canonicalThreshold_pos

This module fixes string compactification radii on the Recognition φ-ladder by writing $R_{\mathrm{comp}} = \ell_{\mathrm{Pl}},\varphi^{-k}$. Near the Planck scale one takes $k=0$; the electroweak scale sits near $k\approx 106$ rungs. The whole development is marked structural (zero sorry, zero axioms).

The golden ratio $\varphi=(1+\sqrt{5})/2$ is the self-similar fixed point forced at T6 of the unified forcing chain. The only upstream fact used here is the elementary tightening $\varphi>1.5$, which follows at once from $\sqrt{5}>2$. The canonical threshold is a φ-native positive scale used by the local cost and certificate definitions (domainCost, StringLengthCert) as a strict lower cut.

One-line wrapper. Unfold the definition of the canonical threshold, then discharge the resulting linear inequality by linarith using the lemma $\varphi>1.5$. No further Recognition identities are required.

Positivity of the threshold is the first arithmetic gate for the string-length certificate package in this file (StringLengthCert, cert, cert_inhabited). Without it the nonnegativity and inhabitance claims for the certificate cannot even be stated cleanly. In the broader RS picture it sits under the φ-ladder mass/length bookkeeping (T6 φ fixed point; rung arithmetic as in the mass formula), specialized here to compactification radii rather than particle masses. The module itself is already closed (structural theorem); this lemma is a tiny but necessary sign check inside that closure. No external downstream edges are recorded.