Anton Rize

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?

The reason I stopped responding to you is that our dialogue lost scientific honesty. You were asking for derivations and then dismissing them using standard definitions without engaging with the actual geometric equations I provided: I have no interest in semantic debates with someone who doesn't read the answers to questions the one asked. I am here for a scientifically honest, rigorous mathematical discussion. If you are willing to actually read the derivations and engage with them directly - whether to mathematically falsify them or explore their consequences - I will gladly continue dialog with you in respectful and friendly manner. Regarding your specific point about the Cosmological Coincidence Problem: it is highly relevant and the full pattern might tell us something more... The hypothesis that lead me to all those predictions: 1. The Geometric Budget The total relational shift: [math]Q^2 = \kappa^2 + \beta^2[/math] 2. The Invariant Partitioning Under closure condition ([math]\kappa^2 = 2\beta^2[/math]), this budget naturally partitions into exact, invariant ratios: [math]\Omega_{potential} = \frac{\kappa^2}{Q^2} = \frac{2}{3}[/math] interpretation: (What standard cosmology labels "Dark Energy") [math]\Omega_{kinetic} = \frac{\beta^2}{Q^2} = \frac{1}{3}[/math] interpretation: (What standard cosmology labels "Dark Matter") 3. Density Scaling via Alpha These macroscopic geometric partitions are rigidly locked to the microscopic radiation floor ([math]\rho_{gamma}[/math]) via EM kinetic limit ([math]\alpha[/math]): [math]\frac{\rho_{max}}{\rho_{gamma}} = \frac{1}{3\alpha^2}[/math] [math]\frac{\rho}{\rho_{gamma}} = \frac{2}{3\alpha^2}[/math] [math]\frac{\rho}{\rho_{max}} = \kappa^2[/math] So it might be that here is no "coincidence epoch". The universe does not require fine-tuning because the 2:1 ratio of "Dark Energy" to "Dark Matter" is simply the native [math]\Omega_{potential}[/math] to [math]\Omega_{kinetic}[/math] projection ratio required to geometrically close the system. Regardless of mind-blowing numerical agreement - my interpretations could be wrong. Still more research is needed...

Just need to point out one important detail: Because of my methodology I could not simply postulate the volumetric nature of mass and density so the terms that I derived are: [math]\rho=\frac{\kappa^{2}\cdot c^{2}}{8\cdot\pi\cdot G\cdot r^{2}} [/math] [math]\rho_{max}=\frac{c^{2}}{8\cdot\pi\cdot G\cdot a^{2}} [/math] when [math]\kappa=1 -> r=R_s [/math] [math]m_0 = 4\pi r^3 \rho [/math] You can find derivation here https://willrg.com/documents/WILL_RG_I.pdf#sec:density So the relation between standard [math]\rho_{crit} \approx 9.5\times10^{-27}[/math] kg/m^3 and my [math] \rho_{max}[/math] is [math] \frac{\rho_{crit}}{\rho_{max}} \approx 3 [/math] so when written in my terms it looks like this [math]\frac{\rho_{gamma}}{3\rho_{max}}=[/math] [math] \frac{4\sigma_{SB}T_{CMB}^{4}}{c^{3}}\ \frac{8\pi GR_{H}^{2}}{3c^{2}}=\alpha^2[/math] My interpretation is that its got to do with total relational shift [math]Q^2[/math]. When system is energetically closed no leakage (we not speculating anything apart of the universe so there's no outside the system is closed) we getting [math] Q^2=\kappa^{2}+\beta^{2}=3\beta^2 -> \beta=\alpha -> 3\alpha^2 -> [/math] [math] -> H_0=\sqrt{8\pi G\ \frac{\rho_{gamma}}{3\alpha^{2}}}[/math] What do you think about it? Does it make sense?

First, I must commend your algebraic reverse-engineering. You perfectly extracted the core relational consequence of the equation: [math]\frac{\rho_{electromagnetic}}{\rho_{critical}} \propto \alpha^2[/math] This is exactly the rigid geometric scaling law I am pointing to. Regarding your skepticism about the fine-structure constant: my methodology does not allow to introduce any standard QM postulates. So instead, the entire atomic structure is derived strictly from the intersection of three geometric closure principles. I do not import the Bohr radius or [math]\alpha[/math]; they emerge as necessary algebraic identities: https://willrg.com/documents/WILL_RG_III.pdf#sec:alpha I am not claiming to have discovered the numerical value of 1/137. What I am showing is its ontological origin. The fine-structure constant identified as the exact kinematic projection ([math]\beta_1[/math]) required to satisfy the geometric closure of a charged fermion on the [math]S^1[/math] and [math]S^2[/math] relational carriers. What is truly striking is the predictions chain that Im doing. Its like all just locked in to placed... Statistically its pretty much impossible to achieve this randomly: Unbroken Chain of 10 Derivations/Predictions. No Fitting Involved. 1. Hubble Constant ([math]H_0[/math]) Derived: 68.15 km/s/Mpc | Empirical: 67.4 [math]\pm[/math] 0.5 km/s/Mpc System: Planck 2018 | Deviation: +1.0% Formulation: Geometric saturation density derived from CMB temperature and [math]\alpha[/math] 2. CMB First Acoustic Peak ([math]l_1[/math]) Derived: 220.55 | Empirical: 220.60 System: Planck 2018 | Deviation: [math]\approx[/math] 0.02% Formulation: Resonant harmonics of an [math]S^2[/math] topology loaded by 4.2% baryonic mass 3. CMB Quadrupole Power ([math]D_{l=2}[/math]) Derived: 0.156 - 0.320 (Corridor) | Empirical: [math]\approx[/math] 0.20 System: Planck 2018 | Deviation: Within predicted corridor Formulation: Vacuum tension acting as a high-pass filter on a tensioned [math]S^2[/math] membrane 4. Galactic Rotation Curves Bias Derived: [math]0.70 \times 10^{-10} m/s^2[/math] ([math]a_k[/math]) | Empirical: -2.26 km/s (Bias) System: SPARC (175 galaxies) | Deviation: RMSE [math]\approx[/math] 0.065 dex Formulation: Structural Projection Resonant Interference with Universal Fundamental Tone 5. Solar Orbital Velocity Derived: 226.4 km/s | Empirical: 229 [math]\pm[/math] 6 km/s System: Gaia DR3 / Milky Way | Deviation: Excellent agreement Formulation: Geometric mean interference between local potential and global horizon 6. Wide Binary Gravity Boost ([math]\gamma[/math]) Derived: [math]\approx[/math] 1.47 | Empirical: [math]\approx[/math] 1.45 - 1.55 System: Gaia DR3 / Chae 2023 | Deviation: Exact Agreement Formulation: Kinetic Resonance Scale ([math]S^1[/math] carrier coupling weight 1/3) 7. Type Ia Supernova Distance Modulus Derived: Offset expected [math]\approx[/math] 0.150 mag | Empirical: Raw residual [math]\approx[/math] -0.151 mag System: Pantheon+ | Deviation: Shape deviation [math]\le[/math] 0.02 mag Formulation: Geometric Energy Budget Partitioning (2:1 ratio of [math]S^2[/math] tension to kinetic mass) 8. Strong Lensing Einstein Radius Derived: 1.46'' | Empirical: 1.43 [math]\pm[/math] 0.01'' System: SDSSJ0946 +1006 (SLACS) | Deviation: [math]\approx[/math] 2% Formulation: Phantom Inertia ([math]Q^2[/math]) acting as universal refractive medium 9. Recombination Epoch Derived: [math]\approx[/math] 364,860 years | Empirical: [math]\approx[/math] 378,000 years System: Standard Cosmological Dating | Deviation: [math]\approx[/math] 3.5% Formulation: Unit Phase Condition ([math]\Omega_{crit} = 1[/math] radian) where arc length equals radius of curvature 10. Electron Mass ([math]m_e[/math]) Derived: [math]9.064 \times 10^{-31}[/math] kg | Empirical: [math]9.109 \times 10^{-31}[/math] kg System: CODATA | Deviation: [math]\approx[/math] 0.49% Formulation: Holographic Projection Principle / Geometric Capacity Resonance -> Mach Principle.

@MJ kihara Forgot to mention: you can test this predictions yourself against 175 galaxies from SPARC database: https://willrg.com/galactic_dynamics/

This is the most important question you could possibly ask. In physics, to measure change, you have exactly two options for a reference point: Option 1: The Standard Method (The Ghost Reference) Standard physics tries to establish a universal background grid. To do this, they invent a mathematical concept called the "Observer at Infinity" (asymptotic flatness). They pretend there is a magical observer infinitely far away, where gravity is zero and space is perfectly flat, and they measure everything relative to that ghost. It has no physical reality, but it makes correct approximations. Option 2: WILL Relational Geometry (The Real Reference) In RG, there is no background grid and no observer at infinity. The observer is always the origin point. Every observer places themselves at the exact center of their own relational coordinate system: [math](\beta, \kappa) = (0,0)[/math]. (its not space and time its state difference) When you observe a star, you are not locating it on a universal 3D grid (x, y, z). You are measuring the energy differential between your state and the star's state. You measure two projections: 1. Its kinematic shift relative to you (Velocity/Direction): [math]\beta[/math] 2. Its gravitational shift relative to you (Potential/Depth): [math]\kappa[/math] Here is the geometric representation of how these two projections define the system's state on their respective relational carriers: The total change - Total Relational Shift - between you and the star is just the sum of squares of these two projections: [math]Q^2 = \beta^2 + \kappa^2[/math] (don't confuse with simple right triangle, [math]Q^2[/math] is NOT constant) Now, here is the beauty of Relational Reciprocity. If that star has an observer looking back at you, they do not see themselves at some coordinate [math](x,y,z)[/math]. They place themselves at [math](0,0)[/math]. They measure your speed and your potential relative to them, and they calculate the exact same total shift: [math]Q^2 = \beta^2 + \kappa^2[/math]. This reciprocity can be visualized on the [math]\beta-\kappa[/math] plane. The observer is always at the center, and the observed object is at the end of the [math]Q[/math] vector: There is no common background arena. There are only mutual total shift magnitudes [math]Q[/math] computed from each observer's own relational origin. This is what I mean by Ontological Minimalism: we do not need to invent an empty box for the universe to exist in. The relationships are the geometry. And this way all extremely complex GR surfaces emerge as simple and beautiful ratios: If you want to interact with this geometrically, I made a very simple interactive graph showing how the Observer, the Object, and the Total Shift Q relate to each other: DesmosQ as Total Relational ShiftTo dive a bit deeper and see the derivations you can read this section https://willrg.com/documents/WILL_RG_I.pdf#sec:DisQ Does this clarify what the "point of reference" is? You are describing "curve fitting" - the process of inventing a free parameter and tuning its value to match the data. If I did that, you would be absolutely right. It would just be Dark Matter under a new name. But WILL Relational Geometry has zero adjustable parameters. I cannot "tune" the Fundamental Tone to make the curves fit. It is strictly derived. Here is exactly what the Fundamental Tone is: 1. The Hubble parameter ([math]H_0[/math]) is not a fitted cosmological parameter in RG. It is derived algebraically from the Fine Structure Constant ([math]\alpha[/math]) and the Cosmic Microwave Background temperature ([math]T_{CMB}[/math]): [math]H_{0}=\frac{T_{CMB}^{2}}{\alpha}\sqrt{\frac{32\pi G\sigma_{SB}}{3c^{3}}}[/math] There is absolutely nothing to adjust here. It relies only on fundamental constants and the single most precisely measured temperature in cosmology. 2. The Fundamental Tone is simply the geometric frequency of the observable universe. It is the speed of causality ([math]c[/math]) divided by the circumference of the Hubble horizon ([math]2\pi\frac{c}{H_{0}}=2\pi R_H[/math]): [math]\text{Fundamental Tone} = \frac{H_{0}}{2\pi}=\frac{c}{2\pi R_{H}}[/math] The Mechanism (Coupling): Because [math]\text{SPACETIME} \equiv \text{ENERGY}[/math], there is no empty void. Therefore, "coupling" is not a mechanical force carried by a particle through space. They was never decoupled in the firs place. No object exists in isolation. Space is determent by relations within. The structural tension of the global horizon (the Tone) geometrically superimposes with the local kinetic state of the star. [math]\beta_{Q}=\sqrt{\beta^{2}+\sqrt{\beta^{2}\frac{R_{s}}{3\pi R_{H}}}}[/math] You can interact with this derivation and confirm it yourself using this pretty Desmos project: https://www.desmos.com/calculator/s9gvcuttqm Because I have zero free parameters, this specific geometric constraint either predicts the exact rotation curves of 10,000+ datapoints of hundreds of galaxies perfectly, or it completely fails. You cannot "curve fit" thousands of different galaxies using a single, rigid equation unless that equation represents the actual structural constraint of the universe. So using the words "curve fitting" towards my research makes no f*** sense. Does this clear's out the misunderstanding for you?

You welcome! And yes this idea is not new. Leibniz, Mach, Einstein, you, me... I like this gang! The Tone depends on Hubble parameter (H_0) and I derived it from alpha and T_CMB. I can't say for sure yet but so far my bet is on the temperature. But its probably wrong to refer to a single parameter... Its all connected. But for intuitive one-liner its fair to say that it might be what we interpret as expansion could be more like cooling down... This is brilliant! You just said yourself that SPACETIME≡ENERGY. The concept force carrier assumes that there's a void through which, one has to "carry" the "force". Can you see the problem? What your intuition tells you? Is this an accusation or misunderstanding? Im not taking lightly calling my research "curve fitting". You have to present the argument properly or take it back. In my humble opinion the best way to develop intuitive and deep understanding is engage with equations. Just play around with them. So I made a desmos project for you https://www.desmos.com/calculator/s9gvcuttqm It calculates H_0 and then using it to predict the rotation speed of the solar system. Also I think Im getting better at it. This one is pretty.

Thank you for pushing me on this. Your critique forced me to look deeper into how my own equations handle the observable light signals, and I have to concede a major point to you: you are absolutely right that we cannot ignore the frequency shift of light. In fact, it holds the exact mathematical answer. I think the friction between our perspectives comes down to the fact that we are looking at two different layers of the Doppler effect. In my Relational Orbital Mechanics (R.O.M.) framework, I strictly separate the observed frequency shift into two components: 1. Observer dependant: the raw optical signal ([math]Z_{raw}[/math]) 2. Systems intrinsic: the invariant systemic state ([math]Z_{sys}[/math]). When you analyze the standard linear Twin Paradox, you are tracking the raw line-of-sight signal ([math]Z_{raw}[/math]). This includes the optical signal delay and the longitudinal Doppler shift (the red/blue shifts you mentioned). It absolutely works for calculating the end result, but the optical delay masks the underlying geometric engine. In R.O.M., the true proper time projection of the system ([math]\tau_{Wo}[/math]) is literally the exact inverse of the systemic frequency shift: [math]\tau_{Wo}(o) = \sqrt{1-\frac{R_{s}}{r(o)}}\cdot\sqrt{1-\beta(o)^{2}} = \kappa_{Xo}(o) \cdot \beta_{Yo}(o) = (Z_{sys}(o))^{-1}[/math] Here, [math]Z_{sys}[/math] is strictly the product of the gravitational redshift ([math]z_k = \frac{1}{\kappa_{X}}-1 [/math]) and the transverse Doppler shift ([math]z_b = \frac{1}{\beta_{Y}}-1[/math]). It dictates the algebraic aging difference instantly and continuously, without relying on signal travel times. [math]Z_{sys}(o) = (1+z_{ko}(o))(1+z_{bo}(o))[/math] This is why I brought up the circular orbit scenario. In a circular orbit, the distance is constant, so the longitudinal line-of-sight Doppler effect drops to zero. The "optical noise" of [math]Z_{raw} = (1+\beta_{int}(\cos(o+\omega_{i})+e \cos(\omega_{i}))\sin(i)) Z_{sys}(o)[/math] strips away, and we are left staring purely at [math]Z_{sys}[/math]. The central observer and the orbiting observer age differently not because a signal takes time to flip from red to blue, but because the continuous, invariant structural tension ([math]Z_{sys}[/math]) geometrically restricts their internal proper time ([math]\tau_{Wo}[/math]). The light just transmits this underlying state. I realize this is much easier to see when you can interact with the math directly. I've built a full Desmos project that contains the closed algebraic system of R.O.M., showing exactly how these invariants generate the orbits and frequency shifts natively: I also have the formal documentation and a massive dataset test explaining how this system rigorously separates the geometric reality from the line-of-sight optical shadow ([math]M \sin(i)[/math]): 1. The complete closed algebraic system R.O.M.: https://willrg.com/documents/WILL_RG_I.pdf#eq:rom 2. Documentation: https://willrg.com/documents/WILL_RG_I.pdf#sec:M_sin(i) 3. Blind Data Test Results: https://willrg.com/msini_test 4. R.O.M Desmos Project: https://www.desmos.com/calculator/n4lmkpsebx I'd love for you to play around with the Desmos graph. When you look at it through the lens of separating [math]Z_{sys}[/math] (the geometric state) from [math]Z_{raw}[/math] (the optical shadow), does this bridge the gap between our two explanations?

Asking questions on public forum about new model with new ontology its like changing pants in the middle of the crowded street. We never in our history had a chance to ask as many and as dumb questions as we want without being judged or annoying. That's why I think AI will do wanders in education. We living at exiting times.

You make a fair point. I accept the critique. "Simplifying" means extracting the core mechanism, and I owe you a direct conceptual answer without making you dig through the full mathematical derivation. Here is the "Why" and "How" of the energetic floor, translated from algebra into physical concepts: The "Why" (The Ccause): Standard physics treats spacetime as an empty container (a void) where objects move and interact. Remove objects - the container will stay. So it postulate its ontological independency like its real thing. There's no empirical data that force this postulate. Here's what Einstein had to say about it: ``There is no such thing as an empty space, i.e., a space without field. . . . Space-time does not claim existence on its own, but only as a structural quality of the field.'' Albert Einstein, \textbf{Relativity: The Special and the General Theory} (Appendix V: ``Relativity and the Problem of Space''), 1952 edition, Methuen (London), p.~155; based on earlier 1920 additions. I complete agree with this quote. But standard physics does not. It speculate this extra primitive: So when standard approach has 2 primitives: 1. SPACETIME 2. ENERGY I have only one: 1. SPACE-TIME-ENERGY and just for convenience of shorter name and a sudle ironic joke I calling it: WILL ≡ SPACE-TIME-ENERGY ⟹ [math]SPACETIME \equiv ENERGY[/math]. The Universe is not a box; it is a closed relational structure. No background. So there's no such thing as just distance. Distance or period can form only between objects/events. Distance is the tension between potential states. Because the topology is strictly closed, any energy perturbation (like a star orbiting a galaxy) cannot just dissipate its influence into an infinite emptiness. It must interact with the boundaries of the system itself (the global horizon). The "How" (The Mechanism): You are the center of your observable Universe with Hubble horizon (R_H) as a boundary of causality. Energy is change. The rate of change is universal and finite. Faster change in space ⟹ slower change in time. The value of "change rate" associated with c - stays conserved. Interactions are not instantaneous, causality propagate just like a wave one thing influence another thing energy transforms but stay conserved. So if we have non instant propagation of cause and effect (c) and we have a boundary as Hubble horizon (R_H). This leads to waves of cause and effect propagation to interfere with themselves. Think like a guitar string or even better think like a drum skin (math says drum). Waves can interact in a good way or bad way. All bad ways just dissolves. So constructive interference only can make a stable propagation. And like with any harmonics the lowest posible would be the fundamental Tone. It means 1 oscillation of this tone will have a wavelength of your observable Universe. And we can calculate it: Its just speed of light divided by the circumference of Hubble horizon expressed as a frequency [math]\frac{c}{2\pi R_{H}}=\frac{H_{0}}{2\pi}=3.5148579278\times10^{-19} 1/s[/math] for the reference its 1 full oscillation in around 90.1565273986 billions years. So we currently still in the yearly days of fundamental Tone cycle. OK, now Imagine a star orbiting the center of a galaxy. In the standard model, the only thing keeping it moving is its local kinetic energy fighting local gravity. When it gets far away, gravity drops, so the math says the star should slow down. When it doesn't, they invent "Dark Matter" to pull it. In RG, the vacuum itself has a structural tension ([math]\rho_{\Lambda}[/math]) between you and R_H. It required to keep the Universe's geometry open against collapse. The star is not moving through an empty void; it is vibrating on a globally tensioned geometric "carrier" (think of a stretched drum skin). The local orbital frequency of the star and the global frequency of the horizon (the Fundamental Tone) superimpose. This creates a constructive geometric interference. This interference adds a constant energetic baseline to the star's kinetic state. As the star gets further from the galactic center, its local gravity drops, but it hits that baseline geometric tension. It cannot slow down past this limit, because slowing down would require it to decouple from the global horizon fundamental Tone, which is topologically impossible. That absolute limit is what we observe as the "flat rotation curve." And to test this hypothesis I used equation from wave mechanics where 2 values are multiplied under the square root to model geometric interference: [math] v_{obs}^2 = \underbrace{v_{N}^2}_{\text{Local Self-Energy}} + \underbrace{\sqrt{v_{N}^2 \cdot (\Omega a_{Mach} r)}}_{\text{Reson. Interference}} [/math] That absolute limit is what we observe as the "flat rotation curve." To make the mechanics of this geometric interference perfectly clear and dimensionally strict, we can express this exact relation through dimensionless velocity (β = v/c). The interference occurs strictly between the local causal limit (the Schwarzschild radius of the central mass, R_s) and the global causal limit (the Hubble horizon, R_H): [math]\beta_{Q}=\sqrt{\beta^{2}+\sqrt{\beta^{2}\frac{R_{s}}{3\pi R_{H}}}}[/math] Here, the term under the inner square root is the direct cross-correlation (interference) between the local kinematic state (β²) and the global structural tension. This tension is expressed as the pure geometric ratio of the two causal horizons: R_s / (3πR_H). It is this exact fundamental resonance that provides the constant energetic baseline and prevents the orbital velocity from dropping. And when I tested against 10000 + datapoints from 3 different datasets I got precise agreement. Regarding what constitutes a "minimal system": a true generative geometry does not require an absolute coordinate grid or a "start point to every location." It only requires relative energy differentials. Spacetime is generated by these relations; it does not pre-exist them. Does this conceptual summary answer your "how and why" more clearly? And this is exactly why I developed WILL-AI. Its free its open and it trained on all 3 main papers of mine. Don't expect him to preform any complex calculations (he cant use any tools) But he can unswear any questions about the model and provide citations. When I was testing him by comparing mine and his answers to the same questions - it was hard to decide do I prefer mine or he's answers. So yeh have fun: https://willrg.com/WILL-AI/ Feedback on he's performance is required so please do share the feedback at some point That is not the sense Id like to propagate. I can't promise anything about your sense (as its in majority your internal processing), but I can guaranty that direct and sound questions will provide you with direct and sound answers. The hard part is to transform the sense in to direct and sound question. Actually I think https://willrg.com/WILL-AI/ could help you with this. He can process half-baked questions.

Both hard questions and in order to unswear them properly you will need to read: https://willrg.com/documents/WILL_RG_I.pdf https://willrg.com/documents/WILL_RG_II.pdf and at least the firs half of https://willrg.com/documents/WILL_RG_III.pdf In total its about 150+ pages and I can imagine what a gatekeeper it might be. So I made you a nice and easy interactive chart: https://willrg.com/LOGOS2 the logical and philosophical core is on the chart it self, and if one wants to see the math - every logical step is clickable and will direct you to the precise section. A more detailed version of the LOGOS chart you can find here https://willrg.com/LOGOS_MAP/ but this one is still in development. For now it covering only https://willrg.com/documents/WILL_RG_I.pdf Please let me know how the LOGOS2 chart communicates the idea.

Why are you here? You not reading my answers you ignoring my derivations after 20+ comments posted you cant even answer the basic question about model under critic. So why are you here? Are you that desperate for self validation that you hanging out in speculations subforum in order to school armatures? And here you got the armature that publicly schooling you... Must be hard for your fragile ego to accept it... I feel sad for you man... From now on I will ignore all your messages. I wish you to find reasons to love and respect yourself.

Yes you did and I provided you with an unswear that you once again completely ignored. You asking questions and then ignoring the answers. Is that the type of communication you want to be a part of? Since you want to talk about equation of state - yes please tell me how I deriving the pressure term and what w=? I end up and why? And this time please try not to make your typical category error where you assuming the Universe is a container filled with fluid. Are you capable?

Guys, hold on, not all at once please!😁

I appreciate your honesty, and I understand why the dynamic of this conversation might look confusing from the outside. Let me address your points calmly and systematically. There is no panic here - only the rigorous execution of a geometric proof. 1. On the Hamiltonian: The difference is fundamental. A standard Hamiltonian describes the total energy of a single, isolated state at one point in space. My equation is a two-point relational difference. It describes the kinetic budget at an orbital point (B) relative to the potential budget at a central point (A). The standard Hamiltonian only appears when you collapse those two distinct points into one ([math]r_A = r_B[/math]). I am not paraphrasing the Hamiltonian; I am showing the deeper geometric architecture it is derived from. 2. On "Predicting the Unknown": Please look closely at the Wide Binary Stars graph I posted and the section https://willrg.com/documents/WILL_RG_II.pdf#sec:wide-binary. The standard "flexible" model (MOND) failed to predict that trajectory. WILL RG predicted that exact curve with zero parameter fitting. Eliminating a 40-year-old hallucination (Dark Matter) by mathematically proving it is just the tension of the macroscopic horizon is a novel, testable prediction. Have a closer look at this web page https://willrg.com/Galactic_Dynamics/ this is 175 unique testable predictions. Also take a look at this section https://willrg.com/documents/WILL_RG_I.pdf#sec:M_sin(i) this is a completely novel method achieving predictions that standard approach consider mathematically impossible. Right now Im going through astronomical databases in order to make predictions about 3D parametrisation (orbital shape and angle relative to us) using only spectroscopy data before the release of astrometric data. This line of unique predictions we will be able to test in the near future with new astronomical data releases. The list can go on... 3. "How do things acquire mass?" Short answer: I don't know. But I'm working on it. 4. On answering "Mainstream Questions" from Mordred: I salute your desire for clarity. However, when Mordred asks me to explain the volume of a 3D sphere, he is asking me to defend a Newtonian container model that my theory explicitly mathematically rejects. It is like asking a quantum physicist to explain electron orbits using gears and springs. I am answering his questions directly, but I will not adopt a broken ontological framework just to make the answers sound "mainstream." I am glad the "energetic floor" concept resonated with you. That is exactly what happens when you let the geometry guide the physics, rather than forcing the geometry to fit the mainstream. Thank you for explicitly confirming that your line of questioning "has nothing to do with physics." This perfectly explains why you are completely ignoring the empirical galactic rotation curves, the Wide Binary data, and the Baryonic Tully-Fisher derivation I provided. You have abandoned the physical territory because you cannot contest the results. Regarding your mathematical claim: the fact that a function [math]f(r) = 4\pi r^2[/math] can be plotted on a 2D graph does not make the expression itself "graphical coordinates." You can graph temperature versus time on a piece of paper; that does not turn temperature into a spatial dimension. Conflating a mathematical plot with an ontological coordinate space is a severe category error. Since you have openly admitted your questions have nothing to do with physics, and you continue to refuse to state your falsifiability criteria... Look... it's just... I'm so disappointed. I was sincerely hoping that I could talk with you about these results. And you have to admit they are far from trivial. I wanted to discuss weak and strong points, potential implications, agreement with other physics domains, ways to empirically test it etc... But instead you are completely ignoring my questions and my derivations and trying to quiz me on high school geometry. Really? Look, if high school homework problems are what you are after - there are subforums on this website dedicated to them specifically.

I notice you completely ignored the empirical data, the SPARC RAR graph, and the Wide Binary tests demonstrating 0-parameter accuracy, choosing instead to retreat to elementary geometry. But I will humor your question. Also you still ignoring my goal pole question. You have to admit it not to me but to yourself: If your believes unfalsifiable then they are NOT scientific. Yes, plain and simple: if you graph [math]y = 4\pi r^2[/math] against [math]r[/math], you get a parabola. Yes, the topological [math]S^2[/math] carrier possesses the exact metric area of [math]4\pi r^2[/math]. If you leading me to: "Since it is a 3D sphere, it must enclose a volume of [math]\frac{4}{3}\pi r^3[/math], therefore your density equation is missing the [math]1/3[/math] factor!" Let me stop you right there and refer you back to the ontology you refuse to read. In classical Euclidean space, a sphere is a boundary enclosing a physical volumetric void that gets filled with a fluid. In WILL Relational Geometry, the [math]S^2[/math] carrier is not a container; it is the geometric capacity of the potential projection itself. The energy state is the surface projection ([math]\kappa^2[/math]). There is no independent "inside volume" to integrate over. That is exactly why the algebraic closure in my model is strictly [math]m_0 = 4\pi r^3 \rho[/math], without Newtonian [math]1/3[/math] fluid coefficient. Now that I have answered your geometry question, are you going to address my goal pole question and the physical rotation curves + the Wide Binary data I just provided, or are we going to continue plotting parabolas?

You asked exactly the right question: "how do you explain the discrepancy between rotation velocity in the outer of the Galaxy and the inner region?" In standard cosmology, because the Newtonian velocity drops off as [math]1/\sqrt{r}[/math], they have to invent a halo of invisible mass (Dark Matter) that grows with [math]r[/math] to keep the rotation curve flat. In WILL Relational Geometry, the explanation requires zero invisible particles and no adjustable parameters. It is a direct consequence of SPACETIME ≡ ENERGY. I will show it in 4 steps: Step 1 The Necessity of Global Resonance Step 2 Deriving [math]H_0[/math] from CMB Temperature and [math]\alpha[/math] Step 3 The Resonant Horizon Interference Step 4 Derivation of the Baryonic Tully-Fisher Relation All this leads us to conclusion: The discrepancy in the outer regions of the galaxy is not caused by "missing mass." It is the physical manifestation of the star hitting the energetic floor (the Fundamental Tone) supported by the global resonance of the Universe. It cannot slow down any further because it is coupled to the horizon. To make sure that everyone could test it against empirical data I developed this page https://willrg.com/Galactic_Dynamics/ Its an interactive LAB where you can calculate rotation speed of any galaxy in SPARC database (175 total). This page development was a peace of work I have to admit. Hope you will find it useful. Also here's RAR graph: Radial Acceleration Relation (RAR) for 175 SPARC galaxies. The green dots shows the density of [math]>3000[/math] individual data points. The cyan line represents the WILL Resonance Interference prediction ([math]g_{obs} = g_{bar} + \sqrt{g_{bar}a_{\kappa}}[/math]) using the [math]H_0[/math] value derived from CMB thermodynamics. The remarkable agreement ([math]RMSE \approx 0.065[/math] dex) without free parameters strongly suggests that galactic dynamics are regulated by the global horizon. Colab notebook link: [RAR test.ipynb](https://github.com/AntonRize/WILL/blob/8967a5fac8a7a42433e9cc66fffbbbbc4e18b6a1/Colab_Notebooks/RAR_test.ipynb) And here's the wide binary stars graph based on Gaia DR3 catalog where we comparing WILL and MOND: The Kinetic Resonance Test. The plot compares the gravity boost factor as a function of Newtonian acceleration. Blue dashed line: Standard MOND prediction ([math]a_0 = 1.2 \times 10^{-10} m/s^2[/math]), which systematically overestimates the anomaly. Red solid line: WILL RG Kinetic Resonance prediction ([math]a_{\beta} = cH_0/6\pi \approx 0.35 \times 10^{-10} m/s^2[/math]) passes precisely through the observational data points, matches the reported trend of Wide Binary observations without any parameter fitting. Colab notebook link: [Wide binary Chae 2023.ipynb](https://github.com/AntonRize/WILL/blob/8967a5fac8a7a42433e9cc66fffbbbbc4e18b6a1/Colab_Notebooks/Wide_binary_Chae_2023.ipynb) You can find all my derivations and all my python scripts on this page: https://willrg.com/results/

There is no single geometric object with an area of [math]8\pi r^2[/math]. The number 8 is simply the result of multiplying 4 by 2. Please look closely at the two lines of the derivation I posted: Line 1: [math]\rho = \frac{1}{4\pi} \left( \frac{\kappa^2 c^2}{2 G r^2} \right)[/math] Line 2: [math]\rho = \frac{\kappa^2 c^2}{8\pi G r^2}[/math] The [math]4\pi[/math] in the denominator is the normalization factor from the [math]S^2[/math] carrier area. The [math]2G[/math] in the denominator comes from the mass scale factor ([math]R_s = 2GM/c^2[/math]). When you multiply the denominators, [math]4\pi \times 2G = 8\pi G[/math]. I am not "switching" terms. I am executing basic algebraic multiplication. Does this clarify the geometry for you?

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

What do you mean " without explaining why" ? "To translate this into a volumetric density, we first adopt the conventional 3D (volumetric) proxy, [math]r^3[/math]. This is not a postulate of RG, but the first step in applying the legacy (3D) definition of density: [math]\frac{m_0}{r^3} = \frac{\kappa^2 c^2}{2G r^2}[/math] This expression, however, is incomplete. Our [math]\kappa^2[/math] "lives" on the 2D surface [math]S^2[/math] (which corresponds to [math]4\pi[/math]), while the [math]r^3[/math] proxy implicitly assumes a 3D volume. To correctly normalize the 2D parameter [math]\kappa^2[/math] against the 3D volume, we must apply the geometric normalization factor of the [math]S^2[/math] carrier by dividing by the area of the sphere, which introduces the factor [math]\frac{1}{4\pi}[/math]. This normalization is the necessary geometric step to interface the 2D relational carrier ([math]S^2[/math]) with the 3D legacy definition of density: [math]\rho = \frac{1}{4\pi}( \frac{\kappa^2 c^2}{2G r^2} )[/math] [math]\rho = \frac{\kappa^2 c^2}{8\pi G r^2}[/math] [math]\text{Local Density} \equiv \text{Relational Projection}[/math]" Please specify what exactly seems unclear to you?

Welcome back @joigus ! I will assume that you came not just to drop a comment but also to engage in conversation. So Ill gladly provide you with transparent derivations of Lagrangian and Hamiltonian. Next in line is Newtons third law you can find here https://willrg.com/documents/WILL_RG_I.pdf#thm:third_law We now demonstrate that the familiar Lagrangian and Hamiltonian formalisms arise as limiting cases of the two-point relational Energy-Symmetry Law. Specifically, they emerge when the relational structure between two distinct observers A and B is collapsed into a single-point local description. This collapse preserves computational utility but reduces the ontological transparency of the underlying relational structure. Definitions of Parameters We consider a central mass [math]M[/math] and a test mass [math]m[/math]. The state of the test mass is described in polar coordinates [math](r,\phi)[/math] relative to the central mass. * [math]r_A[/math] --- reference radius associated with observer A (e.g., planetary surface). * [math]r_B[/math] --- orbital radius of the test mass [math]m[/math] (position of observer B). * [math]v_B^2 = \dot r_B^2 + r_B^2 \dot\phi^2[/math] --- total squared orbital speed at B. * [math]\beta_B^2 = v_B^2/c^2[/math] --- dimensionless kinematic projection at B. * [math]\kappa_A^2 = 2GM/(r_A c^2)[/math] --- dimensionless potential projection defined at A. The Relational Lagrangian Instead of a relational energy, we define the clean relational Lagrangian [math]L_{rel}[/math], which represents the kinetic budget at point B relative to the potential budget at point A: [math]L_{rel} = T(B) + U(A) = \frac{1}{2} m(\dot r_B^2 + r_B^2 \dot\phi^2) + \frac{GMm}{r_A}[/math] In dimensionless form, using the rest energy [math]E_0 = mc^2[/math], this is: [math]\frac{L_{rel}}{E_0} = \frac{1}{2}(\beta_B^2 + \kappa_A^2)[/math] This two-point, relational form is the clean geometric statement. First Ontological Collapse: The Newtonian Lagrangian If one commits the first ontological violation by identifying the two distinct points, [math]r_A = r_B = r[/math], the relational structure degenerates into a local, single-point function: [math]L(r,\dot r,\dot\phi) = \frac{1}{2} m(\dot r^2+r^2\dot\phi^2) + \frac{GMm}{r}[/math] This is precisely the standard Newtonian Lagrangian. Its origin is not fundamental but arises from the collapse of the two-point relational Energy Symmetry law into a one-point formalism. Second Ontological Collapse: The Hamiltonian Introducing canonical momenta, [math]p_r = \frac{\partial L}{\partial \dot r} = m\dot r[/math] [math]p_\phi = \frac{\partial L}{\partial \dot\phi} = mr^2\dot\phi[/math] one defines the Hamiltonian via the Legendre transformation [math]H = p_r \dot r + p_\phi \dot\phi - L[/math]. This evaluates to the total energy of the collapsed system: [math]H = T+U = \frac{1}{2} m(\dot r^2 + r^2 \dot\phi^2) + \frac{GMm}{r}[/math] Interpretation In terms of the collapsed WILL projections ([math]\beta^2 = v^2/c^2[/math] and [math]\kappa^2 = 2GM/(rc^2)[/math], both strictly positive), the match to standard mechanics becomes explicit: [math]L = \frac{1}{2} m v^2 + \frac{GMm}{r} \longleftrightarrow \frac{1}{2} m c^2(\beta^2 + \kappa^2)[/math] [math]H = \frac{1}{2} m v^2 - \frac{GMm}{r} \longleftrightarrow \frac{1}{2} m c^2(\beta^2 - \kappa^2)[/math] Here the "+" or "-" signs do not come from [math]\kappa^2[/math] itself, which is always positive, but from the ontological collapse of the two-point relational energy law into a single-point formalism. In WILL, both projections are clean and positive; in standard mechanics, the apparent sign difference arises only after this collapse. Both the Lagrangian and Hamiltonian thus emerge from the same relational Energy-Symmetry Law under the identification [math]r_A = r_B = r[/math]. The apparent sign difference between [math]L[/math] and [math]H[/math] is not a fundamental feature but an artifact of this single-point collapse: in the full two-point relational law, both projections [math]\beta^2[/math] and [math]\kappa^2[/math] are strictly positive. The Lagrangian and Hamiltonian arise as single-point limiting cases of the two-point relational Energy-Symmetry Law [math]\Delta E_{A\to B} + \Delta E_{B\to A} = 0[/math]. They remain computationally valid within their domain of applicability. RG provides the more general two-point relational structure from which these formalisms can be systematically derived.

You have raised a crucial philosophical point, but I must respectfully disagree with your underlying premise. You are conflating mathematical flexibility with physical constraint. Discoveries like Neptune were not made because the model was "flexible." They were made precisely because classical mechanics was rigid. When the orbit of Uranus violated predictions, Le Verrier did not have the "flexibility" to alter Newton's gravitational constant or invent a new force distance parameter. Because the model's constraints were absolute, the anomaly strictly demanded a localized physical explanation. That rigidity is exactly what pointed the telescope at Neptune. If a model is "flexible" enough that you can just add a mathematical parameter every time an observation deviates from theory (like adding epicycles, or adding an invisible "Dark Matter" halo to fix a galactic rotation curve), you stop looking for physical reality. You just fit the curve. Flexibility is often the enemy of discovery; it hides structural failures under mathematical patches. This brings me to your comment about my math. You suggested that my model "incorporates Hamiltonian, Klein Gordon equations... and manipulation of physics units/SI units in a smart way" to give short answers. I need to be absolutely clear: it does not. There are no hidden quantum mechanical operators, no Hamiltonians, and no dimensional juggling under the hood. I'm keeping everything dimensionless at all times and units used only when its absolutely necessary. The reason the derivations are so short and direct is that the framework relies entirely on generative geometric constraints, not phenomenological equations. Hamiltonian same as Lagrangian are derived: Lagrangian and Hamiltonian as Single-Point Limits of the Relational Energy-Symmetry Law A rigid, zero-parameter model like WILL RG does not prevent the prediction of the unknown. On the contrary: because it has zero flexibility, any future observation that deviates from its predictions cannot simply be "patched" with a new parameter. It will immediately expose the exact location of actual new physics or exact limitation of the current model. There's no place for dark sector in RG. Ok I will copy and paste and convert the formatting of math for you, but you have to explain what difficulties are you facing when clicking on to direct link like this?: https://willrg.com/documents/WILL_RG_I.pdf#sec:density the same derivation from this link Ill provide below: Translating RG (2D) to Conventional Density (3D). In RG [math]\kappa^2[/math] is the 2D parameter defined in the relational carrier [math]S^2[/math]. In conventional physics, the source term is volumetric density [math]\rho[/math], a 3D concept defined by the "cultural artifact" (a Newtonian "cannonball" model) of mass-per-volume. To bridge our 2D theory with 3D empirical data, we must create a "translation interface". We do this by explicitly adopting the conventional (Newtonian) definition of density, [math]\rho \propto m_0/r^3[/math], as our "translation target". From the projective analysis established in the previous sections: [math]\kappa^2 = \frac{R_s}{r}[/math] where [math]\kappa[/math] emerges from the energy projection on the area of unit sphere [math]S^2[/math], and [math]R_s = \frac{2Gm_0}{c^2}[/math] links to the mass scale factor [math]m_0 = \frac{E_0}{c^2}[/math]. This leads to mass definition: [math]m_0 = \frac{\kappa^2 c^2 r}{2G}[/math] To translate this into a volumetric density, we first adopt the conventional 3D (volumetric) proxy, [math]r^3[/math]. This is not a postulate of RG, but the first step in applying the legacy (3D) definition of density: [math]\frac{m_0}{r^3} = \frac{\kappa^2 c^2}{2G r^2}[/math] This expression, however, is incomplete. Our [math]\kappa^2[/math] "lives" on the 2D surface [math]S^2[/math] (which corresponds to [math]4\pi[/math]), while the [math]r^3[/math] proxy implicitly assumes a 3D volume. To correctly normalize the 2D parameter [math]\kappa^2[/math] against the 3D volume, we must apply the geometric normalization factor of the [math]S^2[/math] carrier by dividing by the area of the sphere, which introduces the factor [math]\frac{1}{4\pi}[/math]. This normalization is the necessary geometric step to interface the 2D relational carrier ([math]S^2[/math]) with the 3D legacy definition of density: [math]\rho = \frac{1}{4\pi}( \frac{\kappa^2 c^2}{2G r^2} )[/math] [math]\rho = \frac{\kappa^2 c^2}{8\pi G r^2}[/math] [math]\text{Local Density} \equiv \text{Relational Projection}[/math] Maximal Density. At [math]\kappa^2 = 1[/math] (the horizon condition (for non rotating systems), [math]r=R_s[/math]), this density reaches its natural bound, [math]\rho_{\max}[/math], which is derived purely from geometry: [math]\rho_{\max} = \frac{c^2}{8\pi G r^2}[/math] Normalized Relation. Thus, our "translation" reveals an identity: the geometric projection [math]\kappa^2[/math] is simply the ratio of density to the maximal density: [math]\kappa^2 = \frac{\rho}{\rho_{\max}} \rightarrow \kappa^2 \equiv \Omega[/math] Self-Consistency Requirement The mass scale factor can be expressed in two equivalent ways. From the geometric definition: [math]m_0 = \frac{\kappa^2 c^2 r}{2G}[/math] From the energy density: [math]m_0 = \alpha r^n \rho[/math] Substituting [math]\rho = \frac{\kappa^2 c^2}{8\pi G r^2}[/math] into [math]m_0 = \alpha r^n \rho[/math] gives: [math]m_0 = \frac{\alpha \kappa^2 c^2 r^{n-2}}{8\pi G}[/math] Equating the two forms: [math]\frac{\alpha r^{n-2}}{8\pi} = \frac{r}{2}[/math] For the mass [math]m_0[/math] to remain a constant independent of the measurement scale [math]r[/math], the exponent must be [math]n=3[/math], yielding [math]\alpha=4\pi[/math]. Hence: [math]m_0 = 4\pi r^3 \rho[/math] which closes the consistency loop between the geometric and density-based formulations.

I have to admit that this is a completely new experience to me - you are the first person in this 250+ comments thread who is not dismissing my ontology... I didn't realize how horribly sad it is until I typed it out... I also would like to emphasize the redefinition of energy wasn't my philosophical choice. The standard "ability to do work" - violates the principle of relational origin. So my definition, same as everything else in my research is forced by the core method that Im committed to. If my research ever finds wider acceptance it is this generative method of mine I consider the biggest contribution. It shows a different way to do science where ontology comes first. No mindless phenomenology no curve fitting no unfalsifiable models. P.S. I would be very curious to hear your "controversial" definition - it seems our intuitions are vibrating on the same frequency.

According to my results in https://willrg.com/documents/WILL_RG_II.pdf The vast majority (only week lensing left unsolved; just haven't touched it yet) of observational phenomena that we interpreting as DE, DM are inevitable consequents of Relational Geometry. We just don't have to speculate any "dark" entities to accurately predict and explain observables. The “missing mass” is not missing mass. It is the observable signature of a star’s resonant coupling to the cosmic horizon inside a closed relational geometry. Dark Matter becomes redundant. CRITICAL NOTE: Regardless of complete publicity of all data and tools and methodological discipline of my Open Research - My results are still remain unverified independently. Until independent verification is confirmed we should avoid such categorical formulations: "dark matter doesn't exist".

That is a very vague redirection. 1. On the Math: Could you be specific? Which exact RG equation violates which rule of inner or cross-product addition? [math]\kappa^2[/math] is explicitly derived as the square of a relational projection (inner product geometry) on an [math]S^2[/math] carrier (cross-product area geometry). Dropping terms like "vector addition rules" without showing a specific algebraic contradiction is not a critique; it is a distraction. 2. On the Logic: You completely avoided my primary question. I will ask it again: What are your specific, objective criteria for a model to be taken seriously? I have derived [math]H_0[/math] with zero parameters, solved the orbital degeneracy problem, and computed the recombination epoch. If these results are not enough, what is? Do you have a defined threshold for evidence, or are you simply going to throw random jargon from a physics textbook every time the derivation holds up? You are attempting to judge a fundamental generative geometry using the secondary mathematical artifacts of a 100-year-old descriptive model. 1. On "Inner Products" and "Vector Rules": These are not physical laws; they are bookkeeping tools designed for 3D Euclidean containers. An inner product requires a pre-defined metric space to exist. RG does not assume a metric space - it generates one from the relational energy projections (alpha, light wavelength) things we can physically measure. You trying to legitimize unphysical unmeasurable model specific mathematical tools. Using vector addition rules to "correct" RG is like trying to use a thermometer to measure the volume of a shadow. It is a category error. 2. Observed Reality vs. Tensors: The Stress-Energy Tensor (Tμν) is an ad-hoc mathematical construct used to balance equations in a fluid-dynamic model that requires Dark Matter to function. RG operates on physical observables: absolute temperatures, coupling constants, and geometric closure limits. 3. The Goalpost Question: You have repeatedly retreated into textbook jargon to avoid a simple fact: RG produces accurate physical results (H0, recombination, orbital tests and many more) with zero free parameters - results your "3 methodologies" can only achieve by "throwing in" invisible dark substances. I will not waste more time debating the commutative properties of your mathematical "toys." If you have a specific physical observation that contradicts the output of my equations, present it. If not, then your critique is nothing more than a defense of a map that is no longer the territory. What is your objective threshold for a model to be valid? Does it have to be true, or does it just have to use your favorite tensors? Is your physical believes based on dogmas or scientific principals?