The Scale Axis as a Spatial Dimension: A uniqueness theorem and an exact heat kernel for spectral dimension three

An algebraic label is not a dimension. A field with two base directions and any number of labels that enter only algebraically has spectral dimension two. The diffusing scale axis is unique. A label becomes a direction exactly when it carries a kinetic term. Locality, a positivity-preserving diffusion, covariance under rescaling and self-adjointness without an extra boundary length then fix the operator uniquely: −∂²_ω + c₀/ω² on the half-line, with c₀ ≥ 3/4. Exact heat kernel. The diagonal kernel is K_t(ω,ω) = (ω/2t) e^(−ω²/2t) I_ν(ω²/2t) with ν = √(c₀+¼). On a window its trace is L/(2√(πt)) − ½ + … with explicit corrections. Spectral dimension three across the window. Two base directions and the scale axis give d_s(t) = 3 + ε(t), with ε ≈ √(πt)/L small and positive at every resolution between the two cutoffs. There is no running and no dimensional reduction. Numerically, the closed form agrees with direct diagonalisation to the discretisation error (10⁻⁴ to 10⁻⁶), and d_s = 3.0000 over nine decades of diffusion time. The classical ingredients (Weber's integral, the limit-point criterion, Courrège's theorem) are named as such. What is new is the characterisation of the scale axis as the unique diffusing direction and the resulting closed-form statement of spectral dimension three across a scale-invariant window. Derivations, verification code and drafting were carried out in collaboration with Claude (Anthropic).