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

Brand positioning research faces a fundamental measurement problem: every observer cohort perceives the same brand differently, yet traditional frameworks treat this variation as noise rather than signal. We propose brand triangulation — a geometric framework borrowing from GPS positioning theory to estimate brand spectral profiles from multiple observer cohorts while jointly solving for observer bias. We make four contributions. First, we formalize the GPS-SBT mapping, showing that observer cohorts function as positioning satellites whose geometric diversity determines measurement precision. Second, we introduce Perception DOP (Dilution of Precision), a computable metric quantifying how well a given set of observer cohorts resolves a brand's eight-dimensional spectral profile — before any data is collected. Third, we propose differential brand measurement, a calibration protocol using reference brands with known spectral profiles to correct systematic observer bias across studies. Fourth, we establish identifiability conditions: the minimum observer configurations required for unique brand positioning. The geometric formulation provides pre-study design criteria that Bayesian heterogeneity approaches lack. We demonstrate the framework computationally using dimensional weight data from six large language models drawn from the R15 dataset (Runs 1-4; 4, 860 API calls across fifteen brand pairs). Perception DOP predicts estimation error (Monte Carlo: R² = 0. 926, log-log slope = 0. 995, Spearman rho = 0. 996, all p < 10^-300), and Brand Function specification — reinterpreted as a DOP improvement — reduces dimensional collapse by 20% (DCI 0. 355→0. 284 for local brands). The framework reframes multi-observer disagreement from a methodological nuisance into a primary source of positioning information. Includes paper. 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.