TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-quarter-clock-20261023/PROOF.txt
Original file SHA-256: ac3e95975adb14c82815731b3d8aa0f6e1caf72d25fb749250a5b1bbcd269884
Inclusive original lines: 149-163
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000149 | 6. Two exchanged E7 slices recover the complete ambient E8
000150 | 
000151 | For alpha=e0+e1 and beta=e0-e1 their root union contains the60 D6
000152 | integer roots, four roots +/-alpha,+/-beta, and all128 half roots.
000153 | The missing48 are +/-e_p +/-e_j with p=0,1 and2<=j<=7.
000154 | For example e0+e2 is the sum of
000155 |   (1,1,1,1,1,1,1,1)/2 and (1,-1,1,-1,-1,-1,-1,-1)/2.
000156 | Each is an E8 root; they belong to different E7 slices and have inner
000157 | product-1. W(D6) times the two A1 Weyl groups preserve the root union
000158 | and act transitively on the missing48, so every missing root is such
000159 | a cross sum. All240 choices of initial alpha are checked in the script.
000160 | The actual ambient Chevalley bracket of these root spaces is nonzero.
000161 | Thus the two actual E7 subalgebras generate e8. Their linear sum is
000162 | dimension133+133-66=200, and the remaining48 are the(12,2,2) sector of
000163 | D6+A1+A1. Their integral root-lattice sum is already all of E8, index1.