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

Theorem: Model 2 record

Model 2: dense TT completion and fixed-number spectral removal › A nonzero witness with a controlled number tail · lines 1054–1086 · proof included or linked

IV-B@1#thm:model2

Back to manuscript

For fixed $a, λ \neq = 0$ , one can choose a single $τ > 0$ , independent of $R, M$ , and $θ = π /2$ , such that

∣Δ p_{+, R, M} ∣ \geq c_{2} - e_{M} - e_{R} (M), c_{2} > 0,

where

e_{M} ⟶ 0, e_{R} (M) \leq ∣ τ λ ∣ δ_{M} (R) ⟶ 0 at fixed M .

For sufficiently large but finite $M$ , the limiting contrast is positive. At $λ = 0$ , the two record laws agree exactly for every $R, M, τ, θ$ .

Proof. Let $d = ∣ λa ∣ d_{2}$ , and let $S_{2}$ be a common bound on the sum of the two generator second moments. Choose $0 < τ \leq d / S_{2}$ . The proof of Lemma lem:contrast gives

∣Δ p_{+, \infty, M} ∣ \geq \frac{τ d}{4} - \frac{τ ∣ λ ∣ E _{M}}{2} .

Take $c_{2} = τ d /4$ and $e_{M} = τ ∣ λ ∣ E_{M} /2$ . The two probability errors from the spectral replacement total at most $e_{R} (M)$ . Finally, $A^{(2)}_{R, M} (0) = I_{g} \otimes F_{0, M}$ , and the scalar preparations are identical.

Displayed formulas in this passage

unlabeled · line 1058
$∣Δ p_{+, R, M} ∣ \geq c_{2} - e_{M} - e_{R} (M), c_{2} > 0,$
unlabeled · line 1063
$e_{M} ⟶ 0, e_{R} (M) \leq ∣ τ λ ∣ δ_{M} (R) ⟶ 0 at fixed M .$
unlabeled · line 1077
$∣Δ p_{+, \infty, M} ∣ \geq \frac{τ d}{4} - \frac{τ ∣ λ ∣ E _{M}}{2} .$

Theorem: Model 2 record

Displayed formulas in this passage

Nearby in this section