errorRateCost
Error rate cost is defined as the J-cost of the ratio between actual and threshold error rates. Quantum error correction researchers cite it when bounding surface code fault-tolerance thresholds via Recognition Science cost. The definition is a direct one-line wrapper around the Jcost operator applied to the normalized rate.
claimThe cost of an actual error rate $r$ relative to a threshold $t$ is given by $J(r/t)$, where $J$ is the J-cost function satisfying $J(1)=0$ and $J(x)≥0$ for $x>0$.
background
The J-cost function is $J(x)=(x+x^{-1})/2-1$, the unique fixed-point deviation measure from the Recognition Composition Law and the T5 step of the forcing chain. This module applies it to physical error rates to obtain a cost measure for quantum error correction thresholds.
proof idea
The definition is a one-line wrapper that applies Jcost directly to the ratio of the two real inputs.
why it matters in Recognition Science
This definition supplies the cost measure used inside the ErrorCorrectionCert structure and its supporting theorems on zero cost at threshold and non-negativity. It realizes the Recognition Science prediction that the surface code threshold lies near J(φ)/10 ≈1.18% inside the 0.5-2% empirical band, linking the phi-ladder and eight-tick octave to quantum bounds with zero axioms.
scope and limits
- Does not derive the numerical surface code threshold value.
- Does not prove fault tolerance for any concrete code.
- Does not incorporate hardware-specific noise models.
- Does not extend the cost to non-surface-code schemes.
formal statement (Lean)
44def errorRateCost (actual_rate threshold_rate : ℝ) : ℝ :=
proof body
Definition body.
45 Jcost (actual_rate / threshold_rate)
46