canonicalThreshold
plain-language theorem explainer
The canonical threshold is the real constant φ − 3/2, with φ the golden ratio. It is the fixed comparison level used inside the CPT-from-J-cost structural development. Anyone working the domain-cost or CPT3 certificate lemmas in this module cites it. The body is a one-line arithmetic definition in the RS constant φ.
Claim. Let $\varphi$ be the golden ratio. The canonical threshold is the real number $\varphi - 3/2$.
background
This module develops CPT invariance as a structural theorem of Recognition Science: combined charge conjugation (σ ↦ −σ), spatial reflection in D = 3, and recognition-tick reversal leave the J-cost unchanged. Status is zero sorry and zero axioms.
The golden ratio φ enters from the constants layer; it is the unique self-similar fixed point forced by the T6 step of the unified forcing chain. The J-cost itself is the unique cost satisfying the Recognition Composition Law, J(x) = (x + x^{-1})/2 − 1. Sibling definitions introduce a domain cost on the same real line and prove it nonnegative, so a fixed positive threshold is the natural comparison scale for certificate statements.
proof idea
Pure definition: the real is obtained by subtracting the rational 3/2 from the imported constant φ. No lemmas, no tactics, no proof obligations.
why it matters
The constant sits inside the CPT Theorem 3 from J-cost module, whose goal is to identify CPT with J-cost invariance under the three discrete reversals. It supplies the numeric cut used by the positivity lemma for the same threshold and by the CPT3 certificate constructors that inhabit the structural certificate type. Via φ it inherits the T6 forcing landmark (self-similar fixed point) and therefore sits on the same foundation chain that later forces the eight-tick octave and D = 3. It does not itself close the CPT argument; it only fixes the comparison level those later certificates employ.
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