TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/CARTAN-FAMILY-DICTIONARY.txt
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Inclusive original lines: 220-253
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000220 | 9. What the existing covers can and cannot repair
000221 | 
000222 | For an ACTUALLY supplied degree-q map f:Sigma->S2 from a closed oriented
000223 | surface, connected K bundles are classified by H^2(Sigma,pi1K). Hence
000224 | 
000225 |   [f^*P_K]=2q mod4.
000226 | 
000227 | Every even q makes this class trivial; q=2 is the smallest positive one.
000228 | S2 has no nontrivial connected unbranched cover. The corresponding repair
000229 | must be branched or a more general degree map, not an unbranched double
000230 | cover of the compact sphere.
000231 | 
000232 | IF the existing original C2 -> P1 is proved to be over THIS SAME normal
000233 | S2 and to pull back THIS bundle, its degree2 removes the topological
000234 | obstruction. The existing C6 and C12 do so too. Thus this argument does
000235 | not select degree12 and does not replace the already proved global
000236 | relative-monodromy criterion for C12. It also does not identify two bases
000237 | merely because both are topological spheres.
000238 | 
000239 | For a supplied branched cover, a given nonflat smooth connection remains
000240 | nonflat on the regular open part after pullback: F_f*A=f^*F_A. Smooth
000241 | nonzero curvature holds on an open set, so it cannot be hidden entirely
000242 | at the isolated branch points. Topological triviality therefore does
000243 | not produce a parallel Cartan subspace for that connection.
000244 | 
000245 | Finally the orbit fibration gives
000246 | 
000247 |   1 --> pi1(K)=Z4 --> pi1(K/N) --> N -->1.
000248 | 
000249 | A loop can carry class2 in its kernel while its N monodromy is1. Its
000250 | image in G31 is then also trivial. Pure little-Weyl monodromy can lose
000251 | this central topological information. The full bundle/clutch data must
000252 | travel with the existing finite clock and cover information.
000253 |