Manuscript Em01-F@3 · Em01-F-v3.tex · lemma · 225e59af

Lemma: A reconstruction-null identity

Translation identity and reconstruction-null test · lines 3202–3229 · proof included or linked

Em01-F@3#lem:null

In one normal coordinate $q$ ,

q δ_{D}^{'} (q) = - δ_{D} (q) .

Consequently, for any distribution $u$ in independent variables,

(q δ_{D}^{'} (q) + δ_{D} (q)) \otimes u = 0.

Every positive-scale pairing constructed here sends this null distribution to zero, regardless of a presentation's polynomial degree assignment.

Proof. For a compact smooth $ψ$ ,

⟨ q δ_{D}^{'}, ψ ⟩ = - \partial_{q} (q ψ (q))_{q = 0} = - ψ (0) .

Tensor products preserve the identity. The positive-scale map depends only on the actual reconstructed distribution, proving the last assertion.

Displayed formulas in this passage

unlabeled · line 3205
$q δ_{D}^{'} (q) = - δ_{D} (q) .$
unlabeled · line 3209
$(q δ_{D}^{'} (q) + δ_{D} (q)) \otimes u = 0.$
unlabeled · line 3219
$⟨ q δ_{D}^{'}, ψ ⟩ = - \partial_{q} (q ψ (q))_{q = 0} = - ψ (0) .$