2342 — Recovery Between Three S3 Shuttle Contacts Lowers the Conditional Data Distance to Three
Verdict
The direct relative-coordinate gate is not a fundamental weight-two primitive. A six-state bus initialized at the identity implements
through three two-body contacts and returns clean. The inverse uses the reversed multiplication convention. All 72 forward/inverse basis cases pass.
With distance-three encoding and nonpropagating recovery of both incident blocks after every contact, all 18 abstract single-fault paths—six contact faults and twelve recovery-output errors—have maximum data-block weight one. The complete conditional sector compiler therefore moves from 45 to 49 gates and from 20 to 26 recovery layers per controlled power, uses fifteen encoded bus rails before syndrome ancillas, and requires data distance three rather than five.
The theorem remains conditional on one-fault-transversal logical S3
multiplication and verified nonpropagating syndrome/recovery circuits. It does
not construct those hardware-level gates.
Durable verification
- Packet:
research/kitaev/s3-recovered-relative-coordinate-shuttle.md - Checker:
python research/kitaev/checkers/check_s3_recovered_relative_coordinate_shuttle.py - Result:
research/kitaev/results/s3-recovered-relative-coordinate-shuttle.json - Exact cases: 72 group-action cases and 18 abstract single-fault paths
- Epistemic graph:
ev-000000003211-258fbae8-6564-49a6-88d6-008787f48084 - Ledger allocation:
seqclaim-f57cbef4537e9a74e946831f