TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/CARTAN-FAMILY-DICTIONARY.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000043 | 3. The original Gaussian four is a Vinberg Cartan subspace
000044 | 
000045 | Work first in the original native field frame. Let g=e8(C), let h_C be
000046 | its original Heisenberg Cartan in weight one, and let theta=Jhat. On h_C,
000047 | 
000048 |   h_C=H+ direct-sum H-,  H+=ker(J-i), H-=conjugate(H+), dim H+=4.
000049 | 
000050 | Every element of H+ is semisimple because it belongs to h_C. Elements
000051 | commute because h_C is abelian. No real E8 root alpha restricts to zero
000052 | on H+: if it did, reality would also annihilate H-, hence all h_C.
000053 | The finite union of its root hyperplanes therefore does not fill H+.
000054 | A point x outside it is regular semisimple, with Z_g(x)=h_C.
000055 | 
000056 | If an abelian semisimple subspace of g_i contains H+, it lies in
000057 | Z_g(x)=h_C. Its intersection with g_i is exactly H+. Thus H+ is maximal:
000058 | it is a Cartan subspace of the theta representation. This is a proof of
000059 | maximality, not an inference from the dimension four.
000060 | 
000061 | Use the EXISTING whole-field conjugacy C kappa C^-1=Jhat, and set
000062 | 
000063 |   c=C^-1(H+) subset g_i(kappa), h'=C^-1(h_C).
000064 | 
000065 | Then c is a Cartan subspace inside one of the actual64 branches. In the
000066 | current signed convention W_m=(16,4) has kappa phase -i, so c lies in its
000067 | dual +i branch. Equivalently reverse both chiral labels and write
000068 | g_1=Delta+ tensor F4. One must keep that simultaneous dual convention.
000069 | The conjugator is existent, not preferred. Its choice selects a concrete
000070 | c in the common K_C orbit; it does not select a physical source plane.
000071 | 
000072 | Liu and Manivel, Sections5.1/5.3, identify the same Z4 theta representation,
000073 | its four-dimensional Cartan subspace, and its little Weyl group G31.
000074 | Their Proposition5.5 gives an explicit example. This is an existing
000075 | mathematical theorem used here to type the recovered TFPT objects.
000076 | 
000077 | 4. Why it is the SAME G31, including its action
000078 | 
000079 | Return to the native frame and put G=E8(C), K_C=G^theta, T=Z_G(h_C).
000080 | The previous conjugacy identifies K_C with(Spin10(C)xSL4(C))/mu4.
000081 | An element of N_KC(H+) preserves Z_g(H+)=h_C and hence normalizes T.
000082 | Its Weyl element commutes with J because it belongs to K_C. The kernel
000083 | of the resulting map is Z_KC(H+)=T^J. Therefore there is an injection
000084 | 
000085 |   N_KC(H+)/Z_KC(H+) --> C_W(E8)(J).
000086 | 
000087 | For surjectivity, take w in the right side and a lift n in N_G(T).
000088 | The element delta=n theta n^-1 theta^-1 belongs to T, since the induced
000089 | Weyl actions commute. Replacing n by tn changes delta into
000090 | 
000091 |   t delta theta(t)^-1.
000092 | 
000093 | The torus homomorphism t -> t theta(t)^-1 has differential I-J, which
000094 | is invertible because J has only eigenvalues+i and-i. It is consequently
000095 | a surjective isogeny of T. Choose t making the displayed expression1.
000096 | Then tn belongs to K_C and lifts w. This proves the equality of actions
000097 | 
000098 |   N_KC(c)/Z_KC(c) = C_W(E8)(J) = G31,
000099 | 
000100 | where the first equality uses the transported chart. The rightmost
000101 | group and its faithful original four-dimensional action were already
000102 | computed by v634. This is stronger than a common order or degree list.
000103 | 
000104 | Accordingly the theta-group restriction theorem gives
000105 | 
000106 |   C[g_i]^K_C = C[c]^G31 = C[F8,F12,F20,F24].
000107 | 
000108 | The existing v690 restricted invariants now have this precise joint
000109 | interpretation on the same Cartan subspace. The quotient parametrizes
000110 | closed semisimple orbits. It does not by itself select a physical orbit,
000111 | give a field state, or classify every nilpotent and mixed orbit by four
000112 | numbers. No independent invariant calculation is being advertised.
000113 |