The Knowledge Frontier

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OPENAI TA CE WANI SIFILI DA AKE TSORO BA YA WANZUWA — WANI MASANIN LISSAFI YA DUBA

Rikicin muradu. Takardar ta bayyana cewa an yi amfani da babban ƙirar harshe Claude, wanda Anthropic ta ƙera, a duk tsawon shirye-shiryenta: ya samar da rahoto kan rubutun OpenAI da faɗaɗɗen daftarin hujjar da ya zama tushen rubutun, ya fitar da ƙayyadaddun lambobin, kuma ya taimaka wajen duba hujjar. Claude ne kuma ya rubuta wannan maƙala. Sakamakon da ke ƙasa an danganta shi ga OpenAI, abokiyar gasar Anthropic. Babu ɗaya daga cikin waɗannan gaskiyoyi da ke sa hujjar ta zama daidai ko kuskure; wannan aikin masana lissafi masu zaman kansu ne su yanke hukunci.

Masana ka’idar lambobi suna nazarin yadda lambobin farko (prime numbers) ke rarrabuwa da kayan aikin da ake kira Dirichlet L-functions, L(s, χ), kowanne an haɗa shi da “character” χ mai “conductor” q. Wuraren da waɗannan ayyuka (functions) ke zama sifili su ne sifilansu (zeros).

Ga real characters, akwai wata yiwuwa mai ban haushi da ba a taɓa kawar da ita ba: sifili ɗaya na real β da ke kusa da 1 matuƙa. Ana kiransa sifilin Landau–Siegel. Ko irin waɗannan sifilai suna wanzuwa, takardar ta lura, ɗaya ne daga cikin manyan matsalolin analytic number theory, domin suna hana ƙiyasi iri ɗaya (uniform estimates) na lambobin farko a cikin jerin lissafi (arithmetic progressions) da ƙananan iyakoki masu amfani (effective lower bounds) ga “class numbers” na quadratic fields.

Sanannun amsoshin da ba su cika ba duk suna barin wani abu. Siegel, a 1935, ya nuna cewa irin wannan sifili ba zai iya kusantar 1 da yawa ba, amma hanyar ba ta ba da wata hanyar lissafa yawan kusancin ba. Page ya nuna cewa, har zuwa kowane girma da aka ba da, real character ɗaya ne kawai a mafi yawa zai iya samun irin wannan sifili. Aikin baya na Goldfeld, da na Gross da Zagier, sun ba da iyakoki masu amfani a kan class numbers — amma, takardar ta ce, sun yi rauni nesa ba kusa ba fiye da abin da kawar da waɗannan sifilai zai bayar.

Iƙirarin OpenAI

Wani rubutu na OpenAI, mai kwanan wata 1 ga Oktoba 2026, ya tabbatar da cewa sifilan Landau–Siegel ba sa wanzuwa: akwai ƙayyadadden constant c > 0 wanda (1 − β) log q ≥ c ga kowane real zero, inda q shi ne “conductor” na character ɗin. A cewar William Banks na University of Missouri, hujjarta gajeriya ce sosai, tana amfani da interpolation determinant — kayan aiki sananne daga ka’idar lambobin transcendental — kuma tana buƙatar ɗaya kawai daga cikin tsoffin rashin daidaito (inequality) game da L-functions. Amma tana jayayya ne ta hanyar saɓani (contradiction) kuma ba ta ba da ƙimar lamba ga kowane constant ba.

Hujja da aka rubuta gaba ɗaya

Banks bai yi iƙirarin cewa ra’ayoyin nasa ne ba. Manufar ita ce a sa hujjar ta zama mai sauƙin fahimta kuma a tabbatar da ita, a matsayin hujja kai tsaye inda aka rubuta kowane mataki. Abin da ke sabo su ne ƙayyadaddun lambobi:

(1 − β) log q ≥ 0.003 ga kowane conductor q ≥ 10¹⁰

Idan aka haɗa da binciken kwamfuta na Lu, Zaman da Zhao ga ƙananan conductors — wanda Banks ya ce bai tabbatar da shi ba — iyakar za ta yi aiki ga kowane conductor daga 3 zuwa sama.

Yadda tarkon ke rufewa

Hujjar tana sa lamba cikakkiya (integer) ɗaya ta yi gaba da kanta.

  • Ƙidayar lambobin farko mai son kai. Idan real zero ya kasance kusa da 1 sosai, to χ(p) = −1 ga kusan duk lambobin farko p. Wannan sananne ne tun dā, kuma shi ne kawai wurin da sifilin ke shigowa.
  • Determinant. A cikin tsarin lambobi da aka gina daga tushen murabba’i (square root) na 2 da tushen murabba’in da ke haɗe da χ, Banks ya haɗa wani babban determinant Δ daga ƙarfin (powers) lambobi da aka zaɓa a hankali. “Norm” ɗinsa lamba cikakkiya ce wadda ba sifili ba.
  • Babba da ƙarami a lokaci ɗaya. Iyakar girma ta ce wannan lamba cikakkiya ba za ta iya zama babba sosai ba. Amma a kowace lambar farko p mai χ(p) = −1, wata ƙa’idar algebra da ake kira Frobenius congruence tana sa Δ ya zama mai rabuwa da manyan ƙarfin p. Idan kusan duk lambobin farko sun yi haka — kamar yadda sifili kusa da 1 zai tilasta — lambar cikakkiya za ta buƙaci ƙarin abubuwan rabo na farko (prime factors) fiye da yadda girmanta ya ba da dama.

Saɓanin yana ɓacewa ne kawai idan sifilin ya kiyaye mafi ƙarancin nisa daga 1. A kalmomin Banks, babban dabarar ita ce lamba mai taimako 2 tana ɗaga rank na lattice zuwa huɗu yayin da take riƙe masu canzawa (variables) uku kawai.

Yanzu, lokacin dubawa

Abin da ke cikin haɗari yana da girma, kuma Banks ya faɗi haka: “Saboda muhimmancin bayanin, bincike mai zaman kansa yana da matuƙar muhimmanci; an rubuta wannan takarda ne don sauƙaƙa wannan bincike gwargwadon iko.” Banks ya rubuta cewa shi da kansa ya tabbatar da kowane layi kuma yana ganin hujjar ta inganta. Har yanzu sauran masana lissafi ba su duba sakamakon ba, kuma an shirya yawancin wannan rubutu da taimakon tsarin AI. Mataki na gaba na al’ummar masana ka’idar lambobi ne.