Channel–Response Araki–BKM Information Flow: Exact Hilbert-Space Decomposition, Coarse-Graining Chain Rules, and Interaction-Sector Specializations

We extend a finite-dimensional Araki–Bogoliubov–Kubo–Mori (BKM) responsecost construction to an arbitrary real Hilbert response space, normal quantumcoarse-graining channels between von Neumann algebras, composable channel chains, and continuous coarse-graining flows. The response map is a boundedoperator Rτ : hτ → K on the BKM tangent Hilbert space at resolution τ , withcovariance operator Γτ = RτR∗τ. For the accessible tangent vτ and responsebτ = Rτ vτ , the exact minimum resolved cost is the closed quadratic form qΓ†τ(bτ ) = ∥Γ−1/2τbτ∥2, rather than the operator expression ⟨bτ , Γ†τ bτ ⟩ unlessthe stronger operator-domain condition holds. A normal channel induces,under explicit second-order regularity assumptions, a contraction of BKMtangent Hilbert spaces. Its defect is positive and composes by an exact chain rule. For a right-continuous coarse-graining evolution the retained BKMinformation is monotone and determines a positive Stieltjes loss measure. These ingredients give the master relationg0(v, v) = qΓ†τ(bτ ) + ∥Pker Rτ vτ∥2hτ + µlossv((0, τ ]).When the loss measure is absolutely continuous, the last term is R τ0 Jinfo,v(s) ds with Jinfo,v ≥ 0. BKM-continuous response functionals allow the same theoremto treat suitable unbounded or smeared QFT observables without requiringthe observables themselves to be bounded operators. We give explicit specializations and first quantitative applications in four sectors. In Yang–Mills