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

Theorem: Fixed-M complete spectral limit

Model 2: dense TT completion and fixed-number spectral removal › Quadratic operators and the free-term correction · lines 982–1009 · proof included or linked

IV-B@1#thm:spectral-channel, IV-B@1#eq:model2-complete

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For fixed finite $M \geq 2$ , put

δ_{M} (R) = M C_{L} R^{- 1} + 2 M (M - 1) C_{P} R^{- 1/2} .

Then

A^{(2)}_{R, M} - A^{(2)}_{\infty, M} \leq ∣ λ ∣ δ_{M} (R),

and

d_{⋄} (I_{A^{(2)}_{R, M}}, I_{A^{(2)}_{\infty, M}}) \leq min {1, \frac{∣ τ λ ∣}{2} δ_{M} (R)} .

The estimate is uniform over states and references.

Proof. Combine Lemma lem:spectral, (eq:B2compression), and Proposition prop:instrument-error.

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IV-B@1#eq:model2-complete · line 995
$d_{⋄} (I_{A^{(2)}_{R, M}}, I_{A^{(2)}_{\infty, M}}) \leq min {1, \frac{∣ τ λ ∣}{2} δ_{M} (R)} .$
unlabeled · line 985
$δ_{M} (R) = M C_{L} R^{- 1} + 2 M (M - 1) C_{P} R^{- 1/2} .$
unlabeled · line 990
$A^{(2)}_{R, M} - A^{(2)}_{\infty, M} \leq ∣ λ ∣ δ_{M} (R),$

Theorem: Fixed-M complete spectral limit

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