Kernel proposition. On a complete relational Fisher manifold satisfying the admissible symmetry and positivity conditions, there exists a unique self-adjoint coarse-graining generator Δ_G[R(Ψ)] in the small-data regime. Its heat semigroup defines the kernel
K_{Q₀}(X,Y) = ⟨X | e^{−Q₀² Δ_G} | Y⟩
whose short-time asymptotics are Gaussian with scale Q₀ fixed by the FRG fixed point. The theorem below upgrades local uniqueness to global.
Uniform contraction theorem. On the admissible Fisher manifold, assume: (i) Δ_G[R] has a uniform spectral gap λ₀ > 0 across the full orbit; (ii) the back-reaction map g ↦ G[R(g)] is globally Lipschitz with constant C_Lip satisfying C_Lip · γ < 1 − e^{−λ₀Q₀²}; (iii) ℛ_clock is non-expansive.
Then T = ℛ_clock ∘ 𝒫_{Q₀[Ψ]}^{G[R]} is a strict contraction on the admissible class with contraction ratio
q_total ≤ e^{−λ₀Q₀²} + C_Lip · γ < 1.
Hence T has a unique fixed point g★, the Fisher attractor, and every admissible iterate converges exponentially to g★. The kernel K is globally unique under these conditions.
