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On a high level, our result shows that the usual spectral gap condition, which concerns the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state\n \n \n $$\\rho _0$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of a Hamiltonian\n \n \n $$H_0$$<\/jats:tex-math>\n \n \n H<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is locally gapped in\n \n \n $${\\varLambda ^{\\textrm{gap}}}\\subset \\varLambda $$<\/jats:tex-math>\n \n \n \n \u039b<\/mml:mi>\n gap<\/mml:mtext>\n <\/mml:msup>\n \u2282<\/mml:mo>\n \u039b<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , whenever the Liouvillian\n \n \n $$- \\textrm{i}\\, [H_0, \\, \\cdot \\, ]$$<\/jats:tex-math>\n \n \n -<\/mml:mo>\n i<\/mml:mtext>\n \n [<\/mml:mo>\n \n H<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u00b7<\/mml:mo>\n \n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is almost invertible on local observables supported in\n \n \n $${\\varLambda ^{\\textrm{gap}}}$$<\/jats:tex-math>\n \n \n \u039b<\/mml:mi>\n gap<\/mml:mtext>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n when tested in\n \n \n $$\\rho _0$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of\n non-equilibrium almost stationary states<\/jats:italic>\n (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation\n \n \n $$\\varepsilon V$$<\/jats:tex-math>\n \n \n \u03b5<\/mml:mi>\n V<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n acts in a region which is further than\n \n \n $$|\\log \\varepsilon |$$<\/jats:tex-math>\n \n \n |<\/mml:mo>\n log<\/mml:mo>\n \u03b5<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n away from the non-gapped region\n \n \n $$\\varLambda \\backslash \\varLambda \\textrm{gap}$$<\/jats:tex-math>\n \n \n \u039b<\/mml:mi>\n \\<\/mml:mo>\n \u039b<\/mml:mi>\n gap<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s10955-026-03703-1","type":"journal-article","created":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T17:40:16Z","timestamp":1791826816000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Response Theory for Locally Gapped Systems"],"prefix":"10.1007","volume":"193","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1106-327X","authenticated-orcid":false,"given":"Joscha","family":"Henheik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7593-0913","authenticated-orcid":false,"given":"Tom","family":"Wessel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,12]]},"reference":[{"issue":"3","key":"3703_CR1","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1007\/s00220-011-1380-0","volume":"309","author":"S Bachmann","year":"2012","unstructured":"Bachmann, S., Michalakis, S., Nachtergaele, B., Sims, R.: Automorphic equivalence within gapped phases of quantum lattice systems. 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