canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold appearing in the FinalModule_1395 domain-coverage milestone is strictly positive. Milestone and certificate authors cite this to discharge the positivity side-condition on that threshold. The proof is a one-line wrapper: unfold the definition and finish by linear arithmetic from φ > 1.5.
Claim. The canonical threshold is strictly positive: $0 < \tau$, where $\tau$ is the threshold constant defined from the golden ratio $\varphi$ in this module.
background
FinalModule_1395 is a Recognition Science structural-certificate module (Plan v7, 109th pass): zero sorry, zero axioms, aimed at a domain-coverage milestone. It imports Mathlib, Constants, and Cost.
The only external fact used here is the tighter lower bound on the golden ratio: $\varphi = (1+\sqrt{5})/2 > 1.5$, recorded as phi_gt_onePointFive (from $\sqrt{5} > 2$). The canonical threshold is a local constant built from $\varphi$; after unfolding, its positivity is an immediate arithmetic consequence of that bound.
Sibling definitions in the same file introduce domain cost, the threshold itself, and the milestone certificate bundle that packages these facts.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then apply linarith to the hypothesis $\varphi > 1.5$. No further lemmas or case splits are required; the goal reduces to a linear inequality in $\varphi$.
why it matters
Supplies the positivity side-condition for the canonical threshold inside the FinalModule_1395 structural certificate. Downstream siblings (MilestoneCert, cert, cert_inhabited) package domain-coverage claims; this lemma is the arithmetic gate that lets those certificates assert a positive threshold without reopening the definition of $\varphi$.
In the broader RS stack it sits far downstream of the forcing chain: only the elementary bound $\varphi > 1.5$ is used, not J-uniqueness (T5), the eight-tick octave (T7), or $D=3$ (T8). No external used_by edges are recorded; the lemma is local glue for the milestone module.
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