Manuscript Em01-A@1 · Em01-A.tex · lemma · 880b67af

Lemma: An intrinsic null relation

Reconstruction descent and canonical-domain agreement › Restriction does not establish global descent · lines 1066–1085 · proof included or linked

Em01-A@1#lem:null-relation

For a normal coordinate $z$ ,

z δ^{'} (z) + δ (z) = 0.

Consequently a presentation splitting that treats these two summands independently need not descend to distributions.

Proof. For a test $φ$ ,

⟨ z δ^{'}, φ ⟩ = - \partial_{z} (z φ (z)) ∣_{z = 0} = - φ (0) .

This is $- ⟨ δ, φ ⟩$ . Each summand is nonzero, although their sum vanishes.

Displayed formulas in this passage

unlabeled · line 1069
$z δ^{'} (z) + δ (z) = 0.$
unlabeled · line 1078
$⟨ z δ^{'}, φ ⟩ = - \partial_{z} (z φ (z)) ∣_{z = 0} = - φ (0) .$