Manuscript IX-B-ST@1 · IX-B-ST.tex · theorem · f6af1523

Theorem: Finite enclosure from the analytic estimate

Finite numerical certificates and a destructive clock › A quantitative enclosure interface · lines 1569–1599 · proof included or linked

IX-B-ST@1#thm:enclosure, IX-B-ST@1#eq:certificateerror

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Suppose supplied numerical approximants $A_{ε}, b$ obey

∣ A_{ε} - A_{ε} ∣ \leq e_{T}, ∣ b - b ∣ \leq e_{r} .

Then

A_{ε} - L_{ε} b - z \leq E_{ε} := e_{T} + ∣ L_{ε} ∣ e_{r} + C_{A, ρ, m} C_{A, a, D} ε (1 + ∣ lo g ε ∣) p_{A, 64 + d} (Φ) .

For $ε = 2^{- n}$ , the analytic term is bounded by the same constant times $2^{- n} (1 + n lo g 2) p_{A, 64 + d} (Φ)$ . It tends to zero.

Proof. Subtract $z$ , insert the two exact pairings, apply the triangle inequality, and use Theorem thm:descendants. The dyadic statement is substitution.

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IX-B-ST@1#eq:certificateerror · line 1578
$A_{ε} - L_{ε} b - z \leq E_{ε} := e_{T} + ∣ L_{ε} ∣ e_{r} + C_{A, ρ, m} C_{A, a, D} ε (1 + ∣ lo g ε ∣) p_{A, 64 + d} (Φ) .$
unlabeled · line 1573
$∣ A_{ε} - A_{ε} ∣ \leq e_{T}, ∣ b - b ∣ \leq e_{r} .$

Theorem: Finite enclosure from the analytic estimate

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