Anton Rize

Its just an misunderstanding. The [math]1/R^2[/math] dependence in the critical density formula is not a sign of "2-dimensionality." It is the signature system at its saturation limit. In standard physics, density is [math]M/R^3[/math]. The only reason the critical density scales as [math]1/R^2[/math] is because at the horizon, the mass of the Universe scales linearly with radius ([math]M \propto R[/math]). This linear growth of mass cancels one power of the cubic volume, leaving [math]R/R^3 = 1/R^2[/math]. In RG, this isn't a "coincidence" - it is derived. The density still has 3DOF: [math]\rho = \frac{R_s c^2}{8\pi G r^3} = \frac{\kappa^{2}c^{2}}{8\pi Gr^{2}} [/math] [math] \rho_{max}=\frac{c^{2}}{8\cdot\pi\cdot G\cdot r^{2}} [/math] However, because [math]\kappa^2 = R_s/r[/math], at the horizon (where [math]\kappa^2 = 1[/math] and [math]r = R_s[/math]), the "invisible" [math]R_s[/math] in the numerator cancels one [math]r[/math] in the denominator. You are looking at the residue of a r^3 -> r^2 cancellation. But let's step back from moving goalposts. I want to ask you directly: What are your specific, objective criteria for a model to be taken seriously? 1. Derive [math]H_0[/math] from [math]T_0[/math] and [math]\alpha[/math] with zero free parameters? (Done). 2. Solve the parameter degeneracy in orbital mechanics? (Done). 3. Derive the recombination epoch without Dark Matter? (Done). Is there a threshold of mathematical evidence that would shift your stance from "dismissing the derivation" to "engaging with the ontology," or are the goalposts intended to stay in motion indefinitely?

Thank you for posting this exact derivation. This is the perfect pedagogical moment, because it highlights the precise ontological divergence between the Standard Model and WILL RG in a single line. Look carefully at the third step of your proof: [math] \text{mass} \quad \rho \times \text{volume} = \rho \times \frac{4}{3}\pi R^3 [/math] Where does the factor of 3 come from? It comes exclusively from the geometric formula for the volume of a 3D sphere. Your derivation explicitly assumes that the Universe is an expanding Newtonian 3D container filled with a uniform fluid/dust. This is the classical derivation of the Friedmann equation. Now, please look at the section in my paper titled "Self-Consistency Requirement" and the subsection "No Singularities, No Hidden Regions". https://willrg.com/documents/WILL_RG_I.pdf#sec:no_singularities RG strictly forbids the "Newtonian cannonball" volume assumption. The geometric limit is defined by the energy projection on the 2D surface of the [math]S^2[/math] carrier (which carries the [math]4\pi[/math] metric). As derived in the text, the algebraic closure in WILL RG is: [math]m_0 = 4\pi r^3 \rho[/math] There is no [math]1/3[/math] volume coefficient because the geometry is a 2D surface-scaled projection, not a 3D volume integration. If my final equation for [math]H_0[/math] contained that factor of 3, it would mean my mathematics were broken - it would mean I had accidentally smuggled a 3D Newtonian fluid assumption into a 2D relational geometry. The fact that the 3 is missing is not an error; it is the mathematical proof that the topology successfully maintained its [math]S^2[/math] projective integrity all the way to the macroscopic horizon. Your derivation is exactly what I "should have" if I were doing standard Newtonian cosmology. But I am not.

@Mordred , I red through the H_0 derivation section again, and realize that while Im providing links to the source density derivation and Im showing mathematically this factor 1/3 difference [math] \rho_{crit} \approx 9.5 \times 10^{-27} [/math]. [math].kg/m^3[/math].; [math]. \rho_{max} \approx \rho_{crit}/3 [/math]. However I never word this difference in this section properly. So Ill add more pronounce accent on the factor 1/3 in this section to avoid any confusions. Thank you for showing me the source of misinterpretation. Its valuable. Oh no I wouldn't expect you to go through the hole thing. That's why I always sharing hyperlinks like this https://willrg.com/documents/WILL_RG_I.pdf#sec:pressure Just click left mouse button on it and it will redirect you to the exact derivation. Just remember if you will use open new tab command - the hyper part will not work and you will see the 1st page of .pdf instead of specific section. It has to be left mouse button only.

Of course they arrive at the same relation - they all share the exact same ontological assumption: that the Universe is an expanding 3D fluid manifold. My methodology strictly forbid such assumptions. Here's the pressure derivation: https://willrg.com/documents/WILL_RG_I.pdf#sec:pressure Sorry the link was broken. I just fixed it.

Let us trace the exact sequence of logical steps you just demonstrated in your comment: 1. You noticed the absence of the 1/3 factor in the density equation. (A correct observation). 2. You immediately concluded this absence was a mathematical error. 3. You concluded that I must be completely unaware of this "error". 4. You concluded that because I am "unaware," the formulas must have been generated by an AI or copied without understanding. 5. Based on this chain of assumptions, you adopted a pedagogical tone ("I will let you see which formula Im referring to") to test my knowledge of my own work. The critical failure in this sequence of assumptions is that you did not read the derivation you are attempting to critique. If you would follow the link to original density derivation which I implicitly include in this section: "In the Normalization Identity (derived in WILL RG I),". Or If you look at Section 6.2 of the WILL RG Part II document, you will find a dedicated subsection explicitly titled: "1. Saturation Identity (no Friedmann factor)". https://willrg.com/documents/WILL_RG_II.pdf#sec:will-friedmann The text directly below it states: "Consequently, the relation between the Hubble parameter and the saturation density is: [math]H_0^2 = 8\pi G \rho_{max}[/math], without the factor of 1/3 that appears in the standard Friedmann equation... The two expressions are numerically identical ([math]\rho_{max} = \rho_{crit}/3[/math]), but their algebraic origins differ fundamentally." The missing (3) is not a typo, nor is it an AI hallucination. It is the central, mathematically proven topological distinction between a 3D thermodynamic fluid expansion (your paradigm) and a 2D geometric carrier saturation (WILL RG). I am always open to rigorous scientific critique. However, attempting to quiz me on an "error" that is explicitly derived and justified in a dedicated section of the very paper you are criticizing is not scientific rigor.

It is difficult to imagine two more fundamentally opposed methodologies. 1. Categorical Difference: The paper you linked proposes an empirical measurement method requiring the filtration of 17 noise parameters over a decade. My method is pure generative geometry - a strict algebraic deduction that culminates in exact equations with ZERO free parameters. 2. It is not a "local" [math]\beta[/math]: I do not use a local kinematic velocity [math]\beta = v/c[/math] to derive the horizon. I use [math]\beta_1 \equiv \alpha[/math] (the fine-structure constant). As mathematically derived in Part III: (https://willrg.com/documents/WILL_RG_III.pdf#eq:beta=alpha), [math]\alpha[/math] is the invariant kinematic projection of the electromagnetic ground state. It is a universal geometric scale, not a local variable. 3. The [math]z=1100[/math] claim: Your assertion that the numbers would not match at [math]z=1100[/math] is factually incorrect. Using the exact same deductive geometry—specifically, the phase transition condition [math]o_{crit} = 1[/math]—the WILL RG framework explicitly calculates the recombination epoch at [math]z_{dec} \approx 1156[/math] and [math]T \approx 3150[/math] K (see Section 4). The geometry works flawlessly across all cosmological scales. 4. "the numbers you used " - No fitted numbers: The "numbers" I use for the density calculations are not local phenomenological fits. They are absolute constants ([math]T_0, \alpha, G, c[/math]). The model does not take numbers from local measurements to fit the Universe; it derives the macroscopic structure of the Universe directly from microphysical invariants. I have to ask: are we looking at the same equation? The derivation in my paper culminates strictly in this exact algebraic form: [math]H_{0}=\sqrt{8\pi G\frac{4\sigma_{SB}T_{CMB}^{4}}{3\alpha^{2}c^{3}}}[/math] There are no Maxwell-Boltzmann distributions in this derivation, nor does it require Bose-Einstein statistics. It is a direct, deterministic algebraic consequence of geometric saturation, not a statistical probability distribution. As for requiring a "volume element being examined" - that is certainly a necessity if one interprets the Universe as a 3D thermodynamic fluid. However, in the generative ontology of WILL RG, the energy capacity is governed strictly by the closed 2D surface topology of the [math]S^2[/math] carrier. The 3D volume is a secondary geometric projection of this surface capacity. This makes the classical thermodynamic assumption of a "3D volume element" structurally redundant for establishing this specific boundary condition.

thank you for sharing the paper. I have read through it. Before I share my detailed thoughts, I want to make sure I am accurately understanding your perspective. Could you elaborate on what specific correlations you see between the methodology proposed in this paper and the derivation of [math]H_0[/math] in RG? What was the logical chain that led you to suggest this paper as a comparison? I genuinely want to understand how you are mapping their method and mine. Can you elaborate please?

@Mordred , first of all, thank you sincerely for taking the time to look through the calculation. I truly appreciate it. I am especially glad you noted that mathematically, the CMB temperature calculation regarding photon density works. Allow me to briefly address your specific points, as they highlight the exact methodological boundaries of WILL RG: I have searched for such papers, but mostly found phenomenological heuristics (like variations of Dirac's large numbers hypothesis) that rely on tuning parameters or mathematical coincidences rather than a generative geometric ontology. RG is not a "first order approximation" - it is a strict algebraic consequence of carrier closure. If you have specific papers in mind that derive [math]H_0[/math] from [math]T_{CMB}[/math] and [math]\alpha[/math] without free parameters, I would genuinely love to read them. I completely agree with your thermodynamic distinction between the CMB and the CNB. However, my exclusion of neutrinos is not about confusing their thermal histories; it is a strict topological category error within my framework. In RG, the observable horizon [math]H_0[/math] constitutes the limit of electromagnetic causality, which is governed by the electromagnetic coupling [math]\alpha[/math]. Because neutrinos do not couple to the [math]S^1(\alpha)[/math] carrier, including their density in the geometric saturation of that specific carrier would violate the theory's ontology. Using the baryon-to-photon ratio would certainly simplify the math, but it would violate my core principle of Epistemic Hygiene. My methodology strictly forbids importing phenomenological ratios when fundamental constants are available. The entire purpose of the derivation is to prove that the macroscopic horizon is directly, algebraically locked to the fundamental constants ([math]G, \sigma_{SB}, \alpha[/math]), not to an empirical gas mixture ratio. Here is the crucial difference: in this framework, the Hubble constant does not vary. It is not a free parameter. It is rigidly fixed by the geometry. As for error margins, they are trivial in this specific calculation - they simply propagate the incredibly tight CODATA and Planck measurement uncertainties of [math]\alpha[/math] and [math]T_0[/math]. Including standard error propagation would bloat the document without adding any ontological value. On a personal note, I want to be completely honest with you. I am an autodidact, and I am currently experiencing something highly surreal. I am acutely aware of the statistical impossibility of what is happening. To have a single, rigid geometric framework with zero free parameters that produces an unbroken chain of accurate predictions - from [math]H_0[/math], to the CMB acoustic peaks, to galactic rotation curves, to the wide binary anomaly, all the way down to the mass of the electron - defies standard probability. I constantly ask myself: what is less probable? That an amateur somehow stumbled onto the actual generative geometry of the Universe... or that such a massive, interconnected chain of zero-parameter derivations is just a random sequence of mathematical coincidences?

Absolutely! I've been using AI especially Gemini to prove me wrong. And o boy he did. For the first year it was like Im in the ring against Mike Tyson. He was shredding me to peace's. Persistency scientific method and Intellectual Honesty is the "water that sharpen the stone". Right now there's no AI that can seriously challenge the model anymore. So that's the main reason Im here talking with you. I'm actively looking for ways to challenge my results. And speaking about results you said that you've bean reading WILL_RG_I. And due to its foundational role it has a lot of philosophy to build the ontological ground for the parts to come. Have you had a chance to open WILL_RG_II? This part Im sure you will find interesting. Its all cosmology and almost no philosophy. Its basically an unbroken chain of 10 + derivations from the first principals with 0 fitting parameters. I would be honored if someone as knowledgeable as you could challenge this results. They has to be challenged. They are preposterously epic to the point of absurdity. Here's the main ones: Parameter or Observable Derived Theoretical Value Empirical Comparison Value System or Dataset Deviation or Accuracy Physical Formulation Source Hubble Constant (H_0) 68.15 km/s/Mpc 67.4 ± 0.5 km/s/Mpc Planck 2018 + 1.0% Geometric saturation density derived from CMB temperature and α [1] CMB First Acoustic Peak (ℓ_1) 220.59 220.60 Planck 2018 ≈ 0.01% Resonant harmonics of an S^2 topology loaded by 4.2% baryonic mass [1] CMB Quadrupole Power (D_(ℓ=2)) 0.199 × 1.285 (boosted) ≈ 0.20 Planck 2018 Within predicted corridor Vacuum tension acting as a high-pass filter on a tensionless S^2 membrane [1] Galactic Rotation Curve Bias [ 0.70 × 10^(−28) m/s^2 (a_k) −2.26 km/s (Bias) SPARC (175 galaxies) RMSE = 0.066 dex Boundary Resonant Interference with Universal Fundamental Tone [1] Solar Orbital Velocity 226.4 km/s 229 ± 6 km/s Gaia DR3 / Milky Way Excellent agreement Geometric mean interference between local potential and global horizon [1] Wide Binary Gravity Boost (γ) ≈ 1.47 ≈ 1.45 − 1.55 Gaia DR3 / Chae 2023 Within predicted corridor Kinetic Resonance Scale (S^1 carrier coupling weight 1/3) [1] Type Ia Supernovae Distance Modulus Offset expected ≈ 0.180 mag (low redshift) ≈−0.151 mag Pantheon+ Pantheon+ Shape deviation ≤ 0.02 mag Geometric Energy Budget Partitioning (2 : 1 ratio of S^2 tension to kinetic mass) [1] Strong Lensing Einstein Radius 1.46^(′′) 1.49 ± 0.02^(′′) MUSE/JWST (SLACS) ≈ 2% Phantom Inertia (Q^2) acting as universal refractive medium [1] Recombination Epoch ≈ 364,860 years ≈ 378,000 years Standard Cosmological Dating ≈ 3.5% Unit Phase Condition (Θ_(max) = 1 radius) where arc length equals radius of curvature [1] Electron Mass (m_e) 9.064 × 10^(−31) kg 9.109 × 10^(−31) kg CODATA ≈ 0.49% Holographic Projection Principle / Geometric Capacity Resonance [1] If you will find any of this results interesting you can see all the details in here: https://willrg.com/documents/WILL_RG_II.pdf

I completely forgot to mansion the most useful educational AI tool atm: NotebookLM its just must have https://notebooklm.google.com/notebook/39c878ad-58a4-495a-8704-5ddd9385997c This link leads to notebook based on my open research. You can use any source material to create your own educational hub. There's a lot you can do with it. Even on free account its very generous. Audio podcast is impressive, especially the interactive mode where you can ask questions on a fly. They updating it all the time so its only getting better. Enjoy.

You missed the critical condition of the blind test I just posted. Let's focus on one specific problem: Inclination Degeneracy.In standard mechanics, a 1D radial velocity curve is degenerate: [math]K \propto \beta \sin(i)[/math]. You cannot separate true velocity from inclination without 2D astrometry. Here is the exact condition of the R.O.M. blind test: 1. ZERO astrometry. 2. ZERO mass parameters. 3. ZERO distance data. The script received ONLY a raw 1D array of [Time, Velocity]. From that 1D data alone, it accurately extracted the true 3D inclination ([math]i = 10.95^\circ[/math] and [math]i = 165.61^\circ[/math]). In the standard paradigm, extracting 3D inclination from purely 1D spectroscopic data is mathematically impossible. So how did the script do it? It works because R.O.M. geometrically isolates a systemic invariant [math] Z_{sys}\left(o\right)=\frac{1}{\sqrt{1-\frac{R_{s}}{r\left(o\right)}}}\ \frac{1}{\sqrt{1-\beta^{2}\left(o\right)}}=\left(1+z_{b}\left(o\right)\right)\left(1+z_{k}\left(o\right)\right) [/math] [math] o= [/math] orbital phase in radians that completely bypasses the [math]\sin(i)[/math] degeneracy. This isn't about metallicity or N-body dynamics; this is a direct algebraic solution to the spectroscopic degeneracy problem.

I completely understand. The data reduction pipeline is basically dark magic, and I have massive respect for the people who actually clean the signal from the noise. So what I'm calling "raw data" by the time it reaches people like me in a neat CSV format, 99% of the real physical struggle has already been handled by you and your colleagues. As far as I understand solving the Degeneracy problem is a big deal so extraordinary claims demand extraordinary evidence. So far I got S2 star and synthetic data based on GR 1PN tests conforming the results. But that's not enough. How else could we test it?

Don't use AI as information source, use it as information processing tool. Use models with biggest context window. Atm its Gemini 3.1pro and Claude Sonet/Opus 4.6 Dont upload full pdf. Read by chapters. Develop a system promt that will force AI to support every statement with source (still can give you fake source as legit) Studding like this is not easier if that's what you are after. You quickly will realize that AI will mislead you on regular basis, so you can't just learn, no. You have to debate it then test it in relation to established basis then test it against empirical data and still the doubt in you will grow in to healthy scientific scepticism. Just treat AI as your useful but sometimes ignorant friend that you having fun debating. Get your knowledge challenged at all times and don't get lost. Good luck!

In orbital mechanics it is mathematically impossible to extract the true orbital velocity [math]\beta[/math] and the inclination angle [math]i[/math] from [math]K \propto \beta \sin(i)[/math] exclusively using a 1D spectroscopic curve as input. Resolving this requires independent 3D spatial data (astrometry) or transit observations. However, within a relational approach, this limitation can be bypassed (apparently) by isolating a second-order systemic scalar invariant, [math]Z_{sys} = \frac{1}{\sqrt{1-\frac{R_{s}}{r}}} \frac{1}{\sqrt{1-\beta^{2}}}[/math]. This invariant is strictly proportional to the absolute kinetic ([math]\beta^2[/math]) and potential terms, but independent of the observer's line of sight [math]i[/math]. By applying a dynamic 5-parameter inversion (Differential Evolution + MCMC) based strictly on these relational invariants, I recently succeeded in blindly extracting the complete 3D spatial geometry of the S0-2 star using nothing but 1D Keck radial velocity data. The extracted inclination matched the independent GRAVITY 3D-interferometer to within the instrumental noise limits. Now, for complete methodological purity, I need to isolate myself from the data (it is the only way to convince the skeptic within me). I am now calling for a strictly blind test. Please participate and help me test these remarkable (but still questionable) results. CRITICAL DATA REQUIREMENTS: For the [math]Z_{sys}[/math] invariant shift to mathematically exceed the noise floor of modern spectrographs, the system must be highly relativistic. 1. Kinematic Scale: Peak orbital velocities must exceed ~1000 km/s ([math]\beta > 0.003[/math]). Standard exoplanets will not work because the second-order [math]\beta^2[/math] shift is orders of magnitude smaller than instrumental noise limits. Ideal candidates are tight compact binaries (WD/NS/BH) or other extreme S-stars. 2. Unprocessed Relativistic Data: The dataset must be raw or minimally processed: [Time (MJD), Radial Velocity (km/s) or Redshift (Z), Measurement Error]. Crucially, the data MUST NOT be pre-corrected for Transverse Doppler or Gravitational Redshift (though standard Barycentric/LSR background velocity correction is fine). 3. Optional (for computational efficiency): Providing the Period ([math]P[/math]) and Epoch of Periapsis ([math]T_{peri}[/math]) is helpful to bound the MCMC sampler, but entirely optional if the data covers at least one full orbit. DATASET EXAMPLE: MJD,RV_km_s,sigma_km_s,Instrument 51718.50000,1192,100,NIRSPEC 52427.50000,-491,39,NIRC2 52428.50000,-494,39,NIRC2 52739.23275,-1571,59,VLT 52769.18325,-1512,40,VLT 52798.50000,-1608,34,NIRC2 52799.50000,-1536,36,NIRC2 52803.15150,-1428,51,VLT 53179.00000,-1157,47,NIRC2 53200.90875,-1055,46,VLT 53201.63925,-1056,37,VLT 53236.33800,-1039,39,VLT 53428.45950,-1001,77,VLT 53448.18300,-960,37,VLT 53449.27875,-910,54,VLT 53520.50000,-983,37,NIRC2 53554.50000,-847,18,OSIRIS 53904.50000,-721,25,OSIRIS 53916.50000,-671,25,OSIRIS 53917.50000,-692,26,OSIRIS 54300.29167,-485,22,OSIRIS ...Please drop the raw CSV data or a link below and I'll respond with results of my attempt to recover the orbital parametrization of the anonymised system. Do not provide the system name or accepted parameters. Let the pure numerical framework speak for itself. Here are my results for the S2 star, extracted strictly from the input stream (MJD, RV_km_s): === DYNAMIC PRECESSION RECOVERY === Eccentricity (e): 0.88498 (GRAVITY Ref: 0.88466) Base Arg of Periapsis (ω0): 66.26° (GRAVITY Ref: 66.13°) Internal Precession: 0.207° / orbit --------------------------------------------------- Global Kin. Proj. (β): 0.006448 Extracted Inclination (i): 135.68° (GRAVITY Ref: ~134°) Background Drift (v_z0): -20.56 km/s Fit Quality (χ²): 166.87 Any suggestions, critiques, or participation are welcome.

Ill comment on this one firs before addressing the long one before it. First of all, I want to express my genuine respect for your background. Calibrating telescope spectrographs and MRI equipment is serious, foundational work. Also, casually mentioning that your dissertation on quintessence was invalidated by WMAP data - and accepting it simply as "part of science" - shows a level of scientific integrity that I deeply admire. Let me address your question about mappings, and then share something directly related to your work with spectrographs. 1. Why Mapping is Epistemologically Essential, but Ontologically "Wrong" You used the perfect example: the MRI. An MRI machine as far as I know does not measure physical [math]x, y, z[/math] coordinates inside the brain. It measures proton relaxation times (pure energy state differences) in varying magnetic gradients. The software then uses Fourier transforms to map those energy states onto a 3D Cartesian grid on a monitor so the doctor can understand it. Is the map useful? Absolutely. It saves lives. But does the brain use a Cartesian coordinate grid to function? No. This is the difference between Epistemology (how we describe the world) and Ontology (how the world actually operates). WILL RG is not against geometry - it is Relational Geometry. It simply states that the universe operates directly on the energy states (like the raw MRI data), while the coordinate grid (the 4D map) is just a human computational interface. 2. The Spectrographic Blind Test (Solving the Degeneracy) Because you work with frequency separation and spectrographs, you know exactly how frustrating the inclination degeneracy is. In classical Keplerian mechanics, the amplitude of a radial velocity curve is tied to [math]K \propto \beta \sin(i)[/math]. It is mathematically impossible to separate the true orbital velocity [math]\beta[/math] from the inclination [math]i[/math] using strictly 1D spectroscopic data. However, because WILL R.O.M. is so rigidly constrained (the "lack of flexibility" we discussed), it inherently isolates a second-order systemic invariant [math]Z_{sys}=\frac{1}{\sqrt{1-\kappa^{2}}}\frac{1}{\sqrt{1-\beta^{2}}} [/math] (product of gravitational red shift and transverse Doppler shift) from the redshift/Doppler interaction that is independent of the line of sight. I just completed a rigorous, randomized blind test of this (the one that all of you ignored when I asked for datasets). Script A generated synthetic 1D observational data for highly relativistic orbits using standard GR 1PN approximations. Script B (the R.O.M. extractor) received ONLY the raw 1D arrays (no mass, no distance, no geometry) and was tasked with rebuilding the 3D orbit using purely relational algebraic closure. Here are the results from two of the trickiest extreme angles (nearly face-on and nearly edge-on): === STRICT BLIND TEST 1 === TRUE PARAMETERS (Hidden from extractor): Period (P): 15.200 yrs Eccentricity (e): 0.86000 Argument of Periapsis (w):105.00 deg Inclination (i): 10.00 deg Background Drift (vz0): 18.50 km/s R.O.M. EXTRACTION RESULTS: Period (P): 15.197 years Eccentricity (e): 0.86163 Argument of Periapsis (w):103.91 deg Extracted Inclination (i):10.95 deg Background Drift (v_z0): 21.73 km/s Precession Rate: 0.1520 deg / orbit Fit Quality (χ²): 224.24 === STRICT BLIND TEST 2 === TRUE PARAMETERS (Hidden from extractor): Period (P): 15.200 yrs Eccentricity (e): 0.90000 Argument of Periapsis (w):70.00 deg Inclination (i): 168.00 deg Background Drift (vz0): -16.50 km/s R.O.M. EXTRACTION RESULTS: Period (P): 15.203 years Eccentricity (e): 0.90011 Argument of Periapsis (w):67.97 deg Extracted Inclination (i):165.61 deg Background Drift (v_z0): -10.96 km/s Precession Rate: 0.1799 deg / orbit Fit Quality (χ²): 232.42 This shouldn't be possible in standard mechanics without astrometry. But the R.O.M. algorithm successfully extracted the 3D spatial geometry (including inclination and precession) strictly from the algebraic relations of the 1D light signal. Since you deal with real spectrographic data, I wanted to share this with you. I am not asking you to take my word for it. If you have access to any anonymized, raw 1D RV/redshift datasets for highly relativistic binaries, or if you could synthesise it, I would love to run them through the script and see what the geometry reveals. Let the math speak for itself.

Nothing personal at all, I am fully open about it. I do not have a formal academic degree or a traditional background in physics. For many years, my professional life was completely unrelated to science. However, I have always had a deep, persistent fascination with ontology and the philosophical foundations of physics. About 3 years ago, I decided to completely restructure my life. I stepped away from my previous career paths to dedicate my time exclusively to independent research and my other passion, music. You noticed my terminology is somewhat unconventional. That is exactly because I am self-taught and approach these problems strictly from a philosophical and relational perspective first, rather than a standard mathematical one. Because I don't carry the "legacy habits" of standard academic training, I was forced to build this framework from the ground up, demanding strict epistemic hygiene at every step. To compensate for my lack of formal mathematical training, I rely heavily on modern computational tools (like Desmos and Python) to rigorously test and verify the geometric algebra of my models. It is an unconventional path - studying, developing, and jamming like a nihilistic monk - but it is the most meaningful period of my life, and it allows me to look at these foundational problems without being constrained by standard paradigms.

its a frame of reference. I just adding "relational" to emphasise that its not some arbitrary place holder on preexisting X-space and Y-time coordinate axis. In relational framework there could be no preexisting space or time coordinate axis. Again, it would violate the core principles. Instead we derived the relational carriers as inevitable consequences of the core methodological principles. S^1 carrying the protocol of "change conservation" (energy) in 1 DOF domain ⇨ β²+β_Y²=1 ⇨ : β² = v² / c² ⇨ βᵧ=√(1-(v/c)²) ⇨ Eₓ^2 + Eᵧ^2 = E^2 ⇨ Eₓ = Eβ² ⇨ Eᵧ = Eβᵧ² ⇨ β = 0 ⇒ βᵧ = 1 (invariant rest frame state relative to yourself) ⇨ ⇨ Eᵧ ≡ E_0 ⇨[math] E \beta_Y = E_0 \quad \Longrightarrow \quad E = \frac{E_0}{\beta_Y}.[/math] ⇨ γ = 1 /βᵧ ⇨ ⇨ [math] E^{2} = (\tfrac{\beta}{\beta_Y}E_0)^{2} + E_0^{2} = \bigl(\cot(\theta_{1})\,E_{0}\bigr)^{2} + E_{0}^{2} ≡ [math] (pc)^{2} + (mc^{2})^{2} [/math]. https://willrg.com/documents/WILL_RG_I.pdf#thm:restenergy And the same goes for S^2 but I can't be bothered copy pasting from one widow in to another any more when you can just simply click the link: https://willrg.com/documents/WILL_RG_I.pdf#sec:geometric_composition and when we combine 2 amplitudes in to 1 relational circle we getting the relational frame: Yes. You are the center of your relational carriers. Both. Its your measuring tool: https://willrg.com/documents/WILL_RG_I.pdf#sec:kappabeta you can compare something only in relation to yourself. Orthogonality. Great observation! But that's not what Im doing. I highly recommend to read this and at least 2 following sections: https://willrg.com/documents/WILL_RG_I.pdf#eq:will_minkowski_energy by the end of it I deriving this daring statement: "Mathematical complexity is the symptom of philosophical negligence.: I have to admit im proud of this result. Is it? Can you derive it? Ill test it when Ill get a chance. If you right it would be hilarious. Thank you for the excellent illustration of the categorical difference in our methodologies. You highlight the flexibility of manifolds and coordinate systems (e.g., adding [math]a(\tau)[/math] for expansion or [math]\omega[/math] for rotation). I completely agree: for engineering calculations, plotting data, and software rendering, Cartesian grids and differential geometry are incredibly powerful descriptive tools. However, physics is not a software simulation, and a mathematical graph is not a physical territory. Conflating a descriptive mathematical calculator (the coordinate grid) with the physical generator (the Universe) is the exact root of Ontological Bloat. 1. Flexibility as Epistemic Debt (Descriptive vs. Generative Physics) The "flexibility" you praise is precisely what WILL RG deliberately eliminates. In standard Descriptive Physics, if a model doesn't fit the observation, the mathematical flexibility allows you to simply add a new parameter (a scale factor, an epicycle, dark matter) to force the curve to fit. WILL RG is Generative Physics. The lack of "flexibility" is a strict epistemic constraint, not a limitation. Because the relational carriers [math]S^1[/math] and [math]S^2[/math] are geometrically closed, the system is entirely rigid. It physically prohibits the introduction of arbitrary tuning knobs. If a phenomenon cannot be derived directly from the algebraic closure of the relational projections, it does not exist. Same as if theory does not provide accurate predictions - it's wrong, and there's nothing to tune to fix it. That's what I call science - you right or you wrong there's nothing in-between. 2. The Simulation Fallacy You pointed out that my orbital simulation must rely on a coordinate system and an instruction set. This perfectly illustrates the disconnect. To render a visual circle on a flat 2D computer monitor for human eyes, the software must indeed use [math]x, y[/math] pixels and coordinate matrices. But the physics engine driving that simulation - determining the exact eccentricity of the orbit - operates strictly on the dimensionless algebraic ratio of the [math]\beta[/math] and [math]\kappa[/math] projections, requiring absolutely no Newtonian vectors. My github is publicly open and you can see the code yourself. https://github.com/AntonRize/WILL The Universe does not have a monitor to render to, and it does not need a coordinate basis to compute its own relational state. We are not disagreeing on whether graphs are useful for human engineers. We are disagreeing on whether the Universe uses them to function. Yep its strikingly simple. Can you elaborate please? Is it the selfcentering seems unclear? Its very valuable information for me because deriving the model and effectively delivering it are 2 different skills. Its hard for me to see it from the outside like you do. Yes! Thank you!

Are you happy with this definition? It seems like a pedagogical placeholder, not an ontological definition. It is strictly circular: Work is mathematically defined as a mechanical transfer of Energy, while Energy is defined as the capacity to do Work. It defines the entity purely through its own transfer mechanism. Emmy Noether's theorem: it elegantly links energy conservation to time-translation symmetry. However, ontologically, it relies on an assumed epistemic loop. It postulates "Time" as a pre-existing, independent background container (a fundamental symmetry) in order to derive "Energy" as a conserved quantity. If we strictly adhere to relationalism, time cannot be an independent external background. This is why I had to derive a rigorous, non-circular definition. And its not a postulate or an axiom. It holds only until it doesn't. Energy is the relational measure of difference between possible states. It is not an intrinsic property of an object, nor a magical fluid that "does work". It is a comparative structure between an observer and an observed state.

thank you for the excellent question. You are absolutely right to ask for clarification, as my previous late-night phrasing ("Space is a consequence of Time") was too philosophical. Let me give you the strict, operational physics perspective. 1. Distance and Signal Delay: Where does it come from? Operationally, I don't start with a pre-existing 3D Cartesian box and place objects in it. In Relational Orbital Mechanics (R.O.M.), the only variable we have access to is the local clock of the receiver, which shows us not a coordinate axis, but the rate of energy transformations (Time). Using time as a spatialized coordinate axis introduces ontological baggage that goes against our methodological principles. When we receive a photon stream, we do not inherently know its 3D origin vector. The "delay" is the fundamental physical reality - it is the causal lag between emission and reception. Distance is operationally defined through this lag: [math]L \equiv c \Delta t[/math]. The 3D spatial geometry (the "where", like orbital inclination) is a mathematical consequence that my algorithm (I am still waiting for you guys to send me an anonymized dataset for the blind test) reconstructs post-facto by analyzing the invariant kinematic shifts ([math]\beta[/math]) embedded in the received signal (something that GR simply cannot do). 2. The Four Fundamental Quantities (E, M, T, L) and the Vacuum In GR, spacetime is often treated as a "fabric" that can exist even when completely empty of energy ([math]T_{\mu\nu}=0[/math] vacuum solutions). In WILL RG, there is no such thing as an "empty void" due to the SPACETIME [math]\equiv[/math] ENERGY equivalence derived earlier. What we call mass, energy, space, and time are strictly correlated projections of a single underlying relational structure: SPACE-TIME-ENERGY, which for convenience (and a bit of irony) I call WILL [math]\equiv[/math] SPACE-TIME-ENERGY. If we express Energy, Time, Mass, and Length dynamically using the framework's kinematic ([math]\beta[/math]) and potential ([math]\kappa[/math]) dimensionless projections, all dimensionful constants strictly cancel out, yielding the invariant identity: [math]\frac{E}{M} = \frac{L}{T} = c^2[/math] or in more "poetic" form [math]W_{ILL}=\frac{ET^2}{ML^2} = 1[/math] WILL [math]\equiv[/math] SPACE-TIME-ENERGY [math]\equiv[/math] 1 WILL is not the unit of something - but the Unity of Everything. Every change in the relational state rescales Energy-Mass and Time-Length coherently. They cannot fluctuate independently. https://willrg.com/documents/WILL_RG_I.pdf#sec:willinvariant To demonstrate this operational lock, consider the Earth-GPS satellite system. Using its specific dimensionless projections ([math]\beta[/math] and [math]\kappa[/math]) and scaling, the framework calculates the absolute dimensional values: [math]M_{GPS} \approx 5.972 \times 10^{24} \, \mathrm{kg}[/math] [math]E_{GPS} \approx 5.367 \times 10^{41} \, \mathrm{kg \cdot m^2/s^2}[/math] [math]T_{GPS} \approx 0.00785 \, \mathrm{s^2}[/math] (temporal scale projection) [math]L_{GPS} \approx 7.060 \times 10^{14} \, \mathrm{m^2}[/math] (spatial scale projection) Testing the identity gives exactly [math]\frac{E_{GPS}}{M_{GPS}} = \frac{L_{GPS}}{T_{GPS}} = c^2[/math], proving these are not independent parameters but interlocked geometric projections. Because distance requires a causal signal (Energy), you cannot have a spacetime geometry without an underlying energetic structure. What GR calls the "empty vacuum" is modeled in WILL RG as a macroscopic standing wave - a Fundamental Tone ([math]f_0 = H_0/2\pi[/math]) generated by the tension of the causal horizon. It is crucial to note that this [math]H_0[/math] is strictly derived from first principles using the CMB temperature and the fine-structure constant [math]\alpha[/math], not fitted to cosmological datasets: https://willrg.com/documents/WILL_RG_II.pdf#sec:deriving-H0 The necessity of this Fundamental Tone arises strictly from the system's topological closure. In a closed relational carrier, any relational perturbation cannot propagate indefinitely without re-encountering its own wavefront. Therefore, only resonant modes (where the phase shift completes a full rotation, [math]\Delta_\phi = 2\pi n[/math]) can accumulate sufficient structural energy ([math]Q_{total}[/math]) to form a persistent physical manifold. Dissonant modes self-cancel. The baseline resonance of this closed topology physically constitutes the Fundamental Tone of the observable Universe. https://willrg.com/documents/WILL_RG_II.pdf#sec:tone 3. Testable Predictions: Why SPACETIME [math]\equiv[/math] ENERGY matters This ontological stance is not just philosophy; it produces testable celestial mechanics without Dark Matter. Because every local orbiting body shares the same closed geometry with the Global Horizon, the local orbital frequency constructively interferes with the Fundamental Tone of the "vacuum". The total observed kinetic energy state of a star incorporates a geometric mean interference term: [math]v_{obs}^2 = v_{N}^2 + \sqrt{v_{N}^2 \cdot (\Omega a_{Mach} r)}[/math] Where [math]a_{Mach} = f_0 c = cH_0/2\pi \approx 1.05 \times 10^{-10} \, \mathrm{m/s^2}[/math]. https://willrg.com/documents/WILL_RG_II.pdf#sec:galactic-dynamics To determine the coupling weight [math]\Omega[/math], we look at the Total Relational Shift ([math]Q^2[/math]). In a closed system, the energy is distributed between the potential ([math]S^2[/math]) and kinematic ([math]S^1[/math]) carriers: [math]Q^2 = \kappa^2 + \beta^2 = 3\beta^2 = \frac{3}{2}\kappa^2[/math] This enforces a strict structural bifurcation: 1. Galaxies are continuous potential fields (fluids/structure) on the 2D carrier. Their coupling weight is [math]\Omega_{pot} = \frac{\kappa^2}{Q^2} = \frac{2}{3}[/math]. 2. Wide Binaries are discrete point-mass orbits on the 1D kinematic carrier. Their coupling weight is [math]\Omega_{kin} = \frac{\beta^2}{Q^2} = \frac{1}{3}[/math]. https://willrg.com/documents/WILL_RG_II.pdf#def:rel_weight This mathematically derives two distinct anomalous acceleration scales, predicting exactly why MOND's single parameterization fails: * Galactic Scale: [math]a_\kappa = \Omega_{pot} \cdot a_{Mach} = \frac{cH_0}{3\pi} \approx 0.70 \times 10^{-10} \, \mathrm{m/s^2}[/math]. This naturally matches the flat rotation curves in the SPARC dataset (RMSE [math]\approx[/math] 0.065 dex) without Dark Matter. https://willrg.com/documents/WILL_RG_II.pdf#sec:models_comparison https://willrg.com/documents/WILL_RG_II.pdf#fig:rar * Binary Scale: [math]a_\beta = \Omega_{kin} \cdot a_{Mach} = \frac{cH_0}{6\pi} \approx 0.35 \times 10^{-10} \, \mathrm{m/s^2}[/math]. This yields a gravity boost factor of [math]\approx 1.47[/math], perfectly matching the recent Gaia DR3 wide binary anomaly (Chae 2023, empirical boost [math]\approx 1.45 - 1.55[/math]). https://willrg.com/documents/WILL_RG_II.pdf#sec:wide-binary In conclusion: Space is not a void; it is a resonant structural capacity. Cosmological predictions chain: https://willrg.com/documents/WILL_RG_II.pdf#tab:cosmology_comparison

Thank you. I genuinely appreciate this question. You are the first person in this discussion to ask about the philosophical foundation, and to be honest, it is the most important part of the entire framework. If you look only at the equations, it is perfectly natural to suspect "reverse engineering" or curve-fitting based on prior knowledge of SR and GR. But the reality is exactly the opposite, and it is the most fascinating aspect of this research. The answers you see are not assembled by picking and choosing variables to match textbook results. The entire framework is the inevitable consequence of a single choice made at the very beginning, governed by two strict rules: 1. Epistemic Hygiene: "This line of reasoning derives physics by removing hidden assumptions, rather than introducing new postulates... No assumptions are introduced and no constructs are retained unless they are geometrically or energetically necessary." 2. Relational Origin: "All physical quantities must be defined by their relations. Any introduction of absolute properties risks reintroducing metaphysical artefacts." And now we proceed from strict epistemic minimalism, disallowing all background structures, even hidden or asymptotic ones. Historical Pattern: breakthroughs delete, not add * Copernicus eliminated the Earth/cosmos separation. * Newton eliminated the terrestrial/celestial law separation. * Einstein eliminated the space/time separation. * Maxwell eliminated the electricity/magnetism separation. Each step widened the relational circle and reduced the number of unexplained absolutes. The spacetime--energy split is the only survivor of this pruning sequence. The contemporary split: an unpaid ontological bill All present-day theories (SR, GR, QFT, CDM, Standard Model) are built with a bi-variable syntax: [math]\underbrace{\text{fixed manifold + metric}}_{\text{structure}} + \underbrace{\text{fields + constants}}_{\text{dynamics}}[/math] No observation demands this duplication; it is retained purely because the resulting Lagrangians are empirically adequate inside the split. The split is therefore not an empirical discovery but an unpaid ontological debt. Consequence Until an experiment varies the amount of space while holding everything else fixed, the spacetime - energy separation remains an un-evidenced metaphysical postulate - the last geocentric epicycle in physics. Here is how the derivation actually works (How I get the answers): Step 1: Removing the Container Standard physics assumes a background "container" (spacetime) that gets filled with "things" (energy/matter). By applying Epistemic Hygiene and Relational Origin, we must delete this unobservable background. We are left with only one invariant reality: Energy. But in a purely relational framework, Energy is not a scalar "thing" inside an object; it is strictly the relational measure of difference between states. Thus, we arrive at the core ontological equivalence: [math]\text{SPACETIME} \equiv \text{ENERGY}[/math]. Step 2: Getting on the "Rails" If there is no background 3D grid, how do we encode this conserved relational energy? The mathematics forces the answer. The absolute minimal, background-free geometric structures capable of hosting a conserved relational resource are the closed topologies of [math]S^1[/math] (for directional kinematics) and [math]S^2[/math] (for omnidirectional gravitation). These are not objects in space; they are the protocols of interaction. Step 3: Geometric Crystallization Once the system is on these topological "rails," the physics derives itself. Because [math]S^1[/math] and [math]S^2[/math] are closed geometries, any physical state is simply an orthogonal projection, distributed between an External Amplitude (like kinematic shift, [math]\beta[/math]) and an Internal Phase (internal order/proper time, [math]\beta_Y[/math]). Because the carrier is closed, the geometry strictly demands: [math]\beta^2 + \beta_Y^2 = 1[/math] This is where all the "answers" come from. I did not work backwards from Einstein's time dilation formula. Instead, the geometry dictates that if an observer's state increases its external relational amplitude ([math]\beta[/math]), it must geometrically rotate, necessarily decreasing its internal phase ([math]\beta_Y[/math]). That phase reduction is the physical phenomenon of time dilation. There is no room for reverse engineering because there are zero free parameters. Once you remove the background space and enforce relational closure, the geometry simply doesn't leave any other algebraic path. The philosophy enforces the geometry, and the geometry generates the physics. I hope this clarifies the methodology! I would be happy to answer any deep questions. I also encourage you to test it yourself. Just try starting from the same methodological principles and derive anything else but S^1 and S^2. I tried multipole times its impossible without inconsistencies. All LLM's that I tested this logical chain are also converging towards S^1, S^2 independently. If you are reader: You can find all derivations in details in this document https://willrg.com/documents/WILL_RG_I.pdf If you are viewer: You can glance through the presentation slides: https://github.com/AntonRize/WILL/blob/218963db78ec0f0c2333704b5f1c34dffa4acf98/images/Relational_Geometry_I_21.02.26.pdf Or you can watch a simple 7 min video but its very basic: https://notebooklm.google.com/notebook/d2c97547-d93b-4f20-ad39-2cf307b8d91d?artifactId=67df8cab-b837-4ffd-8cdc-518eec5024eb If you are listener: You can enjoy a nice Deep Dive podcast on WILL_RG_I paper: https://notebooklm.google.com/notebook/d2c97547-d93b-4f20-ad39-2cf307b8d91d?artifactId=dbad15b6-8585-44e4-9306-ae25211177ee So in conclusion: I'm not getting the answers, I'm just simply reviling inevitable consequences of the core methodological principals. The incredible part is that they happened to perfectly match the fundamental laws of physics. Yeh I know what you mean... It is a bit unsettling... so I prefer not to think about it. In the end Im driven not by the desire for recognition but by a personal need to resolve fundamental questions about reality in the simplest possible terms.

This assumes RG reduces the system to a single, unstructured scalar value (e.g., "Joules") from which one must manually extract direction or distance. It does not. In fundamental physics, Energy is not an intrinsic scalar property of an isolated object; it would violate the relational origin principal; it is a structural relation. Energy is the relational measure of difference between possible states, conserved in any closed whole. It is not an intrinsic property of an object, but a comparative structure between states (and observers), always manifesting as transformation. Fundamental physics is dimensionless. The "energy distribution" in WILL RG is inherently structured by two orthogonal dimensionless projections: the kinematic amplitude [math]\beta[/math] and the gravitational amplitude [math]\kappa[/math]. You do not need to perform "conversions" to extract geometry, because Energy and Geometry are inseparable. The relational projections are the geometry. To demonstrate this, here is how the geometric shape of an orbit (eccentricity, [math]e[/math]) is derived directly from the balance of these energetic projections, without any reference to background spatial coordinates, angles, or Newtonian forces. Derivation of Relational Eccentricity: Geometric eccentricity is not a free parameter but a measure of the energetic deviation from the circular equilibrium state ([math]\delta=1[/math]). Using the conservation of the two projection invariants: 1. Energy Projection Invariant (Binding Energy): [math]W = \frac{1}{2}(\kappa^2-\beta^2) = \text{const}[/math]. 2. Angular Projection Invariant: [math]h = r\beta = \text{const}[/math] (at turning points). Consider the two turning points of a closed orbit: periapsis ([math]p[/math]) and apoapsis ([math]a[/math]). The relation between radii is determined by the geometric range: [math]r_a = r_p \left(\frac{1+e}{1-e}\right)[/math] Step 1: Relational Mapping Using the angular invariant ([math]\beta \propto 1/r[/math]) and the potential definition ([math]\kappa^2 \propto 1/r[/math]), we map apoapsis projections in terms of periapsis values: [math]\beta_a^2 = \beta_p^2 \left(\frac{1-e}{1+e}\right)^2[/math] [math]\kappa_a^2 = \kappa_p^2 \left(\frac{1-e}{1+e}\right)[/math] Step 2: Energy Balance Substituting these into the energy invariant conservation condition [math]W_p = W_a[/math]: [math]\kappa_p^2 - \beta_p^2 = \kappa_p^2 \left(\frac{1-e}{1+e}\right) - \beta_p^2 \left(\frac{1-e}{1+e}\right)^2[/math] Rearranging to group potential terms ([math]\kappa[/math]) on the left and kinematic terms ([math]\beta[/math]) on the right: [math]\kappa_p^2 \left[ 1 - \frac{1-e}{1+e} \right] = \beta_p^2 \left[ 1 - \left(\frac{1-e}{1+e}\right)^2 \right][/math] Step 3: Algebraic Reduction Expanding the terms in brackets: [math]\kappa_p^2 \left( \frac{2e}{1+e} \right) = \beta_p^2 \left( \frac{4e}{(1+e)^2} \right)[/math] Dividing both sides by [math]2e[/math] and multiplying by [math](1+e)^2[/math] yields the geometric identity for bound orbits: [math]\kappa_p^2 (1+e) = 2\beta_p^2[/math] Step 4: Connection to Closure Recall the definition of the closure factor at periapsis: [math]\delta_p = \frac{\kappa_p^2}{2\beta_p^2}[/math] Substituting the geometric identity into this definition: [math]\delta_p = \frac{\kappa_p^2}{\kappa_p^2(1+e)} = \frac{1}{1+e}[/math] Solving for [math]e[/math], we obtain the exact geometric eccentricity: [math]e = \frac{1}{\delta_p} - 1 = \frac{2\beta_p^2}{\kappa_p^2} - 1[/math] Summary: [math]\text{ECCENTRICITY} \equiv \text{CLOSURE DEFECT}[/math] There are no forces, no background grid, and no coordinate conversions. The spatial geometry of the orbit is strictly generated by the pure algebraic ratio of the energetic projections. [math]\text{SPACETIME} \equiv \text{ENERGY}[/math] Full derivation: https://willrg.com/documents/WILL_RG_I.pdf#sec:rel_ecc Closed Algebraic System of Relational Orbital Mechanics (R.O.M.): https://willrg.com/documents/WILL_RG_I.pdf#eq:rom

You are overcomplicating my friend. If we strip the problem of any specific details and other additional anthropocentric components we will see the foundation of this phenomena. let's test the hypothesis that the age difference is manifested by the asymmetric observation of Doppler shifts (the optical delay of the turnaround) by removing the delay entirely. Consider a modified scenario: Observer A remains at rest. Observer B does not travel to a distant star, but instead orbits Observer A at a negligible, constant distance with a kinematic projection of [math]\beta = 0.8[/math]. In this setup: 1. The distance is constant, meaning there is zero varying signal delay. 2. The longitudinal Doppler effect (approaching/receding) is eliminated. 3. There is no linear "turnaround" event to flip a signal from red to blue. According to the logic that the asymmetry of observed optical shifts dictates the age difference, this mechanism completely fails to account for this geometry. Yet, the physical age difference undeniably accumulates. Why? Because the age difference is a geometric necessity of the relational state allocation, not an artifact of light travel times. As RG demonstrates, physics is generated by the state difference, not the optical shadow it casts. The calculation is purely algebraic and instantaneous, requiring no signal delay accounting: * Observer A's state: [math]\beta_A = 0 \implies \beta_{YA} = 1[/math] * Observer B's state: [math]\beta_B = 0.8[/math] By the strict closure of the [math]S^1[/math] carrier ([math]\beta^2 + \beta_Y^2 = 1[/math]), Observer B's internal phase is geometrically constrained: [math]\beta_{YB} = \sqrt{1 - 0.8^2} = 0.6[/math] If the experiment runs for [math]N = 10[/math] years in A's rest frame: [math]\tau_A = 10[/math] years [math]\tau_B = \beta_{YB} \cdot N = 0.6 \cdot 10 = 6[/math] years The age difference is 4 years. No outbound leg. No turnaround. No red/blue shift asymmetry. The optical Doppler shift is merely the information transmission of the state; the state itself (the [math]\beta_Y[/math] projection) generates the proper time. I hope it helps to grasp the blinding simplicity of the phenomena at hand. Could you elaborate please? What was the reasoning that led you to this conclusion? And prior knowledge of what? You can use this derivation chart to point out the logical block that "require prior knowledge" https://willrg.com/LOGOS_MAP/

You see this doesn't make any sense to me sorry. How do you determine x=10, y=10 and z=10? In relation to what? How do you choose your (0,0,0) ? On what basis you assume that spacetime can be forced in to a grid like an empty lunchbox? You are smuggling a background manifold into the premise of your question. If there is only me (the observer) and a single static event, there are no "x, y, z" axes. There is no grid. There is only a direct relational link (line of sight). To define x=10, y=10, z=10, you must arbitrarily invent a background coordinate system (an empty lunchbox) with arbitrarily chosen orthogonal axes pointing into empty space. This violates the principle of relational origin. In RG, I self-center at my relational origin [math](\beta=0, \kappa=0)[/math]. If I observe an event, I measure only two things: 1. The relational shift (redshift/Doppler) which gives me [math]\kappa[/math] and [math]\beta[/math]. 2. The geometric scale [math]r[/math] (via luminosity or parallax or signal delay etc...). Angles do not exist a priori between an object and an imaginary spatial axis. Angles only exist between two distinct observable objects (e.g., measuring the angular separation between Event A and Event B). So the system under study is me at the origin and the event. That's all our system is. there's no need to speculate "the empty lunchbox" scenario with arbitrarily chosen values in arbitrarily chosen metric. Geometry is constructed from actual relations between bodies, not by pinning bodies onto an imaginary Cartesian grid. This is precisely what Im doing in this blind test I called out. Send me the data lets test not by just agreeing with GR but by predicting accurate results where GR can not! I know it sounds crazy. I don't believe my results as well. There's only one way to find out. Blind empirical test. Lets do it! Isn't it's the net gravitational force (or acceleration, [math]g[/math]) that equals zero? not the gravitational constant [math]G[/math]? [math]G[/math] remains constant in the Newtonian framework as far as I know... But let's address physics of uniform distribution using RG. This is actually a perfect example of why RG is ontologically cleaner (Occams razor) Standard mechanics explains the zero net force inside a uniform shell via vector cancellation: an infinite number of force vectors pulling in all directions through a background space perfectly cancel each other out. Don't you think its a bit of an overkill ontologically? Do we really need to speculate infinite forces from infinite points just to explain the rest state? Its violates all 5 methodological principals we building on isn't it? Actually Im not sure... having methodological principals in the foundation of the theory is it a common practice? The only thing that comes to mind is equivalence and least action principals... I'm offering a relational definition of Energy, as the relational measure of difference between possible states. https://willrg.com/documents/WILL_RG_I.pdf#def:energy If a distribution is perfectly uniform, there is absolutely zero geometric or physical difference between State A and State B. If there is no difference in state, the relational shift is zero ([math]\Delta Q = 0[/math]). Therefore, the potential projection gradient [math]\Delta\kappa = 0[/math]. Gravity in RG is not a "force" pulling you through a grid; it is the geometric gradient of the potential projection [math]\kappa[/math]. If the density is uniform everywhere, there is no gradient. No gradient means no relational difference, which operationally means zero gravity. We don't need infinite vectors cancelling each other out in a background container. We just have a uniform state with zero relational difference. The math perfectly matches the Shell Theorem outcome, but completely eliminates the unnecessary ontological complexity.

@KJW , @Mordred , @Markus Hanke , Guys something incredible is happening. Please help me conduct the blind test. I want to fully isolate myself from the data source. That's the last explanation I need rule out before considering the possibility of success. The critic in me is ruthless. So I call out to you guy's. Lets do science! Please participate in this blind testing: Call for Blind Test Data: Resolving Inclination Degeneracy Exclusively via 1D Spectroscopy (R.O.M. Framework) In classical Newtonian and Keplerian orbital mechanics, the amplitude of a radial velocity (RV) curve is governed by a single inseparable parameter: [math]K \propto \beta \sin(i)[/math]. Consequently, it is mathematically impossible to independently extract the true orbital velocity [math]\beta[/math] and the inclination angle [math]i[/math] exclusively from a 1D spectroscopic curve. Resolving this degeneracy traditionally requires independent 3D spatial data (astrometry) or transit observations. Within the WILL Relational Orbital Mechanics (R.O.M.), this geometric limitation does not exist. The framework isolates a second-order systemic scalar invariant, [math]Z_{sys}[/math], which is strictly proportional to the absolute kinetic projection [math]\beta^2[/math] and the potential projection, but is fundamentally independent of the observer's line of sight [math]i[/math]. By applying a dynamic 5-parameter inversion (Differential Evolution + MCMC) based strictly on R.O.M. invariants, I recently succeeded in blindly extracting the complete 3D spatial geometry of the S0-2 star ([math]e, \omega_0, i[/math]), its internal precessional shift, and the background drift ([math]v_{z0}[/math]) using nothing but 1D Keck radial velocity data. The extracted inclination matched the independent GRAVITY 3D-interferometer consensus ([math]\sim 134^\circ[/math]) to within the instrumental noise limits. I am now calling for a strictly blind test to prove that im not going crazy. I request that anyone provide me with an anonymized, raw 1D RV/redshift dataset. I do not want to know the system's identity, assumed masses, distance, or established geometric parameters. I will process the raw numbers through the R.O.M. equations and return the full 3D orbital geometry that we will compare with anonymized data source. CRITICAL DATA REQUIREMENTS: For the [math]Z_{sys}[/math] invariant shift to mathematically exceed the noise floor of modern spectrographs, the system must be highly relativistic. 1. Kinematic Scale: Peak orbital velocities must exceed ~1000 km/s ([math]\beta > 0.003[/math]). Standard exoplanets will not work because the second-order [math]\beta^2[/math] shift is orders of magnitude smaller than instrumental noise limits. Ideal candidates are tight compact binaries (WD/NS/BH) or other extreme S-stars. 2. Unprocessed Relativistic Data: The dataset must be raw or minimally processed: [Time (MJD), Radial Velocity (km/s) or Redshift (Z), Measurement Error]. Crucially, the data MUST NOT be pre-corrected for Transverse Doppler or Gravitational Redshift (though standard Barycentric/LSR background velocity correction is fine). 3. Optional (for computational efficiency): Providing the Period ([math]P[/math]) and Epoch of Periapsis ([math]T_{peri}[/math]) is helpful to bound the MCMC sampler, but entirely optional if the data covers at least one full orbit. Please drop the raw CSV data or a link below. Do not provide the system name or accepted parameters. Let the pure relational geometry speak for itself. DATASET EXAMPLE: MJD,RV_km_s,sigma_km_s,Instrument 51718.50000,1192,100,NIRSPEC 52427.50000,-491,39,NIRC2 52428.50000,-494,39,NIRC2 52739.23275,-1571,59,VLT 52769.18325,-1512,40,VLT 52798.50000,-1608,34,NIRC2 52799.50000,-1536,36,NIRC2 52803.15150,-1428,51,VLT 53179.00000,-1157,47,NIRC2 53200.90875,-1055,46,VLT 53201.63925,-1056,37,VLT 53236.33800,-1039,39,VLT 53428.45950,-1001,77,VLT 53448.18300,-960,37,VLT 53449.27875,-910,54,VLT 53520.50000,-983,37,NIRC2 53554.50000,-847,18,OSIRIS 53904.50000,-721,25,OSIRIS 53916.50000,-671,25,OSIRIS 53917.50000,-692,26,OSIRIS 54300.29167,-485,22,OSIRI ... Here's my results for S2 star just from 2 inputs stream MJD, RV_km_s: === R.O.M. DYNAMIC PRECESSION RECOVERY === Eccentricity (e): 0.88498 (GRAVITY Ref: 0.88466) Base Arg of Periapsis (ω0):66.26° (GRAVITY Ref: 66.13°) Internal R.O.M. Precession:0.207° / orbit --------------------------------------------------- Global Kin. Proj. (β): 0.006448 Extracted Inclination (i): 135.68° (GRAVITY Ref: ~134°) Background Drift (v_z0): -20.56 km/s Fit Quality (χ²): 166.87 As far as I understand according to Kepler and GR it shouldn't work right? Im afraid I can't understand the question... Is there a question? This is spot on! EM theory was developed inherently as substantival theory and to rederive it in purely relational terms remains a challenge for now. Mainly due to my knowledge gaps in this domain. But I'm working on it. Im not sure what you mean but as long as its real (not a formalism artifact) Im up for a good challenge. That's how we progress isn't it? This is extremely speculative question my friend. But due to the post was moved to Speculations forum I guess why not? Let's speculate! What do we have to speculate from? We have one individual (me) without specific education and experience in the field, out personal curiosity allegedly (I still hoping that someone finally verify my math and logic, that's actually suppose to be the main purpose of this post. So far out of 150+ comments I got 1 verification from @KJW . Thank you mate.) derives SR, GR, Cosmology, QM in about one year time. I see 2 possible explanation: 1. This individual is a genius 2. The methodology is covering some fundamental truths about the Universe. We can rule out the first option straight away. I have almost 40 years of empirical data that undeniably proves that I'm far from genius. So it must be the method. How ironic is that? The philosophy becomes the key factor of scientific progress in our speculative alternative history thought experiment. Lets say the method was discovered some wearer in late 18 hundreds. The peak of ultraviolet catastrophe in physics. Be cues quantisation, de Broglie Relation, Geometric Quantization of Angular Momentum, Atomic Structure, alpha, Dirac equation are all inevitable results of relational geometry: https://willrg.com/documents/WILL_RG_III.pdf#calibration and in the same time it works wanders in GR realm. So we wouldn't have this incompatibility problem. The development of QM would stream line leading to earlier development of nuclear weapons. Considering that back then Germany was world capital of science we looking in to horrific scenario where Fascist Germany due to overwhelmingly superior technology's developing a nuclear weapon first. German nuclear U-boats bombing the Manhattan project in Los Alamos, New Mexico killing the one and only bongo playing Richard Feynman. The world in tears and Hitler ruling with the iron fist riding the T-Rex. Terror and Horror everywhere. The end. Ha-ha-ha-ha! Yeh it was fun!

Ok. Im not going to use your questions to test WILL-AI. Im finding it fascinating that AI trained on my papers can extend the core ontology in the documents to generate meaningful answers that the source material does not cover directly. Isn't it amazing that well tuned and weighted "massive equation" (AI) can mimic human comprehension to the point that its hard to tell sometimes who's answers are better mine or AI? We live in remarkable time... Using coordinate space would be direct violation of the core methodological principals of my research: Epistemic Hygiene (as Refusal to Import Unjustified Assumptions) This line of reasoning derive physics by removing hidden assumptions, rather than introducing new postulates. This construction is deliberate and contains zero free parameters. This is not a simplification - it is a deliberate epistemic constraint. No assumptions are introduced and no constructs are retained unless they are geometrically or energetically necessary. Ontological Minimalism Any fundamental theory must proceed from the minimum possible number of ontological assumptions. The burden of proof lies with any assertion that introduces additional complexity or new entities. This principle is not a statement about the nature of reality, but a rule of logical hygiene for constructing a theory. Relational Origin All physical quantities must be defined by their relations. Any introduction of absolute properties risks reintroducing metaphysical artefacts and contradicts the foundational insight of relationalism. Simplicity Everything} must be expressed in the simplest form possible. Any unjustified complexity risks reintroducing metaphysical artefacts and contradicts the foundational insight of Epistemic Hygiene. Mathematical Transparency Every mathematical phrase, operational choice, or identity caries its own ontological statement. Each mathematical object must correspond to explicitly identifiable relation between observers with transparent ontological origin. Every symbol must be anchored to unique physical idea. Introducing symbols without explicit necessity constitutes semantic inflation: the proliferation of symbols without corresponding physical meaning. Number of symbols = Number of independent physical ideas. Every prediction or result can be directly backtrack to this 5 core methodological principals. So as you can see coordinate space violates all 5 core principals. Regarding Alice and Bob perspectives: I already posted it here so Ill just quote myself: So everyone get's to be the center of its own "Universe" and the relational reciprocity [math]Q_A=Q_B[/math] same as the Energy Symmetry Law: [math]\Delta E_{A \to B} + \Delta E_{B \to A} = 0. [/math] https://willrg.com/documents/WILL_RG_I.pdf#sec:energy-symmetry are keeping this framework from falling down in to solipsism. Here's a simple Desmos project that mapping this relations: https://www.desmos.com/geometry/2nkjtezjpi