Manuscript En01-B@2 · En01-B-v2.tex · theorem · f226d1eb

Theorem: Uniform small-smearing certificate

Gaussian regularization: sign, margin, and validated integration › A sign theorem for every finite Gaussian width · lines 1752–1791 · proof included or linked

En01-B@2#thm:small, En01-B@2#eq:small

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For all $α \geq 0$ ,

∣ b (α) - b (0) ∣ \leq 48 α^{2} .

Hence, on $0 < r_{c} \leq R /100$ ,

- B_{r_{c}} \geq K_{G} (\frac{5041}{81920} - \frac{48}{10000}) > \frac{K _{G}}{20} .

More generally,

∣ b (α) - b (β) ∣ \leq 48∣ α^{2} - β^{2} ∣.

Proof. Because $\sum_{i} w_{i} = 0$ , $∣ S (u) ∣ \leq \sum_{i} ∣ w_{i} ∣ = 16$ . Parseval's identity for the uniform unit-ball density gives

\int_{0}^{\infty} u^{2} F (u)^{2} d u = \frac{3 π}{2} .

Indeed, $\int μ^{2} = 3 M^{2} / (4 π R^{3})$ ; applying the stated Fourier normalization proves the identity.

Use

∣ e^{- α^{2} u^{2}} - e^{- β^{2} u^{2}} ∣ \leq ∣ α^{2} - β^{2} ∣ u^{2}

in (eq:b). The bound is $(2/ π) 16 (3 π /2) ∣ α^{2} - β^{2} ∣$ . Set $β = 0$ and apply Corollary cor:exactcontrast. The final rational inequality is direct.

Displayed formulas in this passage

En01-B@2#eq:small · line 1759
$- B_{r_{c}} \geq K_{G} (\frac{5041}{81920} - \frac{48}{10000}) > \frac{K _{G}}{20} .$
unlabeled · line 1755
$∣ b (α) - b (0) ∣ \leq 48 α^{2} .$
unlabeled · line 1766
$∣ b (α) - b (β) ∣ \leq 48∣ α^{2} - β^{2} ∣.$
unlabeled · line 1775
$\int_{0}^{\infty} u^{2} F (u)^{2} d u = \frac{3 π}{2} .$
unlabeled · line 1782
$∣ e^{- α^{2} u^{2}} - e^{- β^{2} u^{2}} ∣ \leq ∣ α^{2} - β^{2} ∣ u^{2}$

Theorem: Uniform small-smearing certificate

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