Explicit Derivation of Nonlocal Memory and the Capacity-Closed Reservoir (SFIT)

SFIT is a self-consistent slow-fast field theory where a slow coherence field \(ψ\) couples to a fast reservoir \(χ\) with finite relaxation time \(τ_χ ≠ 0\) and finite capacity \(C[χ] < ∞\). We prove that coarse-graining \(χ\) yields a nonlocal memory kernel \(K ≠ δ\). The capacity is defined as a functional of \(χ\), closing the feedback loop \(χ → C → K → ψ → V → χ\). The FLRW reduction gives a generalized Langevin equation with capacity-regulated memory and fluctuation-dissipation consistency.