Brand Space Geometry: A Formal Metric for Multi-Dimensional Brand Perception

Brand theory has lacked a rigorous distance function capable of quantifying perceptual closeness while respecting the compositional, relative, and observer-dependent nature of consumer attention. This paper constructs the first observer-warped metric space grounded in Cencov’s uniqueness theorem for multi-dimensional brand perception, building on Spectral Brand Theory. Three spaces are defined. The brand signal space equips the positive orthant with the Aitchison metric, justified by Weber-Fechner scaling and subcompositional coherence; metric axioms are proved and geodesics derived. The observer weight space equips the probability simplex with the Fisher-Rao metric, uniquely invariant under Markov morphisms; closed-form distance and geodesic entropy bulge are established. These combine in a warped product generalizing the empirically validated INDSCAL model, reducing to weighted clr distance for fixed observers. Concentration-of-measure analysis at eight dimensions supplies a null baseline of approximately .44 for observer pairwise distance. The positive-octant restriction compresses brand space to 1/256 of the full 7-sphere, quantifying geometric constraints on differentiation. Jacobi-field analysis yields a spectral sensitivity index linking static geometry to dynamic trajectory vulnerability, with singularities at structural-absence boundaries. Application to calibrated brand profiles illustrates observer-dependent distance reversals invisible to traditional methods. The framework supplies a mathematical foundation for differentiation measurement, cohort definition, and positioning optimization. Includes zharnikov-2026d-r1.yaml (Paper Spec v0.1.0) – a machine-readable specification of the paper's claims, assumptions, and dependencies. The paper's full machine-first bundle (the SPINE claim/dependency graph and the ONTOLOGY term module) lives in the public repository; 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.