canonicalThreshold_pos

The ambient module is the RS Gap-45 exact package (v2). Its stated goal is the structural identity gap-45 = 45 = D²(D+2) = 9·5, forced once spatial dimension is D = 3, with zero sorry and zero axioms. The same module packages a small certificate around a domain cost and a canonical threshold built from the golden ratio φ.

φ is the unique positive root of x² = x + 1, written (1 + √5)/2. The only upstream lemma used here is the tighter numerical bound φ > 1.5, which follows at once from √5 > 2. The canonical threshold is the named real constant in this file whose positivity is required before the certificate can be inhabited; the present theorem records that sign.

One-line wrapper. Unfold the definition of the canonical threshold, then invoke linarith on the single hypothesis φ > 1.5. No further case splits or Recognition-cost identities are required; positivity is pure real arithmetic once the definition is expanded.

Gap-45 is one of the small integer landmarks the framework ties to D = 3 (alongside the eight-tick octave and the forcing chain T0–T8). This positivity lemma is a local hygiene fact inside that structural package: the certificate and its inhabited instance sit as siblings, and they presuppose a positive threshold. The declaration does not itself prove the identity 45 = D²(D+2); it only clears the sign obligation so those certificate objects can be formed cleanly. No external used-by edges are recorded, so its role is internal to the Gap-45 exact module.