Established in scope
E7 pairs on all lattice fields and complete correction torsor
In the question: Active mathematical program: marked seam monodromies and lattice lifts
Unchanged eps_e8, ordered integral basis and original lattice VOA. All-lattice equality follows from the character lemma and eight generators; all240 roots and13680 nonzero brackets per line checked separately. Individual lifts do not provide a global group section.
Read the exact result, assumptions and evidence
Result
For all60 native Gaussian root lines, r=s_alpha s_Jalpha is the original complex reflection. Its actual Tits lift T=n_alpha n_Jalpha has T²=D_k, k(lambda)=<alpha+Jalpha,lambda>mod2, negative on128 roots. T=S_r D_chi holds on every lattice field with explicit input character. Exactly eight corrections per line are involutive and commute with the fixed J lift, a torsor under three-dimensional A^{<r,J>}. They agree on the actual neutral4/5/20 diagram and differ on charged sectors. The full G31 extension remains nonsplit.
Required assumptions
- Full original G31action and inherited lattice sign extension
The60actual original complex reflections act on the same original even E8lattice. Study exactly their full preimage in the automorphism group of the lattice group defined by unchanged eps_e8. Kernel A=Hom(E8,Z2), with g.a=a composed g^-1. No scalar replacement, enlarged coefficients or extra square relation for lifted puncture meridians.
- Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
Original evidence
- FIELDS.txt lines 1–133
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/e-series-compiler-closure-20261023/FIELDS.txt
1ad4bbda485ff2e703a5fe5db9de09ce463103d5791933e9bf24dfa68d2d3b50
- THEOREMS.txt lines 20–49
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/e-series-compiler-closure-20261023/THEOREMS.txt
361450df688c5dc8fa3b8c69ff6a8d0f35156772bee5ec0acab494eb32d3587f
finding:e7-pair-full-lattice-field-character-torsor