{"schema":"pubphys.bundle/1","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},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"17bcecc7156c9bf11b5898b22366c3a562f3dea6a1eae9a4f6a52a51b085c909","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"5fca068c7ffcbbf66e28ea21596fa34258660d839981d6707681e5617918d1fc","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"Wd9UvMiGcE2LmWUhtQYxwVlhrJIgoXEUCMW8T_VANoSUS6-mfyRkR2yGkocpiK0IDTE_E4e3G2GSUMx9nmf-Dw"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIX8oGjH_8u_ZuKOohWW-jQlhmDYOZgdZwdoHlYXkY0fzwIF_gXuQ_FwGo9GkKpUsD_IbZxLEHT0mnYynwqR5D-GrpCPEgXPKSfnRsLjmhwWC3s7TVVlYH3e1Za9M8iBQjjdUI4FII8SBYrkfudHH_DixsxtI0HtNfRzJaCqYPV76rTYhIu01YjwjxIKSlnxHgk70AIqOa02_QuNM34qD9y3744QmKABD00ganCPAgbazkGMiLoo4H6CszIxGkH2v7vHw3uZG61lj6cBPGfFMI8SCoIDI38o3m8gxPq9ipueT8_RYayaZzNWFftCYfZWwP6wjxIPz45zKf3oui_6fsPFoesAdn8c7jCNtvma5ZIef8PtuLCPEg-b8Lkm9aIcNajW2D_tonI2BpSvWgXJkN2LygUbv3ObsI8CCFjG9pSMiq3eVzHAmZsjRKj0iAZic3nIa3eyIChZggbAjwIOv_x0B6HyZ6WMHKeIf_d_EUIvwPGdZeVY-hXN3RLDM5CPEgHd9u6OIGCvUV12A1B1xdiOprSJ-4w0cqwfe7CzVX5HsI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIbKmlxS7KQZLDBy2rxHyciG8FJvzESWgQ1ebFIr5kZQ_wIGyrWJNEEn5AnC9YeLletTgbzIwPZJHr3GRdSTlxCzRpCPEgvvfFCtGedjTdzfIGqbcurYq0ikZkXU-SCwDuUYK9B2AI8CBCvFrI6T_6XcutCqSuvO4H6jDDTnMb41LmMXXu2lFYbQjwIPV4Ig_oXk-yU2B78XFBUxgLmUo_nGb9MpmZUZv3TxQGCPAgUmBzncrixG7TDnwHURz5xiwPIz2OdAbrkUa9jXf9DaEI8SD_5zFJHCmnJMVG3y8g48CODsGJwzCSeSjMybN-JiGepwjwICRMULf4Qk7B4Ixs-w5_AnqKLFYrm25Y_JDm1-0Q4j1tCPEgV483a5w_VdTK7my82rqJCPIXOvGhZNeRkwqfWifoP4MI8SBcP9hQb3hxUiLf8dFx-DuqZHA2xkmG-c_qCFt8QfQHCAjwIOv_x0B6HyZ6WMHKeIf_d_EUIvwPGdZeVY-hXN3RLDM5CPEgHd9u6OIGCvUV12A1B1xdiOprSJ-4w0cqwfe7CzVX5HsI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"]},"log":{"leaf_index":"3022","tree_size":"3804","proof":["46c213e24c3abd76b6e85720b50b536bdaae00ebdabd1d9bcd49b0d68813c56d","ba57156f3f7d4943c103c0c73e191c5d88970bd6460acd8365c4fb5c43001ad2","c2a00ed3511093a3f6edb6a48d2518e26a09375f427df199ecdaf1ce24a3f696","79800e7602e307f2b018c20945b70a5bc2f4467e7e818417fb432d764e18dbff","0398f58beb90d4c301909d8a14a3658d9b51a7fff182a6d33438ed92bb029a86","4144ec28590f19e0fc54750765647a0ad1a4721ff1d1d388d77adcb6308cc31a","4d342745f225337b5e11d27fcecea80f1e780419083c24bd0ca3f37687fe6b16","b54387979d97d9abde30a291e115dc6733875d3a816ad7f767bc7b5a22c80a0c","56960156727d0f461a1b3857ab83f18526eaef848c820c927094afc1f989d56c","b0b232078d071c92816af6274529aef23a2ea1f41246694fec32d768cfa09dc8","325287f5a43e4f3ad4072ec4bd53d43c50c1393021a056122e9cb0fad83b868f","3c578dccc15502c5f3efe228cadc99fee76ab536ed01f66b14d316fdcc157429"],"checkpoint":"pubphys.com/log/v1\n3804\nAK6t4qgu96+M9wScTrIlsJ10d/zhKVEtYguyM3aeJvo=\n\n— pubphys.com/log/v1 JFPwIPP8EtoxgcsQg7V7CwIdL4UKPqJtpv+eB8sFNJjyQHIdyA1VabmBsPyLqq2bLxJPn57rA6fQi8y0QDDjRXHvlQg=\n","promise":"pubphys.com/log/v1 promise\n9cb99c00b89d635e948b9682c4433ec1516a2f425244ab396fde47627e50c952\n3022\n2026-10-15T23:43:57Z\n2026-10-16T01:43:57Z\n\n— pubphys.com/log/v1 JFPwIAR6UJrHWY+5GW2FqbY+nkZWDpPb8WDWb5DTmKDY7c3IdaX5ODVz3Y/E2iIMfNa/XXcmW/yFbqdg/VYGgAe55gQ=\n"},"orcid_key_evidence":[],"attesting":null}