bibkey: beiglbocknutz2014martingale authors: Mathias Beiglböck and Marcel Nutz year: 2014 title: Martingale inequalities and deterministic counterparts doi: 10.1214/EJP.v19-3270 url: https://arxiv.org/abs/1401.4698v2 claim: Finite-support martingale values are iterated concave envelopes, and their increasing limit is the least fixed point majorizing the terminal payoff for an update preserving zero increments. strata_touched:

• D5/S3/Quantum/Recovery/FiniteLocalBellmanEnvelope
• D5/S3/Quantum/Recovery/FiniteLocalFrontierMoment license: citation-only triage: anchor

Finite-support Bellman envelopes

Lemma 2.2 identifies the concave envelope with the supremum of expectations under finitely supported laws having a specified barycenter. Proposition 2.1 and Lemma 2.3 connect envelope iteration with finite-horizon martingale values. Theorem 3.3 treats a state update preserving zero increments: the increasing iteration limit is the least fixed point majorizing the payoff. The framework allows extended real payoffs and does not require continuity of the limit.

The Bloch-ball construction restricts each node to one coordinate. Its five-point representation uses Carathéodory in the three active coordinates and one reward coordinate. Joint upper semicontinuity at finite stages follows from compact truncated hypographs. The retained Lean proof directly constructs finite trees and grafts finitely many continuations with a common error; it does not use the cited paper as a kernel premise. The source-specific frontier gap and the physical recovery interface are additional mathematical obligations.

Verified locator

The declared DOI is https://doi.org/10.1214/EJP.v19-3270. Crossref’s record for that DOI identifies Beiglböck and Nutz, the title above, Electronic Journal of Probability and publication year 2014. The version URL https://arxiv.org/abs/1401.4698v2 locates the cited Lemma 2.2, Proposition 2.1, Lemma 2.3 and Theorem 3.3; the full text is https://arxiv.org/html/1401.4698v2. These locators support the general envelope attribution and do not identify a theorem for the five-ray quantitative gap.

Finite local threshold splitting

Matthias Kleinmann, Hermann Kampermann and Dagmar Bruß, Asymptotically perfect discrimination in the LOCC paradigm (2011), DOI: 10.48550/arXiv.1105.5132, provide the interpolation and recovery construction used for the complete common-threshold cut.

Section III.1, equations (5a), (5b) and (6), constructs an interpolating local measurement followed by recovery copies of the original branches. Section III.2, Protocol splitting, applies this construction at the first crossing of a prescribed regular-deviation threshold. The primary version is arXiv:1105.5132v2; the corresponding full text supplies these locators.

For a normalized product prefix effect $Q$ with defect $d>\epsilon$, let $E_i=w_i{Q}_i$ be an original local split, with ${\sum }_i{E}_i=Q$ and unchanged inactive factor. Write $d_i=\mathrm{Tr}(P_{-}{Q}_i)$ and set

$${\alpha }_i=\frac{w_i(\epsilon -d_i{)}_{+}}{d-\epsilon },\beta =\frac{1}{1+{\sum }_i{\alpha }_i}.$$

The interpolation rows are $\beta (E_i+{\alpha }_{i}Q)$. Their masses are $\beta (w_i+{\alpha }_i)$ and, on positive rows, their normalized defects are $\max (\epsilon ,d_i)$. Recovery copies have joint effects $E_{ij}=\beta ({\delta }_{ij}+{\alpha }_i)E_j$. Each column sums to $E_j$. Copying every complete original continuation therefore preserves each terminal effect, endpoint and label, hence every real function of the original terminal leaf under one common leaf map. A zero row has an identity continuation and requires no conditional division.

At an exact-threshold row the recovery copies remain below the cut. Above threshold, ${\alpha }_i=0$ and only the original child $i$ has positive mass; recursion enters that strictly shorter original child. The complete prefix-free cut consists of exact-threshold nodes and terminal nodes above threshold. Each original level introduces at most two refined levels, so a tree of depth $D$ becomes a tree of depth at most $2D$.

The source defect is a regular deviation for the two-state ensemble ${\gamma }_{-}=P_{-}/4$, ${\gamma }_{+}=P_W/4$, with average state ${\gamma }_{-}+{\gamma }_{+}=I/4$. The minus marginal depends continuously on joint classical probabilities, is preserved by postprocessing, and is a weighted average of its conditional marginals under staged postselection. At a normalized product effect $Q$, its conditional value is $\mathrm{Tr}(P_{-}Q)=d$. A non-identity or singular root is a product prefix filter; child effects lie on its support, and the local square-root/pseudoinverse realization acts there with identity completion on the unused kernel. Zero branches carry no conditional value requirement.

The finite frontier proof uses this known construction inside its source-specific moment and failure-defect estimate. The cited discrimination conclusions are not recovery premises. The quantitative five-ray gap and the forward map from actual CP protocols to the source Bellman value are separate conclusions.