Why it is a question
THEORY-744 holds Coifman and Lafon’s α-normalized limit on a compact manifold without boundary: (I − P_{ε,α})/ε → Δf + 2(1 − α)⟨∇U, ∇f⟩ uniformly on smooth functions at rate O(ε). That is minus the generator of a reversible diffusion with invariant density q^{2(1−α)}. LIT-1246 also claims two things the promotion review could not grant.
- The boundary. On a manifold with boundary, the source bounds the operator by O(1) on a strip of measure O(ε^γ) and concludes an L² remainder of O(ε^γ). The bound gives O(ε^{γ/2}). Convergence survives, but the printed rate does not.
- The semigroup. Proposition 11 replaces (I − εΔ)^{−1}(I − εΔ − εR_ε) by I − εR_ε. That needs R_ε to commute with Δ. It also drops a factor ‖(I − εΔ)^{−t/ε}‖ that grows like e^{tν_K²}, and it does not show that P_{ε,1} keeps the band-limited space invariant. So P_{ε,1}^{t/ε} → e^{−tΔ} is not proved as written.
An answer is a complete proof of either, from a later paper read in full or re-derived step by step, or a counterexample. Numerical agreement on manifolds without boundary does not bear on it, since the interior expansion is settled.