A Quantitative Borchers–Wiesbrock Theorem: Windowed Stability of Half-Sided Modular Inclusions

Half-sided modular inclusions are the mechanism by which modular data of a nested pair of von Neumann algebras generates a positive-energy translation group — the algebraic seed of horizons and null translations in quantum field theory. A nontrivial half-sided modular inclusion forces type III₁ behavior; consequently no finite von Neumann algebra admits one, and in settings where local algebras are finite (type II₁ gravitational algebras at finite entropy, as for the de Sitter static patch), the hypotheses of every modular reconstruction theorem are unsatisfiable exactly. This paper develops the corresponding stability theory: windowed approximate half-sided modular inclusions, an elementary stabilization theorem with all constants read off the data, a covariance-restoration lemma for the non-unimodular ax+b assembly, and a quantitative form of the Borchers–Wiesbrock structure theorem with an explicit error budget — linear in the defect up to a logarithm — together with a windowed positivity theorem, exactly rigid at zero defect. Every constant in the error budget is certified against lattice modular data of a free-fermion chain (reproduction package included).