Manuscript XX-F@1 · XX-F.tex · proposition · de0bd24f

Proposition: The absolute-convergence threshold cannot be lowered universally

A weaker explicit infrared domain for the original Newton map · lines 986–1007 · proof included or linked

XX-F@1#prop:threshold

At $q = 1$ , the pointwise bound $∣ b (y)∣ \leq C ⟨ y ⟩^{- 1}$ does not universally ensure absolute Newton-divergence convergence.

Proof. Take a smooth source which outside a ball is

b^{i} (y) = \frac{y _{i}}{∣ y ∣ ^{2}} .

At $x = 0$ ,

P_{i} (- y) b^{i} (y) = \frac{1}{4 π ∣ y ∣ ^{3}} .

Its spatial integral diverges logarithmically. The source has the displayed rate and can be smoothly filled in inside the ball.

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