canonicalThreshold
plain-language theorem explainer
Defines the canonical threshold as φ − 3/2 (≈ 0.118), which equals the J-cost J(φ). Cited by the PMNS reactor θ₁₃ structural certificate when comparing domain cost to a fixed RS scale. Pure definitional assignment; no proof obligations.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ denotes the golden ratio fixed by self-similarity.
background
In Recognition Science the cost functional is $J(x) = (x + x^{-1})/2 - 1$. For the golden ratio $\varphi$ one has $J(\varphi) = \varphi - 3/2 \approx 0.118$, so the present constant is simply that value written without invoking $J$.
The module treats the PMNS reactor angle $\theta_{13}$, whose measured $\sin^2\theta_{13}\approx 0.0218$. Several algebraic combinations of $J(\varphi)$ (powers, ratios by $\varphi^2$, etc.) sit near that figure; the module records a structural rather than numerical match.
Local setting: RS-native constants and the cost layer are imported; the threshold is the scale against which a domain cost for the mixing angle is later compared.
proof idea
Definitional abbreviation. The body is the closed-form real expression $\varphi - 3/2$; no lemmas or tactics are involved.
why it matters
Supplies the fixed RS scale used when the module builds the PMNS $\theta_{13}$ certificate (siblings such as domain-cost non-negativity and the inhabited certificate). Ties the reactor angle discussion to the T5/T6 landmarks: $J$-uniqueness and $\varphi$ as the self-similar fixed point. Does not close a forcing-chain step; it is scaffolding for a structural comparison of $\sin^2\theta_{13}$ to powers of $J(\varphi)$.
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