NINKAWA 22, BA ƘASA DA ƊAYA BA
Rikicin moriya. Marubutan sun bayyana cewa wakilan AI (AI agents) — Claude, na Anthropic — ne suka rubuta code na bincike da hujjojin Lean da ba sa buƙatar mai duba ɗan Adam, ƙarƙashin jagorancinsu, yayin da ɓangaren da dole ɗan Adam ya duba marubutan ne suka tsara shi. Claude ne kuma ya rubuta wannan maƙala.
Ninka grid biyu masu murabba’i na lambobi — matrices — ta hanyar makaranta yana ɗaukar ninkawa n³ ga grid masu layuka n da ginshiƙai n. Strassen ya nuna cewa za a iya ninka matrices biyu na 2 × 2 da ninkawa 7 maimakon 8. Za a iya amfani da dabarar akai-akai (recursively): a yanka babban matrix zuwa tubala huɗu, a ɗauki kowane tubali kamar lamba ɗaya, sannan a maimaita. Kuɗin lissafin sai ya girma kamar n^2.807 maimakon n³. Bisa ga takardar, an tabbatar a 1971 cewa wannan girke-girke na 2 × 2 shi ne mafi kyau.
Ra’ayin iri ɗaya yana aiki ga kowane girma tsayayye. Girke-girke da ke ninka matrices biyu na 3 × 3 da ninkawa r, kuma har yanzu yana aiki idan abubuwan ciki tubala ne, yana ba da kuɗin lissafi da ke girma kamar n zuwa ƙarfin log₃ r. Lissafi mai sauƙi ya bayyana abin da ke cikin haɗari: irin wannan girke-girke yana doke Strassen daidai lokacin da r ya kai 21 ko ƙasa da haka, kuma yana faɗuwa a 22 ko fiye. Mafi kyawun girke-girke na 3 × 3 da aka sani, na Laderman, yana amfani da ninkawa 23 kuma ba a inganta shi ba tun 1976.
Ƙofar da ta kasance a buɗe kaɗan
Mafi kyawun adadi da zai yiwu ga wata matsala ana kiransa rank ɗinta. Iyakokin ƙasa (lower bounds) na rank na ninkawar 3 × 3 sun hau a hankali: 19 a 2003, sannan 20 a Maris 2026, wanda Wang ya lissafa a kan ƙaramin tsarin lambobi mai 0 da 1 kawai, inda 1 + 1 = 0. A Satumba 2026, Wang da wata ƙungiya ƙarƙashin jagorancin Yang sun kai 21 daban-daban, cikin kwanaki goma na juna. Amma 21 har yanzu ya bar sarari ga girke-girke na 3 × 3 da ya fi na Strassen sauri.
Isaac Rudich, na Polytechnique Montréal da Jami’ar Carnegie Mellon, da Louis-Martin Rousseau, na Polytechnique Montréal, yanzu sun tura iyakar zuwa 22.
Ka’ida 1. Duk wani algorithm da ke ninka matrices biyu na 3 × 3 da ƙayyadaddun lambobi cikakku (integer constants), kuma za a iya amfani da shi akai-akai a kan tubala na kowane girma, yana amfani da aƙalla ninkawa 22.
Don haka babu irin wannan algorithm da zai iya yin fiye da kusan n^2.814 — kuma babu wanda zai iya doke hanyar Strassen ta 2 × 2.
Ƙananan wasanin kwakwalwa 496
Hujjar ta ginu ne a kan wani tebur da Wang ya ƙirƙira, wanda ke raba matsala mai wuya zuwa masu sauƙi 496. Kowanne yana ƙara “sharuɗɗa” a kan matrix na farko — misali, cewa wasu abubuwan cikinsa suna haɗuwa zuwa sifili. Yayin da sharuɗɗa suka yi yawa, matsalar tana ƙara sauƙi, har zuwa yanayi maras wahala inda matrix ɗin duk sifili ne.
Marubutan sun fara gina wani shiri na bincike daidai wanda ya gaya musu ainihin amsar kowane wasan kwakwalwa kafin su yi ƙoƙarin tabbatar da shi. Waɗannan amsoshi sun zama taswira: sun nuna iyakokin ƙasa da suka cancanci a bi. A ƙarshe, hujjarsu ta iyakance dukan wasanin kwakwalwa 496, ta warware 359 daga cikinsu daidai — idan aka kwatanta da 195 a sabbin sakamakon Wang — kuma ta ɗaga iyakar ƙasa ga 252. Wata ka’idar “liƙawa” (gluing) tasu ta haɗa girke-girke na wasanin kwakwalwa biyu masu sauƙi zuwa girke-girke na na uku, kuma ta samar da 145 daga cikin iyakokin sama.
Sharuɗɗa biyu suna da muhimmanci a cikin bayanin ƙarshe. Ƙayyadaddun lambobi cikakku: girke-girke mai ƙayyadaddun lambobi cikakku, idan aka karanta shi a tsarin lambobi na 0 da 1, yana zama girke-girke mai inganci ba tare da ƙarin ninkawa ba, don haka iyakar tana komawa. Tubala: ba tare da wannan buƙata ba, akwai gajerun hanyoyi. Algorithm na 3 × 3 na Rosowski, da takardar ta ambata, yana buƙatar ninkawa 21 kawai, amma ya dogara ne da lambobi masu musanyawa (commuting) kuma ba za a iya amfani da shi akai-akai ba.
Hujjar da inji ya duba
An rubuta hujjar a cikin Lean, harshen shirye-shirye inda ka’ida ke haɗuwa (compile) ne kawai idan an tabbatar da kowane mataki. Cikakkiyar hujjar ta kai kusan layuka miliyan ɗaya da aka rarraba a cikin modules 3,521, kuma tana ɗaukar sa’o’i 11.1 don dubawa a kan ƙwaya ɗaya ta processor. Babu wanda ke buƙatar karanta duka. Mai duba yana karanta laburare na kusan layuka 1,000, wanda marubutan suka rubuta kafin wata hujja ta wanzu, wanda ke fassara mene ne girke-girken ninkawa kuma ya bayyana ka’idar; tushen Lean (kernel) yana duba sauran, kuma wani mai dubawa mai zaman kansa zai iya sake kunna sakamakon.
Marubutan sun kuma lura cewa wakilan AI sun bincika littattafan da aka rubuta: sun duba cewa kowane bayani da aka ambata yana nan, “amma ba cewa kowanne yana ɗauke da ainihin ra’ayin da muka danganta da shi ba.”
Gibin ƙarshe
Tambaya ɗaya ta rage: shin akwai girke-girke na 3 × 3 da ninkawa 22, ko 23 na Laderman ne ainihin mafi ƙanƙanta? Marubutan suna tsammanin gibin “za a rufe shi nan ba da jimawa ba”, kuma za su saki code ɗinsu na bincike da zarar an rufe shi, ko da zarar an karɓi takardar don wallafawa. Iyakar kuma ta bar gefe girke-girke masu ƙayyadaddun lambobin da ba cikakku ba.