canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold constant appearing in the domain-coverage milestone is strictly positive. Anyone assembling or citing the FinalModule_1396 structural certificate needs this inequality. The proof is a one-line wrapper: unfold the definition, then finish by linear arithmetic from the bound φ > 1.5.
Claim. The canonical threshold is strictly positive: $0 < T_{\mathrm{can}}$, where $T_{\mathrm{can}}$ is the real constant obtained by unfolding the milestone's canonical-threshold definition (an expression in the golden ratio $\varphi$).
background
Module FinalModule_1396 is a Recognition Science structural certificate for a domain-coverage milestone (Plan v7, 109th pass). Status is fully proved: zero sorry, zero axioms. Sibling definitions introduce a domain cost, the canonical threshold itself, and a MilestoneCert bundle that packages the certificate.
The only external numeric fact used here is the golden-ratio lower bound $\varphi > 1.5$. That bound is elementary: $\varphi = (1+\sqrt{5})/2$ and $\sqrt{5} > 2$, so $\varphi > 3/2$. In RS units $\varphi$ is the self-similar fixed point forced at T6 of the unified forcing chain; positivity statements about thresholds built from $\varphi$ routinely reduce to such elementary inequalities.
No J-cost, defect distance, or mass-ladder apparatus is required for this line. The local setting is purely the arithmetic of the milestone constants imported from Constants and Cost.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, exposing an explicit real expression in $\varphi$. Discharge the resulting strict inequality by linarith supplied with the upstream lemma phi_gt_onePointFive ($1.5 < \varphi$). No case splits, no induction, no cost-functional identities.
why it matters
Supplies the positivity half of the numeric side-conditions for the FinalModule_1396 domain-coverage certificate. Downstream in the same module the MilestoneCert record and its inhabited instance package this fact with the domain-cost identities; without $T_{\mathrm{can}} > 0$ the certificate would not type-check as a structural theorem.
In the broader RS stack the result is bookkeeping rather than a new physical law: it confirms that a threshold built from the T6 fixed point $\varphi$ sits in the positive reals, consistent with other positive RS scales (Berry threshold $\varphi^{-1}$, eight-tick period, etc.). It does not itself force dimension, the octave, or the fine-structure band. used_by is presently empty outside the module, so its role is local to closing the milestone cert.
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