Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

A nonzero witness with a controlled number tail

Model 2: dense TT completion and fixed-number spectral removal › A nonzero witness with a controlled number tail · lines 1013–1053

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Choose a smooth compact-annular TT coherent label whose classical field $k_{*}$ is nonzero somewhere. Choose compact smooth scalar data so that $L_{k_{*}} φ_{*}$ is nonzero there. The same Hessian construction used in Model 1 provides such data. Then $ψ_{*} = S_{t_{0}} (k_{*}) φ_{*}$ is not zero. Choose a nonnegative compact bump $b$ supported where it is nonzero and set

f_{*} = b ψ_{*}, d_{2} = \int b ψ_{*}^{2} d^{4} x > 0.

Let $z_{g}, z_{φ}$ be the corresponding coherent labels and take

Ψ_{0} = Ω_{g} \otimes Coh (z_{φ}), Ψ_{a} = Coh (a z_{g}) \otimes Coh (z_{φ}), a \neq = 0.

Expanding the four legs gives exactly

Δ ⟨ B_{\infty} ⟩ = a d_{2} .

For total coherent mean number $ν$ , define

t_{M} (ν) = [e^{- ν} n > M - 2 \sum (n + 2)^{2} \frac{ν ^{n}}{n !}]^{1/2} .

Number bandwidth two and (eq:B2graph) imply

∥ (Π_{M} B_{R} Π_{M} - B_{R}) Ψ ∥ \leq 2 C_{B} t_{M} (ν) .

Therefore

Δ ⟨ Π_{M} B_{\infty} Π_{M} ⟩ = a d_{2} + ϵ_{M}, ∣ ϵ_{M} ∣ \leq E_{M}, E_{M} ⟶ 0.

The same number estimates bound $A^{(2)}_{R, M} Ψ_{s}$ uniformly in $R, M$ .

Displayed formulas in this passage

unlabeled · line 1021
$f_{*} = b ψ_{*}, d_{2} = \int b ψ_{*}^{2} d^{4} x > 0.$
unlabeled · line 1026
$Ψ_{0} = Ω_{g} \otimes Coh (z_{φ}), Ψ_{a} = Coh (a z_{g}) \otimes Coh (z_{φ}), a \neq = 0.$
unlabeled · line 1031
$Δ ⟨ B_{\infty} ⟩ = a d_{2} .$
unlabeled · line 1036
$t_{M} (ν) = [e^{- ν} n > M - 2 \sum (n + 2)^{2} \frac{ν ^{n}}{n !}]^{1/2} .$
unlabeled · line 1041
$∥ (Π_{M} B_{R} Π_{M} - B_{R}) Ψ ∥ \leq 2 C_{B} t_{M} (ν) .$
unlabeled · line 1046
$Δ ⟨ Π_{M} B_{\infty} Π_{M} ⟩ = a d_{2} + ϵ_{M}, ∣ ϵ_{M} ∣ \leq E_{M}, E_{M} ⟶ 0.$

A nonzero witness with a controlled number tail

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