Spectral Metamerism in Brand Perception: Projection Bounds from High-Dimensional Geometry

Scalar brand metrics compress high-dimensional perceptual signals into single numbers, yet the information cost of this compression has never been formally quantified. This paper introduces spectral metamerism to brand theory: structurally distinct brand profiles that produce identical scalar evaluations. Drawing on Spectral Brand Theory (SBT), which models brands across eight typed dimensions perceived by heterogeneous observers, the paper proves metamerism is a geometric inevitability of dimensionality reduction. Applying the Johnson-Lindenstrauss lemma, it establishes that projecting \(\mathbb{R}^8\) to \(\mathbb{R}^1\) requires distortion exceeding 152% for 10 brands and 198% for 50 brands. Any such projection creates a 7-dimensional null space of “invisible” brand differences. Information-theoretically, a 5-point grade captures 2.32 bits of a \(\sim 20\)-bit spectral profile, retaining 11.6% of available information. Monte Carlo simulations confirm that 31–39% of brand pairs are metameric under random projection. The analysis yields a fundamental distinction between rasterized brand management – human projection through cognitive and communicative bottlenecks – and vectorized brand management, in which the full spectral profile serves as single source of truth and channel outputs are computed projections with known, bounded loss. These results provide the first formal geometric and information-theoretic lower bounds on the fidelity of scalar brand grades. Includes zharnikov-2026e-r2.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.