Anton Rize

Do you understand that distance can be relational without metric involved? Also try google sometimes: "Areal radius is a term primarily used in cosmology and general relativity to describe a specific radial coordinate that measures the "area" of a spherical surface centered on an observer within a curved spacetime. It is defined so that the surface area of such a sphere is A=4πR2, where R is the areal radius." I understand that its different from what you used to, but if everything would be the same as GR it wouldn't be a new idea isn't it? Its not that hard. You can define distance between n number of objects without a "grid" just relational. I already showed you set of equations and 2 demos projects with numerical conformations. I don't know what else I can do... Means that relational properties comes first and they determent geometry. Are you aware that you haven't wrote a single equation here? We can exchange opinions with you forever. In the end its numbers that meters.

@Markus Hanke I made 2 desmos projects for you: https://www.desmos.com/geometry/nrtnjramrl - calculates aphelion of Mercury using the set of algebraic equations I listed above r_{a}=\frac{-R_{s}-\sqrt{R_{s}^{2}-8E_{d}\left(-h^{2}\right)}}{4E_{d}}=6.9762118617\times10^{10} m. empirical value r_a = 6.982×10^10 m (discrepancy due to estimated input values, but you got the point) https://www.desmos.com/geometry/hkxjqfkchp - calculates perihelion precession of Mercury \Delta_{WILL}=\frac{2\pi Q_{Merc}^{2}}{\left(1-e_{Merc}^{2}\right)}\ = 5.0208724126\times10^{-7} radians/orbit. empirical value \Delta_{Merc}=5.02 \times10^{-7} radians/orbit.

@Markus Hanke I'm very disappointed. I genuinely was hoping to finally have a meaningful conversation... I took your question seriously and spend time to provide you with an answer. You on the other hand didn't even try to understand which I can see clearly by your comment "Some of these things are also observer-dependent" - its a relational framework most of "these things" are observer-dependent. You state that concepts like "gradient," "orthogonality," and "conserved quantity" cannot "meanfully exist" without a spacetime manifold and a metric (gμν). My position is that these concepts emerge algebraically from the relational geometry (S¹ × S²), before any metric is postulated. You seem to be mistaking my algebraic definitions (the physics) for their metric descriptions (the coordinate overlay). To move this from opinion to mathematics, please identify which of my exact algebraic definitions you claim is (or illicitly requires) a metric tensor. Objection 1: "You use 'r', 'gradients', 'orthogonality', and 'areas' which require a metric." Response: I define these algebraically. Which of the following definitions is a metric tensor? A) r ≡ Rₛ / κ² (My definition of the areal radius 'r' as an algebraic output of a field measurement κ, not an a priori coordinate.) B) ∇κ (The physical gradient of the scalar potential field κ. I define "radial" as this direction.) C) The intrinsic orthogonality on the S² manifold. (I define "tangential" as the direction orthogonal to ∇κ on my base manifold S², which is the source of geometry, not its product.) Objection 2: "You use 'conserved quantities' which require Killing vectors (metric symmetries)." Response: My invariants are algebraic, not differential. Which of these algebraic budgets do you claim requires a Killing vector? A) ΔE = 0.5 * (β² - κ²) (My fundamental trajectory invariant, derived from the Energy-Symmetry Law, not a ∂ₜ symmetry of a metric.) B) h = r₀ * βₜ,₀ (The definition of angular momentum, which I noted is conserved only because spherical symmetry was a given input - i.e., ∇κ is purely radial.) Objection 3: "You use a 'nebulous scalar' and cannot encode an EM field or non-uniform source." Response: I explicitly defined the general case in my "Case B." Which of these formalisms is an invalid source model? A) κ²(r) = Rₛ / r (The single scalar only for the spherically symmetric case.) B) κ(r, Ω) (The scalar field on S² I use to encode any non-uniform source, such as your EM field, as a directional map. This is not a single scalar.) Objection 4: "You don't account for non-linearity." Response: My model is algebraically non-linear at its foundation. Which of these core equations do you claim is linear? A) ΔE = 0.5 * (β² - Rₛ/r) (The invariant energy budget itself is a non-linear relation between β and r.) B) (2ΔE) · rₚ² + Rₛ · rₚ - h² = 0 (The final algebraic equation I derived to solve the trajectory.) You are assuming these concepts imply a metric. I am deriving them from an algebraic foundation that precedes it. The burden is now on you to show which definition is faulty, not which word sounds like GR. P. S. And regarding your comment about LLM-AI: I'm using AI to translate in to English and reformat Latex in to Unicode. If I would allow AI to wright answers for me this would be a very short thread. Do you think there's an AI who can handle tasks like this? Im using Gemini and it cant. If you know one that can please give me a link Id love to talk to it.

@pinball1970 And you call it "pull it apart"? I cant even understand your argument. Can you express yourself a bit clearer? @swansont Two clarifications: (i) I don’t claim β is Lorentz-invariant; it’s an operational, dimensionless primitive within a chosen frame. What I call “invariant” is the form of the closure identity (e.g. β^2+βY^2=1), not each component separately. (ii) x,t become “measurable” only after conventions (synchronization, frame, units). They are representational choices, not primitives. MH targets exactly this: primitives first (operational, dimensionless), coordinates later as derived bookkeeping. Pedagogy is a separate (and open) question; ontology shouldn’t be outsourced to convenience.

Thanks for jumping in @Markus Hanke Great question! Your premise is a bit flowed. Within WILL RG we not dealing with non physical imaginary regions. Region is not a physical object - you cant measure it. You'll have to postulate one and we don't want to postulate non physical made up stuff. Instead we can work with relations within given system. Hit-or-miss in WILL (short, coordinate-free) • We do not use metrics or geodesics. Dynamics = a single invariant budget: ΔE = 0.5*(β² − κ²). • “Radial” is defined without coordinates: it is the direction of ∇κ (meridian on S²). “Tangential” is orthogonal to it on the relational sphere. • Case A (stationary spherical source): the field is one scalar, κ²(r) = Rₛ / r, equivalently r = Rₛ / κ². Given initial state at r₀ with total speed β₀ (where β₀² = βᵣ,₀² + βₜ,₀²) and tangential part βₜ,₀, two constants follow: ΔE = 0.5*(β₀² − κ₀²), and h = r₀ * βₜ,₀ (specific angular momentum; conserved by spherical symmetry). • Periapsis is where βᵣ = 0, hence βₚ = βₜ,ₚ = h / rₚ and κₚ² = Rₛ / rₚ. Invariance gives a single algebraic equation for rₚ: 0.5*(h² / rₚ² − Rₛ / rₚ) = ΔE, i.e. (2ΔE)·rₚ² + Rₛ·rₚ − h² = 0, solve for rₚ > 0. Hit if rₚ ≤ R_body. • Case B (non-uniform/radiation field): replace the scalar by a directional map κ(r, Ω). ΔE remains invariant; “radial” stays ∇κ. There is no global conservation of h without symmetry, so propagate the tangential budget step-by-step relative to ∇κ. The same hit test applies once rₚ is obtained. Takeaway: WILL computes trajectories by conserving algebraic budgets (ΔE, and h only when symmetry allows). A metric can be added later as a descriptive overlay, not as the engine. PARAMETER GLOSSARY • c - speed of light (units: length/time). • G - gravitational constant. • M - central mass (Case A). • R_s - Schwarzschild radius, R_s = 2 G M / c^2 (units: length). • r - areal radius (units: length), defined by the observable area A via A = 4π r^2. In stationary spherical exterior: r = R_s / κ^2. • r_0 - initial areal radius (units: length). • r_p - periapsis (closest-approach) radius (units: length). • R_body - physical radius of the extended body (units: length). • β - total kinematic projection (dimensionless), β = v / c, with 0 ≤ β ≤ 1. • β_r - radial component of β, defined relative to the local “radial” direction. • β_t - tangential magnitude of β, β_t = √(β^2 − β_r^2). • β_0 - initial β; β_t0 - initial tangential component of β. • β_p - total β at periapsis; at periapsis β_r = 0, so β_p = β_t,p. • κ - potential projection (dimensionless). - General (non-uniform) field: κ = κ(r, Ω), a directional map on S². - Stationary spherical exterior: κ^2(r) = R_s / r (so 0 < κ^2 < 1 for r > R_s). • κ_p - κ at periapsis; in spherical exterior: κ_p^2 = R_s / r_p. • Ω - direction on the unit sphere S² (solid angle label for anisotropic fields). • ∇κ - spatial gradient of κ; defines the “radial” direction locally (coordinate-free). • h - specific angular momentum (units: length), h = r · β_t. - Conserved only when spherical symmetry holds (Case A). - Not a global invariant in generic non-uniform fields. • ΔE - relational energy budget (dimensionless), invariant along the trajectory: ΔE = 0.5 · (β^2 − κ^2). (In spherical exterior this yields the periapsis equation 0.5 · (h^2 / r_p^2 − R_s / r_p) = ΔE.) Notes on domains and use: • Exterior of a compact spherical source: r > R_s ⇒ 0 < κ^2 < 1 and 0 ≤ β ≤ 1. • “Radial” and “tangential” are defined operationally: radial ≡ direction of ∇κ; tangential ≡ any direction orthogonal to ∇κ on S². • Conserved quantities: ΔE always; h only when symmetry (spherical) supplies a corresponding Killing vector. @joigus I think there may still be a misunderstanding here. In a Minkowski diagram the axes X and Y are physical coordinates (ct, x) defined within a metric. In the S¹–S² formulation of WILL, X and Y are dimensionless relational projections – amplitude- and phase-like components of a single conserved quantity. They are not spacetime coordinates; no metric is postulated at that stage. Could you please specify which part of this distinction remains unclear?

Thank you for jumping in @KJW I understand it might feel unconventional at first. Try this page https://antonrize.github.io/WILL/relativistic-foundations/ I wrote it specifically to ease understanding. There's a lot of visual and interactive elements, definitely better than scrolling through long forum debates. No. I'm doing exactly opposite: removing assumptions and anthropocentric elements. If you will follow the link you will see it immediately. Good point! I like your grounded approach. But lets have a look what exactly we measuring: When we “measure distance,” what we actually record is not some absolute spatial entity but a relational configuration of events - a count of oscillations, light pulses, or clock ticks exchanged between systems. Likewise, when we “measure time,” we don’t access a primitive flow; we compare periodic processes and define one as our reference. Both quantities - distance and time - are abstractions distilled from relational data. So in WILL, the goal is not to deny measurements, but to clarify that what is measured are relations, while what is inferred (like space or time) is representational. The relational structure comes first; coordinates appear only after we project that structure into human-friendly form. So yes coordinate space and time not primitive ingredients but the result of our conventional choice of representation. And that what you can call: Universe doesn't know and doesn't care about any coordinate system of ours. Completely agree with you on this. And this is precisely what im trying to show here. --- The remarkable thing is when you adopt this relational view the math is just fits in to places naturally! Its simpler cleaner requires no unphysical constructions (like coordinate system) and perfectly allying with experimental results. And I cant believe it myself! That's why I posted it here.

Thank you for jumping in @swansont Mathematical equality ≠ same ontological clarity. Replacement of ontology with mathematical artifacts is a well known problem in physics raised by Sean Carol, Nima Arkani-Hamed and many more. Exactly! the ease of use and how it’s learned/taught directly depends on amount of abstract coordinates like t, x that require interpretation. They are not primitive observables but representational choices: they depend on frame and synchronization conventions. What’s operationally invariant are proper-time intervals and relational, dimensionless ratios. MH targets exactly this distinction between representational coordinates and observable structure. Two equally correct formulations are not pedagogically equivalent - the one with fewer abstract, non-operational symbols conveys the physics more transparently. Ironically the standard form is the one who has to be deciphered to put it in physical terms (coordinate constructs like t and x). This is a great example of replacing ontology with mathematical artifacts. You just proved my point. Absolutely - that’s why this is framed as an experiment, not an assertion. you’re right that not all physics is naturally dimensionless. That’s exactly why I framed gravity as an open task under the same MH constraints - to see whether equally “hygienic” primitives can be identified there. In other words, the SR example is step 1, not the conclusion. Finally comment with substance not just personal attacks. Thank you @swansont

your critic is based on elementary category error that you once again fail to admit. I already answered this: --- "The Nobel Prize could be just around the corner." - I was hoping to meat likeminded people here who are like me passionate about physics. I was hoping for deep and meaningful discus and collective search for Truth. But all Im getting is just uncalled personal attacks. Im verry disappointed...

@studiot @exchemist @joigus It’s been three days since my last detailed response, and so far none of the critics have addressed or verified a single one of the derivations presented above - despite shown earlier confidence and patronising tone that should not be used in scientific discus. This silence is quite telling: once the discussion reached the level of explicit formulas and transparent definitions, the enthusiasm for debate seems to have disappeared. For readers genuinely interested in the content rather than posturing, all derivations, unit checks, and equivalences have been provided in full detail above. Unless new, substantive objections are raised, I will take the current pause as tacit agreement with the correctness of the presented results. It was far from the level of discussion I was hoping for.

This is exactly what this post is made for. Considering the fact that its your 4th message here and you still not sure about the purpose of this post - I highly doubt that you can. But will be happy if proven wrong - that's the hole point. Be my guest.

Is it all you are capable of? Shallow view and overgeneralisation without any substance? If so - please leave.

Post is written by me but I use AI to translate it in to English and reformat latex in to Unicode. I understand that there's a lot of AI generated crap around but if you assuming that every well structured argument is AI generated the again you overcomplimenting AI's abilities and underestimating mine. You didn't answer my question:

Good point Hahaha! I guess we limited by our language or at I am. You overcomplimenting AI's abilities and underestimating mine. "steaming shit" is not a constructive argument isn't it? "Your theory" is a misplaced term in this context. There's a difference between theory and methodology I suggest you google it. In the end its just 4 premises: A1 A2 A3 A4 Which one you finding resemble a "steaming shit"?

You already showed your glaring incomitance in other thread, so im not surprised. Goodbye.

I am proposing a rigorous thought experiment at the intersection of physics and the philosophy of science. The goal is to test whether we can, by applying a set of strict methodological constraints, arrive at a single, fundamental formulation of physical laws. This challenge is not based on new physics, but on "Mathematical Hygiene" (MH) - a principle demanding absolute correspondence between mathematical symbols and their operational, relational meaning. Part 1: The Rules of the Experiment (The Principles of "Mathematical Hygiene")For this experiment to be pure, we must temporarily agree to follow four disciplines. (I will refer to them as A.1–A.4). * A.1. The Principle of Ontological Economy: A theory must be founded on the smallest possible set of primitive, operationally defined concepts (e.g., observable relations or transformations). All other constructs (fields, potentials, curvature) must be derivative functions of this primitive basis. * A.2. The Discipline of Minimal Representation: The number of symbols in a "hygienic" formula should equal the number of independent physical ideas. Using, for example, a tensor T_μν (10+ components) to describe the simple idea of "energy density" is "semantic inflation." * A.3. The Principle of Hierarchical Sufficiency: All derived constructs must be unambiguously derivable from the primitive basis. New physics cannot be "smuggled in" under the guise of mathematical convenience. * A.4. The Discipline of Epistemic Hygiene: Every symbol in a theory must carry an "interpretive passport" linking it to an empirical, relational (dimensionless) meaning. Symbols without passports (e.g., t=0 or x=0 in an empty universe) are "grammatical artifacts" and must be eliminated. Part 2: Example Application of MH (Kinematics)Let's apply this filter to an icon of physics: the invariance of the SR interval. Test Subject: dτ² = dt² - dx² (with c=1)MH Verdict: This equation is fundamentally "unhygienic." Justification: 1. Violation A.4 (Passport): The symbols dt and dx are not primitives. They are differentials of t and x. 2. Violation A.1 (Container): t and x themselves imply the existence of an a priori coordinate grid - a "hidden container" we impose on reality. This container has no "interpretive passport" in the absence of events. 3. Violation A.4 (Units): Even if we accept dt and dx as "measurable," they are measured in seconds and meters - arbitrary human conventions, not fundamental relational (dimensionless) quantities. "Hygienic" Derivation (Kinematics): To "cleanse" this law, we must use only dimensionless, relational primitives: 1. Amplitude Primitive (Motion): The ratio of an object's velocity to the universal speed limit. This is β = v/c. It has a clean "passport." 2. Phase Primitive (Time): The ratio of an object's proper time to the observer's time. This is β_Y = Δτ / Δt. It also has a clean "passport." The physical law is the Lorentz factor: β_Y = √(1 - β²). If we write this as a closure identity, we get the "hygienic" form of the law of kinematics: β² + β_Y² = 1 Note: no d's, no t or x, no meters or seconds. Just two pure, dimensionless primitives in an algebraic identity. Part 3: The Challenge (Gravity)Now, here is the task for you. We must apply the exact same MH logic to gravity. We must completely reject the formalism of GR (G_μν, g_μν, differentials) as the peak of "semantic inflation" and "hidden containers." The Task: Find the fundamental "hygienic" closure identity for gravity, analogous to β² + β_Y² = 1. Guiding Questions: 1. How do we define the "hygienic" Amplitude Primitive for gravity (let's call it κ)? * It must be dimensionless. It must describe the intensity* of gravity. 2. How do we define the "hygienic" Phase Primitive for gravity (let's call it κ_X)? * It must be dimensionless. It must be the analogue of β_Y (i.e., it must describe the observable gravitational time dilation*). 3. What is the necessary algebraic closure identity that connects κ and κ_X? Rules of Engagement:For the sake of a pure experiment, I ask all participants to adhere strictly to the MH methodology (A.1–A.4). The goal is not to prove "GR is wrong," but to test a hypothesis: Will strict epistemological discipline force all participants toward the same, inevitable algebraic structure? I await your derivations.

Thanks, @studiot and @joigus and @exchemist I think now I see were the misconception is coming from. Let me clarify step by step so that we keep the structure transparent and consistent. Methodology The WILL framework begins with the strictest possible discipline: we postulate nothing and assume no background structure. A Minkowski metric, by contrast, is introduced a priori as a fixed geometric container. Within WILL, such a metric cannot be fundamental - it arises only as an anthropocentric artifact of how we choose to parameterize relational transformations. In other words, distance and time are not primitive ingredients; they are convenient coordinates that appear after the relational structure is already defined. Two geometries, two meanings of X and Y In a Minkowski diagram the axes X and Y are physical coordinates with a postulated metric: ct and x, and boosts are hyperbolic isometries of that metric. In the S¹–S² formulation of WILL, X and Y represent dimensionless relational projections - amplitude-like and phase-like components of one conserved quantity. These projections have no intrinsic metric; they describe ratios, not distances. Emergence of metric and hyperbolic structure The metric-like relations appear only when we reintroduce scale - for example, when we define  r = Rs / κ² with Rs = 2GM / c². At that stage, the familiar hyperbolic form a² − b² = c² naturally emerges as a particular parameterization of the same invariant relation. It is not postulated but derived from closure. This is also how the classical results - the Kepler energy relation, Newton’s third law, the Lagrangian and Hamiltonian structures - appear as necessary consequences, not as inputs. Foundational distinction Traditional relativity assumes a metric and derives dynamics within it. WILL derives dynamics purely from relational closure; metric descriptions then appear as a representational convenience. This reverses the usual order of reasoning and removes the hidden background from physics entirely. On the concept of spacetime It helps to move away from the picture of spacetime as a container holding energy. In WILL, spacetime and energy are two complementary descriptions of a single relational entity -structure and dynamics viewed from different sides of the same relation. Empirical correspondence and next steps Every relation in the framework produces testable, quantitative results - GPS time shifts, orbital radii (photon sphere 1.5 Rs, ISCO 3 Rs), and more. If anyone is interested, we can go through the classical derivations - Keplerian energy, Lagrangian, Hamiltonian, and Newton’s law - one by one in upcoming messages. The only reason for taking them sequentially is to preserve the internal logic of the model, since everything unfolds directly from the same first principles. --- Short conceptual note on what S1 and S2 mean in the WILL framework S1 and S2 are not spatial circles or spheres placed inside spacetime. They are relational architectures – minimal topological structures that guarantee closure, conservation, and isotropy of the relational resource we call energy. Energy here is not a substance but a measure of difference between possible states – a kind of bookkeeping of transformations. S1 and S2 are the internal rules that make such bookkeeping consistent: • S1 encodes directional relations (kinematics) • S2 encodes omnidirectional relations (gravitation) Applied view – two observers Imagine two observers, A and B: A is the center of their own relational framework. B lies on A’s S1 (for motion) and S2 (for gravity). At the same time, B is the center of their own framework, with A lying on B’s S1 and S2. Each observer defines their own relational coordinates, yet the mutual consistency of these structures produces the shared phenomena we call space and time. So S1 and S2 are not in space – they are the patterns of relation that generate its appearance. Please let me know if it clears things out and if you want to see the derivation of classical mechanics results. It's absolutely transparent and I find it beautiful.

Thanks, @studiot - that’s a fair question. The symbols E, p, and m refer to the same physical quantities in both equations. The only difference is the system of units used to express them. In theoretical physics it’s completely standard to set certain constants, like c = 1, to simplify notation. This doesn’t change the meaning of any quantity - it just removes conversion factors that would otherwise clutter the algebra. So:  • In natural units (c = 1), we write E² = p² + m².  • In SI units, we restore dimensions explicitly: E² = (p·c)² + (m·c²)². They are two notational layers of the same invariant relation. If you substitute SI dimensions into the second form, it checks out exactly. The first one is simply a shorthand used universally in theoretical physics to keep the focus on structure rather than bookkeeping. To avoid any further confusions - from now on I'll write all my derivations explicitly in SI units.

I really appreciate that, @studiot . It’s encouraging to see someone open to exploring new formulations. The S² projection is indeed the gravitational analogue of the same closure logic — I’ll outline it briefly below so you can see how it parallels the S¹ derivation: Potential Energy Projection on S² (SI Formulation) (Important note) S¹ and S² are relational manifolds of conservation, not physical spacetime surfaces. They describe how a single energy resource divides between two complementary projections - internal (phase) and external (amplitude). 1. Relational Conservation on S²The components  κₓ — internal (phase) projection  κᵧ — external (amplitude) projection obey  κₓ² + κᵧ² = 1. An increase in κᵧ corresponds to a higher gravitational potential (external relation), and a decrease in κₓ reflects contraction of proper length and slowing of proper time. These geometric redistributions generate all known gravitational effects. 2. Gravitational Tangent FormulationAnalogous to the SR relation using β = v / c, we define for the gravitational case:  κ = vₑ / c  where vₑ = √(2GM / r) is the escape velocity. Thus, κ is a dimensionless gravitational ratio, and κₓ = √(1 − κ²) defines the internal projection. (a) Kinematic relation (S¹)β = v / c = cos θ₁ E = E₀ / sin θ₁  p = (E₀ / c) cot θ₁  → E² = (p c)² + (E₀)² where E is total energy (J), E₀ = m c² (J), p is momentum (kg·m/s). (b) Gravitational relation (S²)κ = sin θ₂  κₓ = cos θ₂ E_g = E₀ / κₓ  p_g = (E₀ / c) tan θ₂  → E_g² = (p_g c)² + (E₀)² Both have the same invariant structure once dimensions are restored. 3. Unified InterpretationSR circle: (E βᵧ = E₀) with (β, βᵧ) = (cos θ₁, sin θ₁) GR circle: (E_g κₓ = E₀) with (κₓ, κ) = (cos θ₂, sin θ₂) Therefore:  E² = (p c)² + (m c²)²  E_g² = (p_g c)² + (m c²)² and  β ↔ κ,  cot θ₁ ↔ tan θ₂. Summary (SI-consistent): All physical quantities retain their standard dimensions:  [c] = m · s⁻¹  [m] = kg  [E] = J = kg · m² · s⁻²  [p] = kg · m · s⁻¹. Relativistic and gravitational relations are duals - two geometric projections of the same conserved quantity E₀ = m c². If you interested in the next post I’ll show how their composition leads directly to the Equivalence Principle, where m_g = m_i appears not as a postulate but as a geometric identity. Hilbert’s variational principle is an elegant route to GR - but it relies on an external time parameter and a freedom to vary trajectories. In the relational framework, both of these assumptions dissolve naturally: change itself defines the temporal order. I’ll show later how this leads to a reformulation of dynamics without equations of motion - a purely geometric self-consistency where energy redistribution replaces the action principle. Its quite fascinating.

@studiot There are two notational layers. In natural units (c = 1) the mass shell is E² = p² + m², (c = 1) and in SI (or any dimensional system) the same statement is E² = (p·c)² + (m·c²)². (SI units) If you want to substitute SI units, use the dimensional form above. Nothing is dimensionally incorrect there. For completeness, the projections used in the derivation are: β := v/c, β_Y := √(1 − β²) E = E₀ / β_Y, with E₀ := m·c² p·c = E₀ · (β / β_Y) ⇒ p = (E₀/c) · (β/β_Y) = γ m v Check: (p·c)² + (m·c²)² = [E₀·(β/β_Y)]² + E₀² = E₀²[(β²/β_Y²) + 1] = E₀²[(β² + β_Y²)/β_Y²] = E₀²/β_Y² = (E₀/β_Y)² = E². Units (SI) for bookkeeping: [c] = m·s⁻¹, [m] = kg, [E] = J = kg·m²·s⁻², [p] = kg·m·s⁻¹, [v] = m·s⁻¹. So: substitute your SI values into E² = (p·c)² + (m·c²)². The shorter E² = p² + m² line I quoted earlier was in c = 1 notation only. But this is really just the tip of the iceberg. The same geometric logic extends beautifully to gravity through the S² projection - that’s where the equivalence principle and curvature emerge directly from the same closure rules. If you’d like, I can show that next - it’s quite elegant and completes the SR/GR connection without introducing tensors.

The interesting thing is that in RG, all primary quantities (β, κ, τ, Q etc.) are dimensionless by construction. It operates entirely through normalized relational ratios, not dimensional measures. That means the equations remain valid under any consistent unit system - SI, Planck, geometrized, or even a hypothetical alien metric. As long as their constants are internally consistent, the relational structure behaves identically. This scale-independence is deliberate: it reflects the idea that physics should describe relations rather than magnitudes. Units are a linguistic layer - the grammar of measurement, not the content of reality. For numerical predictions or comparison with empirical data, it’s always practical to restore SI units for bookkeeping: • [c] = m·s⁻¹ • [m] = kg • [E] = J = kg·m²·s⁻² • [p] = kg·m·s⁻¹ • [v] = m·s⁻¹ Nothing unconventional here - all formulas reduce to standard dimensional identities once SI is restored. I must admit I didn’t expect the discussion to turn to unit conventions - I was hoping for comments on the derivation I posted earlier. The structure itself is what matters; units are just the language we use to read it.

Thank you, noted. Here’s the formal glossary for clarity, as all quantities are defined geometrically within the relational circle: E – total energy of the closed relational system E₀ – invariant rest energy (vertical projection of E) p – momentum (horizontal projection of E) m – rest mass, identical to E₀ in c = 1 normalization βₓ, βᵧ – normalized projections on S¹, satisfying βₓ² + βᵧ² = 1 θ₁ – angular coordinate on S¹, βₓ = cos θ₁, βᵧ = sin θ₁ KINETIC ENERGY PROJECTION ON S¹ Since S¹ encodes one-dimensional displacement, the total energy E of the system must project consistently onto both axes: Eₓ = E·βₓ , Eᵧ = E·βᵧ. THEOREM - Invariant Projection of Rest Energy: For any state (βₓ, βᵧ) on the relational circle, the vertical projection of the total energy is invariant: E·βᵧ = E₀. PROOF: When βₓ = 0, closure enforces βᵧ = 1, yielding E = E₀. Since closure applies for all θ₁, the vertical projection E·βᵧ remains equal to this rest value in every state. COROLLARY - Total Energy Relation: E = E₀ / βᵧ = E₀ / √(1 − βₓ²) REMARK - Lorentz Factor: The historical Lorentz factor γ is simply 1 / βᵧ. No additional structure is introduced: all content is already present in E·βᵧ = E₀. SUMMARY: γ = 1 / βᵧ. REST ENERGY AND MASS EQUIVALENCE Within normalization c = 1, the invariant rest energy equals mass: E₀ = m. PROOF: From the invariant projection E·βᵧ = E₀ and closure of S¹, no additional scaling parameter is required. Hence conventional bookkeeping identities E₀ = m·c² or m = E₀ / c² reduce to tautologies. Mass is therefore not independent, but the rest-energy invariant itself. REMARK: In a framework that is genuinely fundamental and free from arbitrary human units, the natural normalization is always c = 1. With this normalization, the identities E₀ = m·c² or m = E₀ / c² lose all significance. They collapse into the only consistent statement: E₀ = m. Thus mass is the invariant projection of total rest energy. ENERGY–MOMENTUM RELATION PROPOSITION - Horizontal Projection as Momentum: On the relational circle, the unique relational measure of displacement from rest is the horizontal projection E·βₓ; hence p ≡ E·βₓ (c = 1). PROOF: The rest state is (βₓ, βᵧ) = (0, 1). A displacement measure must (i) vanish at rest, (ii) grow monotonically with |βₓ|, and (iii) flip sign under βₓ → −βₓ. The only relational candidate satisfying (i)–(iii) is E·βₓ. Thus the identification is necessary rather than conventional. COROLLARY - Energy–Momentum Relation: With p identified by the previous definition and m = E₀, the closure identity yields E² = p² + m² (c = 1). Equivalently, restoring c: E² = (p·c)² + (m·c²)². PROOF: By closure, (E·βₓ)² + (E·βᵧ)² = E². Substituting p = E·βₓ and m = E₀ proves the claim. Restoring c is dimensional bookkeeping: p → p·c and m → m·c², while E remains E, yielding the standard form. REMARK - Geometric Forms: E² = ( (βₓ / βᵧ)·E₀ )² + E₀² = ( cot(θ₁)·E₀ )² + E₀². These are equivalent renderings of the same geometric necessity. REMARK - Units sanity check (bookkeeping): Using βₓ = v/c, the identification p ≡ E·βₓ gives p·c = E·(v/c) ⇒ p = (E·v) / c². With E = (1/βᵧ)·m·c² = γ·m·c², this reduces to p = (βₓ/βᵧ)·m·c = γ·m·v, the standard relativistic momentum. No new parameters introduced. TABLE - Representation of relativistic effects: βₓ = β , β = v/c , θ₁ = arccos(β) Algebraic form: 1/βᵧ = 1/√(1−β²) = 1/√(1−(v/c)²) βᵧ = √(1−β²) = √(1−(v/c)²) Trigonometric form: 1/βᵧ = 1/sin(θ₁) = 1/sin(arccos(β)) βᵧ = sin(θ₁) = sin(arccos(β)) SUMMARY: The energy–momentum relation E² = p² + m² is a geometric identity of S¹. If you’re interested, next I can show how the same logic extends from the kinetic S¹ projection to the gravitational S² projection, yielding the equivalence principle directly from symmetry. It’s quite elegant, and it closes the SR/GR bridge naturally.

@studiot @studiot Thanks. Let me address your points precisely and ask you to pinpoint the exact step you dispute. 1) “E² = p² + m² is derived but not as you said.” The derivation I gave uses only: (i) the circle identity βₓ² + βᵧ² = 1, (ii) E₀ ≡ E·βᵧ (rest-invariant projection), (iii) p ≡ E·βₓ (displacement projection). Then (E·βₓ)² + (E·βᵧ)² = E² ⇒ p² + E₀² = E². If you think this is illegitimate, please say which identification you reject: A) E₀ ≡ E·βᵧ, or B) p ≡ E·βₓ, or C) the use of βₓ²+βᵧ²=1. “Generally wrong” doesn’t help; name the line and I’ll fix it or show the algebra. 2) On Gμν = (8πG/c⁴)Tμν “not meaning spacetime = energy.” Agreed: the equation does not identify kinds. It states a structural constraint: curvature equals (constant)×stress-energy. My “≡” denotes ontological equivalence of descriptions (mutual definability), not sameness of substance. Your analogies “F = ma ⇒ force is mass”, “distance = speed×time ⇒ distance is time” actually make my point: equality of relations ≠ identity of categories. That’s exactly how I’m reading EFE. 3) “Spacetime is not even a real space.” Yes. And in WILL neither S¹ nor S² are treated as “real space”; they are bookkeeping manifolds enforcing closure/isotropy of a conserved relational resource. Ordinary spacetime is an emergent coordinate representation. So here we agree. 4) “Relational geometry” caricature and the sugar–honey–vinegar example. Composing arbitrary predicates (“sweet”, “brown”) has no algebra and no invariant - of course it doesn’t compose. RG, relations compose because they share a common invariant budget: • Kinematics: β²+βᵧ²=1, • Gravity: κₓ²+κ²=1, • Exchange (closed orbits): κ²=2β². Those are quadratic constraints with a well-defined composition law (projections on one conserved quantity). That is why A–B and B–C imply a determinate A–C. 5) “Tensorial equation quoted inappropriately.” Nothing inappropriate was inferred beyond what I’ve just clarified: EFE links geometry and energy-momentum by equality of relations, not by identity of entities. That’s exactly my usage. So the crux is simple: if you accept (i)–(iii) above, the mass shell follows as an identity. If you reject one of (i)–(iii), which one and why? Point me to the exact line you find unsound, and I’ll respond with the full derivation you prefer (metric, rapidity χ, or purely algebraic). Let me take a brief pause from the line-by-line debate to clarify what kind of question this discussion actually belongs to. What we are discussing here is not really whether one can write “spacetime ≡ energy” or which algebraic symbol is more proper. It is the continuation of a philosophical debate that began more than three centuries ago - between Newton’s substantivalism and Leibniz’s relationalism, later echoed in Mach and Einstein. Newton’s view: space and time exist as independent containers - an absolute background in which events unfold. Leibniz’s view: space and time are nothing over and above the relations among entities and their changes. General Relativity kept Newton’s mathematics but never fully escaped his ontology: spacetime in GR still exists even in the absence of matter (the vacuum solution). That is substantivalism in disguise - geometry treated as an independent thing. Einstein himself was never satisfied with this. In his later correspondence with Schilpp and Besso, he called GR “a provisional synthesis” and wrote explicitly that a truly Machian theory would eliminate spacetime as an independent entity altogether. RG continues that unfinished line - it is not a denial of GR, but the completion of Einstein’s own relational intent. Its principle (Structure ⇔ Dynamics) simply removes the last residue of substantivalism: geometry and energy are two projections of a single relational resource. So when someone objects that “spacetime = energy makes no sense,” they are unknowingly taking Newton’s side of the same 300-year-old debate. The question is not whether GR’s equations work - of course they do. The question is whether they describe the world as it is or a mathematical prosthesis that still carries Newton’s metaphysics. That is the level at which RG operates. Everything else - tensors, topology, units - follows from that one epistemic decision.

@ exchemist Ha! Good reference - though self-satisfied toasters could probably solve most of my philosophical dilemmas. I already posted the short version of this derivation earlier, but I’ll quote it again here for clarity: Let me know if any step seems unclear or if you’d like to see the full version. I’m happy to go through it line by line - that’s exactly what makes discussions like this valuable. Like all of you, I just want to get closer to the Truth, and the fastest path forward is by correcting mistakes; the slowest is by confirming biases. There’s no loss for me in this debate as long as I stay scientifically honest - to you guys, and most importantly, to myself.

@joigus Thanks for jumping in - appreciate the GR-specific angle. A few clarifications to address your points directly: First, on infographics: These aren't just visuals - they illustrate the logical flow of derivations (e.g., from ontological blind spot in separating structure/dynamics to constraints leading to S¹/S² carriers, then projections yielding identities like E² = p² + m²). They're backed by theorems/lemmas in the docs (e.g., Appendix I derives EFE as κ² = κ² identity). If they seem superficial, happy to post the equations here instead - let me know which part (e.g., Schwarzschild equivalence?). On EFE not saying "spacetime = energy": You're right - EFE (Gμν = (8πG/c⁴)Tμν) is a relation (curvature proportional to energy-momentum), not literal equality. But in WILL, "≡" is ontological equivalence: Structure (geometry) and dynamics (energy) are mutually defining aspects of one relational system, not separate. EFE emerges as the tt-component identity (G_tt ⇔ κ², T_tt ⇔ ρ), derived algebraically without metrics (see Appendix I: d/dr(κ² r) = (8πG/c²) r² ρ(r) → κ² = ρ/ρ_max). It's not a misreading - it's a reinterpretation resolving GR's substantival assumptions. Which brings me to the key: This debate is substantivalism (spacetime as independent substance) vs. relationalism (spacetime as emergent relations) - and it's very much open in physics/philosophy. Standard GR leans substantival (vacuum solutions imply independent spacetime), but Einstein himself saw it as relational (influenced by Mach's principle) and spent his later years trying to make GR fully Machian (no absolute effects; e.g., letters to Schilpp). Relational views (e.g., Barbour, Rovelli, Smolin) argue substantivalism adds unnecessary ontological burden (a "container" without content), while relationalism minimizes it (Occam's razor). Judging WILL from GR axioms embeds that burden circularly - it's a category error. Burden of proof: Substantivalists must justify the extra entity (independent spacetime), as relationalism explains the same data with less (e.g., WILL derives EFE, conserves energy relationally via ΔE_{A→B} + ΔE_{B→A} = 0 but globally redistributes via projections - no global non-conservation issue, as no independent arena). On energy in GR: Yes, not globally conserved in general metrics (no Killing vector), but WILL avoids this by making spacetime emergent - energy is conserved in relations (closures like κ²=2β²), with "loss" as phase redistribution (e.g., redshift/cosmo in Part II). It's a useful concept everywhere in WILL, not just local patches. If this line of reasoning makes sense, you can see the formal derivations on the website (https://antonrize.github.io/WILL/) or via the archived DOI (https://doi.org/10.5281/zenodo.17115270). Alternatively, if you'd prefer, I can post the algebraic sequence for a specific case right here - for example, how κ² = 2β² emerges from the angular relations. Open to any concrete test or line-by-line examination.

@studiot, thank you for the follow-up and the detailed critique. You've clearly articulated several key objections regarding: (1) the use of the '≡' symbol, (2) a perceived tautology (that the physics is "embedded" beforehand), and (3) a false association with other non-mathematical posts. These points clearly represent the core misunderstanding of the model's foundations, and I will address them in order. 1. On the notation “SPACETIME ≡ ENERGY” You are formally correct that “≡” denotes a mathematical identity, and I understand the discomfort. My choice of this symbol was deliberate, precisely because standard mathematics lacks a specific notation for “ontological equivalence” or “mutual definition” (at least as far as I know - if you can suggest a more accurate one, I’m open to it). The foundational principle (W1) is that “Spacetime” and “Energy” are not two separate entities that happen to interact; they are two perspectives on a single, self-consistent relational structure. What we perceive as spacetime geometry is the structure of energy relations. This leads directly to your second point. You wrote: “This is especially misleading as the terms energy and spacetime do not refer to identical entities, and there is no Physics or Mathematics in which they may be related by the equals sign either.” I must strongly object to this specific claim, as it is demonstrably incorrect. The very foundation of 20th-century physics is built on precisely such an equation: Gμν = (8πG / c⁴) Tμν On the Left-Hand Side (LHS): The Einstein tensor Gμν (Geometry / Spacetime). On the Right-Hand Side (RHS): The Stress-Energy tensor Tμν (Energy / Matter). This is the mathematical equation that directly relates spacetime geometry and energy distribution through the equals sign. Relational Geometry (RG) merely takes this established physical principle to its logical conclusion. As demonstrated in Appendix I, the geometric field equation is proven to be mathematically equivalent to the corresponding component of the Einstein Field Equations. Here is the derivation summarized from Appendix I: The standard form for the tt-component of Einstein’s field equations (one of the TOV equations) is: (1 / r²) d/dr [r (1 - 1/g_rr)] = (8πG / c²) * ρ(r) where g_rr (a component of spacetime geometry) is related to the enclosed energy m(r) by: 1 - 1/g_rr = 2Gm(r) / (r * c²) The RG defines its potential projection κ using this exact geometric term: κ²(r) ≡ 2Gm(r) / (r * c²) Substituting this definition into the EFE yields: (1 / r²) d/dr (r κ²(r)) = (8πG / c²) * ρ(r) which simplifies to the differential form of the RG field equation: d/dr (r κ²) = (8πG / c²) r² * ρ(r) Furthermore, RG defines energy density algebraically as ρ(r) = κ²(r) c² / (8πG r²). This leads to the central equivalence: κ²(r) = ρ(r) / ρ_max(r) As shown in the Appendix, differentiating this algebraic form confirms its consistency with the differential EFE form. Therefore, the apparent complexity of the Einstein Field Equation is mathematically equivalent to a simple algebraic identity: (EFE differential form) ⇔ (κ²(r) = ρ(r) / ρ_max(r)) ⇔ (κ² = κ²) This demonstrates that “SPACETIME ≡ ENERGY” is not a metaphysical claim but the underlying simplified algebraic reality already encoded within General Relativity. Therefore, my use of “≡” was intended to capture this fundamental equivalence - which is at the heart of modern physics - expressed in its most direct, algebraic form. 2. On the charge of “Tautology” (that the physics is already embedded)This is indeed your central objection, and it deserves a clear answer. You argue that the diagrams are meaningless because: and that I am merely “verifying the arithmetic.” That interpretation is not correct. The model is generative, not descriptive. Relational Geometry (RG) does not assume the E² = p² + m² relation - it derives it. Here is the explicit derivation. It follows a strictly logical sequence, beginning from only one methodological principle: 1. Methodological minimalism - we start with nothing beyond epistemic hygiene: a refusal to accept unjustified assumptions. 2. Identifying the ontological blind spot - applying this minimalism to modern physics reveals a hidden separation between “Structure” (the spacetime container) and “Dynamics” (the energy content). 3. Deriving the W₁ principle - removing this single unjustified separation forces the logical conclusion that Structure and Dynamics are mutually defining: Structure ⇔ Dynamics. This becomes the foundational principle (W₁), derived, not postulated. “WILL ≡ SPACETIME ≡ ENERGY” is the shorthand for this unified relational structure. All physically meaningful quantities are internal relations within WILL; no external container is permitted. 4. Derived constraints - from (W₁), three necessary conditions follow: • Closure: the system is self-contained - no external reservoir of “energy” exists. • Conservation: within WILL, the total relational transformation resource is conserved. • Isotropy: if no external background is allowed, no direction can be privileged; hence the relational geometry must be maximally symmetric. 5. Derivation of the carrier (S¹): applying minimalism again, the simplest possible closed 1D structure satisfying the closure condition is the circle S¹. 6. Geometric law: S¹ geometry has an intrinsic property - its projections obey the Pythagorean identity βₓ² + βᵧ² = 1. This is pure mathematics, not a physical assumption. 7. Physical identification: we now interpret these projections according to their invariant roles: • βᵧ → the component equal to 1 at rest corresponds to the rest invariant, E₀ = E·βᵧ. • βₓ → the component vanishing at rest corresponds to displacement, p = E·βₓ. 8. Generated theorem: substituting these identifications into the geometric identity gives: (E·βₓ)² + (E·βᵧ)² = E² ⇒ p² + E₀² = E² Thus, E² = p² + m² is not an assumption within the framework; it is a derived theorem. The physics of Special Relativity emerges as a necessary algebraic consequence of the derived geometry itself. 3. On the association with “similar non-mathematical proposals”You also compared this work to another post: (“The Fundamental Interrelationships Model”) and implied that the two share the same nature. With respect - that comparison is misplaced. The cited thread discusses a speculative synthesis of physics, biology, and sociology. It makes no quantitative claims, provides no mathematical derivations, and does not reproduce any empirical results. Its aim is philosophical speculation. The RG, in contrast, reproduces established relativistic and gravitational relations (Lorentz factor, Schwarzschild radius, perihelion precession, GPS corrections, and many more) to full numerical accuracy using only algebraic geometry. Every result can be verified directly with a calculator, without invoking tensors or metaphysical language. If two works both use the word “framework,” that does not make them equivalent - any more than the word “field” makes Newton and astrology the same discipline. My model stands or falls by calculation, not by vocabulary. I appreciate your caution - skepticism is essential - but precision demands that criticism target the content of the equations, not the titles of other threads. I trust that in our further discussion, you will apply that same skepticism to substance rather than association - with full respect.