Manuscript XIII-A@1 · XIII-A.tex · section · f56e91c7

The copied witness and a quantifier correction

One frozen region and the observation-qualified defect set › The copied witness and a quantifier correction · lines 593–633

XIII-A@1#eq:copied-witness, XIII-A@1#eq:strict-detection

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For the detecting data of XII-F, with a compact spatial realization,

m = 0, ψ = t^{2} + (x^{1})^{2}, γ = ψ η, ϕ = ψ, b = (- 2 t, 2 x^{1}, 0, 0), c = c ˉ = 0.

The parameter is a compact realization of $a (t) \partial_{t}$ , with $a \geq 0$ a nonzero bump. XII-F proves

B_{ξ} (u_{*}) = Δ_{ξ} Φ^{(2)}_{D} (u_{*}) = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.

It also proves that this lift has nonzero full-shell carrier class XIIF (Theorem on the nonzero auxiliary pairing). The value and its factor $8$ are copied, not recomputed.

The witness can be included in fixed restriction data by choosing $W$ to contain its compact parameter support and choosing the switch to cover the relevant causal compact sets before evaluation. The stronger uniform plateau requirement below is compatible with that realization: the switch may be chosen one near $J^{-} (supp f) \cap J^{+} (W)$ as well as near $W$ . This is still among XII-F's allowed witness switches, since it covers the smaller causal set generated by $supp ξ$ .

For such detecting fixed data,

C_{W} \subseteq C_{def} (W; f, g) ⊊ C_{sol}, u_{*} \in / C_{def} (W; f, g) .

The first inclusion is not asserted to be equality.

Displayed formulas in this passage

XIII-A@1#eq:copied-witness · line 605
$B_{ξ} (u_{*}) = Δ_{ξ} Φ^{(2)}_{D} (u_{*}) = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.$
XIII-A@1#eq:strict-detection · line 627
$C_{W} \subseteq C_{def} (W; f, g) ⊊ C_{sol}, u_{*} \in / C_{def} (W; f, g) .$
unlabeled · line 596
$m = 0, ψ = t^{2} + (x^{1})^{2}, γ = ψ η, ϕ = ψ, b = (- 2 t, 2 x^{1}, 0, 0), c = c ˉ = 0.$

Nearby in this section

remark: Remark: Strictness is not automatic for every observation