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 salience weights. This paper models a brand’s signal portfolio s ∈ ℝ+8 as read by observer cohorts whose weight vectors w(c) ∈ Δ7 lie on the probability simplex, with perceived value ⟨w(c), s⟩ net of separable convex cost. Five results follow. First, optimal investment is proportional to cohort weight over marginal cost, generalizing Dorfman-Steiner (1954) to eight perceptual dimensions (Theorem 1). Second, where the founder’s weights are at least as concentrated as the cohort’s, the resulting loss – the alignment gap – is bounded below by a quartic function of Fisher-Rao distance; outside that regime it is not sign-definite (Theorem 2, corrected). Third, one portfolio efficiently serves multiple cohorts only if their weights lie inside a Fisher-Rao ball of radius r < π/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): the optimization layer brand-tracking systems lack. Includes zharnikov-2026k-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 github.com/spectralbranding/paper-spec for the standard. This PDF is generated programmatically from that machine-first source under a research-as-repository model.