Renormalization of Windowed Quantum Field Theory on Finite Spacetime Domains

Quantum field theory is conventionally formulated on globally extended spacetime domains, yet physical interactions are instantiated only within finite spacetime regions. This work develops a systematic analysis of how standard renormalization procedures are affected when a smooth, non-dynamical localization window function (x) 0, 1 restricts the domain of physical instantiation without modifying the underlying field theory. The central results are as follows. (1) Within the window interior, where = 1 and = 0, standard renormalization prescriptions, bulk counterterms, and renormalization group flow are unchanged from conventional QFT. In the boundary layer B_ where 0, the window acts as a spacetime-dependent coupling and generates boundary-local contact terms that do not modify bulk beta-functions but require separate treatment; these are classified explicitly and demonstrated by a scalar-field worked example. (2) Apparent violations of covariant conservation at localization boundaries are resolved through a compensator stress-energy T^comp_, required by the contracted Bianchi identity and selected uniquely by the minimal retarded prescription. The compensator acts as the direct bridge from the renormalized QFT sector to the gravitational sector: it is the boundary-bookkeeping object that allows localized renormalized stress-energy to satisfy the contracted Bianchi identity and thereby source the Einstein field equations. (3) Finite localization implies an effective window-resolution scale _ 1/ set by the boundary-layer thickness ; this is an operational suppression scale, not a hard UV cutoff or a modification of loop-integral structure. (4) When combined with the localization-based grounding of Jacobson's thermodynamic derivation of the Einstein equations established in companion work~hackett2026emergent, the formalism shows that finite-domain perturbative QFT closes with general relativity without requiring an independently quantized metric. The quantum sector is the renormalized matter sector; the gravitational sector is the geometric consistency condition imposed by causal accessibility, entropy, and global Bianchi closure. No perturbative graviton sector is introduced. This is established as a formal Finite-Domain Closure Proposition. Four explicit calculations demonstrate these results: renormalized stress-energy in curved spacetime (Sections~sec: curved and~sec: stressₑnergy), Unruh--DeWitt detector response (Section~sec: udw), trans-Planckian mode non-instantiation in FRW cosmology (Section~sec: frw), and Casimir energy (Section~sec: casimir). In each case, the windowed formalism reproduces established results while providing a principled account of the effective resolution scale as a localization artifact.