Spectral Resource Allocation: Demand-Driven Investment in Multi-Dimensional Brand Space

Brand investment decisions are conventionally guided by founder intuition, competitive benchmarking, or aggregate market research – 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 brand’s signal portfolio 𝑠 ∈ ℝ8+ is evaluated by observer cohorts with heterogeneous weight vectors 𝑤 ∈ Δ7. We define the spectral value function 𝑉 (𝑠, 𝑐) = ∑𝑖 𝑤𝑖(𝑐) ⋅ 𝑠𝑖 and the alignment gap 𝒜︀(𝑓, 𝑐) = 𝑉 (𝑠∗𝑓, 𝑓) − 𝑉 (𝑠∗𝑓, 𝑐), which measures the expected value loss when a founder optimizes for their own spectral profile rather than the target cohort’s. We prove four main results. First, the optimal signal portfolio for a single cohort allocates investment proportionally to the cohort’s 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 𝑘 cohorts with a single signal portfolio is efficient only when the cohorts’ weight vectors lie within a Fisher-Rao ball of radius 𝑟 < 𝜋/4 on Δ7 (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 − 𝑘)-dimensional family of solutions when only 𝑘 < 8 dimensions are weighted by the cohort (Theorem 4). We connect these results to five established strategy frameworks – Blue Ocean Strategy, Jobs to Be Done, Lean Startup, Porter’s Five Forces, and the Resource-Based View – 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 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?