The universal thickening of the field of real numbers

We define the universal 1-adic thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield T R. Then we prove that the archimedean Witt construction in the context of hyperfields allows one to recover a field from a hyperfield, and we obtain the universal pro-infinitesimal thickening R∞ of R. Finally, we provide the real analogues of several algebras used in the construction of the rings of p-adic periods. We supply the canonical decomposition of elements in terms of Teichmüller lifts, we make the link with the Mikusinski field of operational calculus and compute the Gelfand spectrum of the archimedean counterparts of the rings of p-adic periods.