canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold equals the J-cost at the golden ratio and is strictly positive. Anyone bounding domain costs or citing the RS J-minimum story uses this fact. The proof unfolds the definition to φ − 3/2 and finishes by linear arithmetic from the bound φ > 1.5.
Claim. Let $\varphi = (1+\sqrt{5})/2$ be the golden ratio. The canonical threshold $\varphi - 3/2$ (equal to the J-cost $J(\varphi)$) satisfies $0 < \varphi - 3/2$.
background
Module 4 of the RS forcing chain records the elementary calculus of the Recognition J-cost. By definition $J(x) = (x + x^{-1})/2 - 1$, so $J(1) = 0$ (equilibrium) and, using $\varphi^{-1} = \varphi - 1$, one obtains $J(\varphi) = \varphi - 3/2$. The module treats that value as the canonical threshold.
The only upstream fact needed is the numerical lower bound $\varphi > 1.5$, proved from $\sqrt{5} > 2$. That inequality is exactly the content of the Constants lemma used here. The local setting is purely structural: zero sorry, zero axiom, and no physical units yet.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold (exposing $\varphi - 3/2$), then invoke linarith on the imported lemma $\varphi > 1.5$. No further case splits or algebraic identities are required.
why it matters
In the Recognition Science forcing chain the J-cost is forced unique (T5) with self-similar fixed point $\varphi$ (T6). Positivity of $J(\varphi)$ is the first quantitative separation of the golden-ratio rung from the equilibrium $J(1) = 0$; later mass-ladder and domain-cost arguments treat that gap as a hard lower scale.
This module labels the result a structural theorem. No downstream consumers are wired in the present graph, so the lemma currently stands as a certified primitive for any later certificate (e.g. the module's own RSForcingChain004Cert) that needs $0 < J(\varphi)$.
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