TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-compiler-closure-20261023/THEOREMS.txt
Original file SHA-256: 361450df688c5dc8fa3b8c69ff6a8d0f35156772bee5ec0acab494eb32d3587f
Inclusive original lines: 20-49
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000020 | 
000021 | 2. E7 pairs on all lattice fields, not only the root chart
000022 | 
000023 | For each of the60 native Gaussian root lines, beta=Jalpha and
000024 | r=s_alpha s_beta is literally the original G31 complex reflection.
000025 | The raw Chevalley/Tits pair T=n_alpha n_beta obeys
000026 | 
000027 |   T^2 E_lambda=(-1)^((alpha+Jalpha,lambda)) E_lambda.
000028 | 
000029 | Its square character is nonzero and negative on exactly128 roots. Thus T has
000030 | order4 on the full lattice fields. The generator-normalized lattice lift S_r
000031 | differs from T by an explicitly computed input character:
000032 | 
000033 |   T=S_r D_chi.
000034 | 
000035 | Equality on the integral lattice generators and the common cocycle proves
000036 | this identity on all lattice vectors, with all240 roots independently
000037 | checked. The Heisenberg action then determines the complete lattice VOA
000038 | automorphism. Every one of the60 lines admits exactly8 character corrections
000039 | making T involutive and commuting with the fixed original lift of J. They
000040 | form a torsor under A^{<r,J>} of dimension3, A=Hom(E8,F2).
000041 | 
000042 | Characters act trivially on the neutral Cartan oscillators. Consequently
000043 | these eight lifts agree on the actual native4/5/20 oscillator, quartic and
000044 | erasure diagram; their differences occur in charged sectors. This20 is the
000045 | original neutral derivative space, not a charged current20. The previously
000046 | proved nonsplitting of the full G31 extension remains in force. Separate
000047 | reflection lifts do not supply a simultaneous group section.
000048 | 
000049 | 3. Complete E6/E7 dictionary relative to the original compiler split