{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"353b97ed18dc22224785ea8d8377f9135074618b5bb1ad8dd4406ddad4f1aaf6","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["15249fcc69fc61f22774f16c72c055f50f86ce9259881666583d2bba031a9574","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"a2c3d34606c1ca0e63c9eb64b7bc61819f995bb2a7dace9cfaea2651f330fa60","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.bose-condensation.third-order-energy","kind":"well-posed","literature_status":"partially-resolved","n":"2","parents":[],"plain":"The energy of a dilute repulsive Bose gas is known rigorously to two terms in powers of the density. The next predicted term, with a logarithm, is proven only from above.","posed_since":"1959","precise":"With $\\hbar^{2}/(2m) = 1$, the ground-state energy density is predicted to be $e(\\rho) = 4 \\pi a \\rho^2 [1 + (128/(15 \\sqrt{\\pi})) (\\rho a^3)^{1/2} + 8 (4 \\pi/3 - \\sqrt{3}) \\rho a^3 \\log(\\rho a^3) + o(\\rho a^3 \\log(\\rho a^3))]$ (Wu, 1959). Prove the matching lower bound for general short-range $v \\ge 0$. Answer: a proof of the two-sided expansion.","problem_ref":"15249fcc69fc61f22774f16c72c055f50f86ce9259881666583d2bba031a9574","references":"","settled_by":"A lower bound on $e(\\rho)$ matching the Wu term.","status_note":"The two-term Lee-Huang-Yang formula is proven (Yau and Yin 2009, upper bound; Fournais and Solovej 2020, lower bound); a third-order upper bound in the thermodynamic limit was proven by Brooks, Oldenburg, Saint Aubin and Schlein (arXiv 2506.04153, 2025).","title":"Third-order term in the dilute Bose gas ground-state energy","topic_ref":null},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ddc0b7425b6e04ffc036508987f2d2eb71b15e20823aa95909c820bfa9d7f6d6","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"c419d4ec0aa039aa14e82d1d86e7dbd5da9dfb6d3d275af8a16a65cd84a20f99","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"IM2hx5SuhKBRwRlaX1qvuf4sUrig60ey7LCQ1nGpdI-XFIhWxhYPPLpcl26u11_qVEE9KEqcCxMUjSpXSvgsBQ"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIxBnU7AqgOaoU6C0dhufb1dqd-209J1r4oWplzYSiD5nwIMQabaBRY3lhJ_Vy214DfRVbdNDzwe8cwArab4UG4ez8CPEgE92d8nVg5NOuT7dWU-Xe_Awy62eQ6AzizzdEocDZaaQI8CDo-7jxWUXoq9y2lFkxCCf3Nkc2js805WEh0TPS3o2dtQjwILWrXmGblJWyjfSHBGBDWXDb_Ass6glGNK-1d8WiL9y_CPEgnn6Vx8qL21den8S6SYnwcPulHpueuBd4s2bmaW_ihOUI8SD1aQNIXJcg4yzy1IZRSeotEz041fd4921eddu6uRHpWgjxIGSBYk34U9ro0Avf9MJgik6rywdM4recJQsShrUyoVGJCPEgraRI2npokG3tsYuZYconfMt0ARYLxAQQuCOiQUqeUM8I8SDArwsGYVuomKH_DJiJGziPwECn7skqn20TGM58Ezco0wjwIIgwugvlgSNOmTWMhquqetKjeLf0x7d1a1MH4KU52JbFCPAg3mczMYDKOd7Nhf3ojA9ZvuTsN66Y1NnyYTp9j8GKQNwI8SCvAiiN-Hrv7IEejd8wdUihoozu5KOvpoBFJ0vg3UnGAQj_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_ieLohOiSlAEIbC5GOpl6z4pwrKGCkhGn7Kj2YUMPCFcwkik_obLRLUjwIGw4z4Z_SkxH2y6Ta_qkKL9IcLwi3NUgZ4KUog7XMSoGCPAgORc7w989NJ4pgH2o-CHaEn-P0zys5HpNp6P7_LxL5ZsI8SAk8F35VROxEFJU7gX8Rq-sMrX20tAy60-yTKFM4w4KrwjxIKx46umFBHMb9jeJryscpBTYyWlmGsifNISNtQ-JEp9gCPEglDXC832G8HGLhtNktYSY4RizOy5maubtMig3lctJqq0I8CC9grccB-zGOdKx1BhxpzHdkgA06-u9tvwHq8GGSc5QmwjwICRMULf4Qk7B4Ixs-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":"3406","tree_size":"3804","proof":["7cb76c13ea74082c237ab3449115b12c2ec3306d91eba478932726dfc1da7473","be431a98e48dae746f0d430e476ffc871b86a4286a53a5b98172a878b595d843","1fd1462de57588a351e2be2a2ea83e3b23925d02ccceeba3b9e48cff0e97427f","64233a076453971cc2b94fc426c1013b594a0856b8082ba4ba4b3b87dd565784","f8054e07d574f827bbe7eaba51dfcf5e57fc972d58b070fa0a72f3c754638d24","9d431ce7742af24a57989023456a175d95e430366c461878574a00aa046e43b8","217966f7b365dc70af6d616a83ae450e616d3fef50ead5372fb26577cf221836","f543c2acee32c00eed64ca6a002c4cdf20cd84f300a60f431863f4556659c7bc","5bc8f1bbb7a36ffa2afc301ec767424419006f77d79dfc7ac3adb0dab9eb36e8","98b5a14b42b3a202b6ef825f66833fa505601404c4c0f14575f3ec922607a858","af34d80d0065c0ab502282775e865492e2cb49c9493f4594d63ba5577922854a","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\n34f78b591ecc3e481222f24c7383bf3989fc79f863c0c7c265a2a3a3a1f0416f\n3406\n2026-10-15T23:43:57Z\n2026-10-16T01:43:57Z\n\n— pubphys.com/log/v1 JFPwIGhwVAH5ciWmgqQaAMXdK73JsBfSq0BrUMYpAbCD4gJdPpDMn0cNDE6Xf7b94Dvkcv8ozY3nelbM4IJxpQVITQ4=\n"},"orcid_key_evidence":[],"attesting":null}