Applying nonconnective K-theory to the enhanced interface

By [F1], the whole common-family diagram takes values in essentially small stable idempotent-complete infinity-categories. The inclusion of small models into this essentially small domain is fully faithful and essentially surjective, hence an equivalence of infinity-categories. Extend 𝕂 along this domain equivalence once, or equivalently compose the whole diagram with one inverse equivalence of domains. This gives a diagram of spectra and retains every coherence. Its different inverse choices are naturally equivalent. Thus no independent choice of a skeleton for each object or morphism is used, and the invariant is independent of enhanced presentation.

Apply 𝕂 to the exact pullbacks in [F2]. A functor preserves their identity and composition coherences, so these give contravariance on schemes and pairs. In particular the inclusion Perf_T(X)→Perf(X) induces the supported-to-ambient map of spectra.

By the tensor universal property in [F3], each separately exact pairing of [F2] is an exact functor out of the corresponding Cat_perf tensor product. Compose its image under 𝕂 with the lax monoidal comparison 𝕂(C)∧𝕂(D)→𝕂(C⊗D). The coherent ambient product and unit make the ambient spectrum an E∞ ring, while the coherent module action makes the supported spectrum a module; the same construction gives the mixed pairings. Coherence follows from the supplied algebra/module operations and the lax symmetric monoidal functor, not merely binary maps in homotopy categories. No unit is inserted into the supported category.

These steps only apply the invariant and its coherent products to exact functors. They do not identify π₀ with a separately defined Grothendieck group, identify negative groups by another construction, or prove that a particular scheme-support sequence is a localization sequence. Thus they establish precisely the stated interface.