O = (X, μ) K_ε : X × X × (0, ∞) → ℝ⁺ such that: (P1) Symmetry: K_ε(x, y) = K_ε(y, x) (P2) Positivity: ∀ finite {x_i}, {c_i}: Σᵢⱼ cᵢ cⱼ K_ε(xᵢ, xⱼ) ≥ 0 (P3) Self-consistent scale: ε = ε[K_ε] where ε[K] is a renormalization-invariant functional of K's spectral data — specifically, the effective spectral dimension: ε[K] := lim sup_{λ → ∞} 2 log N(λ) / log λ This functional satisfies ε[T] = ε[T²] because the dimension exponent is invariant under squaring of eigenvalues: λᵢ ↦ λᵢ² is a reparametrization of the spectrum that preserves the asymptotic counting exponent. everything derives and emerges from this.