Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

The fixed witness

The fully specified convolution contract › The fixed witness · lines 447–474

X-A-conv-ST@1#eq:witness

Define

ψ (x) = ⎩ ⎨ ⎧ exp (1 - \frac{1}{1 - 64 r ( x ) ^{2}}), 0, r (x) < 1/8, r (x) \geq 1/8, N = \int_{M} ψ (x) d^{4} x > 0.

Freeze

f = \frac{ψ}{N}, ξ_{*} = ψ \partial_{1} \in g_{C}, ρ_{0} = x^{1} .

The free equation $□_{η} ρ_{0} = 0$ holds. Its unboundedness at spatial infinity is allowed by the declared smooth domain; no Hilbert-space normalizability is being asserted.

The same $g, f, ξ_{*}, ϱ$ are used for every allowed fixed width. No witness is removed after its defect is found.

Displayed formulas in this passage

X-A-conv-ST@1#eq:witness · line 460
$f = \frac{ψ}{N}, ξ_{*} = ψ \partial_{1} \in g_{C}, ρ_{0} = x^{1} .$
unlabeled · line 450
$ψ (x) = ⎩ ⎨ ⎧ exp (1 - \frac{1}{1 - 64 r ( x ) ^{2}}), 0, r (x) < 1/8, r (x) \geq 1/8, N = \int_{M} ψ (x) d^{4} x > 0.$