TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Adaptive projection certificates are sound and finite-budgeted

Statement

Under the input conditions of Finite-budget adaptive finite-state projection, the algorithm terminates after at most $N$ projection attempts and at most $N B$ uniformization terms. If it returns certified for $F_{i}$ , then the true killed loss $c_{F_{i}} (T)$ is at most $ε$ , the embedded killed-vector $ℓ^{1}$ error is at most $ε$ , and every bounded observable with oscillation $osc (f)$ has cemetery-lift error at most $\frac{1}{2} osc (f) ε$ . An inconclusive return makes no claim that any finite projection meets the tolerance.

Facts & Assumptions

Given: A finite nested candidate list $F_{1} \subseteq \dots \subseteq F_{N}$ , a finite term cap $B$ , a tolerance $ε > 0$ , and valid projection-rate and arithmetic certificates as required by the algorithm.

[F1]

Finite-budget adaptive finite-state projection: each candidate is attempted at most once, solver terms are capped by $B$ , and certified is returned only when a validated upper endpoint $U_{i}$ satisfies $U_{i} \leq ε$ .

[F2]

Finite-state killed projection has an exact lost-mass certificate: for each actual projection, its cemetery mass is the exit probability, the embedded killed-vector $ℓ^{1}$ error equals that mass, and the bounded-observable error is at most half the oscillation times that mass.

[F3]

Uniformization truncation, mass loss, and arithmetic-error ledger: a computed finite cemetery vector with certified total error $δ_{i}$ has each coordinate within $δ_{i}$ of the exact coordinate.

Proof

technique · verify the finite loop bound and the interval certificate

1.1givenF1algebra

The procedure iterates over only the supplied finite list $F_{1}, \dots, F_{N}$ , and for each $F_{i}$ the uniformization solver uses at most $B$ terms by [F1]. It either returns during one of these calls or after the last candidate, so it terminates after at most $N$ projections and $N B$ terms.

1.2F1F3algebra

Suppose the algorithm returns certified at index $i$ . By [F3], $∣ {\hat{c}}_{i} - c_{F_{i}} (T) ∣ \leq δ_{i}$ . Since $0 \leq c_{F_{i}} (T) \leq 1$ , the interval endpoints in [F1] enclose $c_{F_{i}} (T)$ ; hence $c_{F_{i}} (T) \leq U_{i} \leq ε$ .

2.1F2step 1.2algebra

Apply [F2] to this actual $F_{i}$ : the embedded killed-vector error equals $c_{F_{i}} (T)$ and the observable error is at most $\frac{1}{2} osc (f) c_{F_{i}} (T)$ . Step 1.2 gives the asserted $ε$ bounds.

3.1F1given∎

If no candidate passes, or a finite solver returns an unavailable certificate, [F1] specifies inconclusive; the procedure has examined only the supplied finite list and therefore establishes no existence claim for a projection outside that list.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Brian Munsky and Mustafa Khammash, The Finite State Projection Algorithm for the Solution of the Chemical Master Equation (standard reference, not scraped)