canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold constant in the FinalModule_1400 domain-coverage certificate is strictly positive. Anyone assembling or citing that milestone certificate needs this inequality before comparing domain costs against the threshold. The proof is a one-line wrapper: unfold the definition, then finish by linear arithmetic from φ > 1.5.
Claim. The canonical threshold (the real constant built from $\varphi$ in this milestone module) is strictly positive: $0 < T_{\mathrm{can}}$.
background
FinalModule_1400 is a Recognition Science structural certificate module (Plan v7, 109th pass) for a domain-coverage milestone. Status is fully proved: zero sorry, zero axioms. It sits in the physics layer and imports the global constants and the J-cost infrastructure.
The golden ratio $\varphi = (1+\sqrt{5})/2$ is the self-similar fixed point forced at T6 of the unified forcing chain. The sibling definition canonicalThreshold packages a simple real expression in $\varphi$ used as the cutoff for domain-cost comparisons in the certificate.
Upstream, phi_gt_onePointFive supplies the tighter numerical bound $\varphi > 1.5$, justified by $\sqrt{5} > 2$ so $(1+\sqrt{5})/2 > 3/2$. That bound is exactly what linear arithmetic needs after the threshold is unfolded.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, exposing a linear inequality in $\varphi$. Discharge the resulting goal by linarith using the single lemma $\varphi > 1.5$. No further case splits or cost identities are required.
why it matters
Positivity of the cutoff is a prerequisite for any certificate that treats the canonical threshold as a strict barrier (domain cost above or below the line). The module presents this as part of the structural theorem package for the domain-coverage milestone; sibling inhabitants of MilestoneCert rely on a well-defined positive scale.
In the broader RS picture the threshold is φ-native, consistent with T6 (φ forced) and with other φ-ladder cutoffs such as the Berry creation scale $\varphi^{-1}$. No downstream theorems currently list this lemma as a direct edge, so its role is local to the milestone certificate rather than a global forcing-chain step.
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