Brand tracking routinely measures static position but rarely quantifies trajectory velocity or acceleration. This paper introduces a differential calculus for multi-dimensional brand perception, extending Spectral Brand Theory’s eight-dimensional space from static profiles to velocity and acceleration vectors, and defines a 16-dimensional phase space combining position and velocity. Three theoretical contributions emerge. First, velocity resolves metric ambiguity: brands with identical spectral profiles but different velocities are distinguishable in phase space, providing a constructive resolution to a problem structurally analogous to the Bonnet pair problem in differential geometry. Second, the paper introduces a directional coherence metric that quantifies alignment between a brand’s actual trajectory and its strategic intent. Third, trajectory clustering groups brands by their dynamic behavior rather than static position, enabling detection of competitive convergence and divergence before they manifest in position. The framework is illustrated using author-assigned illustrative profiles for Dove across four strategic periods (2003-2023); computed velocity and acceleration vectors are illustrative of the method, not validated empirical measurements. The framework connects state-space models in marketing science to the geometric structure of brand perception space, embedding existing Kalman filter approaches within a unified kinematic theory. Includes zharnikov-2026z-r18.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.
Dmitry Zharnikov (Sun,) studied this question.