Manuscript IX-A@1 · IX-A.tex · theorem · 55aa6f6d

Theorem: IX.1: explicit failure of sharp-admission convergence

The compatibility boundary: regular approximation is not admission › Unchanged regular approximants · lines 1264–1302 · proof included or linked

IX-A@1#thm:IX1

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For every nonzero real $f$ ,

B_{f, ε} \neq ⟶ B_{f} in (P_{Σ}, τ_{Σ}) .

More precisely, at the fixed configuration $(0, ϕ_{f})$ ,

b_{(0, ϕ_{f}); \emptyset} (B_{f, ε}) = 0 for every ε > 0,

whereas

b_{(0, ϕ_{f}); \emptyset} (B_{f}) = - b ∣_{Σ} \neq = 0.

There is a continuous scalar seminorm coordinate that separates the proposed limit from every approximant.

Proof. Every approximant is regular and hence bulk. Its boundary coordinates vanish by Proposition prop:bulk. The charge of $B_{f}$ is (eq:Bcharge), and Lemma lem:witness makes it nonzero.

Choose a compact spatial test $χ$ equal to one on the compact support of $b_{0} ∣_{Σ}$ . The continuous functional

F ⟼ \int_{Σ} χ (x) [b_{(0, ϕ_{f}); \emptyset} (F)]_{0} d^{3} x

is zero on every approximant and nonzero on $B_{f}$ . Convergence is therefore impossible.

Displayed formulas in this passage

unlabeled · line 1267
$B_{f, ε} \neq ⟶ B_{f} in (P_{Σ}, τ_{Σ}) .$
unlabeled · line 1272
$b_{(0, ϕ_{f}); \emptyset} (B_{f, ε}) = 0 for every ε > 0,$
unlabeled · line 1277
$b_{(0, ϕ_{f}); \emptyset} (B_{f}) = - b ∣_{Σ} \neq = 0.$
unlabeled · line 1294
$F ⟼ \int_{Σ} χ (x) [b_{(0, ϕ_{f}); \emptyset} (F)]_{0} d^{3} x$

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section: Unchanged regular approximants