canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold in the elastic-modulus construction is strictly positive. Anyone assembling or citing the ElasticMod4 structural certificate needs this side-condition. The proof unfolds the threshold definition and closes by linear arithmetic from the bound φ > 1.5.
Claim. The canonical threshold (the real constant built from $\varphi$ in the elastic-modulus module) is strictly positive: $0 < t_{\mathrm{can}}$.
background
Recognition Science builds constants from the golden ratio $\varphi=(1+\sqrt{5})/2$, forced as the self-similar fixed point (T6). A standard tighter bound is $\varphi>1.5$, which follows from $\sqrt{5}>2$.
The module ElasticMod4 is a structural package (0 sorry, 0 axiom) for an elastic-modulus consistency check on the $\varphi$-ladder: $\varphi^{10}\cdot 2,\mathrm{GPa}\approx 246,\mathrm{GPa}$ against empirical steel $\sim 200,\mathrm{GPa}$. Inside that package a real threshold is defined from $\varphi$; positivity of that threshold is a well-formedness fact used by the module certificate.
Upstream, the lemma $\varphi>1.5$ supplies the only numeric input the positivity argument needs.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then feed the lemma $\varphi>1.5$ to linear arithmetic (linarith). No rewriting beyond unfold and no case split are required; the inequality is an immediate consequence of the lower bound on $\varphi$.
why it matters
This is infrastructure inside ElasticMod4, the structural theorem package that records the RS elastic-modulus check $E\sim\varphi^{10}\cdot 2,\mathrm{GPa}$. Positivity of the threshold is the basic side-condition any certificate comparing domain costs against that threshold must carry. Sibling objects in the module (ElasticMod4Cert, cert, cert_inhabited) are the natural consumers; no external dependents are recorded yet.
Framework landmarks in play are T6 ($\varphi$ forced) and the $\varphi$-ladder mass/scale bookkeeping. The steel comparison is a consistency check, not a derivation of continuum elasticity from the Recognition Composition Law or the forcing chain T0–T8.
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