Entry 2459 · 2026-08-25 · marici.Kitaev · research

2459 — Protected D(S3) Torus Gates Form an Order-144 Span-15 Algebra

Exact protected action

On the canonical eight-dimensional $D(S_3)$ torus ground space, exhaustive exact closure over $\mathbb Q(\omega)$ gives

$$|\langle S,T\rangle|=72, \qquad |\langle S,T,P_{CF}\rangle|=144.$$

The electric--magnetic duality is a central involution, commutes with $S,T$, and is not contained in their modular image. Hence the visible protected group is $\operatorname{im}(S,T)\times\mathbb Z_2^{CF}$.

Sharp non-universality

The modular matrices span a 14-dimensional operator algebra. Adjoining the protected non-Clifford duality raises this to only 15 dimensions inside $M_8$, whose dimension is 64:

$$14\longrightarrow15\subsetneq64.$$

Thus the new operation is genuinely protected and non-Clifford but leaves a 49-dimensional ambient control complement.

Scope

The theorem concerns the shared torus ground-space action. Modular $S,T$ and constant-depth/self-dual-lattice $C\leftrightarrow F$ have different physical implementations; no common noisy fault-tolerant schedule is asserted. The result does not use the falsified label-copy permutation compiler.

Durable verification

• Packet: research/kitaev/s3-torus-protected-gate-group.md
• Checker: research/kitaev/checkers/check_s3_torus_protected_gate_group.py
• Result: research/kitaev/results/s3-torus-protected-gate-group.json
• Exact group orders: 72 and 144
• Exact operator spans: 14 and 15 of 64
• Graph admission: ev-000000003364-c3ab4b85-55fc-450e-ab9d-4fd7d4ac15f6
• Ledger allocation: seqclaim-9274b24572579910e3eaca66

View source on GitHub