The framed correspondences $T$-spectrum of a smooth affine scheme is a $T$-spectrum of Nisnevich sheaves. We construct the motivically equivalent model of the $T$-spectrum representable in the category of pairs of smooth affine ind-schemes in the case of a base scheme of a finite Krull dimension. In other words, the motivic spaces of $(\mathbb{P},\infty)^{\wedge \infty}$-loops in the relative motivic sphere $\mathbb{A}_Y^{\infty+l}/(\mathbb{A}_Y^{\infty+l}-0)$ are represented in the category of pairs of smooth affine ind-schemes. The construction in not functorial on the category of smooth affine schemes, but it is so on the category of smooth affine framed schemes, that is defined in the text.