exact shape parameter (eccentricity) for a massive body: [math]e_m = \frac{2\beta_p^2}{\kappa_p^2} - 1[/math] https://willrg.com/documents/WILL_RG_I.pdf#sec:rel_ecc 2. Light (Single-Axis Collapse) By the Single-Axis Transformation Principle, a photon's kinematic projection completely saturates the carrier ([math]\beta = 1[/math]). This forces the internal phase component to vanish entirely ([math]\beta_Y = 0[/math]). Because the Y-axis is absent, the entire relational resource is concentrated on the single X-axis. The 1/2 partitioning factor is strictly eliminated: [math]W_{\gamma} = \kappa^2 - \beta^2 = \kappa^2 - 1[/math] This gives the unpartitioned effective potential for light: [math]\Phi_\gamma = \kappa^2 c^2[/math]. The gravitational effect on light is exactly twice that on massive particles at the geometric level. This yields the photon shape parameter: [math]e_\gamma = \frac{1}{\kappa_p^2} - 1[/math] https://willrg.com/documents/WILL_RG_I.pdf#sec:nature_of_light 3. The Exact Deflection Equations To find the deflection, we use the exact relational phase state equation: [math]\kappa_o^2 = \kappa_p^2 \frac{1 + e\cos(o)}{1 + e}[/math]. Distant observer ([math]\kappa_o \to 0[/math]) yields the angle [math]\cos(o_{obs}) = -1/e[/math]. Using the total angular phase [math]o_{obs} = \frac{\pi}{2} + \frac{\Delta\varphi}{2}[/math], we get the geometric relation: [math]\sin(\frac{\Delta\varphi}{2}) = \frac{1}{e}[/math] Substituting the respective shape parameters gives the exact, non-linear deflection angles without a single approximation: For a massive body (partitioned budget): [math]\Delta\varphi_m = 2 \arcsin(\frac{\kappa_p^2}{2\beta_p^2 - \kappa_p^2})[/math] For light (unpartitioned budget, [math]\beta=1[/math]): [math]\Delta\varphi_\gamma = 2 \arcsin(\frac{\kappa_p^2}{1 - \kappa_p^2})[/math] The historical "factor of 2" discrepancy does not require to speculate a curved 4D spacetime manifold. It is the direct algebraic signature of the axis count in relational space: massive bodies distribute energy across two axes (requiring the 1/2 factor), while light collapses the geometry to a single axis, experiencing the full unpartitioned geometric effect. As you can see everything is absolutely transparent and intuitive. Let me know what you think.

*Laughing hysterically* Only just now I realized that you where asking for nonzero mass object. Ok no worries it shouldn't hard at this point. Ill be back.

@KJW, I'm so grateful that you decided to invest your time in my research. I highly appreciate it. Thank you! So you got me all exited and I spend pretty much all night looking for error. No success... Maybe I just cant see it? Could you elaborate please? But most important during this sleepless night I finally derived methodologically pure ontologically minimal and I think truly beautiful one input solution for light deflection. No mass, no G, no metric, no 4D spacetime curvature - pure relational geometry at it's best. Maybe its lack of sleep talking but Im feeling like its yet another triumph of Relational Geometry. Please have a look and let me know what you think: Algebraic Derivation of Light Deflection https://willrg.com/documents/WILL_RG_I.pdf#sec:light_deflection In General Relativity, the deflection of light is obtained by integrating the null geodesic equations over a curved spacetime manifold, often relying on weak-field approximations and Taylor expansions. Within WILL Relational Geometry (RG), we reject both the background manifold and the use of mathematical approximations as non-operational ontological artifacts. The system consists exclusively of its participants: the Source, the Lens (at periapsis [math]p[/math]), and the Receiver. The total deflection angle must be derived as a strict, exact algebraic difference between their measurable relational phase states, without resorting to series expansions. Theorem: Algebraic Deflection of Light Let [math]o[/math] be the orbital phase (true anomaly) representing the exact geometric angle between the lens periapsis and an observer. For an observer at local potential state [math]\kappa_o[/math], this angle is strictly determined by the algebraic identity: [math]\cos(o) = \frac{\kappa_o^2 - \kappa_p^4}{\kappa_p^2 \kappa_{Xp}^2}[/math] where [math]\kappa_p[/math] is the potential projection at periapsis, and [math]\kappa_{Xp}^2 = 1 - \kappa_p^2[/math] is the corresponding phase parameter. Proof: Step 1: The Photonic Closure Defect (Shape Parameter) By the Single-Axis Transformation Principle (link: https://willrg.com/documents/WILL_RG_I.pdf#sec:nature_of_light) for light, the kinematic projection saturates the carrier ([math]\beta = 1[/math]), forcing the internal phase to vanish ([math]\beta_Y = 0[/math]). This eliminates the 1/2 partitioning factor inherent to massive bodies. Consequently, the closure defect at periapsis for a photon is defined exclusively by the total projections: [math]\delta_\gamma = \frac{\kappa_p^2}{\beta_p^2} = \kappa_p^2[/math] The geometric eccentricity (link: https://willrg.com/documents/WILL_RG_I.pdf#sec:rel_ecc) (shape parameter) of the light trajectory emerges directly from this closure defect: [math]e_\gamma = \frac{1}{\delta_\gamma} - 1 = \frac{1}{\kappa_p^2} - 1[/math] Step 2: Relational Phase State Equation In Relational Orbital Mechanics (link: https://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom), the local potential [math]\kappa_o[/math] at any orbital phase [math]o[/math] is related to the periapsis potential [math]\kappa_p[/math] by the exact topological scaling: [math]\kappa_o^2 = \kappa_p^2 \frac{1 + e_\gamma \cos(o)}{1 + e_\gamma}[/math] Substituting the photonic shape parameter [math]1 + e_\gamma = \frac{1}{\kappa_p^2}[/math]: [math]\kappa_o^2 = \kappa_p^2 \frac{1 + (\frac{1}{\kappa_p^2} - 1)\cos(o)}{1 / \kappa_p^2} = \kappa_p^4 (1 + (\frac{1-\kappa_p^2}{\kappa_p^2})\cos(o))[/math] Expanding the bracket: [math]\kappa_o^2 = \kappa_p^4 + \kappa_p^2(1-\kappa_p^2)\cos(o)[/math] Recognizing that [math]1-\kappa_p^2 = \kappa_{Xp}^2[/math] (the phase component at periapsis), we solve for [math]\cos(o)[/math]: [math]\cos(o) = \frac{\kappa_o^2 - \kappa_p^4}{\kappa_p^2 \kappa_{Xp}^2}[/math] This completes the exact algebraic link between the measured potentials and the geometric angle. Total Deflection (No Approximations) In a purely flat geometry without gravitational phase ([math]\kappa_p = 0[/math]), this angle would be exactly [math]\frac{\pi}{2}[/math] (a straight line from a distant point to periapsis covers exactly a quarter of a circle). The presence of the gravitational gradient increases this angle by an exact one-sided deflection amount, which is half of the total deflection ([math]\frac{\Delta\varphi}{2}[/math]). Let [math]o_\infty = \frac{\pi}{2} + \frac{\Delta\varphi}{2}[/math]. Applying the fundamental trigonometric identity [math]\cos(\frac{\pi}{2} + x) = -\sin(x)[/math]: [math]-\sin(\frac{\Delta\varphi}{2}) = -\frac{\kappa_p^2}{\kappa_{Xp}^2} \rightarrow \sin(\frac{\Delta\varphi}{2}) = \frac{\kappa_p^2}{\kappa_{Xp}^2}[/math] Solving directly for the total deflection angle [math]\Delta\varphi[/math] gives the absolute, exact, and non-linear equation for light deflection in Generative Physics: [math]\Delta\varphi = 2 \arcsin(\frac{\kappa_p^2}{\kappa_{Xp}^2})[/math] Desmos Project: Algebraic Light Deflection (one input derivation) https://www.desmos.com/calculator/ldynwowqvi Epistemological Triumph: This result is achieved without a single differential equation, without background manifolds, and without Taylor series approximations. The sine of the one-sided deflection angle is strictly equal to the ratio of the potential amplitude ([math]\kappa_p^2[/math]) to the potential phase ([math]\kappa_{Xp}^2[/math]) at periapsis. For weak fields ([math]\kappa_p^2 \ll 1[/math]), the phase component [math]\kappa_{Xp}^2 \rightarrow 1[/math] and [math]\arcsin(x) \approx x[/math], recovering the empirical value [math]\Delta\varphi \approx 2\kappa_p^2[/math] (equivalent to legacy ontologically inflated form [math]4GM/rc^2[/math]). However, this equation remains exact and structurally unbroken across all interaction scales, demonstrating that spacetime curvature is simply the algebraic shadow of relational energy projections. The WILL RG formula and GR exact agree to 2 × 10⁻⁷ arcseconds - six orders of magnitude below measurement uncertainty. Both sit comfortably within the observational error bars. Second-order structure (honest disclosure): The series expansions in κ_p² differ at second order: WILL RG gives coefficient 2.0, GR gives (15π/16 − 1) ≈ 1.945. The discrepancy is ~2.8% of the second-order term, which translates to ~0.2 μas for the Sun - unmeasurable with any current or near-future technology. This could mean I missed something or in some future it could become another unique and falsifiable prediction from RG. So what we got: GR needs mass, G, differential formalism, tensors, 4D curvature of spacetime and lots of other questionable ontological baggage... RG needs z_Sun. That's it! @KJW , thank you again mate! You gave me motivation to finally derive it.

Look I don't want to shot you down. This is not the way I prefer to communicate. It's just the fact that no matter how many times Ill tell you: "I have a methodology I follow in my research," And I explicitly showed the principals. There's no container (x,y,z) - because it violates the core methodological principals that I follow. For once its shocking that you can't understand the difference between relationalism and substantivalism. The Irony is that you unknowingly making way more philosophical claims than I do. And that's exactly the reason why I started this research. You think that you not doing philosofy - you doing physics but you wrong: (x,y,z) - this is enormous philosophical claim that you not only just postulate silently but you also completely blissful about it! Objects is a substance in empty (x,y,z) - another huge unproven speculation. instead of the single SPACE-TIME-ENERGY relational structure you creating extra primitive by dividing it SPACETIME as empty box and ENERGY inside of this box. - not derived not empirically proven just postulated blindly By thinking that you are not doing philosophy you end up making this wild assumptions blind. Physics without philosophy = engineering (no offence to engineers out there, great respect) And with all this you keep trying to fit my research in your made up empty box that you not even awear of - THAT is really frustrates me. And it would be ok if it would happened once or twice but its just non stop! You saying "You model dont have empty box there for it cannot predict a" - Im showing you mathematical prove that you are mistaken and explicitly showing the prediction of a, but you ignore it and demand prediction of b, I demonstrate, we moved to c. Are you planning to go through the hole "alphabet"? Do you ever learn anything when proven wrong? I need a clear answer to this question: without the clear answer we can't move any further.

Great idea! there's around 10 derivations I posted including the density derivation, and you ignored all of them. So yeh I agree lets get in to mathematics. You can start from any of my derivations. You have no idea what are you talking about and this is a major barrier in our communication. In logic this fallacy called The Straw Man. Look it up. Hold on! This is a big deal. Are you saying that you can't see the problem in your logic that I pointed out? Its just if we can't agree even at the most basic level it should be an alarm sing for both of us.

Look, you having major problems at the basic logic level: If you can't see something does it mean it doesn't exist? You assertion rests on a fallacy in formal logic known as argument of ignorance. Google it. The second layer of your assertion is based on cycle reasoning: GR uses formalism x and its true. there for everything that is not x is false. Until you will learn the basis of logical reasoning - we have nothing to talk about.

My first reaction was to provide the explicit derivations for these scenarios. However, you have consistently moved the goalposts and dismissed derivations by reverting to standard definitions instead of engaging with the math. Instead of just throwing more equations at you, let's analyze your reasoning. Can you explain on what exact basis you are making the assertion that this framework cannot describe these dynamics? Are your assertions actually falsifiable, or is the necessity of standard tensor formalism a dogmatic position you are willing to defend against empirical and mathematical facts?