= Closed constant-equation-of-state conformal lifetime
{title2=$\Delta\tau=2\pi/(1+3w)$}
For positive curvature and zero <cosmological constant>, the single-fluid Friedmann solution has a bang-to-crunch <conformal time> interval $2\pi/(1+3w)$ for $w\geq0$. A <radial null geodesic in FLRW spacetime> travels that same great-circle comoving distance. Relative to a three-sphere circumference $2\pi$, it is $1/(1+3w)$ circuits: one for dust and one half for radiation, with the final point reached only in the singular-endpoint limit.
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