Established in scope
E7 pairs are the original Gaussian reflections
In the question: Active mathematical program: marked seam monodromies and lattice lifts
Fixed native Gaussian J and actual root subsystems; no arbitrary Lie-subalgebra embedding classification, new lattice-lift class or physical identification.
Read the exact result, assumptions and evidence
Result
In the given E8 root system J with J²=-I exchanges the120 root-normal E7 slices in60 pairs. Their intersections are D6 with60 roots and their normal planes A1+A1. The operator s_alpha s_Jalpha is literally the original complex Gaussian root reflection of v634; the60 distinct operators generate the existing G31 realization of order46080. Operator equality, not group order alone, supplies the dictionary. The two actual E7 subalgebras contain192 roots; their mixed brackets generate all48 missing roots in one round. Their integral lattice sum is E8 at index1. The60 pairs form a transitive G31 orbit.
Required assumptions
- Active purely mathematical marked-seam program
Within the stated mathematical scope, marked pairs, lattice transport systems, reflections, clocks and cohomology are independent mathematical objects. Earlier origin/completion tasks are historical, not active. Distinguish new definitions, derived theorems and optional markings.
- 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.
Original evidence
- PROOF.txt lines 1–62
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/e-series-quarter-clock-20261023/PROOF.txt
ac3e95975adb14c82815731b3d8aa0f6e1caf72d25fb749250a5b1bbcd269884
- checker.py lines 1–200
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/e-series-quarter-clock-20261023/checker.py
31e9fcf230273955bbf69536c6594255e3c0c53034a87e13ca1d3c56815b0562
- PROOF.txt lines 149–163
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/e-series-quarter-clock-20261023/PROOF.txt
ac3e95975adb14c82815731b3d8aa0f6e1caf72d25fb749250a5b1bbcd269884
finding:e7-quarter-pairs-literal-native-g31-reflections