SurfaceCode
SurfaceCode defines a 2D lattice quantum error correction structure with side length L, L squared physical qubits, one logical qubit, and distance L. Physicists studying fault-tolerant quantum computing cite it when mapping Recognition Science eight-tick redundancy onto surface code thresholds near one percent. The declaration is a direct structure definition that assigns the three derived fields by default.
claimA surface code on a square lattice of side length $L$ consists of $L^2$ physical qubits that encode one logical qubit with code distance $L$.
background
The module INFO-007 derives quantum error correction from the eight-tick redundancy of Recognition Science. Errors appear as phase shifts across the eight phases; correction restores alignment. The surface code is presented as the leading scalable approach whose 2D lattice matches the holographic boundary structure in the framework. Upstream structures supply the J-cost calibration and the spectral emergence that forces three spatial dimensions and the eight-tick octave.
proof idea
The declaration is a structure definition. It introduces the field L and then supplies default values n_physical := L * L, n_logical := 1, and distance := L.
why it matters in Recognition Science
This definition supplies the surface-code instance inside the eight-tick QEC derivation. It directly implements the 2D lattice that the module links to holographic boundaries and the T7 eight-tick octave. No downstream theorems yet consume it, so the explicit derivation of the one-percent threshold from phi constants remains open.
scope and limits
- Does not derive the one-percent threshold from Recognition Science constants.
- Does not specify stabilizer measurements or decoding algorithms.
- Does not prove fault tolerance or connect to the eight-tick code explicitly.
formal statement (Lean)
195structure SurfaceCode where
196 /-- Lattice size -/
197 L : ℕ
198 /-- Number of physical qubits: ~L² -/
199 n_physical : ℕ := L * L
proof body
Definition body.
200 /-- Number of logical qubits: 1 for simple surface code -/
201 n_logical : ℕ := 1
202 /-- Distance: L -/
203 distance : ℕ := L
204
205/-- Surface code threshold: p_threshold ≈ 1%.
206
207 Below this error rate, arbitrarily long computation is possible.
208 Above it, errors accumulate faster than correction. -/