Spectral Resource Allocation: Demand-Driven Investment in Multi-Dimensional Brand Space
Year: 2026
Type: preprint
Abstract: Brand investment decisions are conventionally guided by founder intuition, competitive benchmarking, or aggregate market research β none of which account for the multi-dimensional structure of brand perception. This paper develops a formal resource allocation model within Spectral Brand Theory (SBT), where a brandβs signal portfolio π β β^8+ is evaluated by observer cohorts with heterogeneous weight vectors π€ β Ξ^7. We define the spectral value function π (π , π) = βπ π€π(π) β
π π and the alignment gap ποΈ(π, π) = π (π βπ, π) β π (π βπ, π), which measures the expected value loss when a founder optimizes for their own spectral profile rather than the target cohortβs. We prove four main results. First, the optimal signal portfolio for a single cohort allocates investment proportionally to the cohortβs dimensional weights scaled by the inverse marginal cost (Theorem 1). Second, the alignment gap is bounded below by the Fisher-Rao distance between founder and cohort profiles, establishing that perceptual distance directly predicts economic loss (Theorem 2). Third, for the multi-cohort case, serving π cohorts with a single signal portfolio is efficient only when the cohortsβ weight vectors lie within a Fisher-Rao ball of radius π < π/4 on Ξ^7 (Theorem 3). Fourth, the metamerism result from Zharnikov (2026e) implies that the cost-minimizing signal portfolio achieving a target perception is generically unique when all eight dimensions are active, but admits a (8 β π)-dimensional family of solutions when only π < 8 dimensions are weighted by the cohort (Theorem 4). We connect these results to five established strategy frameworks β Blue Ocean Strategy, Jobs to Be Done, Lean Startup, Porterβs Five Forces, and the Resource-Based View β showing that each framework implicitly operates on a low-dimensional projection of the spectral resource allocation problem. Application to five case-study brands demonstrates that the alignment gap ordering corresponds to known product-market fit outcomes. The model provides, to the best of our knowledge, the first formal bridge between multi-dimensional brand perception measurement and operational resource allocation, answering the question: given a measured cohort profile, where exactly should the next dollar of brand investment go?Includes paper.yaml (Paper Spec v0.1.0) β a machine-readable specification of the paper's claims, assumptions, and dependencies. See https://github.com/spectralbranding/paper-spec for the standard.
Author: Zharnikov, Dmitry
Language: English
Subfield: Marketing
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7135178594