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: 172-219
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000172 | 7. The full Cartan normalizer is finite
000173 | 
000174 | For the original integral E8 lattice the existing Smith form is
000175 | 
000176 |   SNF(I-J)=(1,1,1,1,2,2,2,2).
000177 | 
000178 | Thus T^J is F2^4, of order16. Section4 yields the exact finite sequence
000179 | 
000180 |   1 --> T^J --> N_KC(c) --> G31 -->1,
000181 |   |N_KC(c)|=16*46080=737280.
000182 | 
000183 | No splitting or centrality of this extension is inferred. In particular
000184 | this is not an alteration or splitting of the earlier sign-lift mu4
000185 | extension. The kernel here is a pointwise torus centralizer of c.
000186 | 
000187 | If a given K connection preserves a whole such Cartan subspace bundle
000188 | parallelly, it reduces to this finite normalizer. Its Lie algebra is
000189 | zero; its curvature must vanish. The natural iota_m map is injective on
000190 | the Lie algebra of U5, so a determinant connection with nonzero degree
000191 | cannot meet this parallelity requirement. The clock alone can still
000192 | commute with it, as the previous connection theorem already proved.
000193 | 
000194 | 8. On the old primitive normal S2 even a smooth global choice is obstructed
000195 | 
000196 | The September22 theorem gives, for EVERY natural m,
000197 | 
000198 |   [P_K]=2 in pi1(K)=Z4
000199 | 
000200 | on the supplied normal S2 with c1(E3)=0,c1(E2)=1. Complexification retains
000201 | this class because K is a maximal compact subgroup of K_C.
000202 | 
000203 | All Cartan subspaces of this theta representation are K_C conjugate.
000204 | An everywhere smooth choice of them is a section of
000205 | 
000206 |   P_KC x_KC (K_C/N_KC(c)).
000207 | 
000208 | By the reduction theorem for principal bundles, such a section is exactly
000209 | an N_KC(c) reduction. Every principal bundle of a finite group over the
000210 | simply connected S2 is trivial. Its extension to K_C is therefore also
000211 | trivial, contradicting class2. This proves that the stated global choice
000212 | does not exist. No connection hypothesis is needed for this obstruction.
000213 | 
000214 | The statement concerns a whole Cartan-subspace orbit, not a single field
000215 | vector or an arbitrary rank4 bundle. A punctured surface, a different
000216 | bundle class or fibers leaving this orbit at defects/degeneracies change
000217 | the premises. The original flavor H1 is on a different punctured base;
000218 | its identification with this normal S2 has NOT been supplied here.
000219 |