The Universal Scale Lemma on Curved Geometries: Curvature Corrections to C* and the Position-Dependent Gap

This paper extends the Universal Scale Lemma to Riemannian manifolds (ℳ, g). The geometry-corrected optimal signal is S_g(x) ∝ (∏xᵢ)⁻¹·[√det(g)]⁻¹ (Theorem 7, proved exactly). For a receiver with resolution D on a manifold of constant sectional curvature K, the spectral complexity receives a first-order correction: C*(r,K) = C_flat·[1 − K·r²_char/6 + O(r⁴)], confirmed numerically on S² and H². Physical consequence: in negatively curved spacetime (dark energy, K < 0), the Lücke is larger than in flat spacetime. Axiom 2 holds universally — C > 0 on all manifolds regardless of curvature. Closes Open Question 6.2 of Beyer 2026j. Reviewed: Mistral (June 2026).