Bounded-Real Leakage Certificates for Soft Observer Boundaries: From Commutator Defects to Finite-Dimensional H∞ Channels

This technical note develops a self-contained finite-dimensional observer-boundary leakage theory for DAGI accessibility sectors. For a finite graph G=(V,E), a graph-local self-adjoint generator H, and a positive observer-accessibility effect K=diag(k), the finite observer-boundary lemma states [H,K]uv = Huv(k(v)-k(u)). A sharp observer cut creates an edge-cut flux defect, while a smooth accessibility kernel controls that defect by a graph-gradient term. The note promotes the commutator defect to a measurable boundary-leakage output C_K=i[H,K] and defines the damped input-output system xdot=(-αI-iH)x+Bu, y=C_Kx+Du. For the canonical realization B=I and D=0, the transfer channel G_K(s)=C_K(sI+αI+iH)^-1 is stable and satisfies the conservative certificate ||G_K||∞ ≤ ||C_K||2/α. The strict continuous-time bounded-real LMI is certified explicitly by P=αγ²I whenever γ > ||C_K||2/α. The paper also gives symbolic two-node and three-node families, path-graph frequency-domain validation, CSV outputs, and reproducibility scripts.