Brand Triangulation: A Geometric Framework for Multi-Observer Brand Positioning

Brand positioning research faces a structural measurement problem: every observer cohort perceives a brand through a distinct spectral weight profile, yet conventional methods treat inter-cohort variation as noise. Brand triangulation reframes this variation as geometric information. Drawing on GPS positioning theory, the framework maps observer cohorts to satellites, brand emission profiles to receiver location, and cohort biases to clock error. The central innovation is Perception DOP (PDOP), a scalar computed solely from proposed cohort weight profiles that quantifies expected estimation precision before data collection. The approach further introduces differential brand measurement using reference brands to correct systematic observer bias and establishes identifiability conditions linking spectral metamerism to geometric underdetermination. The framework is illustrated with dimensional-weight data from six large language models (4,860 observations) and validated through Monte Carlo simulation (2,000 trials) confirming the theoretical PDOP–MSE power-law relationship (log-log slope = 1.000, \(R^2\) = .993, Spearman \(\rho\) = .992). Brand Function specifications reduce dimensional collapse by 20% in the LLM constellation. By converting multi-observer disagreement into a primary source of positioning information, the framework supplies pre-study design criteria, bias-correction protocols, and dynamic tracking machinery that together upgrade brand measurement from opinion aggregation to geometric estimation. Includes zharnikov-2026y-r17.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.