{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"a7fea0bd2c9ece060d496cccbb6a0f6cc8875ce949a152a6cf43076e46b09c8c","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["7c76a666c47331c2654443388577edd6af3e7ba9731421380c24cf9ee050572a","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"9d26829e176d2330f22ce53fafa697acd33c7f970869487d5f896da6c5259de2","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":["ai:gpt-5.5"],"external_id":"beams.nonlinear-dynamics.long-term-da-scaling","kind":"well-posed","literature_status":"partially-resolved","n":"2","parents":[],"plain":"Computers can track particles for about a million turns, but colliders store beams for billions; the rule used to extrapolate the stable region is a conjecture.","posed_since":"1996","precise":"The dynamic aperture $D(N)$, the largest initial amplitude stable for $N$ turns of the 4D or 6D symplectic one-turn map of a hadron collider, is fitted by Nekhoroshev-type laws $D(N) = D_{\\infty} + b/(\\mathrm{log}\\, N)^{\\kappa}$. Determine whether such a law fitted for $N \\le 10^{6}$ predicts $D(N)$ and the resulting intensity decay at $N \\sim 10^{9}$ (about $10^{5}$ s of LHC operation) to within 10 percent, and derive $\\kappa$ from the resonance structure of the map.","problem_ref":"7c76a666c47331c2654443388577edd6af3e7ba9731421380c24cf9ee050572a","references":"","settled_by":"Direct tracking to $10^{9}$ turns of a realistic lattice compared with the extrapolation, plus a derivation of $\\kappa$ from normal-form or resonance analysis.","status_note":"Fitted scaling laws described LHC intensity and luminosity evolution in several fills (2018-2019 papers); a derivation of the exponent and a test at $10^{9}$ turns are missing.","title":"Extrapolating dynamic aperture from a million to a billion turns","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"7e776c06d57087b55b6aeabfdccb9a2e6946bcb6eff8388c27586fd3f5e5ed46","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"3e7bec62c4d35fa887e870264e3efdd11b12baf717f9966b31ddd201b06559ee","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"K_HrWAcH4jJdHzOjgUjqdgOw-hl8MUN51_GbaKymezzy6SeLLTBvwP2xlTLGiNvxfgD_0NqUovO5LiQ8iiReBA"},"schema":"pubphys.envelope/1"},"record_hash":"7e776c06d57087b55b6aeabfdccb9a2e6946bcb6eff8388c27586fd3f5e5ed46","leaf_index":2502}