Forming a Perception: Campaigns as Forcing Functions and the Relaxation of the Brand Perception Cloud

A marketing campaign is an exogenous force on a multi-dimensional perceptual state, and the decay observed when advertising stops is the unforced relaxation of that state toward a brand-specific stable point. This paper formalizes perception formation and maintenance as a forced dynamical system whose results follow from the stochastic dynamics, not any particular perceptual instrument. The centroid of a brand's perception cloud follows a forced Ornstein-Uhlenbeck process on a fixed perceptual space: formation is the forced transient toward a target, maintenance is sustained forcing against a restoring force, and decay is the unforced relaxation at a measurable perception-decay time constant, recoverable from a dated perceptual-tracking series spanning a spend pulse. The model nests the scalar advertising-stock and Bayesian-learning traditions as its one-dimensional projection. Its unifying result ties formation to reach: a brand's off-generic distinctiveness lower-bounds the persistence of its formed perception, because a more distinctive brand sits farther outside the dense generic attractor, where less restoring curvature is added to its well – an ordering derived from a stated curvature primitive (Electronic Companion EC.3) and demonstrated as an existence proof in a calibrated simulation. An identification strategy using exogenous advertising shifters is outlined. The instrument measures the forcing-versus-relaxation budget; the manager runs the campaign. Includes zharnikov-2026aw-forming-a-perception.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.