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. Radiopharmaceutical and RS chemistry work cites it as the irreducible labeling-failure floor (~0.118), whose square is ~1.4%. It is a one-line definition from the RS constant φ.
Claim. The canonical threshold is the real constant $\varphi - 3/2$, where $\varphi$ is the golden-ratio fixed point of Recognition Science.
background
Recognition Science forces the unique cost $J(x)=(x+x^{-1})/2-1$ (T5). At the self-similar fixed point $\varphi$ (T6) one has the algebraic identity $J(\varphi)=\varphi-3/2$. The module treats radiopharmaceutical labeling yield as $1-J(\varphi)^2$, so the squared threshold is the structural recognition floor (~1.4%) and the complementary yield is ~98.6%.
Local setting (module doc): structural package with zero sorry and zero axioms; clinical requirement is >90% yield, good syntheses exceed 95%, and $J(\varphi)^2$ is identified as the recognition-theoretic minimum labeling loss. Constants and Cost are imported; $\varphi$ comes from the RS constants module.
proof idea
Pure definitional abbreviation: the real phi - 3/2 with phi from IndisputableMonolith.Constants. No lemmas, no tactics, no proof body.
why it matters
Anchors the chemistry module's yield story to the forcing chain: T5 supplies J-uniqueness and T6 forces φ, so the threshold is not an empirical fit but a derived constant. Sibling positivity and certificate lemmas (domain cost non-negativity, RadioYieldCert) compare against this value when stating that labeling yield sits above the recognition floor. It converts the abstract J-cost into the concrete ~1.4% failure budget quoted for radiopharmaceutical labeling.
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