jake
@yetanotheruseless.com
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he/they /in/jakemannix 📷: runmeatrobotrun fka @pbrane professionally: Tech Fellow, AI/Relevance, $BigCo here: bad math/physics jokes, AI++, puns, OSS ML news, ultras/MTB/outdoorsy stuff, DL papers, shitposting, the fall of democracy Abolish ICE FALSC
![(the start of) A proof of the classification of all groups of order 8, in Arcane Lean:
spell + Eightfold'+ Roster'+ <$ « Essence®> [Veyr $1 [Counted $1<ward + Tallyğcensus $=/\»*
# Indivisible*+ Cubic*+ Veyrs*+ ember st $ → cast channel '† Indivisible + Cubic+ Veyrst+ gexhaustive*+ census st $
clarify solely [Tally'+ Two*+ Ways*+ to'† Take'+ the'+Census + • proclaim KEt + Tallies) ^ * = /\ cast numerology] through ward
spell 't Five*† Eightfold'† Forms'+ «
Tallyğcensus «Unchanted «* Indivisible' Cubic*+ Veyrs*+ cinder st>= /\ A Tallyğcensus «Unchanted «Clockwork KEt ^ Et>» * Unchanted «Clockwork st»» = /\ A Tallyğcensus «Unchanted «Clockwork Et> * Unchanted «Clockwork Et» * Unchanted «Clockwork =t»»
=八A
Tallyğcensus «* Mirror*+ Veyr:+ R9» = /\ л
Tallyğcensus «* Hamilton's + Veyrt st> = A\» cast hone 1?0 • ?0 ?0?0?0
& numerology [Tally'† Two'† Ways:†to*† Take'†the'† Census*+•
# Indivisible'+ Cubic + Veyrs:+cinder]
¿ numerology [Tally'+ Two'+ Ways'+ to'+ Take:+ the + Census +• Counted*'+Census t of +Pairs*+ 1
• numerology [Tally:+ Two:+Ways:+to+ Take:+ the + Census t. Counted + Census t oft+Pairs +]
* clarify solely Eproclaim (Et « Tallies> ^ * = /\ cast numerology] through
* Indivisible'+ Cubic*+Veyrs:+ *tinder
• clarify solely [proclaim KEt « Tallies» ^ :
* = / cast numerology] through
* Indivisible*+ Cubic:+ Veyrs:+ wisp
spell Five'+Distinct+Forms*+ «
chanted](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:bjaceaxgoto5p7gor2rcov3g/bafkreiczmwu2pg32ia2ldfk33z6gqmnfay3xkkzdxpjnbovkfxrdd7wq7e)
![theorem mathoverflow_1973 :
∃ atlas : ChartedSpace (EuclideanSpace ℂ (Fin 3)) (unitSphere 6),
letI := atlas
IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) 1 (unitSphere 6) := by
exact SixSphereComplexAtlas.exists_complex_atlas
#print axioms mathoverflow_1973
-- 'Mathoverflow1973.mathoverflow_1973' depends on axioms: [propext, Classical.choice, Quot.sound]
end Mathoverflow1973
end](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:bjaceaxgoto5p7gor2rcov3g/bafkreiaxof2o26pelcilzk3zi4z7rjttbptes24lz3ir7aoozjx74j7d5u)
![The Main Theorem is incompatible with a published result. CDP proved
in [CDP98] that a compact complex threefold X with bz(X) = 0 and a (X) > 0 has c3(X) = e(X) = 0, and deduced that a compact complex threefold homeomorphic to S° has algebraic dimension O. That proof rests on [CDP98, L1.5], which is incorrect as stated; the corrigendum says so and re-proves the results in two cases. [CDP20, T2.1] treats the case in which there exists a non-constant meromorphic map g: X -- P' which is not holomorphic (this case is written out in detail in [LRS,
Thm. 3.1, Cor. 3.2]), while [CDP20, T2.2] treats the case H' (X, Z) = H2(X,Z) = 0,
a (X) = 1 with holomorphic algebraic reduction f: X → C, concluding (a) H' (X, Tx 8 M) = 0 for i ‡ 1 and generic M є Pic (X), (b) x (X, Tx 8 M) ≤ 0 and (c) c3 (X) ≤ 0; together these give [CDP20, Cor. 2.3]: if X
is homeomorphic to S' then a (X) = 0. The manifold of the Main Theorem has H' (X, Z) = H-(X,Z) = 0,
a(X) = 1, holomorphic algebraic reduction f: X → P1, and c3(X) = 2 > 0, so that conclusion (c) fails; so does conclusion (b), since c (X) = c2(X) = 0 and Pic(X) = Pic® (X) give, by Riemann-Roch,
x (X, Tx @ M) = 5c3 (X) = 1 > 0 for every M e Pic(X). It therefore contradicts [CDP20, Thm. 2.2] and
[CDP20, Cor. 2.3], as well as [CDP20, L4.2] (the special case 8 = 0, b1 = 0 of [CDP98, L
3.21), [CDP20, C5.2] and [CDP20, Prop. 5.5] (= [CDP98, Thm. 3.1, Rem. 3.8(1)]), whose conclusions fail for X (their printed proofs presuppose trivial monodromy); this failure is repairable for the purpose of the proof of Cor. 2.3 at X (§10), unlike the reduction at the non-normal fibre, which is decisive. It does not contradict [CDP20, Thm. 2.1] or [LRS], whose hypothesis is the existence of some non-constant non-holomorphic meromorphic map X -- P', and no such map exists on our X: by item (5) of the Main Theorem M (X) = f*C(t), so every non-constant meromorphic map X --> P' is of the form h.f with h.a non-constant rational function on B, hence is holomorphic.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:bjaceaxgoto5p7gor2rcov3g/bafkreie6ybxglvg4g4nxlsiywhbm2rmkac5brr4ycolpt5is7jfjbbxkoe)
