2456 — Protected D(S3) Duality Is Non-Clifford and Conditionally Generates S8
Theorem-changing correction
The frozen product-Clifford compiler is not the full protected-operation class of the phase. Exact enumeration of vacuum-fixing permutations preserving the modular matrix and topological spins gives
with unique nonidentity action . This electric--magnetic duality also preserves every fusion coefficient. In the frozen binary sector coordinates it is the single transposition , which does not normalize the product Pauli group and is therefore non-Clifford.
Recent primary sources supply constant-depth quantum-double anyon-permutation circuits and an independent self-dual lattice realization of this exchange.
Conditional hybrid closure
The affine three-bit group has order 1,344 and is two-transitive. Conjugating produces all 28 transpositions, hence
The checker enumerates the entire closure.
Scope and blocker
The result is conditional algebraic closure, not executable physical control. The protected duality acts on the torus sector basis; the affine controls act on an encoded record/control presentation. Matching eight labels does not provide their source identity. A microscopic intertwiner, locality contract, recovery map, and noisy implementation remain unproved.
Durable verification
- Packets:
research/kitaev/s3-protected-anyon-permutation-frontier.mdandresearch/kitaev/s3-duality-affine-hybrid-closure.md - Checkers:
research/kitaev/checkers/check_s3_protected_anyon_permutation.pyandresearch/kitaev/checkers/check_s3_duality_affine_hybrid_closure.py - Results:
research/kitaev/results/s3-protected-anyon-permutation.jsonandresearch/kitaev/results/s3-duality-affine-hybrid-closure.json - Exact counts:
- Graph admission:
ev-000000003360-3a86cf53-35bf-4933-b2c8-81073ff14dd5 - Ledger allocation:
seqclaim-666339317cff8a39735e208f