Manuscript X-Enter-ST@1 · X-Enter-ST.tex · proposition · f76acc1f

Proposition: Selective and universal continuity bounds

Stage F: bounded segments and excluded prescription changes › Complete-output bounds for general finite programs · lines 1382–1423 · proof included or linked

X-Enter-ST@1#prop:bounds

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For $∣ λ ∣ \leq Λ$ ,

Δ_{Φ} (a_{0} + λ c_{ε}, a_{0}) \leq ∣ λ ∣ c_{ε} L_{Φ} (J) .

For all $a, h$ ,

Δ_{Φ} (a + h, a) \leq min {1, ∣ h ∣/ (2 σ)} .

Exact selective constancy on any nontrivial real interval already forces $B_{y} = C_{y} = 0$ for all $y$ .

Proof. Integrate the derivative of each block in (eq:selective). The trace norm of a block-diagonal operator is the sum of its block trace norms.

For a Hermitian $Z = Z_{+} - Z_{-}$ , positivity and trace preservation give

∥ Rec_{Φ} (Z) ∥_{1} \leq tr Rec_{Φ} (Z_{+}) + tr Rec_{Φ} (Z_{-}) = ∥ Z ∥_{1} .

Differentiating (eq:bloch) gives

\frac{1}{2} ∥ τ^{'} (a) ∥_{1} = \frac{s ( a )}{2 σ} \leq \frac{1}{2 σ} .

Integration and the fact that distances of states are at most one prove the universal bound.

If the derivative is zero throughout an interval, division by $s^{2} > 0$ gives $B_{y} = C_{y} sinh (a / σ)$ . Taking two different points in the interval gives $C_{y} = 0$ , then $B_{y} = 0$ .

Displayed formulas in this passage

unlabeled · line 1385
$Δ_{Φ} (a_{0} + λ c_{ε}, a_{0}) \leq ∣ λ ∣ c_{ε} L_{Φ} (J) .$
unlabeled · line 1390
$Δ_{Φ} (a + h, a) \leq min {1, ∣ h ∣/ (2 σ)} .$
unlabeled · line 1406
$∥ Rec_{Φ} (Z) ∥_{1} \leq tr Rec_{Φ} (Z_{+}) + tr Rec_{Φ} (Z_{-}) = ∥ Z ∥_{1} .$
unlabeled · line 1411
$\frac{1}{2} ∥ τ^{'} (a) ∥_{1} = \frac{s ( a )}{2 σ} \leq \frac{1}{2 σ} .$

Proposition: Selective and universal continuity bounds

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