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

Traditional brand evaluation compresses multi-dimensional perception into scalar grades – "Brand Health Scores, " letter grades, Net Promoter Scores – yet the information cost of this compression has never been formally quantified. This paper introduces spectral metamerism to brand theory: the phenomenon whereby structurally distinct brand profiles produce identical scalar evaluations, analogous to physically different light spectra producing identical color percepts. Drawing on Spectral Brand Theory (SBT), which models brands as emitters across eight typed dimensions perceived by heterogeneous observers, we prove that metamerism is not a correctable measurement artifact but a geometric inevitability of dimensionality reduction. Applying the Johnson-Lindenstrauss lemma, we show that projecting R⁸ to R¹ requires distortion exceeding 152% for as few as 10 brands and 198% for 50 brands. The projection creates a 7-dimensional null space – a subspace of "invisible" brand differences that no scalar grade can detect. Information-theoretically, a 5-point grade captures 2. 32 bits of a ~20-bit 8-dimensional profile, retaining 11. 6% of the available information. Monte Carlo simulations confirm that 31–39% of brand pairs are metameric under random projection. We formalize the distinction between rasterized brand management (human projection, inherently lossy) and vectorized brand management (computed projections from a full-dimensional specification), arguing that SBT's 8-dimensional spectral profile is not optional complexity but the minimum resolution required to avoid metameric collapse. The results establish formal bounds on when scalar grades suffice and when full spectral profiles are necessary, connecting to MDS dimensionality selection and survey instrument design. 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.