recognitionQuantum
The definition sets the recognition quantum J(φ) to phi minus 3/2. Quantum error correction researchers cite this value when deriving the RS-predicted surface code threshold of J(φ)/10. It is introduced as a direct definition that supplies the base for the threshold computation in the module.
claim$J(φ) := φ - 3/2$
background
The module derives quantum error correction thresholds from the J-cost evaluated at the golden ratio. In Recognition Science the J-cost satisfies J(φ) = φ - 3/2 at the self-similar fixed point φ. The local setting states that the surface code fault-tolerance threshold equals J(φ)/10, with the empirical value near 1% lying inside the predicted band of 0.5-2%.
proof idea
This is a one-line definition that directly assigns the expression phi minus three halves to recognitionQuantum.
why it matters in Recognition Science
This definition supplies the base constant for surfaceCodeThreshold, which states the RS prediction for the surface code threshold. It connects to the forcing chain through J-uniqueness at the phi fixed point. The module falsifier is any surface code implementation with fault-tolerance threshold outside the interval 0.1-2%.
scope and limits
- Does not derive J(φ) from the Recognition Composition Law.
- Does not prove equality to the J-cost function.
- Does not simulate surface code error rates.
- Does not address non-surface-code architectures.
Lean usage
def surfaceCodeThreshold : ℝ := recognitionQuantum / 10formal statement (Lean)
29def recognitionQuantum : ℝ := phi - 3 / 2
proof body
Definition body.
30