Manuscript Em02-D@4 · Em02-D-v4.tex · theorem · fcf2fac7

Theorem: Complete leading-order result

Main result and scope · lines 129–161

Em02-D@4#thm:main, Em02-D@4#eq:main-expansion, Em02-D@4#eq:main-remainder

For the fixed model of Sections sec:model and sec:rows, there is a continuous distribution $C_{seven}$ , given explicitly in (eq:Cseven), such that

t_{δ}^{ζ} = δ^{- 3} C_{seven} + R_{δ} .

For every fixed compact joint-test support $H$ ,

∣ R_{δ} (Φ) ∣ + ∣ (δ \partial_{δ} R_{δ}) (Φ) ∣ \leq C_{H} δ^{- 2} ∥Φ ∥_{17, H}

for sufficiently small $δ > 0$ .

The leading coefficient is nonzero. One fixed compact test supported away from $z = a$ satisfies

t_{δ}^{ζ} (Φ_{A}) = c_{A} δ^{- 3} + O (δ^{- 2}), c_{A} \neq = 0.

Consequently the complete raw family has no weak distributional limit, and no correction family supported on $z = a$ restores convergence.

Displayed formulas in this passage

Em02-D@4#eq:main-expansion · line 135
$t_{δ}^{ζ} = δ^{- 3} C_{seven} + R_{δ} .$
Em02-D@4#eq:main-remainder · line 142
$∣ R_{δ} (Φ) ∣ + ∣ (δ \partial_{δ} R_{δ}) (Φ) ∣ \leq C_{H} δ^{- 2} ∥Φ ∥_{17, H}$
unlabeled · line 152
$t_{δ}^{ζ} (Φ_{A}) = c_{A} δ^{- 3} + O (δ^{- 2}), c_{A} \neq = 0.$