Quantitative Recovery Bounds from Vacuum Clustering in Finite-Mode Gaussian States (A Regularized CCR Blueprint Motivated by Split Inclusions)

We prove a quantitative clustering—recovery bound for centered quasi-free (Gaussian) states in a finite-mode bosonic CCR (Weyl) setting. Motivated by split inclusions in algebraic quantum field theory, we work in a regularized framework where Gaussian states are parametrized by finite covariance matrices and a recovery map admits an explicit covariance block formula. Using a perturbative Gaussian fidelity input and explicit coercivity bounds for inverse covariances, we control the recovery error in terms of a vacuum cross-correlation factor, a cross-correlation perturbation parameter, and a recovery-error matrix norm ||DeltaGamma||_HS with an explicit quadratic+quartic structure. In a distinguished class (Family A, X = X0), this reduces to a bound in terms of the cross-block error ||Delta12||_HS. We include ancillary numerical sanity checks verifying the perturbative regime, a collar-envelope decay model, a dimension sweep n1 = n2 in {1,2,3}, and phase-diagram checks of the perturbative domain. v2 adds: a collar-suppressed recovery corollary making the "clustering suppresses recovery error" mechanism a single displayed inequality; an upgraded discussion of the fidelity constants (the local coefficient 1/8 is shown numerically to be a directional benchmark, not a uniform bound, and an empirical constant is certified on the sampled domain); an independent verification suite in pure NumPy/SciPy implementing the Banchi—Braunstein—Pirandola fidelity formula with closed-form anchors, with all proved inequalities tested on random draws (zero violations); positioning remarks relative to the companion notes 2512.0060 and 2512.0101; and, aligning with 2512.0060 v2, corrected Petz-identification language (the recovery rule is conditional reattachment, with Petz agreement only in the factorized case) plus an explicit admissibility hypothesis for the recovered covariance.