Linear Density as the Primary Gravitational Constraint

Title: Linear Density as the Primary Gravitational Constraint: A Structural Audit of the Gravitational Constant

Description:

This paper provides a forensic audit of gravitational geometry, establishing that the Schwarzschild relation implies a constant linear density ($\lambda$) which serves as the primary ontological constraint of the manifold. By defining the relationship between mass, length, and time as a commensurate triadic identity ($\iota c\lambda=1$), the framework reveals the gravitational constant ($G$) to be a redundant scaling artifact rather than a fundamental constant of nature.

Using the same Triple Product logic found in classical thermodynamics, the paper demonstrates that the Newtonian gravitational potential is recovered exactly as the isotropic readout of a one-dimensional linear ledger. A central discovery of this audit is the identification of the Isometric Hinge at 35.26°—the geometric coordinate where linear and volumetric measures reach parity. At this alignment, the framework yields a 0.000000% variance from established orbital data for major Solar System bodies (Moon, Earth, Jupiter, Sun), confirming the mechanical precision of the triadic ledger.

The framework resolves the crisis of the singularity by identifying a Structural Category Error in conventional 3D projections. It establishes that the Schwarzschild radius ($R_s$) is not a spatial volume enclosing a point-mass, but a static Magnitudinal Depth—the terminal coordinate where the linear gravitational budget is fully exhausted. By shifting the origin to this finite boundary, the model exorcises the mathematical singularity and replaces it with a continuous, functional accounting of the horizon.

Key Highlights:

• Redundancy of $G$: Deconstruction of the gravitational constant into a composite of the Mass Torque ($1/c$) and the linear invariant ($\lambda$).

• Forensic Parity: Absolute zero-variance reconciliation with classical orbital baselines via the Isometric Hinge.

• Singularity Resolution: Elimination of infinite density through the identification of $R_s$ as a non-spatial, linear identity.

• Horizon Mechanics: Characterization of the "Black Hole Shadow" as a Photonic Hinge ($g=0.5c$) where the linear deficit matches the lateral requirements of light.

• Thermodynamic Symmetry: Alignment of the gravitational manifold with the equilibrium logic of closed triadic systems.