Part 3The Geometric Origin of Inverse-Square Scaling in the Gravitational Constantᵥer₁₀

In this work, we present a geometric interpretation of the inverse-square scaling associated with the gravitational constant, derived within the SRCD (Self-Regulated Curvature Dynamics) framework. Rather than postulating the inverse-square law as a fundamental dynamical principle, we show that this scaling emerges naturally from the shell hierarchy formed by discrete spherical units—JS (Junctional Spheres) and their composite organization into SH (Shell Hierarchies). By constructing a discrete shell structure and examining the ratio between adjacent shells, we obtain a quartic shell-mapping relation. When interpreted in three-dimensional space, this relation implies an effective radial scaling proportional to the square root of the shell index. Independently, the requirement of conserved geometric throughput (or flux) across spherical shells fixes the radial density to scale inversely with the surface area. Taken together, these two ingredients—shell geometry and conservation across shells—lead directly to an inverse-square radial dependence, without assuming any force law, field equation, or phenomenological input. Importantly, this work does not propose a modification of Newtonian gravity, nor does it introduce a new coupling constant. Instead, it provides a structural explanation for why inverse-square scaling appears so robust and universal across gravitational phenomena. The result suggests that the inverse-square form traditionally attributed to gravity can be understood as a geometric consequence of discrete shell organization and conservation, rather than as an independent postulate. This paper is intended as a geometric and structural analysis. Broader conceptual interpretations of JS and SH within the SRCD worldview will be presented separately. ===================================================================================== Limitations of Existing Theories (Precise Diagnosis) Conventional physics has proceeded in the following sequence. It assumes that spacetime is continuous.Within that continuous space, equations are constructed to fit observed phenomena.In this process, constants are fixed as input values. This approach has been successful within the present dimension,but because it avoids questioning the structure of spacetime itself,it loses explanatory power the moment the dimensional context changes. In other words,without ever asking the question“What is the structure of spacetime?”,it has carried out calculations under the assumption“Spacetime is already given in this form.” Modern physics provides highly successful mathematical answers to the question of how physical phenomena operate.Yet, the deeper structural question of why these interactions take their observed forms remains open.But, This is the fundamental error of the continuous-space assumption. Point of Departure of Our Equations (Decisive Difference) Our equations were not constructed to fit results. They start from the minimal structure of spacetime.They take discreteness, not continuity, as the fundamental premise.The equations are not tools designed for a specific dimension,but computational rules that naturally rise upward as the structure expands. That is,we did not construct equations that are valid only within a single dimension;we constructed equations in which the same computational rules are preserved even as one moves across dimensions. The Most Important Difference (One Sentence) Existing equations are result equations tuned to a single dimension,whereas our equations are structural equations that remain computable across dimensions. That difference is everything. ====================================================================================