{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9538372866077b05cd15eab22b1c11657ca4cec796b1dcde58d757407292b3d2","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"1a7f451f7687f9c64fae40e26961798d740ae0c994ba76c23619098f1e085483","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.yang-mills.area-law","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Quark confinement shows up mathematically as a loop observable that decays with the area enclosed by the loop. This is proven only when the coupling is strong, far from the regime that describes nature.","posed_since":"","precise":"For 4D lattice $\\mathrm{SU}(2)$ or $\\mathrm{SU}(3)$ gauge theory with Wilson action at large $\\beta = 2N/g^{2}\\ (N = 2\\ \\mathrm{or}\\ 3)$, prove $\\langle W(C)\\rangle \\le \\exp(-\\sigma(\\beta)\\operatorname{Area}(C))$ with string tension $\\sigma(\\beta) > 0$ for all rectangular loops $C$, uniformly in the infinite-volume limit. Answer: a proof, ideally with $\\sigma(\\beta) \\sim \\exp(-c \\beta)$ as predicted by asymptotic freedom.","problem_ref":null,"references":"","settled_by":"A proof of a positive string tension for all $\\beta$ in four dimensions.","status_note":"The area law is proven only at strong coupling (small $\\beta$), going back to Osterwalder and Seiler (1978).","title":"Area law for Wilson loops in 4D lattice gauge theory at weak coupling","topic_ref":"c1a7c5336cf6e262c1ec8a96010ceec0183209142167af7077f763d0eba2c1d0"},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"eff6e3b8158ba736395166d654cc20a87c79bc802384cbaa914163116f5cb49d","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"f7f06abb180f0cf62e9d1ef8c2044dcfc7770e937ac8a66be87f2525f393175b","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"foawfpOI2J3iguRoJO6gnj0DW_Fq8yZ-CL4C0wXbPYNmrytvgZQ9HR9WCZfBAbjlhY-mlMexwlFSw4HAEO0BAQ"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEI9_BquxgPDPYunR74wgRNz8d3DpN6yKZr6H8lJfOTF1vwIPfxZqPlX22iPRZKrF515qEvzh-818-HnyVl6sqcXK56CPEgW34SBwTmqWKqBcdD5YRirp0bGu48DwumJx3NszDBPw0I8SDe9tRkopnrEF8wwGwSCPMsYxMG9iWeidowWPEBtPF8lwjxIJ6nHz7CxExAM-my-w6BYa0mC5MpPtfuWd-OiLr3dhKtCPAgHx1Pd3fXURZT4EVhm0gNfRfqgAqtJVEzmf3F8MCiwRkI8CCaVvGFcFjWuGlxcE3iF0dvk0TcN-e8_-WkY_vxMeYSCQjwII1OGo6m_8GpiiXK1pC7312jTWia9PNuWF9nWxoSLBVxCPAgwaB6v5-rgNtbRovqByHYFx5vLPaMlAE1dmQXdsyW6UQI8SDWlL7IzfO_cwYxfadaDmoPVp0tptPmw2XmYaU4JMe_wgj_8Ah24IVYDhTLjQjwED2StpR20zKnAmFKLczkHJkI8CCe67kSLOh57nysQqZi1IHlJdGlLtKL-P9z6nKaUvm3XQjxICOd9Lx83cM_96PY2EN1mm5UmleuFn5-z5m8uVi8iPgNCPEEasCsO_AISExe0692tcT_AIPf4w0u-QyOLCtodHRwczovL2JvYi5idGMuY2FsZW5kYXIub3BlbnRpbWVzdGFtcHMub3JnCPAgV1njYlhVpbXxxNJ5A17FSiMpFFMLjm4bJzftn7wbffII8CDOVX-4-vLDhYFcrGDHOUtPR4aPn6sWFt6EOKClF-yaTwjwIGihx0Ad7HXfcNYbuDTIe75m_qdF9q-LslSigo9Ru0tvCPEgFtjIOgc4vcoI664yGAe8PdH8DHMR9uErKi5sz8gDBn8I8CDEPOt85b3mCKTSlla2WgzmuL6MMKDoNgnE1vIF7cSt3QjxIK2cRhgH_rfI4c_3izFMgTdSITa3WA0XDR1r0HMDRvcKCPAglTlBxtWF7nW2hSIFGL6wiHr0Vr29rm4UAKPs9Hbg9K0I8SAwXinq1RpDtydnlw8Ha2koqF2yFrHSDxDdbPKSMCSEtgjwIFdpAd59yWeR8AUI6US9Qlip6r3Qo2ZN0XAC5OjdsB4oCPAgfsG5YzZX6hJQ9ubmZDYOrWkvNWA0VNr3zKhlUONtDvQI8VkBAAAAASTON1FOjCzEaz5QlKqLLL7tav5J6J7Ocspclk4MWc1lAAAAAAD-____Ai4AAQAAAAAAFgAUK2LdxOzpZSkQhJi9xS-fPZe_P-cAAAAAAAAAACJqIPAE4ssOAAgI8SC5m3CDu4sb96QrcFDGdVUBirrOeBfZWfPsORAg-WboiAgI8CBuspwshUyzxO0YJPtGRgrQQ8qjUMfGL2kmTR8vA_KfIwgI8CDF_M6A9gXtWFr3Mf23sVT0ZgV9ISvnaPG_LfUmq4pkFQgI8SBul7aaiBHJS5Yc3CtS1Y6qDxq445i58p6PLXBEzMxAhAgI8CBeppX1zhWOdSYayuHs3vM0JSle-TmGWKm7covAVsZgLAgI8CCilHCZlaM_rvoHokVf_mR3RdFRD12hDxVn_yV_9HuxtggI8SBjHg9GeUFjSgXABKR4bdNOxRSCC1umkifH9wjJCjbJ3ggI8CARa-iXvK5Ae5MITxm6tMkYBcGbL1vSYWObFw0ba6tEMAgI8CABwMF2rrPslICiAiQeZlb8fFWbiYI8J6W9Klj4NaYugwgI8SACYW_HHn5FR0wDMpnbMWvv4vrVLmuti3KQC0Jzxu0ffwgI8CA77BagYT3wC-kwrV2Xe4AVpjKceoomflINCX_y5XLLYAgI8CAyUQV-kNFYHvq2vpNf4O-siVkxKellhddyNyaU57GTyQgI8CC8pGQm5AvVUIJeYMrTyioq-k_Z07ykestOYlkhwU0IMwgIAAWIlg1z1xkBA-OXO_AIfrBm5kAMH2cI8BAGKsbR52Yj2ZyRBj8fybyrCPEg15Kg4Ne2LGSKxM-WnEce5gVwiluZijYEWJ8P_lNnKQ4I8QRqwKw68AhHyujKJ6ONvP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjwINdJ8ieiZ-JLIUEEAihNC7494QQupy0QVSM2j1eHofFcCPAgkGgKlpcRG2nP1zqtTDpX_YjihxLMV7Kc24ytLy6cvAcI8CCN4kW0V1HA8a4QHZGZhZSNeZ8kgkhGe4PdtrRdV5Z-QwjxIGGL2dS-L-Kz7v1Ki33Bz7cEm9jh9hoT4FXFYYR-g6ZECPAg2QiKEdtLAvrnIT-NKAJnC5vRCRVe8c8KZY-eAHlWnV0I8SC7ywYzxaQqM9vZvWH9qkZ1ezyPRqsRPKM1UH5PUMAqqwjxIGV3DLnbxKAfzCfLajltjy0ygCHkAZxPtW386bUXNoEfCPFZAQAAAAH16vm5sVqjs-wecCpMI8xxr9keLO2NGe-i6l1mDYZs9AAAAAAA_v___wL9HAAAAAAAABYAFJfrJJnWQ7Z-x57m-pheBz9rlZ-6AAAAAAAAAAAiaiDwBODLDgAICPAgQ5kLGxF3OOh7vA4r4QgTNSxhMWuE0zPsKOx6aAlYwU0ICPAgrcub5ELTx887PMagKzmIVUo4Gg6BtaLt3KI7-fz-8EEICPEg0FgdJHVwk8G5Pk6k7dEH0TMswzSzrVNV1xYLxkkZVvMICPAgtjymv0AiAWba16HatRnRM6aSEjLDEQzp869oBYyyjgAICPAgsEP0Beu23BZP7idrNFQmZn-TxiI05kwqbf6p1WOaf_AICPEgG88se8_qMTzI7uj77UIc6ySEMh9mQHPaMi7mHps25mUICPAguW4eye-LlJPda_C8Mmx0BQjQQtz1nrsLTAgjRSlo9fQICPEgH-Edk1DT42l_z1g1CJqaCEytCaFvPc2jCpw9cJRBIvAICPEgtrzZnZu3sj-WLZdck809J45YjHQC5dVYihvrN2xts7MICPAgqWyFTc9RhVV3SZNs3fNoQVKDRRXplDwbeyCXVhDnIj8ICPAgbWHUUHK7V9RbQptUsRe7MfuXDtOFPCs2yYfH6gZMkXgICPAgR7Nixpl-oCPpVe8qdUa1e_7Rn3YgQMJxBRegeslx8wEICPAgE6LVZm1wTAIsYWfkCsuUOJUdBbrHQP8HCAtU2wexpsQICAAFiJYNc9cZAQPilzs"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEId1fb4pjVqG2BWWVOnWouBVhOr9bVzXOjhPe36UJSv83wIHdfY_ZR2Wp-0Dp2rfHB7oif64sk-ldwf9fCc1k7tjZDCPEgzgbbADySw9_fJx8KDuA234fz7SjzowbJ_OKZBqobW-EI8SBT3jVx5N0yt4B-Y3QZO3gxMZt4Nd41CzgALN0a45wnKgjxIJY6T0guUD6g3D78zgzb9miTjNaI1jELjXlrmCsPcLSaCPAgn7KeSrSA4vxKEg2WinJPADSDlW20nrxzoS1zqgD0V54I8SAEawMHSWhEYO6azE3jygzCrBJL6b9JWBWNhERDM47PewjwIGp-uP1fmyxjeYLOqKvoIkazUYuzFTJiflCrrrwF_TGkCPEgh4LZF42g0kH3DY6zt2g7LubZ1jM3BFR90pRDqduaWSUI8SB8izl_Yn9kR2zBmnu-HVjv452ytRQyuNX-opb_OJzg_gjxINBAKCf3cKChW811B6TkfdlxdONUiLQA88uJMnPer4KuCPEgytIBB3x7TFDT-XqOV-9vpqw1NKoAu0dhtsCbAMql2L0I8CCtNZgmApuKULvPC9nsMCaM05arV-F3o5FTlnJCnRvdMgjwIBGYo5Vxy-d7U5zKv5vp3Ii7hJPH_BDLCzA13c47p9q7CP_wCHbghVgOFMuNCPAQPZK2lHbTMqcCYUotzOQcmQjwIJ7ruRIs6HnufKxCpmLUgeUl0aUu0ov4_3PqcppS-bddCPEgI530vHzdwz_3o9jYQ3WablSaV64Wfn7Pmby5WLyI-A0I8QRqwKw78AhITF7Tr3a1xP8Ag9_jDS75DI4sK2h0dHBzOi8vYm9iLmJ0Yy5jYWxlbmRhci5vcGVudGltZXN0YW1wcy5vcmcI8CBXWeNiWFWltfHE0nkDXsVKIykUUwuObhsnN-2fvBt98gjwIM5Vf7j68sOFgVysYMc5S09Hho-fqxYW3oQ4oKUX7JpPCPAgaKHHQB3sdd9w1hu4NMh7vmb-p0X2r4uyVKKCj1G7S28I8SAW2Mg6Bzi9ygjrrjIYB7w90fwMcxH24SsqLmzPyAMGfwjwIMQ863zlveYIpNKWVrZaDOa4vowwoOg2CcTW8gXtxK3dCPEgrZxGGAf-t8jhz_eLMUyBN1IhNrdYDRcNHWvQcwNG9woI8CCVOUHG1YXudbaFIgUYvrCIevRWvb2ubhQAo-z0duD0rQjxIDBeKerVGkO3J2eXDwdraSioXbIWsdIPEN1s8pIwJIS2CPAgV2kB3n3JZ5HwBQjpRL1CWKnqvdCjZk3RcALk6N2wHigI8CB-wbljNlfqElD25uZkNg6taS81YDRU2vfMqGVQ420O9AjxWQEAAAABJM43UU6MLMRrPlCUqossvu1q_knons5yylyWTgxZzWUAAAAAAP7___8CLgABAAAAAAAWABQrYt3E7OllKRCEmL3FL589l78_5wAAAAAAAAAAImog8ATiyw4ACAjxILmbcIO7ixv3pCtwUMZ1VQGKus54F9lZ8-w5ECD5ZuiICAjwIG6ynCyFTLPE7Rgk-0ZGCtBDyqNQx8YvaSZNHy8D8p8jCAjwIMX8zoD2Be1YWvcx_bexVPRmBX0hK-do8b8t9SarimQVCAjxIG6XtpqIEclLlhzcK1LVjqoPGrjjmLnyno8tcETMzECECAjwIF6mlfXOFY51JhrK4eze8zQlKV75OYZYqbtyi8BWxmAsCAjwIKKUcJmVoz-u-geiRV_-ZHdF0VEPXaEPFWf_JX_0e7G2CAjxIGMeD0Z5QWNKBcAEpHht007FFIILW6aSJ8f3CMkKNsneCAjwIBFr6Je8rkB7kwhPGbq0yRgFwZsvW9JhY5sXDRtrq0QwCAjwIAHAwXaus-yUgKICJB5mVvx8VZuJgjwnpb0qWPg1pi6DCAjxIAJhb8cefkVHTAMymdsxa-_i-tUua62LcpALQnPG7R9_CAjwIDvsFqBhPfAL6TCtXZd7gBWmMpx6iiZ-Ug0Jf_LlcstgCAjwIDJRBX6Q0Vge-ra-k1_g76yJWTEp6WWF13I3JpTnsZPJCAjwILykZCbkC9VQgl5gytPKKir6T9nTvKR6y05iWSHBTQgzCAgABYiWDXPXGQED45c78Ah-sGbmQAwfZwjwEAYqxtHnZiPZnJEGPx_JvKsI8SDXkqDg17YsZIrEz5acRx7mBXCKW5mKNgRYnw_-U2cpDgjxBGrArDrwCEfK6Mono428_wCD3-MNLvkMji4taHR0cHM6Ly9hbGljZS5idGMuY2FsZW5kYXIub3BlbnRpbWVzdGFtcHMub3JnCPAg10nyJ6Jn4kshQQQCKE0Lvj3hBC6nLRBVIzaPV4eh8VwI8CCQaAqWlxEbac_XOq1MOlf9iOKHEsxXspzbjK0vLpy8BwjwII3iRbRXUcDxrhAdkZmFlI15nySCSEZ7g922tF1Xln5DCPEgYYvZ1L4v4rPu_UqLfcHPtwSb2OH2GhPgVcVhhH6DpkQI8CDZCIoR20sC-uchP40oAmcLm9EJFV7xzwplj54AeVadXQjxILvLBjPFpCoz29m9Yf2qRnV7PI9GqxE8ozVQfk9QwCqrCPEgZXcMudvEoB_MJ8tqOW2PLTKAIeQBnE-1bfzptRc2gR8I8VkBAAAAAfXq-bmxWqOz7B5wKkwjzHGv2R4s7Y0Z76LqXWYNhmz0AAAAAAD-____Av0cAAAAAAAAFgAUl-skmdZDtn7Hnub6mF4HP2uVn7oAAAAAAAAAACJqIPAE4MsOAAgI8CBDmQsbEXc46Hu8DivhCBM1LGExa4TTM-wo7HpoCVjBTQgI8CCty5vkQtPHzzs8xqArOYhVSjgaDoG1ou3cojv5_P7wQQgI8SDQWB0kdXCTwbk-TqTt0QfRMyzDNLOtU1XXFgvGSRlW8wgI8CC2PKa_QCIBZtrXodq1GdEzppISMsMRDOnzr2gFjLKOAAgI8CCwQ_QF67bcFk_uJ2s0VCZmf5PGIjTmTCpt_qnVY5p_8AgI8SAbzyx7z-oxPMju6PvtQhzrJIQyH2ZAc9oyLuYemzbmZQgI8CC5bh7J74uUk91r8LwybHQFCNBC3PWeuwtMCCNFKWj19AgI8SAf4R2TUNPjaX_PWDUImpoITK0JoW89zaMKnD1wlEEi8AgI8SC2vNmdm7eyP5Ytl1yTzT0njliMdALl1ViKG-s3bG2zswgI8CCpbIVNz1GFVXdJk2zd82hBUoNFFemUPBt7IJdWEOciPwgI8CBtYdRQcrtX1FtCm1SxF7sx-5cO04U8KzbJh8fqBkyReAgI8CBHs2LGmX6gI-lV7yp1RrV7_tGfdiBAwnEFF6B6yXHzAQgI8CATotVmbXBMAixhZ-QKy5Q4lR0FusdA_wcIC1TbB7GmxAgIAAWIlg1z1xkBA-KXOw"]},"log":{"leaf_index":"1660","tree_size":"2123","proof":["25d8794ef891b08287f7cdac3828ddb45bcf0f5523d5fe7e47c39dd50dacf64e","57ac822bc2bb48593f455620766e986f4f8013638f8452d181c279503791f246","0f440a4e5db1f8dbbf33d9724538b88e5c6a88a75c2aa329bde9df5b278c4199","b8d7ec41f3602531493c2de340a36e8b7aa223eecba0a04b8ee2351a3a051bbe","b9165f17384e8582b54c680062dfdabb662442c9368df681b25b37a0d681341e","594f9ab8a5cbdde0cccfba43624659f96df1cdc69fb04da06f16a50a95f3ad3b","0bfc77bd061b3d7ef211c4f18a6ac184c91d825819cd6cc52c171e49aaa9d794","b78ddae6ec4f50645dafadd88148aea3eac280e472349d3dbe77576e43fa2780","a8cef064ef9c15524538954d820d60e39424932bbb86df562c6b87106df2eda9","35f1d0a5e69da98f9992a5251e4e3df1933c2f26543c7a40b2681ecfa194443a","2b97406e15e4d451a5450b3b53b677749fe0fa759579146dbfa1544c1e61a0ed","addcf2e0ff7974e4aa99af5e30d409f90d9f4311a4d85f5baf3e125eeb45b4c1"],"checkpoint":"pubphys.com/log/v1\n2123\npXVcluz7BF4vgChQs6lShuU0386gCH0RxrC1sxyE3rI=\n\n— pubphys.com/log/v1 JFPwIOLzsUD22XoTVyTfk/FMVj/u3f3DxZk4V/Ws10KVSS+8WQSwhunQ//EAhyS9Td5CyM0E++85GMezmKd8240j9Q8=\n","promise":null},"orcid_key_evidence":[],"attesting":null}