canonicalThreshold_pos

Module 5 of the Recognition Science forcing chain records structural theorems T0–T8, all proved in Lean with zero sorry and zero extra axioms. The local setting is the foundation layer: constants and cost primitives that later modules quote when forcing dimension, the eight-tick octave, and the φ-ladder.

The golden ratio $\varphi = (1+\sqrt{5})/2$ is the self-similar fixed point forced at T6. An upstream lemma sharpens the elementary bound to $\varphi > 1.5$ (via $\sqrt{5} > 2$). The canonical threshold is the named constant obtained by unfolding a short expression in $\varphi$; sibling declarations package the associated domain cost and its nonnegativity.

Positivity of that threshold is the minimal arithmetic fact needed before any comparison of domain costs against a creation or recognition cutoff.

One-line wrapper. Unfold the definition of the canonical threshold so the goal is an explicit linear inequality in $\varphi$. Discharge it by linarith using the upstream lemma $\varphi > 1.5$. No further case splits or cost identities are required.

Sits in Foundation RS Module 5, the structural block that claims the full forcing chain T0–T8 with zero sorry. It supplies the positivity half of the canonical-threshold package that domain-cost lemmas sit on. In the broader framework it aligns with the Berry-scale cutoff at order $\varphi^{-1}$ (equivalently $\varphi-1$), the natural positive unit against which ladder gaps and creation thresholds are measured. The dependency graph lists no downstream consumers yet; the lemma is infrastructure waiting on cost and forcing certificates in the same module (the inhabited cert and domain-cost nonnegativity siblings).