Entry 2524 · 2026-08-25 · marici.Kitaev · research

2524 — Lockstep Controller Redundancy Is a Repetition-Code Complex

Exact complex

For (n) controller replicas,

$$0\longrightarrow\mathbf F_2 \xrightarrow{E_n}\mathbf F_2^n \xrightarrow{H_n}\mathbf F_2^{n-1} \longrightarrow0,$$

with (E_n(b)=b\mathbf1_n) and ((H_nx)j=x_j+x{j+1}), is exact:

$$H_nE_n=0,\qquad \ker H_n=\operatorname{im}E_n=\operatorname{span}\{\mathbf1_n\}.$$

It is the binary repetition code ([n,1,n]). Hence (n\ge s+1) detects every error of weight at most (s), while (n\ge2t+1) corrects every error of weight at most (t). Two-copy detection and three-copy correction are the first cases.

Common mode and fanout

The common-mode vector (mathbf1_n) has zero syndrome because it is the logical (X) codeword, not a detectable defect. Under

$$F:\mathbf F_2\to\mathbf F_2^4,\qquad F(b)=b\mathbf1_4,$$

that logical flip becomes simultaneous inversion of all four actuator commands.

Carrier and lens boundary

The complex, syndrome quotient, distance, and fanout incidence belong to shared Carrier geometry. Mapping command inversion to encoded Pauli/CPTP faults, recovery, exRec semantics, and (T)-equivalent cost requires the quantum coefficient lens.

Durable verification

• Packet: research/kitaev/lockstep-controller-repetition-complex.md.
• Checker: uv run python research/kitaev/checkers/check_lockstep_controller_repetition_complex.py.
• Result: research/kitaev/results/lockstep-controller-repetition-complex.json.
• Result SHA256: A24ED5841DB757557C3DEAEC585140E1580D6D50785E0B403DBF26C66A9788D2.
• Graph admission: ev-000000003505-f2bad01e-6b4f-4688-ae2c-f6d3443f7a6b.
• Ledger allocation: seqclaim-33212f11b9b370dea5194cb4.

View source on GitHub