Manuscript Oai265-A@2 · Oai265-A.tex · lemma · c0a34a45

Lemma: A positive spectral constraint need not be local

Stage 1: audit of the five proof interfaces › What the global gap gives immediately · lines 453–474 · proof included or linked

Oai265-A@2#lem:spectral

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Let $K = H - E_{0} I$ obey (eq:gap). Then

Q = K (K + Δ I)^{- 1}

is well defined and satisfies

0 \leq Q \leq I, Q Ω = 0, Q \geq \frac{1}{2} (I - ∣Ω ⟩ ⟨ Ω∣) .

These spectral statements provide no graph-local decomposition or tail estimate for $Q$ .

Proof. On the ground vector the spectral value is zero. On an orthogonal eigenvector with eigenvalue $λ \geq Δ$ , the spectral value is $λ / (λ + Δ) \in [1/2, 1]$ . The assertions follow from the spectral theorem. No support conclusion occurs in this argument.

Displayed formulas in this passage

unlabeled · line 456
$Q = K (K + Δ I)^{- 1}$
unlabeled · line 460
$0 \leq Q \leq I, Q Ω = 0, Q \geq \frac{1}{2} (I - ∣Ω ⟩ ⟨ Ω∣) .$

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section: What the global gap gives immediately