The Correspondence Principle of Brand Management: When a Brand-Health Score Is a Sufficient Statistic, and the Decision Cost When It Is Not

Managers steer brands by a single brand-health score, yet a brand is read by fragmenting audiences, by faster-moving culture, and now by artificial observers that summarize it from text. This paper asks when one number still suffices. It models a brand as an observer-completed probability measure on an eight-dimensional perception manifold and represents each incumbent framework – Aaker, Keller, Kapferer, the Ehrenberg-Bass distinctiveness school, single-figure valuation – as a projection-and-aggregation operator that Blackwell-garbles the full measurement. Three results follow. The correspondence theorem proves the aggregate score is a sufficient statistic for the decision exactly when four estimable regime conditions hold jointly: a tight unimodal perception cloud, slow dynamics, a human-only observer set, and a firm-dominated signal; the “brands differ little in image” finding becomes a regime diagnosis, not a law. The decision-efficiency theorems show that outside that regime the finer measurement weakly dominates the score for every decision-maker, with a decision-loss bound monotone in four regime-departure parameters; a closed form gives the gap as off-axis perceptual variance. A dataset-calibrated simulation validates the closed form, and a theorem-derived managerial procedure translates it into action. Includes zharnikov-2026au-brand-correspondence.yaml (Paper Spec v0.1.0) – a machine-readable specification of the paper's claims, assumptions, and dependencies. See https://github.com/spectralbranding/paper-spec for the standard. This PDF is generated programmatically from that machine-first source under a research-as-repository model.