canonicalThreshold
plain-language theorem explainer
Defines the canonical recognition-cost threshold as φ − 3/2, which equals J(φ) for the unique J-cost. Gravity and geodesic analysts cite it as the first nonzero cost level where recognition geodesics leave the straight-line ground-state orbit. The body is a one-line real constant built from the RS golden ratio.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ is the golden ratio. Equivalently it is the J-cost evaluated at $\varphi$, i.e. $J(\varphi) = (\varphi + \varphi^{-1})/2 - 1$.
background
Recognition Science forces a unique nonnegative cost $J(x) = (x + x^{-1})/2 - 1$ (T5) and the self-similar fixed point $\varphi$ (T6). On the recognition manifold, geodesics are curves that minimize $\int J(r(t)),dt$. The module treats the zero-cost locus $J=0$ as the recognition ground-state orbit (straight-line trajectories) and identifies the first departure from that orbit with the cost level $J(\varphi)$.
A short algebraic identity shows $J(\varphi)=\varphi-3/2$: because $\varphi+\varphi^{-1}=2\varphi-1$, one has $(\varphi+\varphi^{-1})/2-1=\varphi-3/2$. The present definition simply names that real constant so later positivity and comparison lemmas can refer to it by a single symbol.
proof idea
Pure definition: the real constant is introduced by the closed-form expression $\varphi - 3/2$. No proof obligations; downstream lemmas (e.g. positivity of the threshold) discharge any needed inequalities separately.
why it matters
Inside the Recognition Geodesic v3 development this constant is the numerical marker for the first nontrivial cost shell. The module status line frames the whole file as a structural theorem (0 sorry, 0 axiom) about geodesics that minimize integrated J-cost; the threshold supplies the concrete scale at which those geodesics leave the $J=0$ ground-state orbit. It therefore sits between the abstract J-uniqueness/forcing chain (T5–T6) and any later gravity or curvature statements that compare domain costs against a fixed positive level. Sibling positivity and certificate constructions are expected to quote it directly.
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