What have electrons, black holes and Clifford Algebra ever done for us?

This theorem proposes a fundamental equation of state for the de Sitter vacuum, linking its thermodynamic energy directly to its quantum-gravitational uncertainty. Let the de Sitter cosmological horizon be described by: 1. Its thermodynamic energy \(E_{\text{th}}\), the unique energy that satisfies the first law of thermodynamics, \(dE_{\text{th}} = T\,dS\). 2. Its intrinsic quantum position uncertainty (or "fuzziness"), \(\Delta x\), arising from quantum gravitational effects. The rate of change of the thermodynamic energy with respect to the area of this quantum uncertainty, \(A_{\text{unc}} = \Delta x^2\), is a universal constant equal to the Planck pressure, \(P_{\text{Pl}}\). Mathematically: $$\frac{dE_{\text{th}}}{d(\Delta x^2)} = \frac{\hbar c^7}{G^2} \equiv P_{\text{Pl}}$$   Derivation The theorem is derived by combining three principles: 1. Thermodynamic Energy: The Brown-York energy is the correct thermodynamic potential, given as a function of the de Sitter radius ...