Established in scope
All product-unitary lifts of the quarter clock excluded
In the question: Historical: common source, state and event selection
Classification of code-preserving product unitaries and stated classical mixtures. Collective operations, ancillas, general LOCC, postselection and protocols leaving the code temporarily are not excluded.
Read the exact result, assumptions and evidence
Result
All six actual code-projector pair marginals are (I+Swap)/4. Every code-preserving U1 tensor U2 tensor U3 tensor U4 therefore has projectively equal factors. The identical local stabilizer is K=U(1)G31 by leakage Gram32I15, a finite determinant slice and NRS6.5. R5 lies projectively outside logical S6, excluding every such product lift. Rank-one Choi also excludes convex native-word-channel mixtures reproducing Ad_R5 on all inputs.
Required assumptions
- Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
- Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
- Existing logical clock and charge marking
Retain the existing logical labels weak0/1, color2/3/4, source point5 and Y5,R5,sigma5. Equivalence under their stabilizer is not automatically an equivalence of lifts commuting with the original native four/P1 clock.
- Actual Clifford identification and finite-overgroup theorem
Retain the actual concrete identification of <G31,zeta8 I4> with complex Clifford X2. Apply NRS6.5 only to the proved finite determinant slice; a group-order match alone does not suffice.
Original evidence
- MARKED-CLOCK-LIFT.md lines 87–121
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/joint-native-algebra-selection-20261016/MARKED-CLOCK-LIFT.md
da0d56aeb94f078f496a0748911ad559fa5835d77757fc7d1879c2eb70123468
finding:actual-quartic-code-product-stabilizer-quarter-exclusion