Entanglement Equilibrium, von Neumann Algebras, and QCD Confinement Boundary Gravity: A Synthesis of the Algebraic Framework with Application to QCD Confinement Entropy and Metric Perturbation Amplitude

The connection between quantum entanglement and spacetime geometry, first articulated by Van Raamsdonk, was formalized by Jacobson through theentanglement equilibrium condition and extended by Cao and Carroll to a bulk derivation of the linearized Einstein equation from the extremizationof generalized entropy over a geodesic ball. A foundational obstruction to this programme is that local algebras in relativistic QFT are Type III1 andadmit no well-defined von Neumann entropy. We resolve this obstruction by reformulating the Cao–Carroll condition in the crossed-product algebra of Wittenand Chandrasekaran–Penington–Witten, producing a Type II∞ setting where entropy and its variation are rigorously defined. We then apply this frameworkto QCD in the confining regime, where the confinement entropy gradient acts as a localized matter perturbation sourcing curvature through the equilibriumcondition. The resulting metric perturbation amplitude is hS ≈ 5 × 10−39, parametrically set by the ratio (ΛQCD/MPl)2. All assumptions are statedexplicitly, and detectability is assessed against current experimental capabilities.The detailed algebraic proof supporting the crossed-product reformulation is presented in the companion paper.