HASASHEN 1959, HUJJA DAGA OPENAI, DA MASANIN LISSAFI DA YA KARANTA TA

Wannan labari game da bayanan da wani masanin lissafi ya rubuta ne don bayyana hujjar wani. Duk abin da aka faɗa a nan game da hujjar OpenAI ya fito ne daga waɗannan bayanan, a hannu na biyu.

Layuka masu lanƙwasa ta cikin ɗigo-ɗigo

Matsalar ta tsiro ne daga wani aikin makaranta da ake kira interpolation: zaɓi ’yan ƙimomi na x, yanke shawarar wace ƙima wani aiki (function) zai ɗauka a kowanne, kuma wani polynomial koyaushe yana yin hakan. A fili, wasa iri ɗaya yana neman layi mai lanƙwasa ya ratsa ta cikin tarin ɗigo-ɗigo, abin da zai iya yi idan matakinsa (degree) ya yi tsayi sosai. Layi mai lanƙwasa na mataki d — wanda ake zana shi da daidaiton polynomial a cikin x da y wanda babban ƙarfinsa (power) shi ne d: madaidaicin layi yana da mataki 1, da’ira 2 — yana da (d+1)(d+2)/2 ma’auni (coefficients), kuma neman ya ratsa wani ɗigo daidaito (equation) ɗaya ne.

Yanzu ka nemi ƙari: dole layin ya ratsa kowane ɗigo da maimaitawa m (multiplicity) — sau da yawa, kamar ƙullin rassa da ke ratsa juna a can. Kowane irin wannan ɗigo yana ɗora daidaito m(m+1)/2. Ga d da m manya, ƙirgawar tana kusan ma’auni d²/2 idan aka kwatanta da sharuɗɗa r m²/2, don haka, ga ɗigo r, ƙirgawar tana tabbatar da layi mai lanƙwasa da zarar d/m ya ɗan wuce √r — a daidai, da zarar ma’auni sun fi sharuɗɗa yawa.

Hasashen Nagata (1959) yana bayyana haka daidai: ga ɗigo r ≥ 10 da aka sanya ta hanya “gama-gari sosai” — ba tare da wani daidaituwa ta musamman a tsakaninsu ba, kamar yadda ɗigo-ɗigo da aka zaɓa ba da tsari ba za su kasance —, kowane irin wannan layi yana da mataki fiye da m√r. Sharaɗin r ≥ 10 ya zama dole: ɗigo tara koyaushe suna kan wani cubic (layi mai lanƙwasa na mataki 3), kuma wannan cubic da aka ɗauka sau m layi ne na mataki 3m da ke ratsa kowanne daga cikinsu sau m — daidai m√9, daidaiton da hasashen ke hana. Nagata ya tabbatar da shi lokacin da r cikakken murabba’i (perfect square) ne.

Misalai biyu daga bayanan: babu layi na mataki 9 da ke ratsa kowane ɗaya daga cikin ɗigo 10 na gama-gari sau uku (ma’auni 55 kawai ga sharuɗɗa 60 — wannan kaɗai bai isa a yanke hukunci ba, domin sharuɗɗa na iya haɗuwa da juna; abin da ke hana shi shi ne iyakar Nagata, tun da 9 ya gaza 3√10 ≈ 9.5), kuma babu layi na mataki 12 da ke ratsa ɗigo 16 na gama-gari sau uku — yanayin daidaito 12 = 3√16.

Hujjar, kamar yadda Tevelev ya ba da labarinta

Abin nuni shi ne “OpenAI, Nagata’s conjecture for plane curves”, wani preprint na OpenAI Math Release mai kwanan wata 23 ga Satumba 2026, tare da rubuce-rubucen abokan tafiya kan tambayoyi masu alaƙa da aka fitar a ranar 23 ga Satumba da 5 ga Oktoba. Jenia Tevelev, na UMass Amherst, ya yi nazarinta don wata lacca kuma ya mayar da wannan bayani zuwa waɗannan bayanan, yana fatan za su “yi amfani, musamman ga waɗanda ba ƙwararru ba”. Tevelev bai yi da’awar wani sabon abu ba kuma ya sake tsara hujjar.

Matakan farko na yau da kullum suna matse filin a kan layin cubic mai santsi kuma suna haɗa ɗigo r zuwa ɗaya. Sabon ra’ayin, a wannan bayanin, shi ne barin cubic ɗin ya samar da ƙulli (node) — wani ɗigo inda layin ke ratsa kansa, yana matse wata zobe ya yanke ta — da kuma bin ayyukan theta (theta functions) ɗinsa, hanyar da aka saba bi don rubuta ayyukan da ke rayuwa a kan irin wannan layi, yayin da wannan matsewar ke rufewa. Sa’an nan wata dualiti tana musanya tambayar “waɗanne layuka ne ke zama sifili a ɗigon?” da madubinta, “waɗanne ƙimomi ne za a iya ƙayyade a can?”, kuma abin da ya rage interpolation ne a kan alwatika ta ɗigon lattice masu lambobi cikakku — tebur mai iyaka na ƙimomin da za a cika, “wanda yake a sarari”.

Yanayin ɗigo goma sha shida masu ninki uku: ɗigon lattice 91 na alwatikan (hagu) da monomials 96, wato ɓangarori na asali kamar x²y da ake gina polynomials da su (dama). Nuna cewa waɗannan ɓangarori 96 za su iya ɗaukar kowace ƙima a kan ɗigo 91 — matrix na 96 × 91 yana da “cikakken rank” (full rank) — yana kawar da layin. — Figure 3, Tevelev (2026), arXiv:2610.30050, CC BY 4.0.

Lissafi iri ɗaya yana warware wata tambaya mai alaƙa, wadda ta fi zurfi a fasaha, inda ake ƙara girman fili a kewayen ɗigo ɗaya da ma’auni daban-daban a alƙibla biyu (“weighted blow-ups”), lokacin da rabon waɗannan ma’aunan ya wuce tara — wani yanayi na hasashen Dumnicki, Harbourne, Küronya, Roé da Szemberg.

Bayanan suna gabatar da hujjar a matsayin hujja. Ba su ambaci wani dubawa na hukuma mai zaman kansa ba, kamar bita na takwarorin aiki (peer review) ko hujjar da kwamfuta ta tabbatar.

“Mai karatunsu na farko ɗan Adam”

Bayanan kuma suna ɗauke da ra’ayin marubucin. Yana karanta rubuce-rubucen da aka fitar, Tevelev ya rubuta: “Na yi mamaki ko ni ne mai karatunsu na farko ɗan Adam. Sun yi kama da kusan ba a gyara su kwata-kwata ba.” Tevelev yana zargin cewa kamfanin ya yi tsammanin masana lissafi za su gyara su, ƙoƙarin edita wanda “yana buƙatar ƙwarewa da lokaci, amma yana kawo ƙaramin yabo da ake haɗawa da bincike na asali”. Kamar wasu da yawa, Tevelev ya yi imanin cewa wannan hanya tana cutar da sana’ar, “musamman ƙwarin gwiwa da horar da matasa”.

“A lokaci guda, ina ganin waɗannan rubuce-rubucen babbar nasara ce mai ban mamaki,” Tevelev ya rubuta: samfurin ya warware matsalolin da ƙwararru da yawa ke tunanin ba za a iya kaiwa ba, kuma “tsohon jejin hasashen lissafi” “yana ƙonewa da sauri a cikin wutar AI”.

Shawarar: gidauniya mai zaman kanta da kamfanonin AI za su ɗauki nauyinta, mai ba da tallafi, gurabe na karatu (fellowships) da damar amfani da mafi kyawun samfura ga matasa masu bincike, tare da bita na takwarorin aiki bisa misalin US National Science Foundation.

Bayanan sun ce an “gyara su kuma an zana musu hotuna da ChatGPT Astra”.