External actual augmented EM and smooth slice tower comparison

PDF p. 9: Actual low-weight motivic projections and their normalization

Assume the Axiom of Choice and the fixed compatible site and universe bounds for the selected perfect K-theory and motivic-filtration realizations. Let F be a prime field. Construction4.34 gives an actual natural morphism of augmented filtered spectra Fil_cla K(X)→Fil_mot K(X) on Sm_F, equal to identity K(X) on stage zero; Corollary6.4 makes it a filtered equivalence. Its graded comparison maps are exactly (4.9), so they commute with the stage-to-layer maps.

For X smooth of finite type over an arbitrary extension field k/F, use the ACTUAL map Fil_mot K(X)→Fil_cdh KH(X) of Construction4.36. X is regular Noetherian, so Theorem6.1(2)/Remark6.3 makes it an augmented filtered equivalence, over K(X)→KH(X). Remark3.6 explicitly states that Fil_cdh KH is the slice filtration of KGL. Thus the comparison is between the tower augmentations and their induced cofibers, not only cohomology groups. The canonical graded maps and EM c₁ normalization must be retained. On any supported pair, applying homotopy fibers to such a natural comparison gives the corresponding supported tower comparison with these same signs.

Given: Choice, the stated compatible site and universe bounds, and the exact maps and field hypotheses just stated; K is the perfect-complex nonconnective invariant used by EM (explicitly precomposed with Perf in Construction5.5), as in the completed run supplier.

Goal: Specify the map-level tower bridge needed when the classical slice calculation is used in the final combined lemma.

[A1] The assumed The Axiom of Choice and size bounds discharge the local realization premises of Nonconnective K theory of perfect complexes and Motivic filtration and image groups whenever the imported maps in [F1] are interpreted as maps of these specified local K-theory and filtration models. These are inherited construction premises, not additional geometric hypotheses attributed to Elmanto–Morrow.

[F1] The exact EM comparison passages listed in the frontmatter give the morphisms of augmented filtered spectra and their equivalence hypotheses.

Import [F1], including the exact status/reading limit: the arbitrary-field cdh-KH/slice identification in Remark3.6 is stated there and attributed to the then-in-preparation joint work [7]; its proof was not read and is not reconstructed. Corollary6.4 by itself is NOT asserted for arbitrary k. Low-weight lemmas below use the direct EM normalized maps of the preceding import; they do not pretend to supply a new proof of this foundational tower bridge. The final supported author must preserve this explicit external-input provenance if using the arbitrary-field bridge.