IndisputableMonolith.QuantumComputing.ErrorCorrectionThresholdFromJCost
The module defines the recognition quantum J(φ) and derives surface-code error-correction thresholds from the J-cost function. Quantum-information theorists working inside Recognition Science would cite it for the explicit link between the phi-ladder and threshold statements. The module is organized as a chain of definitions (recognitionQuantum, errorRateCost) followed by positivity and certification lemmas.
claimThe recognition quantum is $J(φ)$ with $J(x)=(x+x^{-1})/2-1$. The surface-code threshold is the value of the error-rate cost function at the J-cost minimum; an ErrorCorrectionCert is inhabited when this cost lies below the derived bound.
background
The module imports the RS time quantum τ₀=1 tick from Constants and the J-cost machinery from the Cost module. It introduces recognitionQuantum as J evaluated at the self-similar fixed point φ, then builds errorRateCost and surfaceCodeThreshold on top of that definition. The local setting is the quantum-computing subdomain of Recognition Science, where the J-cost supplies the native energy scale for error processes.
proof idea
This is a definition module, no proofs. The argument proceeds by successive definitions of recognitionQuantum, errorRateCost and ErrorCorrectionCert, followed by direct statements of non-negativity and threshold positivity.
why it matters in Recognition Science
The module supplies the J-cost bridge into quantum error correction and therefore feeds any later theorem that invokes surfaceCodeThreshold or ErrorCorrectionCert. It realizes the step that converts the T5 J-uniqueness result into a concrete threshold statement inside the eight-tick octave framework.
scope and limits
- Does not compute numerical threshold values.
- Does not treat codes other than the surface code.
- Does not incorporate hardware-specific noise models beyond the J-cost.
- Does not address fault-tolerant gate sets.