canonicalThreshold
plain-language theorem explainer
The canonical recognition threshold is the real constant φ − 3/2, with φ the golden-ratio fixed point of the forcing chain. Domain-cost comparisons and positivity certificates in the 2026 state module treat this value as the cut between admissible and inadmissible defects. The declaration is a one-line arithmetic definition in the RS-native constant φ.
Claim. Define the canonical threshold by $\mathrm{canonicalThreshold} := \varphi - \tfrac{3}{2}\in\mathbb{R}$, where $\varphi$ is the golden ratio (self-similar fixed point forced by the Recognition Science chain).
background
The ambient module is the Recognition Science 2026 state-of-the-art certificate: a structural package claiming a complete T0–T8 forcing chain, derived constants, and zero sorries. It imports the global RS constants (including φ) and the cost layer.
In RS, φ arises as the unique self-similar fixed point (forcing step T6) of the cost functional whose unique minimizer is the J-cost $J(x)=(x+x^{-1})/2-1$. Domain costs built from J are nonnegative and vanish only at the identity scale; thresholds on those costs separate regimes in which a recognition event is accepted or rejected.
The sibling positivity lemma for this constant, and the nearby domain-cost identities, indicate that φ − 3/2 is the concrete numerical cut used for those comparisons inside the certificate.
proof idea
Pure definition: the real is introduced by the arithmetic expression φ − 3/2. No proof obligations, lemmas, or tactics are involved; downstream facts (positivity, comparisons with domain cost) are separate declarations.
why it matters
Inside the 2026 structural certificate this constant is the named numerical gate for domain-cost arguments. It sits next to domain-cost nonnegativity and the positivity fact for the same threshold, so any certificate clause that asserts a cost lies above or below the cut refers here.
Framework-wise it is an RS-native scale built from φ (T6), not an external fit parameter. It is smaller than the Berry creation scale φ^{-1} and far below the eight-tick and dimensional landmarks (T7–T8), so it functions as a fine local acceptance threshold rather than a global period or dimension constraint. The module presents the whole package as closed structural mathematics; this definition is one of the elementary constants that package relies on.
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