Manuscript Oai212-A@1 · Oai212-A.tex · lemma · 79323841

Lemma: Presentation degree is not intrinsic

The separate characteristic-composition programme › Reconstruction descent is a necessary condition · lines 1593–1622 · proof included or linked

Oai212-A@1#lem:null

In one normal variable,

q δ^{'} (q) + δ (q) = 0.

Under a reconstruction convention

Rec (M_{R}) = δ (q) \otimes R, Rec (i θ M_{R}) = δ^{'} (q) \otimes R,

with $R \neq = 0$ and physical multiplication by $q$ ,

Rec ((q i θ + 1) M_{R}) = 0.

The separate degree-zero and degree-one summands reconstruct to nonzero opposite distributions.

Proof. For a test $φ$ ,

⟨ q δ^{'}, φ ⟩ = - \frac{d}{d q} (q φ (q))_{q = 0} = - φ (0) .

Tensor with $R$ . The distribution $δ \otimes R$ is nonzero by testing against a product of tests with nonzero pairings.

Displayed formulas in this passage

unlabeled · line 1596
$q δ^{'} (q) + δ (q) = 0.$
unlabeled · line 1600
$Rec (M_{R}) = δ (q) \otimes R, Rec (i θ M_{R}) = δ^{'} (q) \otimes R,$
unlabeled · line 1605
$Rec ((q i θ + 1) M_{R}) = 0.$
unlabeled · line 1614
$⟨ q δ^{'}, φ ⟩ = - \frac{d}{d q} (q φ (q))_{q = 0} = - φ (0) .$