canonicalThreshold_pos

Astrophysics RS Module 9 treats the interstellar-medium dust fraction as a structural prediction: $J(\varphi)^2 \approx 1.39%$, set against the empirical $\sim 1%$ level. The module is recorded as a structural theorem (zero sorry, zero axiom).

Here $J$ is the unique cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$, and $\varphi$ is the golden-ratio fixed point of self-similarity (forcing steps T5–T6). The canonical threshold is the non-negative scale extracted from that cost evaluation for the dust-fraction comparison.

The sole upstream lemma is the tighter bound $\varphi > 1.5$, which follows at once from $\sqrt{5}>2$.

One-line wrapper. Unfold the definition of the canonical threshold, then discharge the resulting inequality by linarith using the lemma $\varphi > 1.5$. No further algebraic identities or case splits are required once the definition is expanded.

Positivity of the threshold is a necessary sign check inside Astrophysics RS Module 9, whose headline claim is the ISM dust-fraction match $J(\varphi)^2 \approx 1.39%$ versus the empirical $\sim 1%$ figure. No downstream theorems are yet recorded, but the sibling certificate bundle packages the module as a structural theorem; the sign fact is required before any comparison or non-negativity argument can proceed. It inherits the forcing-chain pedigree (T5 $J$-uniqueness, T6 $\varphi$ fixed point) rather than an ad-hoc astrophysical fit.