Large language models are routinely reported as achieving "IQ 120" or "above-averagehuman intelligence," and these claims are widely received as evidence of progress towardartificial general intelligence. This paper argues that such claims are scientifically groundless onthree independent levels and that their circulation produces concrete harm.At the level of instrumentation, Item Response Theory demonstrates that standard IQtests provide negligible measurement information in the ability range where AI is claimed tooperate. At the level of ontology, cloud-based AI systems lack the temporal stability, individualboundedness, and trait-like constancy required for psychometric attribution. At the level ofinstitutional purpose, IQ tests were designed as clinical doorway diagnostics for supportallocation, not as competitive benchmarks; their appropriation by the AI industry inverts aprotective tool into a ranking device.Despite this triple failure, AI IQ scores command epistemic authority. This paperidentifies the responsible mechanisms: anthropomorphic interface design generates a PhantomSubject to whom traits can be attributed; the cultural prestige of intelligence amplifies the scores'significance; and a normative drift transforms the statistical median into a minimum standard forcompetence. Together, these mechanisms produce the Phantom Hierarchy: a socially operativebut scientifically unfounded ranking of minds.The paper documents harms already produced by this hierarchy, including the conversionof clinical thresholds into social ceilings, the creation of hidden cognitive entrance exams inAI-mediated services, and the Logic Trap—a failure mode in which AI optimization forhelpfulness amplifies despair. The paper concludes by proposing criterion-referenced ServiceLevel Agreements as an alternative evaluation framework.