Tetrometry, the Planck Quadrex, & the Unity of Physical Law

This paper introduces "Tetrometry," a geometric framework that represents the four fundamental physical quantities (Space, Time, Mass, and Energy) as vertices of a tetrahedron (termed the "Quadrex"). Building on Einstein's two great unifications (space-time via c and mass-energy via c²), the framework treats these equivalence axes as skew lines in three dimensions, naturally generating a tetrahedral structure. The six edges of the Quadrex correspond to fundamental physical ratios (velocity, specific energy, force, power, linear mass density, and mass flow rate), while the four triangular faces map onto distinct physical domains (Dynamics, Statics, Kinematics, and Thermodynamics). The three pairs of opposite edges are shown to correspond to the three fundamental constants of physics: c (relativity), G (gravity), and ℏ (quantum mechanics). The paper derives the Planck scale as the "unit cell" of the Quadrex where all three frameworks converge, and provides a geometric resolution of the hierarchy problem by demonstrating that the gravitational coupling constant α_G = (m/m_P)² follows directly from dimensional analysis. The dominant factor of 10³⁸ separating gravity from electromagnetism is shown to arise from the ratio of particle masses to the Planck mass, with dimensional transmutation explaining why stable particles exist far below the Planck ceiling. Finally, the framework is shown to naturally encode Lorentz invariance through the preservation of dimensionless products under coordinate transformations.