  We construct geometric models for the $\mathbb P^1$-spectrum $M_{\mathbb
P^1}(Y)$, which computes in Garkusha-Panin's theory of framed motives
\cite{GP14} a positively motivically fibrant $\Omega_{\mathbb P^1}$ replacement
of $\Sigma_{\mathbb P^1}^\infty Y$ for a smooth scheme $Y\in \Sm_k$ over a
perfect field $k$. Namely, we get the $T$-spectrum in the category of pairs of
smooth ind-schemes that defines $\mathbb P^1$-spectrum of pointed sheaves
termwise motivically equivalent to $M_{\mathbb P^1}(Y)$.
