Manuscript IV@1 · IV.tex · theorem · 351cb2d7

Theorem: Model 1 record: fixed plane and removable edge

Model 1: a fixed detector and its contrast › An explicit positive detector coefficient · lines 1310–1369 · proof included or linked

IV@1#thm:model1

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The fixed plane, detector, and states just constructed have the following property. For each fixed $λ \neq = 0$ , choose

μ_{0} = ∥ z_{*} ∥^{2}, μ_{*} = μ_{0} + \frac{1}{2},

S_{1} = C^{2} s \in {0, *} \sum (μ_{s}^{2} + 5 μ_{s} + 4),

and fix

θ = π /2, 0 < τ \leq min {\frac{1}{8 C}, \frac{∣ λ ∣ d _{0}}{S _{1}}} .

For the instruments of $A^{(1)}_{f_{*}, ε} (λ)$ ,

∣ p_{+, ε} (Ψ_{*}) - p_{+, ε} (Ψ_{0}) ∣ \geq c_{1} - e_{ε}, c_{1} = \frac{τ ∣ λ ∣ d _{0}}{4} > 0,

where

e_{ε} = \frac{1}{2} [E_{ε} (μ_{*}; τ) + E_{ε} (μ_{0}; τ)] ⟶ 0.

At $ε = 0$ the contrast is at least $c_{1}$ . The plane is fixed; no gravitational-sector limit is involved.

For $λ = 0$ , the two probabilities agree exactly for every $ε, τ, θ$ .

Proof. The graph bound and the Poisson moments give

s \in {0, *} \sum A^{(1)}_{f_{*}} (λ) Ψ_{s}^{2} \leq S_{1} .

For $θ = π /2$ , formula (eq:prob) reads the imaginary part of the unitary expectation. Equations (eq:Taylor) and (eq:mean1) give

∣Δ p ∣ \geq \frac{τ ∣ λ ∣ d _{0}}{2} - \frac{τ ^{2} S _{1}}{4} \geq c_{1} .

Proposition prop:edge-limit and the probability norm bound in Lemma lem:Ramsey give the stated $e_{ε}$ .

At $λ = 0$ , both the regulated and unregulated generator act only on the scalar factor, which is identical in the two preparations. This proves exact equality.

Displayed formulas in this passage

unlabeled · line 1314
$μ_{0} = ∥ z_{*} ∥^{2}, μ_{*} = μ_{0} + \frac{1}{2},$
unlabeled · line 1317
$S_{1} = C^{2} s \in {0, *} \sum (μ_{s}^{2} + 5 μ_{s} + 4),$
unlabeled · line 1322
$θ = π /2, 0 < τ \leq min {\frac{1}{8 C}, \frac{∣ λ ∣ d _{0}}{S _{1}}} .$
unlabeled · line 1330
$∣ p_{+, ε} (Ψ_{*}) - p_{+, ε} (Ψ_{0}) ∣ \geq c_{1} - e_{ε}, c_{1} = \frac{τ ∣ λ ∣ d _{0}}{4} > 0,$
unlabeled · line 1337
$e_{ε} = \frac{1}{2} [E_{ε} (μ_{*}; τ) + E_{ε} (μ_{0}; τ)] ⟶ 0.$
unlabeled · line 1352
$s \in {0, *} \sum A^{(1)}_{f_{*}} (λ) Ψ_{s}^{2} \leq S_{1} .$
unlabeled · line 1359
$∣Δ p ∣ \geq \frac{τ ∣ λ ∣ d _{0}}{2} - \frac{τ ^{2} S _{1}}{4} \geq c_{1} .$

Theorem: Model 1 record: fixed plane and removable edge

Displayed formulas in this passage

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