toricCode
The toricCode definition supplies a string label for the Kitaev toric code in which qubits sit on torus edges and logical information is carried by homology classes, with error correction performed through local syndrome measurements. Researchers in quantum information theory and Recognition Science cite it when mapping eight-tick phase redundancy onto topological codes. The implementation is a direct string constant with no computation or proof obligations.
claimThe toric code places qubits on the edges of a torus such that logical qubits correspond to homology classes and error correction proceeds via local syndrome measurements extracted from phase patterns.
background
The Information.ErrorCorrectionBounds module derives error-correction bounds from the eight-tick structure. The fundamental time quantum is the tick, equal to one in RS-native units. Phases are defined as kπ/4 for k ranging over Fin 8, supplying periodic redundancy that encodes bits across correlated phases. The module opens by recalling Shannon capacity and then shows how eight-tick phase correlations furnish natural redundancy for error recovery.
proof idea
This definition is a one-line string assignment that records the standard description of the Kitaev toric code.
why it matters in Recognition Science
The definition supplies a concrete topological example inside the eight-tick error-correction program of INFO-005. It links the homology-based protection of the toric code to the phase correlations generated by the eight-tick octave. No downstream theorems are recorded, yet the entry supports the module goal of obtaining bounds from Recognition Science mechanisms rather than external noise models.
scope and limits
- Does not encode the toric lattice or stabilizer operators inside Lean.
- Does not derive the one-percent error threshold from the Recognition Composition Law.
- Does not connect the toric code to the phi-ladder or mass formula.
- Does not supply a formal theorem relating syndrome extraction to eight-tick phases.
formal statement (Lean)
167def toricCode : String :=
proof body
Definition body.
168 "Qubits on torus edges, information in homology"
169
170/-! ## The 8-Tick Syndrome -/
171
172/-- Error syndromes from 8-tick phases:
173
174 If phase pattern is (0, 0, 1, 0, 0, 0, 0, 0):
175 - Error detected at phase 2
176 - Syndrome = [0, 0, 1, 0, 0, 0, 0, 0]
177
178 Recovery: Flip phase 2 back to 0.
179
180 The 8-tick structure naturally supports syndrome-based EC! -/