eight_tick_corrects_3
The eight-tick code has minimum distance 7 and therefore corrects up to three errors via the standard relation t = (d-1)/2. Researchers deriving quantum error correction bounds from Recognition Science phase structure would cite this equality when bounding channel capacity. The proof is a direct reflexivity check on the distance parameter already assigned in the code definition.
claimFor the error-correcting code built from the eight-tick phases, the minimum distance $d$ satisfies $(d-1)/2 = 3$.
background
The module derives error correction bounds from the eight-tick structure. The fundamental time quantum is one tick, and the eight phases are defined as $kπ/4$ for $k=0,…,7$, giving the periodic octave that supplies natural redundancy through phase correlations. Upstream results fix the tick as the RS-native time unit and establish the phase function over Fin 8.
proof idea
The proof is a one-line wrapper that applies reflexivity to the pre-defined minimum distance of the eight-tick code.
why it matters in Recognition Science
This equality supplies the concrete distance for the eight-tick code, supporting the module claim that eight-tick phases furnish natural quantum error correction. It sits inside the eight-tick octave step of the forcing chain and connects to quantum channel capacity results that use the same phase structure.
scope and limits
- Does not derive the minimum distance from first principles.
- Does not treat continuous or non-discrete error models.
- Does not prove physical realizability of the code.
formal statement (Lean)
123theorem eight_tick_corrects_3 :
124 -- 8-tick code corrects up to 3 errors
125 (eightTickCode.d - 1) / 2 = 3 := by rfl
proof body
Term-mode proof.
126
127/-! ## Quantum Error Correction -/
128
129/-- Quantum error correction is different:
130
131 - Can't measure without disturbing
132 - No cloning
133 - Errors are continuous (not just bit flips)
134
135 Yet QEC is possible! Using entanglement and syndrome measurement.
136
137 In RS: 8-tick phases provide natural QEC through phase correlations. -/