TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-quarter-clock-20261023/PROOF.txt
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Inclusive original lines: 64-91
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000064 | 3. E6 slices and the quarter-clock obstruction
000065 | 
000066 | For alpha,beta with (alpha,beta)=-1, their6 roots form A2. Its orthogonal
000067 | root complement in E8 is E6 with72 roots. All1120 A2 root subsystems
000068 | are enumerated without random sampling. None is J invariant; J forms
000069 | 560 pairs. The corresponding E6 intersections have12 roots in640
000070 | subsystem cases and24 roots in480 cases. Those counts alone are not
000071 | used to claim an intersection type or an isometry classification.
000072 | 
000073 | A stronger all-root-subsystem obstruction needs no enumeration. If J
000074 | preserved an E6 root subsystem, it would preserve its root lattice K,
000075 | its dual, and its discriminant group K*/K=Z3. Every automorphism of Z3
000076 | squares to identity. But J^2=-I induces nontrivial inversion on Z3, a
000077 | contradiction. The same argument applies to A2. Thus no E6 ROOT subsystem
000078 | can be fixed by this J, including any other root-subsystem embedding.
000079 | This says nothing about arbitrary non-root Lie-subalgebra embeddings.
000080 | 
000081 | 4. A compatible marked flag still exists
000082 | 
000083 | In the standard5+3 chart choose alpha=e5-e6, beta=e6-e7. Then
000084 |   D5 roots on coordinates0..4
000085 |     subset E6=Phi intersect alpha-perp intersect beta-perp
000086 |     subset E7=Phi intersect alpha-perp
000087 |     subset E8.
000088 | Root counts are40,72,126,240. This actual nesting is a mathematical
000089 | unfolding, with the root/A2 flag explicitly chosen. It is not a single
000090 | J-invariant chain. Therefore nesting and clock-compatible transport
000091 | are distinct requirements in the marked-seam category.