{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"f63ceadb3ef863c3ca443130f971d75f8a391e06ba85e73422f8f425b0670a15","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["0f28e8121133821cb89a2a4f055f11d912c5e23ac5039fb2605de121bf9bbb64","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"265e1f2f3f56092940aec3deb3b2d03415e7447155077a4d84d46ce4da8b49d3","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"cm.sign-problem-numerics.analytic-continuation-limits","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Most electron simulations work in imaginary time, and converting their output into measurable spectra is mathematically unstable: tiny noise allows wildly different answers. The best possible resolution for a given noise level, and a way to certify an answer, are not established.","posed_since":"","precise":"Given $G(\\tau) = \\int \\mathrm{d}w\\, K(\\tau, w) A(w)$ with $K = \\exp(-\\tau w)/(1 + \\exp(-\\beta w))$ on $0 \\le \\tau \\le \\beta$ and Gaussian noise of size $\\sigma$, determine the optimal resolution of the spectral function $A(w)$ (minimal resolvable width and amplitude of a feature at frequency $w$) as a function of $\\sigma, \\beta \\text{ and } w$, and construct estimators (maximum entropy, stochastic, Nevanlinna, Bayesian) with certified error bounds that reach it. An answer is proven bounds plus an estimator attaining them on benchmarks.","problem_ref":"0f28e8121133821cb89a2a4f055f11d912c5e23ac5039fb2605de121bf9bbb64","references":"","settled_by":"A theorem bounding achievable resolution from below and an estimator whose certified intervals match that bound on test spectra.","status_note":"Nevanlinna interpolation (Fei, Yeh and Gull 2021) enforces causality exactly but gives no certified bounds for noisy data.","title":"Resolution limits of analytic continuation from imaginary-time data","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"17bcecc7156c9bf11b5898b22366c3a562f3dea6a1eae9a4f6a52a51b085c909","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"5fca068c7ffcbbf66e28ea21596fa34258660d839981d6707681e5617918d1fc","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"Wd9UvMiGcE2LmWUhtQYxwVlhrJIgoXEUCMW8T_VANoSUS6-mfyRkR2yGkocpiK0IDTE_E4e3G2GSUMx9nmf-Dw"},"schema":"pubphys.envelope/1"},"record_hash":"17bcecc7156c9bf11b5898b22366c3a562f3dea6a1eae9a4f6a52a51b085c909","leaf_index":3022}