The Computational Incompleteness of the Kardashev Scale in the Era of Artificial Superintelligence

The Kardashev scale ranks technological civilizations by the power they command. This paper argues that a single energy axis is incomplete once a civilization's cognition is carried by an artificial superintelligence, because the number of useful operations obtained per joule can vary by many orders of magnitude at fixed power. Building on the capability index Ξ = K + B of Gurses (2026), the paper defines a two-coordinate descriptor Ψ = (K, Ξ_ASI) that adds the fraction of power devoted to computation and an algorithmic multiplier. For a civilization using one solar luminosity (3.828 × 10^26 W, K = 2.06), it shows that the rate of irreversible bit operations is fixed by the outer radiator temperature alone, about 1.3 × 10^48 per second at 30 K, and that nesting shells in a Matrioshka-brain arrangement adds nothing to this budget. Biological and ASI-directed civilizations at the same power differ by four to seven orders of magnitude in operations per second through allocation and hardware alone. The same offset holds at Type I and Type III, where planetary heat rejection and galactic light-travel time set the additional constraints. On the observational side, a 30 K radiator around a Sun-like star has a radius of about 170 AU and peaks near 97 µm. It is outside the WISE bands used by current Dyson-sphere searches but bright in existing far-infrared surveys (about 70 Jy at 100 µm from 100 pc). The paper proposes a four-step search of archival IRAS, AKARI, Herschel and Planck data, including a distance-independent angular-size test that separates an opaque shell from a natural dust cloud. It also gives predicted fluxes for complete and partial shells, and labels which parts of the argument are established physics, exploratory, or speculative