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We give a fully faithful integral model for spaces in terms of$\mathbb{E}_{\infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The keytechnical input is the development of a homotopy coherent Frobenius action on acertain subcategory of $p$-complete $\mathbb{E}_{\infty}$-rings for each prime$p$. Using this, we show that the data of a simply connected finite complex $X$is the data of its Spanier-Whitehead dual as an $\mathbb{E}_{\infty}$-ringtogether with a trivialization of the Frobenius action after completion at eachprime.  In producing the above Frobenius action, we explore two ideas which may be ofindependent interest. The first is a more general action of Frobenius inequivariant homotopy theory; we show that a version of Quillen's$Q$-construction acts on the $\infty$-category of $\mathbb{E}_{\infty}$-ringswith "genuine equivariant multiplication," which we call global algebras. Thesecond is a "pre-group-completed" variant of algebraic $K$-theory which we callpartial $K$-theory. We develop the notion of partial $K$-theory and give acomputation of the partial $K$-theory of $\mathbb{F}_p$ up to $p$-completion.

Integral Models for Spaces via the Higher Frobenius