recognitionQuantum
plain-language theorem explainer
Recognition quantum is the real number phi minus three halves, which equals the J-cost evaluated at the golden ratio. Quantum error correction modelers would cite this constant when computing the Recognition Science surface code threshold prediction. The declaration is a direct one-line abbreviation with no lemmas or tactics.
Claim. The recognition quantum is the real number $J(phi) := phi - 3/2$, where $J$ denotes the J-cost function and $phi$ is the golden ratio.
background
The J-cost satisfies the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ and takes the explicit form $J(x) = (x + x^{-1})/2 - 1$. The golden ratio $phi$ is the unique fixed point greater than one obeying $phi^2 = phi + 1$, so that $J(phi) = phi - 3/2$ holds by direct substitution. This module works inside the Recognition Science derivation of quantum error correction thresholds from the J-cost at $phi$.
proof idea
One-line definition that directly assigns the value $phi - 3/2$ to recognitionQuantum.
why it matters
The definition supplies the base constant used by recognitionQuantum_eq_Jph, recognitionQuantum_pos, and surfaceCodeThreshold in the same module. surfaceCodeThreshold is defined as recognitionQuantum / 10 and is presented as the RS prediction for the surface code fault-tolerance threshold, stated to lie near 1.18% and consistent with the empirical band 0.5-2%. It instantiates the J-uniqueness step (T5) and the forcing of $phi$ as self-similar fixed point (T6) from the IndisputableMonolith forcing chain. The module falsifier is any surface code implementation whose threshold lies outside the interval (0.1%, 2%).
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