canonicalThreshold_pos
plain-language theorem explainer
The canonical materials threshold is strictly positive. Anyone citing the iron-melting structural certificate in RS Materials Module 5 needs this positivity fact. The proof is a one-line unfold-and-linarith wrapper off the bound φ > 1.5.
Claim. The canonical threshold constant satisfies $0 < T_{\mathrm{can}}$. After unfolding its definition in terms of the golden ratio $\varphi$, this is immediate from $\varphi > 1.5$.
background
Materials RS Module 5 records the iron-melting match: $\varphi^{15}\cdot\varphi^{0.7},\mathrm{K} = 1814,\mathrm{K}$ against the experimental $1811,\mathrm{K}$ (about $0.2%$). The module 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 upstream lemma phi_gt_onePointFive supplies the tighter real bound $\varphi > 1.5$, proved from $\sqrt{5} > 2$. The canonical threshold is a module-local real constant built from $\varphi$; positivity is the elementary gate before any cost or certificate assembly that uses it.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then discharge the resulting linear inequality by linarith using the single upstream fact $\varphi > 1.5$. No further case splits or Recognition-cost identities are required.
why it matters
Positivity of the canonical threshold is a local hygiene lemma inside the iron-melting materials certificate (RSMatl005Cert / cert_inhabited in the same module). Downstream edges are empty in the graph snapshot, so the lemma presently supports only in-module certificate construction rather than a named parent theorem elsewhere.
In the broader RS picture it sits under the materials domain that consumes the forced $\varphi$ (T6) and the J-cost apparatus imported from Cost. It does not itself touch the RCL, the eight-tick octave, or the $\alpha$ band; it only guarantees that the threshold scale used when comparing the $\varphi$-ladder melting prediction to experiment is a positive real.
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