We construct a lift of the degree filtration on the integer-valued polynomials to (even \({{\,\textrm{MU}\,}}\) -based) synthetic spectra. Namely, we construct a bialgebra in modules over the evenly filtered sphere spectrum which base-changes to the degree filtration on the integer-valued polynomials. As a consequence, we may lift the Hilbert additive group scheme to a spectral group scheme over \(\mathbb {A}^1 / \mathbb {G}_m\) . We study the cohomology of its deloopings, and show that one obtains a lift of the filtered circle, studied in [25]. At the level of quasi-coherent sheaves, one obtains synthetic lifts of the \(\mathbb {Z}\) -linear \(\infty \) -categories of \(S^1_{{{\,\textrm{fil}\,}}}\) -representations. Our constructions crucially rely on the use of the even filtration of Hahn–Raksit–Wilson; it is linearity with respect to the even filtered sphere that powers the results of this work.
