# Verification report — OWR-15212-005 (Wolansky's question on mutually dominating multiphase measure spaces, Oberwolfach Report 7/2017)

Verification date: 2026-10-02.

**Verdict.** The answer is no as posed: two atomless vector-valued measure spaces that dominate each other in the multiphase-transport order need not admit deterministic maps T and S with T_#σ_i = η_i and S_#η_i = σ_i. Explicit atomless examples with k = 2 on [0,1] and [0,1]² show that S can exist while T does not, and that neither may exist. The answer becomes yes if the colour fibres are atomless, for example for atomless spaces with k = 1 or with finitely many colours. For finite spaces the existence of T and S is classified exactly; for general spaces only necessary and sufficient conditions are proved, and they do not meet. The result is a folklore-level consequence of sufficiency and Blackwell equivalence of experiments. Two independent AI-assisted verification runs re-derived all proofs and reproduced the computations with separate code; neither found a mathematical error. The note is unrefereed.

## Statement checked
- **Primary source.** G. Wolansky, "Multiphase Optimal Transport", in *Applications of Optimal Transportation in the Natural Sciences* (organised by J.-D. Benamou, V. Ehrlacher, D. Matthes), Oberwolfach Reports 14 (2017), no. 1, Report No. 7/2017, pp. 388–390, doi:10.4171/OWR/2017/7.
  - The PDF of the report was read (78 pages, SHA-256 400d411ff8310f28bd79aab8211bb60d57789fb54f61538dc26396689c7ade05). The finder's copy and the copy used by the independent verification run, both fetched anonymously, are byte-identical.
  - The second verification run fetched the PDF again anonymously (SHA-256 unchanged), rendered pages 50 to 52 and compared the text with the definitions below.
  - **Definitions as printed.** A multiphase transport from (X, σ) to (Y, η) is a measurable family x ↦ P_x(dy) with ∫ P_x(dy) σ_i(dx) = η_i(dy), 1 ≤ i ≤ k; this is a Markov-kernel definition. If it exists, (σ, X) dominates (η, Y). The measures are vector-valued (R^k_+), atomless, on σ-algebras. The report states an equivalent convex-function (Jensen-type) criterion, ∫ F(dσ/d|σ|) d|σ| ≥ ∫ F(dη/d|η|) d|η| for convex F, and attributes it to its reference [1], D. Blackwell, "Comparison of experiments" (Proc. Second Berkeley Symp., 1951). The norm in |σ| is not specified. Gover–Wolansky (arXiv:2007.03982, Sect. 5) state that kernels of total mass one are meant.
  - **Question.** "Open Question: If both (σ, X) ≽ (η, Y) and (η, Y) ≽ (σ, Y), then there exist deterministic transports T : X → Y, S : Y → X such that T_#σ_i = η_i and S_#η_i = σ_i for i = 1, …, N?" Two harmless misprints: the second domination is printed with Y instead of X, and the index range uses N instead of k. The report gives no answer.
- **Corpus record.** ulamai/UnsolvedMath, OWR-15212-005 (status `open`). Its statement ("Must there be measurable maps T : X → Y and S : Y → X such that T_#σ_i = η_i and S_#η_i = σ_i for every component i?") is faithful to the source.
- **Reading is natural.** The examples use genuine probability kernels, atomless measures (Lebesgue measure times a density) on compact metric spaces, and in Examples 2 and 3 the total mass measure is Lebesgue measure on both sides. No misprint and no trivial literal reading is exploited. The obstruction lives in the atoms of the conditional measures on colour fibres, which atomlessness of the total mass does not exclude.

## Readings
| Reading | Answer | Witness |
|---|---|---|
| Wolansky's definition: kernels of probability measures; atomless; standard Borel spaces | no | Thm 3.1 (Examples 1–3) |
| Deterministic maps measurable only for the completion of the σ-algebra | no (stronger: non-existence is proved for m- and n-measurable maps) | Thm 3.1 |
| Domination with a weaker notion of kernel (the set of dominating pairs can only grow) | no (the examples dominate each other with probability kernels) | Thm 3.1 |
| Any norm in \|σ\| (sum or Euclidean) | no change: domination is defined by kernels, and equality of colour laws does not depend on the norm | Remark 2.1 |
| k = 1, or finitely many colours (colour law purely atomic), atomless standard Borel spaces | yes | Cor. 4.3 |
| Colour fibres of X atomless (resp. of both spaces) | T exists (resp. T and S exist) | Thm 4.2 (ii), (iv) |
| Finite spaces (not atomless) | exact classification | Prop. 4.5 |
| Weak approximation by deterministic maps instead of exact equality | yes in Example 1 | Remark 3.2 |
| General standard Borel spaces whose fibres have atoms | open to us: necessary and sufficient conditions are proved, and they do not meet | Thm 4.2 (i), (iii); Remark 4.4 |

## Results in the paper
- **Lemma 2.2 (Blackwell equivalence).**
  - (a) If σ ≽ η then the total masses agree and ∫ F∘λ dn ≤ ∫ F∘κ dm for continuous convex F on the simplex.
  - (b) Mutual domination implies equal colour laws ν_σ = ν_η.
  - (c) If Y is standard Borel and ν_σ = ν_η then σ ≽ η (kernel P_x = n_{κ(x)}).
- **Lemma 2.3 (rigidity).** If η ≽ σ and T_#σ_i = η_i, then λ∘T = κ almost everywhere. It is proved by an L² computation: ∫|κ − λ∘T|² dm = ∫|κ|² dm − ∫|λ|² dn ≤ 0. In statistical language, T is a sufficient statistic.
- **Theorem 3.1 (counterexamples), k = 2, all spaces atomless and compact.**
  - (1) X = [0,1], σ = (x dx, (1−x) dx); Y = [0,1]², η = (y_1 dy, (1−y_1) dy): S(y) = y_1 exists, no T (the image of T would sit on a graph, a Lebesgue-null set).
  - (2) X = Y = [0,1], σ = (x dx, (1−x) dx), η = ({2y} dy, (1−{2y}) dy): S = {2y} exists, no T.
  - (3) X = Y = [0,1], σ = ({3x} dx, …), η = ({2y} dy, …): neither T nor S (a density would have to equal 1/2 but takes values in {0, 1/3, 2/3, 1}, resp. 1/3 but takes values in {0, 1/2, 1}).
  - The same holds for every k ≥ 2 by repeating a component.
- **Remark 3.2.** In Example 1, T_N(x) = (x, {Nx}) satisfies (T_N)_#σ_i → η_i weakly, so the failure is one of exactness.
- **Theorem 4.2 (fibres).** For standard Borel spaces with equal colour law and fibres m_p, n_p: (i) T exists ⇒ T_#m_p = n_p; (ii) m_p atomless ⇒ T exists (quantile construction, jointly Borel); (iii) T and S exist ⇒ the fibres have the same atom masses with multiplicities (Lemma 4.1); (iv) both fibres atomless ⇒ T and S exist.
- **Corollary 4.3.** k = 1 and purely atomic colour laws (for example finitely many colours): T and S exist.
- **Proposition 4.5 (finite spaces).** T exists iff the masses in each colour class can be packed exactly into the masses of the other space; T and S both exist iff the multisets of masses coincide in every colour class, iff there is a bijection f with η(f(x)) = σ(x).
- **What is not claimed.** The converses of Theorem 4.2 (i) and (iii) for general standard Borel spaces are not proved; the classification is exact for finite spaces only.

## Computations (exact; scripts and outputs in reproducibility/)
The proofs for infinite spaces do not rely on computation; the computations check the elementary identities and the finite classification.
- **Lead** (`lead/lead_checks.py`, standard library only, about 2 s; ALL CHECKS PASSED).
  - The kernel identities of Examples 2 and 3 in both directions and for both components on all intervals with endpoints in (1/d)Z, d ≤ 8 (960 identities), with a negative control (a kernel with wrong weights is detected).
  - The second moments ∫|κ|² dm = ∫|λ|² dn = 2/3, and the value sets {0, 1/3, 2/3, 1}, {0, 1/2, 1}, {0, 1} of the densities in the non-existence arguments.
  - Lemma 4.1 on all 12,647 ordered pairs of integer partitions of n ≤ 12: measure-preserving maps in both directions exist iff the multisets coincide.
  - The exact errors of the weak approximation in Remark 3.2 (for f = y_2 and f = y_1y_2 the first-component error is exactly 1/(12N)).
- **Finder** (`finder/`).
  - `finite_exhaustive.py` (about 20 s): finite spaces with integer vector masses (k = 2: entries ≤ 2 and ≤ 5 points, entries ≤ 3 and ≤ 4 points; k = 3: entries ≤ 2 and ≤ 4 points). For all 43,619 ordered pairs with equal colour law it certifies mutual domination by explicit kernels (exact), brute-forces all maps T and S, compares with the packing criterion, and asserts "both exist iff the fibre multisets coincide". For k = 2, entries ≤ 3, at most 4 points, the four cases (T and S; T only; S only; neither) occur 3875, 882, 882 and 1422 times. All assertions pass.
  - `analytic_checks.py`: exact checks of the kernel identities for A = B = [0, t], t rational with denominator ≤ 12, and of the value sets.
- **First independent verification run** (`independent_run/`, AI-assisted, code written separately).
  - It re-ran both finder programs on its own copies with identical output.
  - `indep_check.py`: for k = 2, vectors in {0,…,3}², at most 4 points, and 6000 random ordered pairs with equal total mass, it decides domination by a linear program (not through Lemma 2.2) and the existence of T and S by brute force over all maps. Mutual domination holds iff the colour laws agree, and the existence of T agrees with the packing criterion, in every case. Outcomes (T and S, neither, T only, S only): 115, 33, 31, 25.

- **Second independent verification run** (`independent_run_2/`, AI-assisted, code written from scratch; see its README).
  - `run2_finite.py` (about 2.5 minutes): domination is decided by a hand-written exact Phase-I simplex over rational numbers (cross-checked against HiGHS on 199 random instances), never through Lemma 2.2; the existence of T and S by brute force over all maps. Exhaustive over all 12,348 ordered pairs with equal colour law, and 47,318 sampled ordered pairs with equal total vector and different colour law, in five blocks (k = 2 with entries ≤ 2 and ≤ 4 points, entries ≤ 3 and ≤ 4 points, entries ≤ 1 and ≤ 5 points; k = 3 with entries ≤ 1 and ≤ 4 points, entries ≤ 2 and ≤ 3 points). Mutual domination holds iff the colour laws agree; T exists iff the packing criterion of Proposition 4.5(a) holds; T and S both exist iff the per-colour mass multisets coincide iff a bijection exists. Outcomes (T and S, T only, S only, neither) = 8406, 1212, 1212, 1518; for k = 2, entries ≤ 3, ≤ 4 points: 3875, 882, 882, 1422, as in the finder's run. Lemma 4.1 for all pairs of partitions of n ≤ 8 with ≤ 6 parts (805 pairs).
  - `run2_blackwell_k2.py`: for k = 2, exact LP domination equals Blackwell's criterion on 17,407 ordered pairs.
  - `run2_examples.py`: exact kernel identities of Examples 2 and 3 on all intervals with endpoints in (1/d)Z, d ≤ 12 (two wrong-kernel negative controls fail as they should), the value sets of h and g, the density identity on 300 random grid sets, second moments 2/3, and the errors 1/(12N) of Remark 3.2 for N ≤ 60.
  - Re-run of `lead_checks.py`, `analytic_checks.py`, `finite_exhaustive.py` and `indep_check.py` from a fresh extraction of the released `source.zip`: outputs identical byte for byte.

## Independent adversarial audits
Two independent verification runs (AI-assisted), both on 2026-10-02.

**First run.** Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (the examples do not exploit a misprint or a trivial reading) |
| Proofs | CONFIRMED (Lemmas, Examples 1–3, Theorem 4.2 re-derived; one minor measurability gap in Example 1) |
| Computations | CONFIRMED (identical re-runs; separate LP-based code agrees) |
| Answer as posed | CONFIRMED (negative) |
| Novelty | folklore-level consequence of sufficiency and Blackwell equivalence; no priority claim |
| Presentation | CONFIRMED_WITH_FIXES |

All required presentation and rigour fixes of the first run were applied:
1. No exact characterisation is claimed beyond finite spaces; for general spaces the paper gives necessary conditions (Theorem 4.2 (i), (iii)) and sufficient conditions (Theorem 4.2 (ii), (iv)), and says that they do not meet.
2. In Example 1 a Borel version of T_2 is used before the graph is formed, and non-existence is proved for maps measurable for the completion.
3. In Lemma 2.2(c) only Y is required to be standard Borel; |σ| is the total mass measure, and the independence of the notion of equal colour laws from the norm is proved (Remark 2.1).
4. The folklore-level nature of the result is stated plainly, with citations, and no novelty is claimed beyond the answer to this question with explicit atomless examples and the atomless-fibre positive case.
5. The positive result is worded as: no as posed; yes if the colour fibres are atomless.
6. The loose attribution of the one-sided finite case to a remark in the abstract was removed.

**Second run (final readiness check).** It re-fetched the report, re-derived every statement with measurability details, wrote new code, re-read the later literature, and read Blackwell's 1951 paper and the statements of Blackwell's 1953 paper first hand. Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (definitions and Open Question as printed on p. 388; no topological hypothesis in the report; later sources by Gover and Wolansky neither pose nor answer the question) |
| Proofs | CONFIRMED (Lemmas 2.2, 2.3, 4.1, Examples 1–3, Theorem 4.2, Corollary 4.3, Proposition 4.5, Remark 3.2; no error) |
| Computations | CONFIRMED (own exact code; earlier programs re-run from the released archive with identical output) |
| Answer as posed | CONFIRMED (negative); positive results correctly scoped |
| Novelty and credit | CONFIRMED_WITH_FIXES: the mechanism is in Blackwell 1951 (standard experiment, Thm 3; independent combination, Thm 11); the claim of no novelty beyond the examples and the fibre analysis is accurate |
| Presentation | CONFIRMED_WITH_FIXES |

Fixes required by the second run, all applied:
1. Credit precision: Lemma 2.2 is contained in Blackwell 1951 (Thms 3, 6, 8) and Blackwell 1953 (Thms 1–3) without the Sherman–Stein converse; the convex-function criterion is credited in Blackwell 1951 to Bohnenblust, Shapley and Sherman (private communication); Example 1 is an instance of Blackwell 1951, Thm 11; Examples 2–3, Theorem 4.2 and Proposition 4.5 were not found in the sources read.
2. The paper now states which sources were read and which were not (Sherman, Halmos–Savage, Le Cam, Torgersen).
3. Wolansky's book (arXiv:1911.04348, Definition 5.3.1, Theorem 5.3) was added, with a remark that Gover–Wolansky call the order a partial order and Gover a preorder.
4. The words "atomless" were added to the clauses on k = 1 and finitely many colours (abstract, Answer paragraph); Theorem 1.2(c) now reads T_# m_p = n_p and "the same atom masses with the same multiplicities"; the paper says that the report imposes no topological hypothesis.
5. Lemma 2.2(c): the exceptional null set of colours is shown to pull back to a null set.
6. The Verification and Scope paragraphs describe this second run and the search statement (arXiv, Crossref, OpenAlex, zbMATH, three web searches in total).
Not applied: a cosmetic change of the bibliography labels for Le Cam ("Cam64"), because the BibTeX style used cannot produce initials for this surname.

## Relation to the literature, novelty and scope
- **Searches (October 2026).** We searched arXiv (author, title and abstract queries on Blackwell equivalence, isomorphism of experiments, Wolansky, Gover, mutual domination), Crossref, OpenAlex and zbMATH, and made three web searches in total over the runs. Nothing contains the examples or answers the question.
  - The report itself, Gover–Wolansky (arXiv:2007.03982, Sect. 5), Wolansky's book (arXiv:1911.04348, Chapter 5) and Gover's thesis (arXiv:2501.13557) were searched in full text for "mutual", "antisymmetric", "isomorphic", "deterministic" and "Blackwell". None poses or answers the question apart from the report. Gover–Wolansky call the order a partial order, the thesis (Claim 3.34) a preorder. Section 3.9 of the thesis gives, under additional hypotheses (Assumptions 3.59 and 3.16, Y complete), a sequential-domination criterion for the existence of some transport map for X ≽ Y; it does not address mutual domination.
- **Known theory (read first hand).** Blackwell, "Comparison of experiments" (Proc. Second Berkeley Symp., 1951), was read in full: every experiment is equivalent to its standard experiment, whose outcome is the posterior (Thm 3); the sufficiency order is described by mean-preserving transformations (Thm 6); sufficiency implies the convex-function inequality (Thm 8); a combination with an independent experiment does not change the standard measure if that experiment does not depend on the parameter (Thm 11); the convex-function criterion (Thm 4) is credited there to Bohnenblust, Shapley and Sherman (private communication). The statements of Blackwell's "Equivalent comparisons of experiments" (Ann. Math. Statist. 1953, pp. 265–267; Thms 1–6) were also read. Fritz–Gonda–Perrone–Rischel, arXiv:2010.07416 v3, Thm 5.6, states the Blackwell–Sherman–Stein theorem for standard Borel spaces with finite parameter set. Mutual domination is Blackwell equivalence of the experiments (σ_i/σ_i(X))_i and (η_i/η_i(Y))_i; Lemma 2.2 does not need the Sherman–Stein converse; Lemma 2.3 is a version of the fact that a statistic whose image experiment is at least as informative is sufficient; Example 1 is an experiment combined with an independent uniform variable.
- **Caveats.** The papers of Sherman (1951), Halmos–Savage (1949), Le Cam (1964, 1986) and Torgersen (1970, 1991) were not accessed; what the note says about them is not taken from their text. The fibre-packing mechanism of Examples 2–3, Theorem 4.2 and Proposition 4.5 were not found in the sources read, which is not a priority claim. This negative search is not a proof of priority.
- **Scope.** The note answers the question as posed for atomless compact spaces and every k ≥ 2, gives a positive result under atomless fibres, and classifies the finite case exactly. The converses of the necessary conditions for general spaces are not proved.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/