Why the rate beats every polynomial

Lemma 9.1 (lem:filter, p. 24) and Lemma 10.5 (lem:rate, p. 28)

What is happening

Lemma 9.1 builds an exact kernel W_γ with Ŵ_γ(E) = χ(E)/E, where χ is smooth, vanishes on [−γ/2, γ/2] and equals 1 outside (−γ, γ). Because Ŵ_γ is smooth and every derivative of order ≥ 1 is integrable, repeated integration by parts gives |W_γ(t)| ≤ ‖Ŵ_γ^(k)‖₁ / 2π|t|^k for every k. That is what the left panel shows: no fixed power describes the envelope, which eventually falls below every power of t. Its local slope, measured below, does not rise steadily; it scatters, with no bound.

Lemma 10.5 sums that kernel against the light-cone geometry. Its rate function splits into an exponential summand and a filter-tail summand. The exponential part dies quickly; the tail is what remains, and it is the part that is bounded by C_N(1+d)^−N for every N.

Read the heights with care. χ is not determined by the lemma — only its support conditions are — and the constants depend on the choice. These curves use one admissible χ, built from the standard exp(−1/u) step. γ enters the kernel itself, so it cannot be a free slider: the three values offered are the ones docs/figures/make_figures.py actually computed. M is an exact prefactor and stays continuous. Ψ_γ is an upper bound on ‖(ω_g − ω_g′)|_A(O)‖, never a measured decay.

An independent check on the numerics: W_γ(t) = W₁(γt) forces ‖W_γ‖₁·γ to be constant, and the three computed values agree to 0.7%.