{
  "$type": "com.whtwnd.blog.entry",
  "theme": "github-light",
  "title": "Relational Kernel",
  "content": "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\n\n    K_{Q₀}(X,Y) = ⟨X | e^{−Q₀² Δ_G} | Y⟩\n\nwhose short-time asymptotics are Gaussian with scale Q₀ fixed by the FRG fixed point. The theorem below upgrades local uniqueness to global.\n\nUniform contraction theorem. On the admissible Fisher manifold, assume:\n  (i)  Δ_G[R] has a uniform spectral gap λ₀ > 0 across the full orbit;\n  (ii) the back-reaction map g ↦ G[R(g)] is globally Lipschitz with \n       constant C_Lip satisfying C_Lip · γ < 1 − e^{−λ₀Q₀²};\n  (iii) ℛ_clock is non-expansive.\n\nThen T = ℛ_clock ∘ 𝒫_{Q₀[Ψ]}^{G[R]} is a strict contraction on the admissible class with contraction ratio\n\n    q_total ≤ e^{−λ₀Q₀²} + C_Lip · γ < 1.\n\nHence 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.",
  "createdAt": "2026-04-24T15:56:03.416Z",
  "visibility": "url"
}