Relative-Entropy Stability for Emergent Geometry: When Information Loss Controls Geometric Drift

Relative entropy is the canonical stability functional in quantum theory, enforcingmonotonicity under restrictions and channels and controlling operational distinguisha-bility. In QFT and holographic settings, relative entropy on a region admits themodular-energy decomposition S(ρA∥σA) = ∆⟨Kσ,A⟩ − ∆SA (where the entropy differ-ence is interpreted in the standard renormalized sense in type III cases). We establish areconstruction-agnostic stability theorem for any “geometry reconstruction” procedurebuilt only from bounded operational data (finite witness sets of bounded observablesand bounded modular-flow functionals): if S(ρ∥σ) is small, then any reconstructedgeometry parameter θ(ρ) can drift at most as O(pS(ρ∥σ)) with an explicit constantdetermined by (i) the Lipschitz constant of the witness map in trace norm and (ii) theconditioning (Lipschitz modulus) of the reconstruction rule. We package the latter asthe Entropy-to-Geometry Modulus κW attached to a witness set W , and prove a sharpobstruction criterion: persistent geometric drift while S(ρn∥σ) → 0 occurs preciselywhen (at the reference state) either the witness map or the reconstruction map failsto be Lipschitz (equivalently, κW = ∞ or a witness component is not trace-normcontinuous). This identifies, in purely operational terms, when a putative emergentgeometry is not a stable coarse variable in the sector under study.