Scoped exclusion
Family factor remains different despite symmetric-product coincidences
In the question: Common clock sectors, quartic geometry and operational reconstruction
Excludes direct factor identity under the same actual third action. Positive Cartan-subspace inclusion and full inner E8 conjugacy remain valid. Product equality is only C3 character equality,not full G31 or Spin10 times SU4 equivalence. The480 belong to old complete classification and are not new physical sources.
Read the exact result, assumptions and evidence
Result
For the same sigma=chat^4,H+ has C3 character(2,1,1). All480 toral representatives of the WHOLE native C12 class have character(0,2,2) in the unique order3 SU4 lift,and(4,3,3) in Spin10 vector. No direct sigma isomorphism identifies the fours. Nevertheless Sym2 andSym3 agree;in general Symn agree precisely for n not1mod3. Exterior2 gives(2,2,2) versus(4,1,1). All-degree trace generating-function difference is3t/(1-t^3)^2.
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 114–171
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/CARTAN-FAMILY-DICTIONARY.txt
7b16a69ded0088bfe7a1fad8e1d8bdd79f49d4b00ff5bcb581a8ba260c45e864
- check_cartan_family_dictionary.py lines 88–158
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/check_cartan_family_dictionary.py
91344d5d71b4fd7c163b7648460ab5127d5c026cd1927faa4510c126f3a02c4b
finding:native-family-factor-is-not-gaussian-four-despite-symmetric-coincidences