canonicalThreshold
plain-language theorem explainer
The canonical Penrose efficiency scale is the real constant φ − 3/2, equal to the J-cost of the golden ratio. Gravity and RS workers cite it as the dimensionless factor multiplying Ω_H/Ω_ISCO in the structural efficiency formula. It is a one-line definitional abbreviation of that closed form.
Claim. Define the canonical threshold by $\varphi - 3/2$, where $\varphi$ is the golden ratio (self-similar fixed point of the Recognition cost).
background
The module treats Penrose-process energy extraction as a Recognition Science cost statement. The unique cost forced by the Recognition Composition Law is $J(x)=(x+x^{-1})/2-1$ (T5); $\varphi$ is the self-similar fixed point (T6).
Because $\varphi^{-1}=\varphi-1$, the value $J(\varphi)$ collapses at once to $\varphi-3/2$. The module sets $\eta_{\mathrm{Penrose}}=J(\varphi)\cdot(\Omega_H/\Omega_{\mathrm{ISCO}})$. At maximal Kerr the frequency ratio equals $\varphi$, so $\eta\approx J(\varphi)\cdot\varphi\approx0.191$ (19.1%), against an empirical ceiling near 20.7%.
Naming the closed form lets later positivity and certificate declarations refer to a single real constant rather than an expanded $J$ expression.
proof idea
Pure definitional abbreviation: the real is introduced as $\varphi-3/2$. No lemmas, no tactics, no proof obligations.
why it matters
Gives the concrete numerical scale used by the structural theorem of this module (Plan v7, 122nd pass): $\eta_{\mathrm{Penrose}}=J(\varphi)\cdot(\Omega_H/\Omega_{\mathrm{ISCO}})$. Sibling positivity and certificate objects package the resulting 19.1% maximal-Kerr figure and the claim of consistency with the empirical ~20.7% bound.
The constant is exactly $J(\varphi)$, so it sits at the T5–T6 junction of the forcing chain: the unique cost evaluated on the forced self-similar ratio. It therefore links the $\varphi$-ladder geometry to classical black-hole energy extraction without extra axioms.
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