{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"cdaf115ba44875dd9807fe38e55deff96f46f43ab61bbefb31209a6cc45db5ba","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"89ee1e055a8e705a0859961a5188658f223e83589678146978786474ec5810ab","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.kam-stability.planetary-kam-real-masses","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"KAM theory proves that planetary systems have many stable quasi-periodic orbits, but only for planets far lighter than real ones. Proving the same with the actual masses of Jupiter and Saturn is open.","posed_since":"1963","precise":"Planetary N-body problem (Sun plus planets with mass ratios $\\mu$). Arnold (1963) and Fejoz (2004) proved positive-measure sets of KAM tori for $\\mu$ below an unspecified tiny threshold. Prove existence of invariant tori near the observed orbits for the actual $\\mu \\sim 10^{-3}$ of the Sun-Jupiter-Saturn system, for example by computer-assisted KAM. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A computer-assisted KAM proof for the full (non-truncated) planar or spatial Sun-Jupiter-Saturn problem with real parameters.","status_note":"Computer-assisted KAM has been done for truncated secular models of Sun-Jupiter-Saturn (Locatelli and Giorgilli, 2000s); Capinski and Gidea (arXiv 2504.09273, 2025) gave a computer-assisted proof of Arnold diffusion in the planar full three-body problem.","title":"KAM tori for the planetary N-body problem with actual masses","topic_ref":"cd1c7cff821fba931816a6faeec94ad21e6f069bd36505c710e41ce43b61e40a"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"e6c6fcacd2ac2c1d13ecd37ff51d2bb1c659097928e52882bb241689ded0bd6a","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"6858e7bd58b0415ee79f772796de7ad27af935635bfbb1e9125b4ffc362fc058","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"lzYDJ_Jo4oM4Hr7EnBToOtvhYvY4PiUxjIZpEgM2AJDAs5_CM4BMNUUDno_2toRMPhaG6HEeT2O3DFL-DMxEBg"},"schema":"pubphys.envelope/1"},"record_hash":"e6c6fcacd2ac2c1d13ecd37ff51d2bb1c659097928e52882bb241689ded0bd6a","leaf_index":1636}