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

Proposition: Large-smearing upper bound

Gaussian regularization: sign, margin, and validated integration › Large smearing reduces the available margin · lines 1814–1839 · proof included or linked

En01-B@2#prop:large

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For the homogeneous sphere,

∣ B_{r_{c}} ∣ \leq K_{G} \frac{47}{5120 π α ^{5}} .

Thus its constructive margin tends to zero as $α \to \infty$ , despite remaining strictly signed for every finite $α$ .

Proof. The quadratic terms cancel and

∣ B_{r_{c}} ∣ \leq D_{r_{c}}^{(4)} i \sum ∣ w_{i} ∣ d_{i}^{4} .

Using $∣ μ (k) ∣ \leq M$ ,

D_{r_{c}}^{(4)} \leq \frac{G M ^{2}}{160 π ℏ r _{c} ^{5}} .

The weighted fourth moment is $47 R^{4} /32$ . Substitute and express the result in units $K_{G}$ .

Displayed formulas in this passage

unlabeled · line 1817
$∣ B_{r_{c}} ∣ \leq K_{G} \frac{47}{5120 π α ^{5}} .$
unlabeled · line 1828
$∣ B_{r_{c}} ∣ \leq D_{r_{c}}^{(4)} i \sum ∣ w_{i} ∣ d_{i}^{4} .$
unlabeled · line 1833
$D_{r_{c}}^{(4)} \leq \frac{G M ^{2}}{160 π ℏ r _{c} ^{5}} .$

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