Manuscript Em01-D@2 · Em01-D-v2.tex · corollary · d3b49151

Corollary: Fixed-test failure and finite part

Exact nonphysical eight-variable diagnostic › All diagnostic coefficients · lines 2299–2329 · proof included or linked

Em01-D@2#cor:diagnostic

There is one fixed compact test $Φ_{*}$ for which

∣ u_{δ} (Φ_{*}) ∣ ⟶ \infty.

Hence the two complete mathematical families cannot both have raw limits.

The explicitly subtracted difference

u_{δ} - δ^{- 3} B_{- 3} - δ^{- 2} B_{- 2} - δ^{- 1} B_{- 1}

converges strongly to $B_{0}$ , with the stated order-eight $O (δ)$ rate.

Proof. Choose $Φ_{*}$ detecting $B_{- 3}$ . Then

δ^{3} u_{δ} (Φ_{*}) ⟶ B_{- 3} (Φ_{*}) \neq = 0.

The difference could not diverge if both original families converged. The finite-part conclusion follows directly from the remainder estimate.

Displayed formulas in this passage

unlabeled · line 2302
$∣ u_{δ} (Φ_{*}) ∣ ⟶ \infty.$
unlabeled · line 2309
$u_{δ} - δ^{- 3} B_{- 3} - δ^{- 2} B_{- 2} - δ^{- 1} B_{- 1}$
unlabeled · line 2322
$δ^{3} u_{δ} (Φ_{*}) ⟶ B_{- 3} (Φ_{*}) \neq = 0.$