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

Brand managers allocate resources across perceptual dimensions – design, storytelling, pricing, heritage – yet rarely ground those decisions in measured customer salience weights. This paper develops a resource allocation model in which a brand’s signal portfolio \(s \in \mathbb{R}^8_+\) is evaluated by observer cohorts whose weight vectors \(w(c) \in \Delta^7\) reside on the probability simplex, and perceived value is \(\langle w(c), s \rangle\) net of a separable convex cost. Five results follow. First, optimal investment is proportional to cohort weight divided by marginal cost, generalizing Dorfman-Steiner (1954) to multi-dimensional perceptual space (Theorem 1). Second, the economic loss from optimizing under founder rather than cohort weights – the alignment gap – is bounded below by the Fisher-Rao distance between weight vectors (Theorem 2). Third, a single portfolio efficiently serves multiple cohorts only when their weights lie inside a Fisher-Rao ball of radius \(r < \pi/4\) (Theorem 3). Fourth, the cost-minimizing portfolio achieving a target perception is unique when all dimensions are active (Theorem 4). Fifth, cohort-specific interaction terms shift investment toward complementary dimensions, generalizing Naik and Raman (2003) (Theorem 5). The framework supplies a diagnostic (alignment audit) and a prescriptive tool (dimension-specific budget ratios), providing the optimization layer missing from current brand-tracking systems. Includes zharnikov-2026k-r7-spectral-resource-allocation.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.