We prove a recognition principle for motivic infinite P 1 -loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E∞-spaces. A framed motivic space is a motivic space equipped with transfers along finite syntomic morphisms with trivialized cotangent complex in K-theory. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties, and in particular for the motivic sphere spectrum, in terms of Hilbert schemes of points in affine spaces. Contents 1. Introduction 1.1. The recognition principle in ordinary homotopy theory 1.2. The motivic recognition principle 1.3. Framed correspondences 1.4. Outline of the paper 1.5. Conventions and notation 1.6. Acknowledgments 2. Notions of framed correspondences 2.1. Equationally framed correspondences 2.2. Normally framed correspondences 2.3. Tangentially framed correspondences 3. The recognition principle 3.1. The S 1 -recognition principle 3.2. Framed motivic spaces 3.3. Framed motivic spectra 3.4. The Garkusha-Panin theorems 3.5. The recognition principle 4. The ∞-category of framed correspondences 4.1. ∞-Categories of labeled correspondences 4.2. The labeling functor for tangential framings 4.3. The symmetric monoidal structure 5. Applications 5.1. Representability of the motivic sphere spectrum 5.2. Motivic bar constructions Date: August 7, 2019. E.E. and A.K. were supported by Institut Mittag-Leffler postdoctoral fellowships. M.H. was partially supported by NSF grants DMS-1508096 and DMS-1761718. M.Y. was supported by SFB/TR 45 "Periods, moduli spaces and arithmetic of algebraic varieties".

We prove that the $\infty $ -category of $\mathrm{MGL} $ -modules over any scheme is equivalent to the $\infty $ -category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$ -loop spaces, we deduce that very effective $\mathrm{MGL} $ -modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$ , $\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$ .

We relate the recognition principle for infinite P 1 -loop spaces to the theory of motivic fundamental classes of Déglise, Jin, and Khan.We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with A n /(A n − 0), and the Gysin transfers defined via Verdier's deformation to the normal cone.We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calmès and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.

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