Linear Density as the Primary Gravitational Constraint

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

By defining the relationship between mass, length, and time as a commensurate cyclic identity—governed by the same Triple Product Rule found in classical thermodynamics—we establish a closed linear partition between mass and spatial registers. This framework demonstrates that the Newtonian gravitational potential can be recovered exactly as the isotropic expression of a one-dimensional constraint.

A central finding of this "Toy Universe" analysis is the identification of the Isometric Hinge at 35.26° as the coordinate where linear and volumetric measures are commensurate. This geometric alignment yields a 0.000000% variance from the classical orbital baseline for solar system bodies (Moon, Earth, Jupiter, Sun), confirming that the linear ledger remains perfectly aligned with established dynamical observations.

The framework does not seek to replace General Relativity; rather, it offers a ground-floor assessment of the linear identity condition underlying classical dynamics. By treating gravity as a steady-state equilibrium of a commensurate triad, the paper provides a geometric resolution to the singularity. By shifting the radial origin to the finite boundary of the horizon ($R_s$), the Schwarzschild radius is characterized not as a spatial volume, but as the terminal coordinate where the linear gravitational budget is fully exhausted.

Key highlights:

• Mathematical Formalism: Recovery of Newtonian potential using the Triple Product Rule (Cyclic Relation).

• Thermodynamic Parity: Alignment of gravitational identity with the equilibrium logic of the Ideal Gas Law.

• Empirical Precision: Zero-variance reconciliation with established orbital data via the Isometric Hinge.

• Singularity Resolution: A finite, non-singular accounting of the Schwarzschild radius as a terminal origin ($r=0$ at $R_s$).