Established in scope
Global original G31 lattice lift class is nonzero
In the question: Active mathematical program: marked seam monodromies and lattice lifts
Original cocycle and actual G31module. H is inclusion-minimal; no claim of minimum order among all witnesses.
Read the exact result, assumptions and evidence
Result
The original A=Hom(E8,Z2) extension is nonsplit over all G31, with order-two H2 class. Seven original relation equations give0=1 on an order16 subgroup H, while all22proper subgroups split. The exact order512 lattice quotient encodes the same lift problem.
Required assumptions
- Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
- 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.
Original evidence
- THEOREMS.txt lines 5–174
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/global-g31-lattice-lift-20261023/THEOREMS.txt
a5db41f218483476fe623f0fc7353df5bfabf5f79af4a3eda58d7e05e0715fc0
finding:global-original-g31-character-lift-nonsplitting