canonicalThreshold
plain-language theorem explainer
The canonical recognition threshold is the real number φ − 3/2, which equals the J-cost at the golden ratio and is the RS-native value of the quantum-to-geometric capacitance ratio ≈ 0.118. Anyone citing the crossover C_Q/C_geo = J(φ) in this module uses this constant. It is a one-line definition, not a proved identity.
Claim. Define the canonical threshold by $\varphi - 3/2 \in \mathbb{R}$, where $\varphi$ is the golden ratio (self-similar fixed point of the recognition cost).
background
The module treats quantum capacitance $C_Q = e^2 D(E_F)$ and asserts that, in Recognition Science units, the ratio of quantum to geometric capacitance at the recognition crossover equals the J-cost of the golden ratio: $C_Q/C_{\mathrm{geo}} = J(\varphi) \approx 0.118$.
The cost is the unique symmetric functional $J(x) = (x + x^{-1})/2 - 1$ forced by the Recognition Composition Law (T5). Because $\varphi$ satisfies $1/\varphi = \varphi - 1$, one has $J(\varphi) = \varphi - 3/2$. The symbol $\varphi$ is imported from Constants; the cost infrastructure from Cost.
Local status is a structural theorem package (zero sorry, zero axioms) that packages this threshold for a certificate of the capacitance claim.
proof idea
Pure definitional abbreviation: the right-hand side is the closed-form real $\varphi - 3/2$. No tactics, no lemmas, no proof obligations. The reason the constant is the right threshold is the elementary identity $J(\varphi) = \varphi - 3/2$, which is not discharged inside this declaration itself.
why it matters
Supplies the numeric anchor for the module claim that the quantum-to-geometric capacitance ratio equals $J(\varphi)$ at crossover. Sibling positivity (canonicalThreshold_pos) and the certificate bundle (QCapacitanceCert, cert, cert_inhabited) read this value. Framework landmarks: T5 (J-uniqueness) and T6 ($\varphi$ forced as self-similar fixed point). The numerical band $\approx 0.118$ is the concrete RS prediction for the capacitance ratio; eight-tick structure and $D=3$ are not used here.
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