{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"c9303ce13ab0ea08828061210b4b51de4d2401e272688eeb6e1749779b2f7e0a","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b","cf126bf977bf78a7d7f9b4dcf3fb05b0611f3281a1ed56cb6f60af31d3e2df66"],"salt":"41e87b89edc10b2e3f5be49a454ea98c2a44d4ad523e391ae86df1f0f15bb89d","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.kam-stability.generic-arnold-diffusion","kind":"well-posed","literature_status":"partially-resolved","n":"2","parents":[],"plain":"Arnold conjectured that in typical nearly regular systems with three or more degrees of freedom, some orbits drift by a finite amount no matter how small the perturbation. This is proven only in a weaker sense of typical, or for low numbers of degrees of freedom.","posed_since":"1964","precise":"$H = h(I) + \\varepsilon f(\\theta, I)$, $(\\theta, I)\\ \\mathrm{in}\\ T^{n} x R^{n}$, h strictly convex, $n \\ge 3$. Prove that for generic f (residual in $C^r$, r large; separately, for an open set of analytic f) and all small $\\varepsilon > 0$ there are orbits with $\\mid I(T) - I(0) \\mid \\ge c$ with c independent of $\\varepsilon$. Answer: a proof for every $n \\ge 3$.","problem_ref":"cf126bf977bf78a7d7f9b4dcf3fb05b0611f3281a1ed56cb6f60af31d3e2df66","references":"","settled_by":"A proof valid for arbitrary n with residual (or analytic open) genericity.","status_note":"Proven for two and a half degrees of freedom for cusp-residual $C^r$ perturbations (Kaloshin and Zhang, Annals of Mathematics Studies 2020; Cheng); arbitrary n in a weaker cusp-residual sense (Bernard, Kaloshin and Zhang, Acta Mathematica 2016); genuinely residual or analytic perturbations remain open.","title":"Arnold diffusion for generic nearly integrable 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