<span class="mw-page-title-main">晓美焰</span>

Contains spoilers for the TV series
According to an interview with Gen Urobuchi at Animagic 2013, Homura Akemi underwent no fewer than 100 loops, with each loop lasting one month~~, so her actual age is estimated to be that of a university graduate in her late twenties, a "magical auntie"~~。《Puella Magi Homura☆Tamura》's age meme Among the many loops, six appear in the anime: the first through fifth loops where she first met Madoka (content from Episode 10), and the final loop of the main storyline (the following part comes from PSP supplements). In the first loop, Homura was an introverted, timid, and self-deprecating sickly girl with braids and glasses. She had a heart condition since childhood. As a transfer student from a mission school in Tokyo to the second year of Mitakihara Middle School, she was in the same class as Madoka Kaname and Sayaka Miki. Madoka, as a health committee member, took the initiative to care for Homura's health, which is how Homura first met Madoka (~~From then on, whether in fanon or official media, and regardless of the loop, the first thing Homura does upon meeting Madoka at school is go to the nurse's office.~~). Due to her physical weakness caused by congenital heart disease, Homura had neglected most of her schoolwork, resulting in poor classroom performance; her physical stamina was also far behind others. On her way home from school, she began to doubt the meaning of her existence and was thus~~Aoi-chanvoiced by~~lured into a barrier by a Witch. She was saved by the magical girls—Madoka and Mami. クラスの皆には内緒だよ☆(Keep it a secret from the rest of the class☆) Being saved by the two made Homura begin to admire the life of a magical girl, and she stayed by the side of Mami and Madoka as they fought Witches. However, Homura later witnessed Madoka and Mami die in battle during Walpurgisnacht. Overcome with grief, under QB's~~shamelessJust as planned~~pyramid scheme: 「私は……鹿目さんとの出会いをやり直したい…」「I... I wish my encounter with鹿目さんcould be redone...」「…彼女に守られる私じゃなくて、彼女を守る私になりたい！」「...not as someone protected by her, but as someone who protects her!」 Homura chose to sign a magical girl~~soul-selling~~contract with QB, with the ability of "Time Manipulation." From then on, Homura began her journey of constant loops, using her ability to rewind time. In the second loop, as soon as Homura finished her self-introduction, she ran up to Madoka and grabbed her hands:「鹿目さん, I've become a magical girl too! Let's work hard together~ and do our very best!」, leaving Madoka so embarrassed she didn't know how to respond. Following Mami Tomoe's suggestion, Homura taught herself how to make time bombs, developing her own unique combat power and fighting style,~~Entering the growth phase~~. And after defeating the Witch, Madoka would also hug Homura with exceptional joy, causing Homura to break into a shy but satisfied smile. But in the end, even though the two defeated Walpurgisnacht, Homura witnessed Madoka's witchification firsthand and learned the cruel truth about magical girls. In the third timeline, Homura privately sought out Mami, Madoka, and Sayaka—who were already magical girls—and tried to tell them~~Injuu~~that Kyubey had deceived them, but no one believed her. Sayaka even complained that her bomb tactics would hinder melee teammates, so Homura had no choice but to go to Shotai Kogyo (~~SHAFT~~) to steal handguns and ammunition. Later, Sayaka turned into a Witch. Kyoko and Madoka, just like in the main storyline, foolishly called out to her hoping she would change back. During this, Madoka was nearly injured by Sayaka, and Homura steeled her heart, using time stop to set up a large number of bombs and kill the Witch that was once Sayaka. Mami Tomoe witnessed Sayaka's witchification and suffered a mental breakdown. She conceived the idea of collective suicide, first restraining Homura—who possessed time magic—with ribbons, then shattering Kyoko's Soul Gem with a single shot. To save Homura, Madoka shattered Mami's Soul Gem with an arrow just as Mami pointed her gun at Homura, and then began to wail: Madoka:「No! I can't take it anymore... not like this...」 Homura:「It's okay, let's do our best together and go defeat Walpurgisnacht together.」 Madoka:「Mmm...」 After defeating Walpurgisnacht, the two were exhausted of magic and on the verge of witchification. Although Homura had given up hope and wanted to become a Witch to forget all the pain, Madoka used her hidden~~Mermaid Witch~~Grief Seed to purify Homura's Soul Gem, wishing for Homura to go back to the past and save her foolish self who hadn't been deceived by Kyubey yet. Homura tearfully agreed to Madoka's request: 「約束するわ！絶対にあなたを救ってみせる！何度繰り返すことになっても、必ずあなたを守ってみせる！」「I promise, I will definitely save you! No matter how many times I have to do it over, I will definitely protect you!」 Madoka's final request to Homura was that although there are many sad and hateful things in this world, there are also many things she wants to protect, so she does not want to become a Witch herself. Homura:「まどか!」 Madoka:「ほむらちゃん, you finally called me by my first name, I'm so happy」 Madoka holds up her Soul Gem, signaling Homura to destroy it... (This timeline left a deep psychological scar in Homura's heart.） In the fourth timeline, because of Madoka's request in the third timeline, Homura resolved to change herself: 「誰も未来を信じない，誰も未来を受け止められない。だったら、私は…」「No one will believe in the future, and no one can accept the future. In that case, I...」(~~Kuriyama Miraigetting caught in the crossfire~~) Homura used her Soul Gem to cure her nearsightedness and heart disease, undid her braids, changed her appearance, and personally warned Madoka not to accept miracles from unknown sources. Homura decided to no longer rely on anyone~~relying on the JSDF's arsenal~~, and no longer let Madoka fight; she no longer expected anyone to understand her, resolving to deal with all the Witches herself—including Walpurgisnacht. Unfortunately, the result was still insufficient strength, and she was unable to defeat Walpurgisnacht alone. Finally, amidst her hoarse screams, Homura could only watch as Madoka made a contract with Kyubey, defeated Walpurgisnacht with a single arrow, and immediately blackened into a Witch due to excessive magic consumption. In the fifth timeline, after being struck once again, Homura remained unwilling to give up. Amidst her disappointment, she became even more obsessed with her original wish: 「繰り返す。私は何度でも繰り返す。同じ時間を何度も巡り、たった一つの出口を探る。あなたを、絶望の運命から救い出す道を」「Repeat. I repeat it over and over again. Cycling through the same time, searching for an exit. A way to save you from a fate of despair.」 And in Homura's heart, Madoka had already become her only friend:「まどか……たった一人の私の友達……」，Chi-samastated that she personally likes this quote~~Essentially a Madoka-con~~, it can be said that everything from Episode 10 is contained within these words, being that important.Friend~~Wife~~It's really nice.~~Aoi-chanAlso said, "It's really nice!"~~ However, Homura broke through her psychological barriers, no longer hesitating because Madoka was holding Kyubey, and decided to save Madoka at all costs: 「あなたの、あなたのためなら、私は永遠の迷宮に閉じ込められでも、構わない……」「If this is... if this is for you, I don't care if I'm trapped in this infinite maze forever!」 Whenever Homura fails to prevent Madoka's death or witchification, she activates her ability to rewind time. The repeated sorrow and despair of failing to save Madoka caused Homura's outward personality to become increasingly cold, becoming focused solely on protecting Madoka alone, with no energy left to care for others; helping others was merely to keep Madoka from being sad. That being said, she still helps others within her power, such as preventing Mami Tomoe from heading to Charlotte's barrier and Kyoko's attempt to wake the witchified Sayaka. When someone other than Madoka is injured or dies, Homura's heart is still hurt, but she hides it all. BeingSayakacommented: Always putting on a look of having given up on everything, while thinking about something else in her heart. In the main timeline, she is beautiful and an all-rounder in academics and sports~~Practice makes perfect?~~, usually expressionless, her warnings to Madoka are always vague but meaningful. This is because of the fragility deep within Homura's heart and her desire to protect Madoka, so when speaking to Madoka, Homura always swallows her words. However, there are two times in the main timeline where she couldn't help but reveal her emotions to Madoka: the first was when Madoka, in order to make Sayaka happy again, almost signed a~~slave contract~~. After stopping time, Homura fired over a dozen shots at Kyubey, riddling him with holes to prevent the contract from being signed. She finally couldn't help but explode at Madoka's behavior of always sacrificing herself just to make herself look less useless: 「貴女はなんで貴女はいつだってそうやって自分を犠牲にして!役に立たないとか意味がないとか!勝手に自分を祖末にしないで!貴女を大切に思う人のことも考えていい加減にしてよ！あなたを失えばそれを悲しむ人がいるって、どうしてそれに気づかないの！貴女を守ろうとしてた人はどうなるの！」「You, why do you! Always sacrifice yourself like that! Saying you're useless, that it's meaningless! Don't belittle yourself like that! Think about the people who value you, and enough is enough! If you were lost, there would be people who would be very sad! Why can't you realize that! What about the person who wants to protect you!」 The second time was when Kyoko had already with the blackened Sayaka~~committed double suicide~~, Homura is the only magical girl left in the city who can fight Walpurgisnacht. Madoka goes to Homura's house and tries to convince Homura to let her join the battle, but Homura flatly refuses and coldly says she can handle it, and that Kyoko was just a sidekick. A perceptive Madoka tells Homura... she really wants to believe Homura isn't lying to her... but she can't... The bitterness Homura accumulated over hundreds of loops suddenly erupts: 「本当の気持ちなんて、伝えられるわけないのよ……だって私は……私はまどかとは……違う時間を生きてるんだもの！」「How could I tell you my true feelings... because I... I andまどかare in different times!」(Homura ran to Madoka and hugged her, as if seeking comfort, with her cheek on Madoka's shouldernuzzlingtwice) 「私ね････未来から来たんだよ。何度も何度もまどかと出会って，それと同じ回数だけあなたが死ぬところを見てきたの。どうすればあなたが助かるのかどうすれば運命を変えられるのか，その答えだけを探して何度も始めからやり直して」「I... am from the future... many, many times I've metまどかyou, and then witnessed your death countless times... how to save you, how to change your destiny, just to find that answer, I've looped countless times.」「それってえ……」「That means... Eh!」(Homura suddenly hugged tighter) 「ごめんねわけわかんないよね気持ち悪いよね!まどかにとっての私は出会ってからまだ1ヶ月も経ってない転校生でしかないものね。だけど私は私にとってのあなたは」「I'm sorry! It's confusing, isn't it! It's disgusting, isn't it! Toまどかyou, I'm just a transfer student who arrived a month ago... but to me... to me... you are...」(Homura was already sobbing uncontrollably by this point) 「繰り返せば繰り返すほどあなたと私が過ごした時間はずれていく……気持ちもずれて言葉も通じなくなっていく……たぶん私はもうとっくに迷子になっちゃってたんだと思う」「The more I loop, the more I drift away from the time I spent with you... our feelings drift apart, and words gradually fail to reach... I think I've probably lost my way...」「あなたを救うそれが私の最初の気持ち，今となってはたった一つだけ最後に残った道しるべ……わからなくてもいい何も伝わらなくてもいい……それでもどうか，お願いだからあなたを私に守らせて！」「"To save you," that was my original intention, and now, it's the only signpost left... It's okay if you don't understand, it's okay if nothing can be conveyed... even so, please, I beg you... please let me protect you!」 > In the end, despite her meticulous planning, Homura was still defeated by Walpurgisnacht. Just as she was about to loop again, she remembered what she had learned from the Incubator: with each loop, the repeated rewinding of time caused all previous timelines to converge. Since a magical girl's potential is determined by the karmic burden she carries, looping again would only further increase Madoka's potential as a magical girl, leading her into even deeper despair. Realizing she was heading toward the unavoidable despair that all magical girls face, Homura shed tears of agony, and just as she was about to~~Darkening~~, she suddenly felt Madoka take her hand: Madoka:「ほむらちゃん, I'm sorry, I'm going to become a magical girl.」 Homura「まどか，How could this...」 Madoka:「I finally know what I want. I've found the wish I truly want to fulfill, so I'm willing to give my life to make it come true.」 Homura「No! If that happens... then what was I doing all this for...」(Homura cries) Madoka:「I'm sorry, I'm so sorry. All this time, I've been constantlyほむらちゃんprotected and hoped for by you, and I think that's why I am who I am now. I'm truly sorry."(Madoka takes Homura's hand) Madoka:「This is the answer that I, as I am now, have finally found. Believe me, I promise,ほむらちゃんeverything you've done for me will definitely not be in vain.」 Homura:「まどか……」 To live up to Homura's long-term protection and the deep hopes placed upon her, Madoka took a deep breath: Madoka：「I want to erase all witches before they are even born! I will personally destroy every witch in every universe, past and future!」 QB:「That kind of wish... if that kind of wish were to be realized, it wouldn't just be a matter of time interference; it would be a defiance of causality itself. Do you really intend to become~~Kami~~!!!?」「神様でも何でもいい。今日まで魔女と戦って来たみんなを、希望を信じた魔法少女を、私は泣かせたくない。最後まで笑顔でいて欲しい。それを邪魔するルールなんて、壊してみせる。変えてみせる。これが私の祈り。私の願い。さあ、叶えてよ！インキュベーター！」「I don't care if I become a god or anything else. Everyone who has fought witches until now, the magical girls who believed in hope—I don't want to see them cry anymore. I want them to be able to smile until the very end. Any law that stands in the way of that, I will break it, I will rewrite it! This is my prayer, my wish! Now! Grant it!Incubator！」 And so, Madoka went back to the past and traveled through the future, taking upon herself theDarkeningpain and despair before they reached the brink: 「It's okay, it's all okay now. You don't have to hate anyone anymore, and you don't have to curse anyone anymore. Before you become a witch, I will take everything upon myself in your place.」(Madoka says to Walpurgisnacht) However, while this wish could create a universe, the curse it brings and the despair it carries are also enough to destroy one. But the wish Goddess Madoka made was to eliminate all witches, including herself, so she majestically destroyed her own corrupted self with her own hands. According to the lewd beast's~~malicious~~explanation, from then on, Madoka's life has no beginning or end, and all her former traces on Earth would be erased; Madoka's existence was elevated to a higher dimension, leaving behind only the concept of the "Law of Cycles" in the original dimension. No one can perceive its physical existence, nor can anyone interact with Goddess Madoka; she is no longer a part of this dimension. Homura:「you~~This lewd beast...~~What do you mean! Is thisまどかthe result of her prayer? Is this fair for everything she's done! Don't joke around! This is even worse than death... it's too much...」 Madokami:「That's not it~ほむらちゃん。Now I can see everything, past and future, universes that could have existed, and universes that might someday exist~ all of them~」 Homura:「まどか?」 Madokami:「So, I know everything now~ throughout the countless timelinesほむらちゃんAll the efforts you made for me, everything. You shed countless tears and were covered in wounds, yet you still did it all for me... I'm sorry I never noticed, I'm so sorry!」 The battle-scarred Homura is finally fully understood by the person she devoted herself to protecting, understanding all the bitterness brought by everything done for her. Homura can finally cry her heart out in Madoka's arms: Madokami:「Because I've become who I am now, I can finally know the real you. To think I had such an important friend, I'm so happy I know now.ほむらちゃん，Thank you! You are my best friend (わたしの、最高の友達) and yet I never knew~」 Homura:「But you can never go back to your original world! You'll be separated from everyone you care about forever! Are you going to be all alone here forever!」 Madokami:「I'm not alone~ Everyone... everyone will always be with me, because from now on, I am everywhere, in every time and space. So even if you can't see me or hear me, I will still always be with youほむらちゃん。」 Homura:「まどかIs that really okay with you... even if I forget you... even if I don't know you're here anymore!」 Madokami:「No, no (that's not it), don't give up so soon~ほむらちゃんEven after all this, you're still with me, so even if you return to the original world, you should still be able to remember me~ It's okay~ It's really okay~ You just need to believe~」 Homura:「まどか……」 Madokami:「After this, the wishes and dreams of magical girls will truly come true~」 In the restructured world, like Kaname Tatsuya, she remembers Madoka (though Gen Urobuchi stated that Tatsuya had only seen Madoka for a short time before she became a god, so it doesn't quite count as "remembering"). Her original headband was also replaced by the one Madoka gave herribbon(In the manga version, it is tied with a ribbonTwin Tails） To protect the world that Madoka loved and gave everything to protect, she continues to fight, representing the inheritance of Madoka's will. Her weapon has also changed from the Shield of Time to a carved bow similar to Madoka's. At the end of the TV anime, she fights alone against a large number of Wraiths in the ruins. Corrosive black wings sprout from her back, and she hears the sound of Madoka saying to her, "Go for it!".

Contains the spin-off "Puella Magi Madoka Magica [Wraith Arc]"
(To be added)

Children of the False Town (Bonus Content)
1.イバリ（＝威張り[Corresponding kanji are for reference only, same below], Arrogance/Showy) はじめに来たのはイバリ。地べたに転がるデキソコナイの頭を踏み越え、勇ましくその声を代弁する。「こうまんちきども。ゆるすものか。この感情はわたしらだけのもの」 The first to arrive is Arrogance. Stepping over the heads of the trash rolling on the ground, she speaks with that brave voice: "Arrogant fools. Unforgivable. This feeling belongs only to us." 2.ネクラ（＝根暗, Gloom) 2番手はネクラ。コソコソと歩み出て、デキソコナイをあざ笑う。「このデキソコナイは、みっともない」この人形どもは、自由だけを師事し欲望に忠実。 The second is Gloom. Sneaking onto the scene, she mocks the trash. "This trash is truly unsightly." These dolls only revere freedom and are faithful to their desires. 3.ウソツキ（＝嘘つき, Deception) 次は3番目のウソツキ。デキソコナイの魂へ捧げる涙の芝居。「哀れなデキソコナイ様。愚かなその魂は、我等が誇り」この人形どもは、愚か者を嘲笑し愚か者を味方する。 Next is the third, Deception. Offering a tearful performance to the soul of the trash. "Poor Master Trash. Your foolish soul is our glory." These dolls mock fools and become their companions. 4.レイケツ（＝冷血, Cold-bloodedness) その後ろからは4人目、レイケツ。デキソコナイの頭につまづいてちょっと不機嫌。「デキソコナイ様をもっと持ち運びやすくするためにちいさく切り分けてしまうのはどうカシラ?」この人形どもは、悪魔の言葉であれば素直に従う。 Following closely is the fourth, Cold-bloodedness. She is a bit unhappy after being tripped by the head of the trash. "How about cutting Master Trash into small pieces to make him easier to carry?" These dolls will obediently follow as long as it is a demon. 5.ワガママ（＝我侭, Selfishness) 更に5番目、ワガママは怒りながら歩く。「葬列はまだなの？わたしいっとうはやく泣けるよう練習したのよ?もうちょっとも待てないわ」とプリプリ。この人形どもは、誰かであって誰でもない。 Next is the fifth, Selfishness, walking angrily. "Has the funeral procession not arrived yet? I've practiced to be the first to cry, you know? I can't wait a moment longer," she says so angrily. These dolls are everyone yet no one. 6.ワルクチ（＝悪口, Slander) 面倒そうに出てくるのが6人目、ワルクチ。憎まれ口は相変らず。「デキソコナイ様は、大変な喜劇役者であらせられますワ」この人形どもは、絡まり極まる因果のママゴト。 Appearing with a look of annoyance is the sixth, Slander. Her words are as biting as ever. "Master Trash, you are quite the magnificent comedian." These dolls are a game of house played with the ultimate entanglement of karma. 7.ノロマ（＝鈍間, Stupidity) ようやく到着した7人目、ノロマ。無口な人形はその瞳で魔女を笑う。この人形どもは、生真面目な兵隊とは違い魔女の願いをそこそこは聞き入れる。 The seventh to arrive at last is Stupidity. The taciturn doll mocks the Witch with its eyes. Unlike a serious army, these dolls only half-heartedly listen to the Witch's requests. 8.ヤキモチ(Envy) 8番目に見えるはヤキモチ。うっとりと空を見る。「箱を用意しましょう。あの輝きをずっと閉じ込めておきたいワ」この人形どもは、色の集合、空にはあらず。 The eighth one seen is Envy. Gazing obsessively at the sky. "Let's prepare a box. I want to lock that radiance away forever." These dolls are a collection of form, yet they are not emptiness. 9.ナマケ（＝怠け, Sloth) 9人目のナマケはあくび混じりに喋る。「お遊戯のお手伝いまでしなきゃいけないの？編み物なんてめんどり面倒」邪魔に転がるデキソコナイの頭をひと蹴り。この人形どもは、葬儀がちっとも始まらないので暇を持て余し街を徘徊する。 The ninth one, Sloth, says amidst a yawn, "Can I not help with this game? Knitting things is such a pain, such a pain," and kicks away the head of a bothersome piece of trash. These dolls wander aimlessly through the streets because there is no sign of the funeral starting. 10.ミエ（＝見栄, Vanity) 10番目に走ってきたのはミエ。デキソコナイの頭を大げさに避けて一言「べとべとした血でケープを汚されてはたまらないワ」この人形どもは、魔女の自傷を馬鹿にする。 The tenth to come running is Vanity. After exaggeratedly avoiding the head of the trash, she says, "I cannot bear to have my shawl stained with sticky blood." These dolls mock the Witch's self-harm. 11.オクビョウ（＝臆病, Cowardice) 11番目、震えているのはおくびょう。「お空へ連れて行かれては、うさぎの首も刎ねられない」この人形どもは、魔法というちからで完全な回帰を実現する。 The eleventh, trembling, is Cowardice. "If I'm taken to the sky, I won't even be able to cut off a rabbit's head." These dolls can achieve complete recovery using the power known as magic. 12.マヌケ（＝間抜け, Foolishness) ふらふら歩いてきたのは12人目、マヌケ。さっき鳥から聞いた事をわざわざみんなに話して回る。「女神様のお話を聞いたの。美しく輝く女神様い。女神様はきっと私たちのことも愛してくださるワ」この人形どもは、手が届くものだけを欲しがる。 The twelfth to come wobbling along is Foolishness. She goes around specifically telling everyone what she just heard from the birds. "I heard about the Goddess. A beautiful and holy Goddess. If it's the Goddess, she will surely love us." These dolls only want what they can obtain. 13.ヒガミ（＝僻み, Perversity) 13番目の足音はヒガミ。「いつかお話で聞いたみたく素敵な葬列にしましょう。綺麗な娘もかわいい動物も一緒にたくさん埋めましょう。」この人形どもは、魔法少女にも劣らぬちからを持つ。 The thirteenth footstep is Perversity. "Someday, let's hold a wonderful funeral like the ones in stories. Beautiful girls, cute animals—let's bury many things together." These dolls possess power no less than that of a magical girl. 14.ガンコ（＝頑固, Stubbornness) そして14番目、ガンコ。地を指し空を拒絶する「此岸の淵こそ我らが舞台」さあ泣き屋達は集まりました。ずいぶん遅くなったけれども大丈夫。後は葬列を待つばかり。 Then comes the fourteenth, Stubbornness, who rejects the sky and points to the earth, "The abyss of this shore is our stage." Now, the mourners have gathered. It is late, but that does not matter. All that remains is to wait for the funeral procession to arrive. 15.アイ（＝愛, Love) 15番目、最後に来るのが、アイ。その悪魔をまだ誰も見ぬ。もう夜は終わらせない。我等は泣き屋、此岸の劇団。 The fifteenth, the last to arrive, is Love. No one has seen this devil yet. The night is not over. The night will never end. We are the mourners, the troupe of this shore.

Akemi Homura's Evolution Process
	Status	Introduction
Braids		Originally attended a certain mission school in Tokyo. Due to poor health, she was hospitalized for a long time, and thus had little contact with others, leading to her being very timid and shy in social interactions. She had poor eyesight, and her academic performance in school was not very good (especially in physical education), leaving her with very little self-confidence.The true "helpless girl" of this work The root of all evil in this work; because she made a wish to meet Madoka again, the story of the "Endless Beginning" commenced. Initially, she was the magical girl with the weakest combat ability, having only time-stop magic as support. She even used a golf club, which lacks destructive power, as a weapon. When using self-made explosives, she was complained about by the melee specialist, Sayaka Miki, because the blast radius was too close. She was also very clumsy when she first started using firearms.
long straight black hair		After experiencing countless loops, she gave up on saving the other magical girls and resolved to save Madoka alone. Using magiclaser surgeryshe cured her eyes and graduated from being a "glasses character." From then on, she became taciturn and her aura became incredibly cold and aloof. In the eyes of her classmates during the loops, she was an "ice queen" goddess with excellent grades and an elegant posture. At times, her cold words would make others feel uncomfortable and displeased. She has become very familiar with the use of magic and weaponry, and is even able to, with the veteran senior Mami Tomoe,confronts them（clash）. The monotonous loops of constantly witnessing her friend's death continuously tempered her spirit and will, and she silently endured the resulting pain alone. At this time, Homura Akemi believed that Kyoko Sakura possessed the personality most suited to being a magical girl.
Red Ribbon	(Pid=20539689)	After Madoka became the Law of Cycles, she wore the ribbon Madoka left behind on her head instead of her hairpins, becoming the only person in the world who still remembers Madoka. Her shield, which originally controlled time, transformed into a longbow similar to Madoka's Rose Bow because she no longer needed to loop to save Madoka. She possesses the ability to subjugate numerous Wraiths single-handedly.Despite expectations for her to be as active as a "Musou" character in Rebellion, this was cut for the sake of the script.At the end of the main anime series, she unfurled black wings~~as if foreshadowing her corruption~~。 For some reason, she took the initiative to talk too much and revealed almost everything about Witches to the Kyubey of the new world, digging a significant hole for herself. In the movie Rebellion, she lay on a bed-like stone monument waiting to be received by the Law of Cycles; she was used as an experimental subject by the Kyubeys and began her slumber, constructing a barrier within her Soul Gem with her consciousness and continuously luring others in, only to end up with self-inflicted amnesia. In the movie Rebellion, as Homura Akemi's Soul Gem reached its limit, the shape of the longbow in her hand also underwent a subtle change: the top curled and bloomed with black roses, and two black leaves sprouted from the bottom. Compared to the obvious curve when used previously, the entire bow body now appears very straight.In the spin-off manga "Puella Magi Madoka Magica: Wraith Arc", there is an explanation for the reason the bow's shape changed, and the ending also mentions the cause of Kyubey's meddling; reading it allows for a better understanding of Homura-chan's demonic nature.
Demon of Love	(Pid=46489673)	Because she affirmed her twistedobsession with Madoka（Ai）, thereby becoming an existence that transcends the limitations of a magical girl. She crushed her own Soul Gem, creating a new Dark Orb through her karmic ties with Madoka Kaname. Her eyes are tinged with a blood-red rose purple, and black wings unfold from her back. Compared to her previously conservative magical girl outfit, she now wears a highly revealing costume. Her bewitching appearance and her actions—tearing apart the Law of Cycles while remaining loyal only to her desires—are met with hostility by Sayaka Miki, who calls her a devil. She easily manipulates the memories of the Incubators and everyone else. When the Devil descended to earth, she wore lizard-shaped earrings, with the pendant below sealing away the power of the Law of Cycles. The familiars from her Witch form still exist and obey Devil Homura herself. ~~She places tea tables on the road.~~ Her tone becomes frivolous and seductive; she dances alone in the park where she talks to Madoka and brutally kills Kyubey.~~Mental anxiety~~Her sadistic and yandere nature is fully exposed. She returned the ribbon to Madoka, who is now human.

Homura Akemi's evolution branches
	Status	Introduction
Twin-tail Homura		Manga version Homura Akemi's Soul Gem In the manga version, after Madoka Kaname becomes a god, Homura Akemi does not tie the ribbon where her headband used to be, but instead ties her hair into twin-tails, and her casual clothes are relatively youthful; to commemorate and maintain consistency with Madoka Kaname's style, this was highly praised for a time after the manga serialization ended.As a result, it didn't catch the twin-tail craze and was eventually overshadowed by the momentum of the orthodox long straight black hair.
Akuma Homura		Akuma Homura suddenly appeared in the world of "Magical☆Express." Faithful to her desires, she likes devilishly high-calorie foods. Although she calls herself a devil, she has a side with deep feelings and is unexpectedly good at taking care of others. Additionally, she only resembles the Akuma Homura appearing in the movie in appearance and is a complete stranger. ^[10]But in reality, she is very good at using memes from "The Rebellion Story"

Episodes	Opponent	Result	Final blow	teammates	Notes	Timeline
TV Version
Episode 1	Walpurgisnacht	failure	-	-	Madoka's dream	Fourth timeline
QB	Interrupted	Rapid fire	-	Kyubey uses telepathy to have Madoka save her	Final timeline
Episode 3	Mami Tomoe	failure	-	-	Warns Mami, but is ambushed by her firstBondage play*1
Episode 3	The "Dessert" Witch	Victory	Weaponry	-	Allegedly blew up Mami's Soul Gem, which had been swallowed by Charlotte. (False)
Episode 8	QB	Unsuccessful	Time stop, rapid fire	-	Instant kill, preventing Madoka Kaname from making a contract. However, QB has spare bodies.
Episode 10	The "Class Representative" Witch	Victory	Time stop, homemade bombs	Madoka Kaname, Mami Tomoe	Cooperated with Madoka Kaname and Mami Tomoe.	Second playthrough
Walpurgisnacht	Unknown	-	Madoka Kaname	Witnessed Madoka Kaname's witchification.
The "Mermaid" Witch	Victory	Time stop, homemade bomb array	Madoka Kaname, Mami Tomoe, Kyoko Sakura	Subsequently, Mami broke down and shot Kyoko, and was stopped by Madoka Kaname shattering her Soul Gem.Tied up by Mami *2	Third Timeline
Walpurgisnacht	failure	-	Madoka Kaname	Defeated along with Madoka Kaname.
The Witch of the "Birdcage"	Victory	Military-grade weaponry	-		Fourth timeline
Walpurgisnacht	failure	-	-	Namely, Madoka's dream in the first episode.
Episode 11	Walpurgisnacht	failure	-	-	Exhausted all military weaponry + time stop.	Final timeline
Episode 12	Wraiths	Victory	-	Mami Tomoe, Sayaka Miki, Kyoko Sakura	Sayaka Miki died in battle.	New World
Wraiths	Presumed victory	Magic Bow	-
Massive swarm of Wraiths	Mutual destruction (pseudo)	Eroding Black Wings	-
Wraith Arc
Wraith Arc	Large Wraith	failure	Circle of Arrows (Modified)^[12]		Bow of memory manipulation stolen by a Wraith
Wraith Kyoko	Victory	-	Mami Tomoe	Wraith Madoka informed of Wraith Kyoko's location via telepathy, successfully repelling her
The "Spinning Wheel" Witch	Victory	Circle of Arrows (Modified)	Kyoko Sakura, Mami Tomoe, Wraith Madoka	Sustained a fatal wound, but through the time-manipulation shield^[13]Restarted the timeline^[14]
Movie [New Edit] Rebellion
Movie [New Edit]	Hitomi's Nightmare	Victory	-	Madoka Kaname, Mami Tomoe, Sayaka Miki, Kyoko Sakura, Bebe	"Holy Quintet", assisting with time stop	Rebellion
Mami Tomoe	failure	-	-	Bound by Mami *3
Miki Sayaka	Interrupted	-	-	Sayaka Miki escapes by swinging her cape
Madoka Kaname Mami Tomoe Kyoko Sakura Miki Sayaka Nagisa Momoe	failure (Feigned surrender)	-	Familiar "Die Wächter" Familiar "Die Leibwächter" Familiar "Die Karies" Familiar "Die Vögel" Familiar "Clara Dolls"	Witchification, fighting alongside familiars
Magia Record TV series
Season 2, Episode 1	The "Class Representative" Witch	Interrupted	-	Madoka Kaname, Sayaka Miki	Saw it during time stopUi Tamaki	Magia Record
The "Class Representative" Witch	Victory	-	Madoka Kaname, Sayaka Miki

A typical "~~lady~~”long straight black hair
At this time, Homura Akemi looks very much like a certaingirl who loves recording videos of her female friends

Fan work "To the Stars"^[16]Some settings from
This fan work was created before the release of "The Rebellion Story" and is set in the new world at the end of TV Episode 12. In the new world, Homura Akemi's innate ability is to deploy a barrier that can slow down others' magic consumption; additionally, she has a pair of wings for flight. Abilities during Madokami's descent: reflecting all attacks, detecting invisible weapons, and unfolding black wings. After reaching adulthood, Homura Akemi gradually demonstrated her potential as an entrepreneur and politician.(Establishing and managing the Magical Girl Guild, establishing multiple enterprises such as Mitakihara Express)。

Skill Name	Effect
Blunt Weapon	"Melee": Striking enemies wildly with a golf club
Time stop	[Support] For 5 turns, the enemy cannot act
Magic Arrow	[Shooting] Fires magic arrows in a line forward for a piercing attack
Space Jump	[Support] Instantly teleports to the starting point of the current floor
Speed Up Spell	[Support] Grants the "Double Speed" buff to all allies
Slow Spell	[Support] Inflicts the "Slow" status on enemies in front
Large-Caliber Handgun	[Shooting] Attacks a line forward with a gun
Blunt Weapon + Blunt Weapon	"Melee": Striking enemies wildly with a golf club
Squad Support Weapon	[Shooting] Fires wildly at a line forward
Anti-Tank Weapon	"Shoot" Attacks with countless rocket launchers
Time Bomb	"Shoot" Throws a bomb forward
Mine Placement	"Support" Sets a mine in front
Flash Grenade	"Support" Throws a flashbang forward, putting enemies into an "Unable to Attack" state
Hazardous Material Class 4	"Melee" Summons a tanker truck in front, causing a massive explosion
Anti-Ship Missile	"Shoot" Fires an anti-ship missile forward, dealing massive damage to enemies
Black Wings of Erosion	"Shoot" Erodes the surroundings with black wings, dealing massive damage to enemies

Character Songs	Vocals	Usage
Mebius Ash	Homura Akemi	Game "ぱちんこ魔法少女まどか☆マギカ》 First included in the album "Puella Magi Madoka Magica Ultimate Best》
ユメおと (Yumeoto)	Madoka Kaname、Homura Akemi	Game "ぱちんこ魔法少女まどか☆マギカ》 First included in the album "Puella Magi Madoka Magica Ultimate Best》
naturally -Rebirth-	Holy Quintet （Madoka Kaname、Homura Akemi、Miki Sayaka、Mami Tomoe、Kyoko Sakura）	Game "スマスロ劇場版魔法少女まどか☆マギカ[前編]始まりの物語/[後編]永遠の物語f-フォルテ-》
arch	Madoka Kaname、Homura Akemi	Game "P 魔法少女まどか☆マギカ3》

Solution 1 (Akemi Homura's method on the blackboard)
From the problem, let $x = 14 q + r (0 ⩽ r < 14), a = 14 s + 6, b = 14 t + 1$ 。 then $\begin{aligned} x^{2} - 2 a x + b & = (14 q + r)^{2} - 2 (14 s + 6) (14 q + r) + (14 t + 1) \\ = 196 q^{2} + 28 q r + r^{2} - 392 s q - 28 s r - 168 q - 12 r + 14 t + 1 \\ = 14 (14 q^{2} + 2 q r - 28 s q - 2 s r - 12 q + t) + r^{2} - 12 r + 1 \\ = 14 (14 q^{2} + 2 q r - 28 s q - 2 s r - 12 q + t - r) + (r + 1)^{2} \\ = 0 \end{aligned}$ It is obvious that $14 ∣ (r + 1)^{2}$ (14 can be $(r + 1)^{2}$ divided exactly), thus $14 ∣ r + 1$ (14 can be $r + 1$ divided exactly). Also because $0 ⩽ r < 14$ , so $r = 13$ 。 Therefore, the remainder of the integer solution of this equation divided by 14 is 13.

Solution 2
Directly take $x^{2} - 2 a x + b = 0$ on both sides $m o d 14$ , then $\begin{aligned} x^{2} - 2 a x + b & \equiv r^{2} - 12 r + 1 \\ \equiv r^{2} + 2 r + 1 \\ \equiv (r + 1)^{2} \\ \equiv 0 (\mod 14) \end{aligned}$ Therefore, $r = 13$ 。 Therefore, the remainder of the integer solution of this equation divided by 14 is 13.

Solution 1
Consider the binomial expansion: $(1 + n)^{p} = n^{p} + C_{p}^{1} n^{p - 1} + C_{p}^{2} n^{p - 2} + \dots + C_{p}^{p - 1} n + 1 .$ Prove below $(1 + n)^{p} - n^{p} - 1 = C_{p}^{1} n^{p - 1} + C_{p}^{2} n^{p - 2} + \dots + C_{p}^{p - 1} n$ is divisible by $p$ . For any natural number $k (1 ⩽ k ⩽ p - 1)$ , we have $(p, k) = 1$ (i.e., $p$ and $k$ are coprime). Thus, we have $(p, k!) = 1$ ， $(p, (p - k)!) = 1$ 。 From the above, we can obtain $(p, k! (p - k)!) = 1$ 。 Also because $p ∣ p!$ , so $p ∣ \frac{p!}{k! (p - k)!} = C_{p}^{k}$ 。 Therefore, $(1 + n)^{p} - n^{p} - 1$ is divisible by $p$ .

Solution 2
By Fermat's Little Theorem, $\begin{aligned} (1 + n)^{p} - n^{p} - 1 & \equiv (1 + n) - n - 1 \\ \equiv 0 (\mod p) . \end{aligned}$ The original proposition is proven.

Solution
First, move the right side to the left side to simplify. $\begin{aligned} (a^{3} + a^{2} - 1) - (a - 1) b \\ = & (a^{3} - 1) + (a^{2} - 1) + 1 - (a - 1) b \\ = & (a - 1) (a^{2} + a + 1) + (a - 1) (a + 1) + 1 - (a - 1) b \\ = & (a - 1) [(a^{2} + a + 1) + (a + 1) - b] + 1 \\ = & (a - 1) (a^{2} + 2 a - b + 2) + 1 \\ = & 0 \end{aligned}$ Namely $(a - 1) (a^{2} + 2 a - b + 2) = - 1$ 。 When ${\begin{cases} a - 1 = - 1 \\ a^{2} + 2 a - b + 2 = 1 \end{cases}$ When , we get the solution $(a, b) = (0, 1)$ 。 When ${\begin{cases} a - 1 = 1 \\ a^{2} + 2 a - b + 2 = - 1 \end{cases}$ When , we get the solution $(a, b) = (2, 11)$ 。 In summary, $(a, b) = (0, 1), (2, 11)$ 。

Solution
Let $a = \sqrt{2 x + 1}, b = \sqrt{2 x - 1}$ . Then $\begin{aligned} f (x) & = \frac{a^{2} + b^{2} + a b}{a + b} \\ = \frac{(a^{2} + b^{2} + a b) (a - b)}{(a + b) (a - b)} \\ = \frac{a^{3} - b^{3}}{a^{2} - b^{2}} (a^{2} - b^{2} = 2) \\ = \frac{1}{2} (a^{3} - b^{3}) \\ = \frac{1}{2} [(2 x + 1)^{\frac{3}{2}} - (2 x - 1)^{\frac{3}{2}}] \end{aligned}$ By telescoping, $\begin{aligned} f (1) + f (2) + \dots + f (60) \\ = & \frac{1}{2} [(3^{\frac{3}{2}} - 1^{\frac{3}{2}}) + (5^{\frac{3}{2}} - 3^{\frac{3}{2}}) + \dots + (12 1^{\frac{3}{2}} - 11 9^{\frac{3}{2}})] \\ = & \frac{1}{2} [12 1^{\frac{3}{2}} - 1^{\frac{3}{2}}] \\ = & 665 \end{aligned}$ Therefore, $f (1) + f (2) + \dots + f (60) = 665$ 。

Problem
Played by $p_{1} = 1$ ， $p_{2} = 1$ ， $p_{n + 2} = p_{n} + p_{n + 1}$ The sequence defined by ${p_{n}}$ is called the Fibonacci sequence, and its general term is: $p_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n} - {(\frac{1 - \sqrt{5}}{2})}^{n}]$ given by the above conditions. Please use this fact to answer the following questions: Define a sequence of natural numbers where each digit is 0 or 1 ${X_{n}}$ , determined by the following rules: (A) $X_{1} = 1$ (B) For $X_{n}$ a digit at a certain position in $α$ , if $α = 0$ , then $α$ replace $1$ ; if $α = 1$ , then $α$ replace $10$ 。 Regarding $X_{n}$ in each row of $α$ , after performing the above replacement digit by digit, the resulting natural number is denoted as $X_{n + 1}$ For example, $X_{1} = 1, X_{2} = 10, X_{3} = 101, X_{4} = 10110, X_{5} = 10110101, \dots$ etc. (1) $X_{n}$ the number of rows of is $a_{n}$ , find $a_{n}$ (2) $X_{n}$ the number of occurrences of the digit sequence “01” therein is $b_{n}$ , find $b_{n}$ (For example, $b_{1} = 0, b_{2} = 0, b_{3} = 1, b_{4} = 1, b_{5} = 3, \dots$ etc.)

Solution (The solution to the 1st sub-question written on the blackboard by the female student)
Let $X_{n}$ The number of 0s and 1s in [it] are respectively $x_{n}, y_{n}$ Then according to rule (B), we have: ${\begin{matrix} x_{n + 1} = y_{n} \dots ① \\ y_{n + 1} = x_{n} + y_{n} \dots ② \end{matrix}$ holds, and from ① and ②, it is known that: ${\begin{matrix} x_{n + 2} = x_{n + 1} + x_{n} \dots \\ y_{n + 2} = y_{n + 1} + y_{n} \dots \end{matrix}$ Adding them term by term: $x_{n + 2} + y_{n + 2} = (x_{n + 1} + y_{n + 1}) + (x_{n} + y_{n})$ Furthermore: $a_{n + 2} = a_{n + 1} + a_{n}$ Also because $a_{2} = 2$ , so $a_{n} = p_{n + 1}$ ， $(n \geq 1)$ Era $a_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n + 1} - {(\frac{1 - \sqrt{5}}{2})}^{n + 1}]$

Solution (Supplementary solution 1 for the unfinished 2nd sub-question on the blackboard by the female student)
Because: $y_{1} = 1, y_{2} = 1, y_{3} = 2$ , and $y_{n + 2} = y_{n + 1} + y_{n}$ Therefore: $y_{n} = p_{n}$ ， $(n \geq 1)$ Also, because of equation ①: $x_{n + 1} = y_{n}$ Therefore $x_{n} = p_{n - 1}, (n \geq 2)$ ， $x_{1} = 0$ So: $x_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n - 1} - {(\frac{1 - \sqrt{5}}{2})}^{n - 1}], (n \geq 2)$ Also because $n = 1$ When , we have: $\frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{0} - {(\frac{1 - \sqrt{5}}{2})}^{0}] = \frac{1}{\sqrt{5}} (1 - 1) = 0$ $x_{n}$ The general formula for $n = 1$ still holds, therefore: $x_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n - 1} - {(\frac{1 - \sqrt{5}}{2})}^{n - 1}], (n \geq 1)$ From rule (B), it can be seen that: (I'm not sure how this was derived either) $X_{n}$ Except for the final $0$ , all $0$ are followed by $1$ if and only if $n$ is even, $X_{n}$ the end is $0$ Therefore, $b_{n} = x_{n} - 1$ ， $n$ when is even $b_{n} = x_{n}$ ， $n$ when is odd Furthermore $b_{n} = x_{n} - \frac{1 + (- 1)^{n}}{2}$ ， $b_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n - 1} - {(\frac{1 - \sqrt{5}}{2})}^{n - 1}] - \frac{1 + (- 1)^{n}}{2}$

Solution (Supplementary Solution 2 for the unfinished second sub-question by the female student on the blackboard)
Because: $y_{1} = 1, y_{2} = 1, y_{3} = 2$ , and $y_{n + 2} = y_{n + 1} + y_{n}$ Therefore: $y_{n} = p_{n}$ ， $(n \geq 1)$ ; From rule (B), it can be seen that, $(n \geq 2)$ When: Because $X_{n}$ all 0s in [1532] come from $X_{n - 1}$ the 1s in [1533] and all 0s are separated by 1s, if and only if $n$ is even, $X_{n}$ the end is $0$ , therefore $b_{n} = y_{n - 1} - 1$ ， $n$ when is even $b_{n} = y_{n - 1}$ ， $n$ when is odd Furthermore $b_{n} = p_{n - 1} - \frac{1 + (- 1)^{n}}{2}, (n \geq 2)$ $b_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n - 1} - {(\frac{1 - \sqrt{5}}{2})}^{n - 1}] - \frac{1 + (- 1)^{n}}{2}, (n \geq 2)$ Also because $n = 1$ when, there is $\frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{1 - 1} - {(\frac{1 - \sqrt{5}}{2})}^{1 - 1}] - \frac{1 + (- 1)^{1}}{2} = \frac{1}{\sqrt{5}} (1 - 1) - 0 = 0$ Therefore the above $b_{n}$ The general formula for $n = 1$ still holds: $b_{n} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{n - 1} - {(\frac{1 - \sqrt{5}}{2})}^{n - 1}] - \frac{1 + (- 1)^{n}}{2}, (n \geq 1)$

Homura Akemi

Long Straight Black Hair Homura pixiv ID: 97690194 Author:SuperPig Braided Homura pixiv ID: 95327486 Author:EO Ribbon Homura pixiv ID: 32223708 Author:Kiriko Devil Homura
Real Name	暁美（あけみ）ほむら (Akemi Homura)
Aliases	Homura-chan, Homu-chan, Braided Homura → Long→Homucifer, Homura~~Homora~~(Homura transliteration), HomuHomu, Transfer Student (Miki Sayakatitle)
Hair Color	Black hair
Eye color	purple eyes
Height	155~158cm
Age	~~Physically~~14 years old (8th grade)
Voice Actor	Chiwa Saito(Japanese VA) Cristina Vee(English VA)
Moe points	Looper：Coodere、Kuudere、Honor Student、Beautiful, strong, and miserable、long straight black hair、Warrior、Flipping hair、Loyal Dog、Ice queen、Dàngyàng、Hime cut、Split long hair Early stage：Red-framed glasses、Braids、Meek、Crybaby、Poor Student、Sickly、Plain Girl Common：close friend、Flat Chest、See-through bangs、Silent girl、Reclusive、Yuri、Wife-protecting maniac、Transfer student、Military Girl、Pant-boots、Sock boots、Shield、Binding、Cat Lover(Drama CD),Black pantyhose、Headband、Ribbon、Darkening、Demon(The Rebellion Story),Wings、Yandere(Rebellion First Press Dub Version),Chijo(Fan-made setting)
Region of Origin	Tokyo → Mitakihara City
School	Mitakihara Middle School
Affiliations	Magical girl~~、RPG Cult、SOX~~
Intrinsic Skills	Time manipulation, Memory manipulation (Wraith Arc)
Weapon	Shield (+ modern firearms and bombs), Bow and arrow (Wraith Arc)
Status	Became a demon (extracted a small part of the Law of Cycles' memories and altered everyone's memories)
Character Design	Ume Aoki
Related People	Best friend~~Wife~~：Madoka Kaname ：Mami Tomoe Other teammates:Miki Sayaka、Kyoko Sakura Teacher:Kazuko Saotome Enemies:QB Multiverse Counterparts:Tomoyo Daidouji^[1]

Personality

Character Setting

Character Image

History

In the original anime

In the spin-off manga

Bond with Madoka

Abilities

Application of Abilities

Abilities in the New World

Witchification

The Witch of the Mortal World

The Nutcracker Witch

Familiars in the Movie

Homura Evolution Chapter

Research: Homura Akemi, Faust, and the Nibelungenlied

Combat record in the work

Moe Tournament Results

Homura Akemi in Fanon

Common Pairings

PSP Version Skills

Character Songs

With the Japan Ground Self-Defense Force 4th Anti-Tank Helicopter Squadron

Appearing in the main workmathematicsTitle

Question 1

Question 2

Question 3

Question 4

Question 5

Notes

External Links

“	Of course not. —youThe same goes for you.	”
——Replied to Kyoko without any delay,Homurasaid calmly.

“	Anime fans in the military believe that Homura Akemi inspires them. She is an expert in various firearms, willing to do whatever it takes to complete her mission, and her will is unstoppable, like cold, hard steel. She prioritized the mission of saving Madoka above all else, with everything else coming second. Given her single-minded dedication, brilliant talent, and terrifying firepower, it is not surprising that the JGSDF 4th Anti-Tank Helicopter Squadron painted her image on one of their "Cobra" attack helicopters.	”
— McAfee, "Apocalypse"^[17]^(p82)