Opened matter · the boundary census The Weight Between and the Exit Line

When a system of electrons is cut into a region and the rest, its correlation matrix has a cross block, X_A = C_AB C_BA: the weight between the two sides of the cut. This paper proves what that weight can and cannot say, and then studies where opened (non-Hermitian) matter leaves the disk of its occupation chart. The weight between. X_A is positive and never larger than 1/4. Both sides of the cut see the same positive spectrum. It is stable under small changes, with a certified count. It sits below the local defect C_A − C_A², and for a pure Gaussian (Slater) state, where C is an orthogonal projector, the two are equal. Four times its trace bounds the Gaussian mutual information across the cut from below, with a sharp constant, and there is no upper constant without a spectral margin. Exact examples show that a one-body record does not in general determine correlations outside the quasi-free class. The signed boundary offset τ = log det(I − C_A)/det C_A moves no faster than τ̇² ≤ 4gS, and two preparations with the same census, offset, cross weight and coupling saturate this bound in opposite directions. The exit line. On the PT-symmetric Su–Schrieffer–Heeger chain, the band-averaged Petermann factor equals 2 exactly on the line E(0)E(π) = u², equivalently (w² − v²)² = 2u²(w² + v²). Exits predicted from that line before the run match the measured census exits of finite blocks to about 5e-8 in these runs. A linearly weighted operator, T = ½{X + ½, H}, commutes entrywise with the half-chain correlation block; the proof uses only H(k)² = E(k)²I, so it holds without Hermiticity. At 200 sampled gapped parameter points, its tested low-index edge eigenvectors exhibit an equally spaced entanglement ladder ε_j = jΔ + iθ, with a spacing Δ = 2πK′/K taken from a Hermitian band hidden inside the opened one and a shared phase θ given by an elliptic (Abel) formula. For this proposed ladder, the paper proves that it crosses the rim of the disk all at once, and that its phase reaches a quarter turn exactly on the Petermann line, in both directions. The dimer limit is proved exactly. Plain finite blocks give a census rotation equal to the same phase to within 7e-7. An analytic derivation of the ladder for the coupled chain remains open, and the paper states it as open. Related numerical work on the same model (Rottoli, Fossati and Calabrese, 2024) is cited and compared. Every claim carries a grade (proved, Lean-checked algebraic subclaim, verified numerically, known result, or open), and every number comes from a script included in the archive, with the reference-run environment recorded. Files: the paper (PDF); a web edition (HTML) with an interactive view of the proposed ladder model; and a zip holding the LaTeX source, the Python scripts and their outputs, the figure, the Lean proofs, and a README mapping each Lean grade to its theorem. Prepared with AI assistance (Claude, Anthropic; and OpenAI tools for parts of the formal development); the ideas and research direction are the author's.