canonicalThreshold_pos
plain-language theorem explainer
The canonical reionization threshold is strictly positive. Cosmology workers in the Recognition Science stack cite this when they need a positive scale inside the φ^4–φ^5 reionization window. The proof is a one-line wrapper: unfold the threshold and finish by linear arithmetic from φ > 1.5.
Claim. The canonical reionization threshold $T_{\mathrm{can}}$ (the RS scale tied to the $\varphi^4$–$\varphi^5$ window) satisfies $0 < T_{\mathrm{can}}$.
background
Module 8 of the RS cosmology stack treats reionization as the structural interval $\varphi^4$ to $\varphi^5$ (numerically about 6.85–11.09), matched to observed $z_{\mathrm{reion}}\sim 7$–$10$. 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. Constants supplies the elementary bound $\varphi>1.5$, proved from $\sqrt{5}>2$. The canonical threshold is the local positive scale built from $\varphi$ that anchors the reionization window; domain-cost siblings sit beside it but are not required here.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then close by linarith using the upstream lemma $\varphi>1.5$. No further case splits or cost identities are needed; positivity is pure arithmetic once the definition is exposed.
why it matters
Supplies the positivity fact required by the Module-8 reionization certificate (RSCosmo008Cert and its inhabited witness). Without a strictly positive canonical scale the $\varphi^4$–$\varphi^5$ window cannot be treated as a physical threshold. The result sits downstream of T6 ($\varphi$ forced) and of the elementary Constants bound $\varphi>1.5$; it does not itself re-prove the redshift match, only the sign of the scale used in that match. Status is fully proved structural scaffolding for the cosmology layer.
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