2 hours ago, KJW said:

I should also remark that the mathematics is stock standard mathematics used in general relativity. That you call it "intimidating" is revealing about your understanding of general relativity. I personally regard the mathematics as beautiful, although actually doing the mathematics, especially the index manipulations, can be quite tedious to do manually.

"reviling"? I was always open about my lack of formal training end mathematical limitations. There's no secret to reveal. And it does look intimidating hahaha!
But I often downplay myself. There's no way I will ever solve something like this with pen and paper, but with python I think few ours of bad coding will do the trick.

Anyway I still fully agree with my result:

The Strict Minimality Theorem

The adoption of substantialist assumptions produces three structural consequences:

1.Inflated Formalism

Equations multiply to compensate for ontological error.

2.Loss of Transparency

Physical meaning becomes hidden behind coordinate dependencies.

3.Empirical Fragmentation

Each domain (cosmology, quantum, gravitation) requires separate constants.

By contrast, Relationalism - as epistemic hygiene - enforces relational closure and yields simplicity through necessity, not through approximation.

The Inflation Chain

Surplus Ontology -> Ontological Duplication -> Mathematical Inflation

GENERAL CONCLUSION

"Mathematical complexity is the symptom of philosophical negligence."

Mathematical complexity is not an inevitable feature of physical law but a structural consequence of surplus ontological assumptions. When those assumptions are removed, the same empirical content is reproduced with strictly fewer primitives.

7 hours ago, KJW said:

I should also remark that the mathematics is stock standard mathematics used in general relativity. That you call it "intimidating" is revealing about your understanding of general relativity. I personally regard the mathematics as beautiful, although actually doing the mathematics, especially the index manipulations, can be quite tedious to do manually.

I was quires and looked in to this mathematical beast of yours. As far as I understood it looks for invariants, so called "Killing vectors" where certain quantities are conserved like angular momentum or energy, right?

I just don't understand why would you do this in such a hard way with that huge equation?
I can do it like this
Energy:
[math]\kappa_o^2(o) = \beta_R^2(o) + \beta_T^2(o) + \beta^2[/math];
[math]\kappa_o^2(o) - \beta_o^2(o) = \frac{\beta^2(1-e^2)}{1-e^2} = \beta^2 = 2W[/math]

momentum:
[math]\frac{\beta_T(o)}{\kappa_o^2(o)} = \frac{\sqrt{1-e^2}}{2\beta}[/math]:
[math]h = R_s c \frac{\sqrt{1-e^2}}{2\beta}[/math]
Is that what it does?

I'll show derivations below:

---

Relational Origin

RED vs BLUE

Orbital dynamics boils down to interplay of frequency shifts. No mass, no G, no manifold, no metric required.

[math]\kappa^2 = 1-\frac{1}{(1+z_{\kappa})^2}[/math] ([math]z_{\kappa}[/math] = gravitational redshift)

[math]\beta^2 = 1-\frac{1}{(1+z_{\beta})^2}[/math] ([math]z_{\beta}[/math] = transverse Doppler shift)

Observational Z Inputs

[math]Z_{sys} = (1+z_{\kappa})(1+z_{\beta}) = \frac{1}{\tau}[/math] (product of gravitational redshift and transverse Doppler shift)

[math]\tau = \kappa_{X}\beta_{Y} = \frac{1}{Z_{sys}}[/math] (product of projectional phase factors on [math]S^1[/math] and [math]S^2[/math] carriers)

[math]z_{\kappa} = \frac{1}{\kappa_{X}}-1[/math] (gravitational redshift)

[math]z_{\beta} = \frac{1}{\beta_{Y}}-1[/math] (transverse Doppler shift)

Everything else is Closed Algebraic System of Relational Orbital Mechanics (R.O.M.)

Please follow this link to see all [R.O.M. equations and definitions](https://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom)

---

Vis-Viva

In elliptical systems, the global closure theorem [math]\kappa^2 = 2\beta^2[/math] is conserved across the entire orbital cycle, while local phase variations manifest as an internal "breathing" of relational projections. We now derive the exact conservation law governing this local dynamic distribution.

Proposition: Phase-Invariant Structural Depth

The difference between the square of the local potential projection [math]\kappa_o^2(o)[/math] and the square of the local kinetic projection [math]\beta_o^2(o)[/math] is a global system invariant equal to [math]2W[/math].

Proof:

The binding parameter [math]W[/math] defines the global relational depth of the system:

[math]W = \frac{\beta^2}{2}[/math]

From the exact definitions of the [math]S^2[/math] and [math]S^1[/math] projections at any arbitrary phase [math]o[/math]:

[math]\kappa_o^2(o) = \frac{2\beta^2(1+e\cos(o))}{1-e^2}[/math]

[math]\beta_o^2(o) = \frac{\beta^2(1+e^2+2e\cos(o))}{1-e^2}[/math]

Subtracting the local kinetic projection from the local potential projection:

[math]\kappa_o^2(o) - \beta_o^2(o) = \frac{2\beta^2 + 2\beta^2 e\cos(o) - (\beta^2 + \beta^2 e^2 + 2\beta^2 e\cos(o))}{1-e^2}[/math]

[math]\kappa_o^2(o) - \beta_o^2(o) = \frac{\beta^2(1-e^2)}{1-e^2} = \beta^2 = 2W[/math]

Thus, the algebraic distance between the omnidirectional and directional relations remains globally constant irrespective of orbital geometry.

Theorem: The Orthogonal Signature of the Orbit

The square of the local potential projection [math]\kappa_o^2(o)[/math] on the [math]S^2[/math] carrier is the strict Pythagorean vector square sum of the local radial [math]\beta_R^2(o)[/math], transverse [math]\beta_T^2(o)[/math] and the global [math]\beta^2[/math] kinetic projections.

Proof:

The total local kinetic projection on the [math]S^1[/math] carrier splits orthogonally into radial and transverse components within the orbital plane:

[math]\beta_o^2(o) = \beta_R^2(o) + \beta_T^2(o)[/math]

Substituting this orthogonal decomposition into the phase-invariant relation [math]\kappa_o^2(o) - \beta_o^2(o) = \beta^2[/math]:

[math]\kappa_o^2(o) - (\beta_R^2(o) + \beta_T^2(o)) = \beta^2[/math]

Rearranging yields the three-dimensional algebraic relational closure:

[math]\kappa_o^2(o) = \beta_R^2(o) + \beta_T^2(o) + \beta^2[/math]

Remark: Ontological Replacement of Vis-Viva

This geometric invariant replaces the descriptive Newtonian Vis-Viva equation ([math]v^2/2 - GM/r = -GM/2a[/math]). Standard mechanics interprets this balance as a scalar subtraction of an abstract potential energy from kinetic energy. In WILL, it is revealed as a strict Pythagorean theorem of relational geometry: the local potential projection ([math]\kappa_o^2[/math]) is generated by the orthogonal summation of the kinetic "breathing" ([math]\beta_R^2 + \beta_T^2[/math]) and the global [math]S^1[/math] projection ([math]\beta^2[/math]).
SPACETIME [math]\equiv[/math] ENERGY.

---

Angular Momentum Conservation

The conservation of specific angular momentum [math]h[/math] is derived as a phase-independent structural invariant of the relational balance.

Proposition: Invariant Ratio of Projections

The transverse kinetic projection [math]\beta_T(o)[/math] is strictly proportional to the local potential projection [math]\kappa_o^2(o)[/math] scaled by the global system constants.

Proof:

From the definition of the local potential projection (based on closure theorem [math]\kappa^2 = 2\beta^2[/math] and geometry of ellipse):

[math]\kappa_o^2(o) = 2\beta^2 \frac{1+e\cos(o)}{1-e^2} \rightarrow (1+e\cos(o)) = \kappa_o^2(o) \frac{1-e^2}{2\beta^2}[/math]

The transverse kinetic projection [math]\beta_T(o)[/math] is defined on the [math]S^1[/math] carrier as:

[math]\beta_T(o) = \beta \frac{1+e\cos(o)}{\sqrt{1-e^2}}[/math]

Substituting the expression for [math]1+e\cos(o)[/math] into the definition of [math]\beta_T(o)[/math]:

[math]\beta_T(o) = \beta (\kappa_o^2(o) \frac{1-e^2}{2\beta^2}) \frac{1}{\sqrt{1-e^2}} = \kappa_o^2(o) \frac{\sqrt{1-e^2}}{2\beta}[/math]

Rearranging to isolate the invariant ratio of the phase-dependent projections:

[math]\frac{\beta_T(o)}{\kappa_o^2(o)} = \frac{\sqrt{1-e^2}}{2\beta}[/math]

Theorem: Conservation of Angular Momentum

The specific angular momentum [math]h[/math], defined as the product of the local scale [math]r_o(o)[/math] and the transverse kinetic projection, is a global phase invariant.

Proof:

Using the local scale [math]r_o(o) = \frac{R_s}{\kappa_o^2(o)}[/math], we define [math]h[/math]:

[math]h = r_o(o) \beta_T(o) c = \frac{R_s}{\kappa_o^2(o)} \beta_T(o) c[/math]

Substituting the invariant ratio [math]\frac{\beta_T(o)}{\kappa_o^2(o)} = \frac{\sqrt{1-e^2}}{2\beta}[/math]:

[math]h = R_s c \frac{\sqrt{1-e^2}}{2\beta}[/math]

Since [math]R_s[/math], [math]e[/math], and [math]\beta[/math] are global invariants, [math]h[/math] does not depend on phase [math]o[/math]. Angular momentum is thus the manifestation of the fixed structural exchange rate between the [math]S^2[/math] and the [math]S^1[/math] carriers and their projections.

---

Eccentricity

Eccentricity is a measure of the projectional deviation from the circular equilibrium state ([math]\delta=1[/math]).

Theorem: Relational Eccentricity

For a closed orbital system governed by the projection invariants of WILL Relational Geometry, the orbital eccentricity [math]e[/math] is strictly determined by the closure factor at periapsis, [math]\delta_p[/math]:

[math]e = \frac{2\beta_p^2}{\kappa_p^2} - 1 = \frac{1}{\delta_p} - 1[/math]

Proof:

Instead of relying on classical force laws, we derive this relation directly from the conservation of the two fundamental projection invariants of the WILL framework:

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_o(o) \beta_T(o) c = \text{const}[/math] ([math]\beta = \beta_T(o)[/math] at turning points).

Consider the two turning points of a closed orbit: periapsis ([math]p[/math]) and apoapsis ([math]a[/math]).

By operational definition of the shape parameter [math]e[/math], the relation between radii is determined by the geometric range:

[math]r_a = r_p (\frac{1+e}{1-e})[/math]

Step 1: Relational Mapping.

Using the angular invariant [math]h[/math] (implying [math]\beta \propto 1/r[/math]) and the field definition [math]\kappa^2 \propto 1/r[/math], we express the apoapsis projections in terms of the periapsis values:

[math]\beta_a^2 = [\beta_p (\frac{r_p}{r_a})]^2 = \beta_p^2 (\frac{1-e}{1+e})^2[/math]

[math]\kappa_a^2 = \kappa_p^2 (\frac{r_p}{r_a}) = \kappa_p^2 (\frac{1-e}{1+e})[/math]

Note: Kinematic projection scales quadratically with the radius ratio, while potential projection scales linearly.

Step 2: Energy Balance.

Substituting these into the energy invariant conservation condition [math]W_p = W_a[/math]:

[math]\frac{1}{2}(\kappa_p^2-\beta_p^2) = \frac{1}{2}(\kappa_a^2-\beta_a^2)[/math]

Cancelling the factor 1/2 and substituting the mappings from Step 1:

[math]\kappa_p^2 - \beta_p^2 = \kappa_p^2 (\frac{1-e}{1+e}) - \beta_p^2 (\frac{1-e}{1+e})^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 ( 1 - \frac{1-e}{1+e} ) = \beta_p^2 ( 1 - (\frac{1-e}{1+e})^2 )[/math]

Step 3: Algebraic Reduction.

Expanding the terms in brackets:

LHS bracket ([math]\kappa[/math] term): [math]1 - \frac{1-e}{1+e} = \frac{(1+e)-(1-e)}{1+e} = \frac{2e}{1+e}[/math]

RHS bracket ([math]\beta[/math] term): [math]1 - \frac{(1-e)^2}{(1+e)^2} = \frac{(1+e)^2 - (1-e)^2}{(1+e)^2} = \frac{4e}{(1+e)^2}[/math]

Substituting back into the balance equation:

[math]\kappa_p^2 ( \frac{2e}{1+e} ) = \beta_p^2 ( \frac{4e}{(1+e)^2} )[/math]

Dividing both sides by [math]2e[/math] and multiplying by [math](1+e)^2[/math]:

[math]\kappa_p^2 (1+e) = 2\beta_p^2[/math]

This yields the geometric identity for bound orbits:

[math]2\beta_p^2 = \kappa_p^2 (1+e)[/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 stated result:

[math]e = \frac{1}{\delta_p} - 1 = \frac{2\beta_p^2}{\kappa_p^2} - 1 = 1 - \frac{2\beta_a^2}{\kappa_a^2}[/math]

Remark:

This result confirms that eccentricity is strictly a measure of the energetic deviation from the circular equilibrium state ([math]\delta = 1[/math]), derived entirely from the conservation of relational projections without invoking mass or Newtonian forces.

SUMMARY:

[math]\frac{2\beta_p^2}{\kappa_p^2} - 1 \equiv e \equiv 1 - \frac{2\beta_a^2}{\kappa_a^2}[/math]

[math]\text{ECCENTRICITY} \equiv \text{CLOSURE DEFECT}[/math]

[math]\text{SPACETIME} \equiv \text{ENERGY}[/math]

---

So now can you see what I mean when I say:
"Mathematical complexity is the symptom of philosophical negligence."?

Edited April 17Apr 17 by Anton Rize

[math]\overline{\underset{^{\large \sim}}{\partial}}{}^{}_{t} \overline{\mathfrak{T}}{}^{r_\lambda}_{s_\lambda}\ \ \ \underset{^{\large \sim}}{\partial}{}^{}_{k} \mathfrak{T}{}^{i_\lambda}_{j_\lambda}[/math]

Edited April 18Apr 18 by KJW