Linear Density as the Primary Gravitational Constraint

This paper presents a minimal toy framework—a "clinical audit" of gravitational geometry—derived from the single ontological constraint that the Schwarzschild relation implies a constant linear density ($\lambda = c^2/2G$) across all scales of gravitational collapse.

By treating the universe as a closed linear partition between commensurate mass and spatial registers, the author demonstrates that the Newtonian gravitational potential can be recovered exactly as the isotropic expression of this one-dimensional constraint. This "Toy Universe" analysis identifies the Isometric Hinge at 35.26° as the coordinate where linear and volumetric measures are commensurate, yielding a 0.000000% variance from the classical orbital baseline for solar system bodies (Moon, Earth, Jupiter, Sun).

The framework does not seek to replace General Relativity or provide a full dynamical theory; rather, it offers a ground-floor assessment of the linear identity condition underlying classical dynamics. By shifting the radial origin from $r=0$ to the finite boundary of the horizon ($R_s$), the paper provides a geometric resolution to the singularity, characterizing the event horizon as the point where the linear gravitational budget is fully exhausted.

Key highlights:

• Recovery of Newtonian potential without volumetric primitives.

• The identification of the Linear Ledger as the primary gravitational identity.

• Zero-variance reconciliation with established orbital data.

• A finite, non-singular accounting of the Schwarzschild radius.