Half-Sided Modular Inclusion ⇒ Translation Law: A Quantitative Stability Principle for Emergent Translation Symmetry

We formalize the reconstruction principle that a standard half-sided modular inclu-sion (N ⊂ M, Ω) canonically generates a positive-energy translation semigroup and anaffine (ax + b) covariance relation between translations and modular flow. The exactextraction is classical (Wiesbrock; Borchers) and forms a core algebraic mechanismbehind half-line/wedge localization. The operational novelty pursued here is stability:real laboratory families are finite-resolution and typically satisfy half-sidedness onlyapproximately. We introduce an approximate half-sided modular inclusion (AHSMI)notion measured by a subalgebra distance d (with the Kadison–Kastler metric as acanonical choice), and formulate a quantitative stability statement: AHSMI implies theexistence of an approximately covariant unitary translation semigroup Uε(a) satisfyingan approximate Borchers relation∆itM Uε(a) ∆−itM ≈ Uε(e−2πta),with defect bounds controlled by the AHSMI parameter. This converts half-sidedmodular structure into an operational diagnostic for emergent translation symmetrycompatible with modularly closed laboratory families.