#this is enough abuse of notations and concepts to last me several years.

[reductionisms]

tama math vol. 1

prop: takasugi is locally homeomorphic* to tama. (⇔takasugi can be "related" with tama, but not all of tama can be "related" with takasugi) *roughly

I outline the proof.

case 1: fuyo arc

(i) clearly tama and takasugi are both missing their left eyes. there are two further relations. (ii) no. 502/professor hayashi "takes control again" of tama/fuyo and uses her to attack shinpachi, just as utsuro "takes control" of takasugi and uses him to attack shouyou. (iii) tama restrains no.502 so that gintoki can kill him, just as takasugi restrains utsuro so that gintoki can kill him. in (ii), the person attacked is someone important to tama/takasugi. this satisfies the "equivalence". (ii) implies that no. 502 and utsuro are also "equivalent" (homeomorphic). further, that no. 502≅utsuro is necessary to prove (iii). without taking the time to discuss this rigorously (it's probably obvious), i introduce it as a lemma. case 1 lemma 1: no. 502≅utsuro (locally)

additional "equivalences": (i) no. 502 and utsuro are both other selves of a "parent figure" (i use this for lack of better terminology) who has been swallowed by them. (ii) fuyo and the final both end in the terminal. (iii) no. 502 and utsuro are seemingly undefeatable because of their regenerative ability; this is solved by pushing them into the terminal's altana. we notice that no. 502 and utsuro don't align perfectly. here, they align just enough to draw a local "equivalence" (an "equivalence" that holds only for certain areas/times/places). some questions remain. Q1.1.1 if tama≅takasugi, then why is takasugi not present with yorozuya and shouyou? A1.1.1 because he's become gintoki/gintoki's sword. he's there by metaphysics and ontology

Q1.1.2 if no. 502≅utsuro, why is utsuro not present with yorozuya and shouyou ⇔ why is utsuro defeated beforehand? A1.1.2 in no. 502's dying moments, he becomes professor hayashi again A.1.1.2 is fairly straightforward. A.1.1.1 can be shown in more detail.

case 1 lemma 2: tama's death in fuyo ≅ takasugi death (locally)

i had no desire (nor, frankly, ability) to grab equivalent moments from the final. suffice to say that anyone well-versed in gintama sword theo-ontology and/or gintama human ontology/philosophical anthropology should understand. takasugi becomes gintoki's sword; gintoki, the human, carries others inside his soul (his sword); takasugi is inside gintoki's soul. this is how he dies. this lemma leads us into the next case: tama's second death.

case 2: kintoki arc

there's little to say. (i) tama first sequence is request->sad face->"pathetic", while takasugi first sequence is sad face->request->"pathetic"; a permutation. (ii) tama second sequence is next-time-we-meet->gintoki's promise->i'll hold you to it and takasugi second sequence is next-time/promise (i'll come to hell for you)->i'll hold you to it; an "equivalence". (i) tama's request is different from takasugi's; she wants to see gintoki's face, who is crying, while takasugi asks for gintoki’s smile. but the elements are the same: calling gintoki “pathetic”, request to see gintoki's face, and gintoki's tears. here tama and takasugi permute. with extra mathematical book-keeping and extreme abuse of notation we can call these some sort of "equivalent" (don't ask me why. or, if you do, give me a few days). (ii) gintoki doesn't promise tama and takasugi the same thing, but again, the elements are the same (this time in the same order): meet again->some sort of promise->affirmation ("i'll hold you to it"/"bring it on"). another sort of "equivalent" with similar mathematical abuses. more abstractly, we have:

case 2 corollary 1: homeomorphism of the "good" death

for both tama and takasugi, there is a proper way to die. gintoki respects this in both cases. their deaths are therefore "equivalent".

case 3: bfy

(i) left eye. (ii) at this point in bfy, gintoki lives in a world where no one knows or remembers him-- that is, he's completely alone-- because he killed himself in the past to save the world; compare with post-cliff (shouyou=gintoki). (iii) tama is now the only person who remembers gintoki and his sacrifice. she remembers him because she "is" a camera, because she "records". ⇔ takasugi is the one who reveals gintoki's past, that is within his left eyelid, that he watched. (iv) from kintoki arc, we know that tama records with her eyes. ⇔ the eye, the eyelid.

case 4: "reincarnation"

when tama's head is recovered, and "resurrected/reincarnated" post-terminal death in fuyo, tama remembers next to nothing; otose says she's "like a baby". after tama's "death" in silver soul, gengai makes a child tama to record what's going on while tama sleeps. so we have that tama deaths generally imply tama births (in 2/3 cases); and then there's the baby in 704. who knows what that baby is, or should be. perhaps one use of the tama/takasugi local homeomorphism is a proof of the baby's plausible existence. i reserve judgement on this case.

case 5: tama

this is half corollary, half unproven proposition. but it deserves consideration. tama's name, in fuyo, is equivocated with egg (tamago), head (atama), testicles (kintama), ball (tama) and soul (tamashii-- the kanji's pronunciation can be shortened to just tama). gintama likes to equivocate several of these tamas whenever it can. this collapses into an eventual equivocation with the soul (which is by turns mundane, lewd, and disgusting); and tama is unique among robots because she has a soul, the seed, a human soul. she is, quite literally, a, the, soul. gintama also loves to say sword=soul. gintoki's sword is his soul, and by equivalence class, gintoki's soul is the soul of everyone. gintoki's sword becomes takasugi⇒ takasugi becomes (was already) gintoki's soul ⇒ tama is the soul ⇒ tama = takasugi. you can't really abuse math and narrative like this. i'll stop for now.

proposition: tama is locally homeomorphic to gintoki

notably, i didn't consider tama quest arc or other tama ≅ gintoki moments. that is for tama math volume 2.

(this romanization of “tama” is almost a permutation of “math”. :) )