Established in scope
Gaussian four is Cartan subspace in whole64 with actual G31 action
In the question: Common clock sectors, quartic geometry and operational reconstruction
Exact fiberwise identification and group action;no numerical fit. Current orientation puts H+ in dual64. Conjugator exists but is not preferred. GIT quotient describes closed semisimple orbits,not physical orbit or state selection. Reuses v634,v690 and whole-clock conjugacy.
Read the exact result, assumptions and evidence
Result
Every real E8 root restricts nontrivially to H+. A generic point is regular semisimple,so H+ is a maximal commuting semisimple Cartan subspace in g_i. Existing whole-clock conjugacy carries it into the appropriate64 half-spin times family branch. Its normalizer quotient is exactly C_W(E8)(J)=G31:Weyl lifts can be corrected through surjective torus isogeny1-J to centralize the quarter clock. Old degrees8,12,20,24 are invariants of this same theta representation.
Required assumptions
- Native Cartan four in whole conjugated source
Given original Cartan h_C,H+=ker(J-i),and whole native clock chat with chat^9=Jhat,chat^4=sigmahat. Existing inner conjugacy transports the entire tuple and Cartan to kappa. H+ is not identified with SU4 family factor. In current W_m orientation,+i64 is its dual branch. No preferred conjugator is selected.
- Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
- Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Original evidence
- CARTAN-FAMILY-DICTIONARY.txt lines 43–113
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/CARTAN-FAMILY-DICTIONARY.txt
7b16a69ded0088bfe7a1fad8e1d8bdd79f49d4b00ff5bcb581a8ba260c45e864
- v690_quartic_half.py lines 6–40
Source checked at this snapshot
Original location and fingerprint
verification/v690_quartic_half.py
489c6c034e8a79cbe281dc09cb17d3d84ffe7e3eb46df4b7b92c46d6ccacb324
- v634_st31_structure.py lines 280–346
Source checked at this snapshot
Original location and fingerprint
verification/v634_st31_structure.py
5ff53ed031ca03103187ac22edf723aabce88bee3df1085c52486ecb0044d37c
finding:native-gaussian-four-is-cartan-subspace-in-whole-sixtyfour