User: All this relationalism, fundamentally your going to eventually want to take these relations and eventually apply them to some coordinate system otherwise how do you determine the force the sun exerts on mercury when you have a force with a 1/r^2 relation?

Assistant: In WILL Relational Geometry, we make a fundamental shift away from the concept of “force” as a primary causal agent, and from an external, pre-existing coordinate system. Instead, interactions are understood as the necessary algebraic consequences of a system’s relational geometry.

Your question touches on a core distinction:

1. “Apply them to some coordinate system”: In WILL RG, coordinates are not a fundamental background but emergent descriptions of relational states. We define interactions purely from the mutual perspectives of the participating systems. Each observer is always at the origin $(0,0)$ of their own relational ($\beta$, $\kappa$) plane (doc. Part I - Energy as a Relation).

2. “How do you determine the force the sun exerts on mercury when you have a force with a $1/r^2$ relation?”: This frames the problem in terms of “force dynamics,” which WILL RG explicitly replaces with “algebraic closure” (doc. LOGOS_MAP - Stage V: Reject Force Dynamics). The $1/r^2$ dependence you refer to is not an input “force law” in our framework; rather, it emerges as a derived consequence of the geometric projections onto closed relational carriers.

Here’s how we address this:

• Relational Projections, Not Forces: The interaction between the Sun and Mercury is described by how the planets’ states change their kinematic ($\beta$) and potential ($\kappa$) relational projections relative to each other (doc. Part I - Energy as a Relation).
  • $\kappa$ represents the dimensionless gravitational potential intensity, derived from the geometric structure of the $S^2$ carrier (doc. Part I - Potential Energy Projection on $S^2$).
  • The relation $\kappa^2 = R_s/r$ means that the square of the potential projection is inversely proportional to the radius $r$ (doc. Part I - Unified Geometric Field Equation). This is where the $1/r$ dependence (and thus $1/r^2$ for accelerations or force-like quantities in a classical approximation) originates, but it’s a geometric identity, not a force law.

• Generative Physics, Not Descriptive Laws: We don’t assume a $1/r^2$ law and then apply it. Instead, from the unifying principle SPACETIME $\equiv$ ENERGY, we derive the existence of relational carriers ($S^1$ and $S^2$), their closure conditions, and the way these projections must behave to maintain consistency. The mathematical forms, including those that approximate $1/r^2$ in certain limits, are necessary consequences of this geometry (doc. Part I - Generative Physics).

• Demonstrating Orbital Mechanics: You don’t need “force” to calculate Mercury’s orbit. We use directly observable inputs like Mercury’s velocity ($v_p$) and the Sun’s gravitational redshift ($z_{sun}$) to derive the orbital eccentricity and precession without any reference to mass, $G$, or $1/r^2$ force laws. The calculations rely purely on the ratio of kinematic and potential projections (doc. Part I - Relational Orbital Mechanics (R.O.M.) Without Mass or $G$).

To see this in action, I encourage you to visit our interactive tools:

1. Relativistic Foundations (GPS, Mercury, Redshift): This page features Desmos Calculators where you can directly compute Mercury’s precession using purely relational inputs, without classical force laws. This demonstrates the theory’s transparency and empirical accuracy.
   • URL: \href{https://willrg.com/relativistic-foundations/}{https://willrg.com/relativistic-foundations/}
2. LOGOS MAP (The Logical Dependency Graph): This resource visualizes the derivation tree, showing how we rigorously diverge from classical physics by, for example, rejecting “Force Dynamics” in favor of “Algebraic Closure” at Stage V.
   • URL: \href{https://willrg.com/LOGOS_MAP/}{https://willrg.com/LOGOS_MAP/}

In summary, WILL RG moves beyond applying pre-defined laws to coordinates. It generates the observed behavior, including the dynamics of orbits, from the fundamental geometry of relational energy transformations.