# Verification report — OWR-15428-015 (convexity of the non-completable region in rank-one probability tensor completion)

Verification date: 2026-09-29 (revised the same day after a second referee report).

**Verdict.** The answer is no. For a pattern E of observed entries, let N_E be the set of nonnegative partial tensors that cannot be completed to θ_1 ⊗ … ⊗ θ_n with probability vectors θ_j. N_E need not be convex, and convexity fails inside the proposers' own setting.
- The bare negative answer is implicit in Kubjas–Rosen (J. Algebr. Stat. 8 (2017), Examples 4 and 9, 3(b)). Two of the proposers wrote that paper, and it predates the question.
- The paper's contribution is the characterisation: for n ≥ 2 and all d_j ≥ 2, N_E is convex if and only if no observed entry is pinned.
- Under the standing assumptions (1) of Kahle–Kubjas–Kummer–Rosen (KKKR), to which the question refers, the convex cases for n ≥ 3 are exactly the ∏ d_j corner patterns, and "always convex" holds only in the formats 2 × m and m × 2.
- Under only the two conditions stated in the Oberwolfach abstract (#E = Σ(d_j − 1) and C_E full-dimensional, without KKKR's slice condition), "always convex" holds only in the format 2 × 2 (Example 7.4, pointed out by the second referee).
- A second independent referee found the mathematics correct and no fatal problem, and recommended acceptance after minor revisions. All nine required fixes were applied (see below).
- The note is unrefereed: the checks below are independent internal verifications, not journal peer review.

## Statement checked
- **Primary source.** M. Kummer (joint work with T. Kahle, K. Kubjas, Z. Rosen), "Rank one tensor completion", in *Algebraic Statistics*, Oberwolfach Report 20/2017, pp. 1270–1271, doi:10.4171/OWR/2017/20.
  - The abstract was read in the report PDF.
  - The second of the two closing questions asks whether the set of nonnegative partial tensors that are not completable to the joint distribution of independent random variables is "always a convex set?".
  - The setting is stated there: the completable set is full-dimensional and #E = Σ(d_i − 1). The slice condition is not stated there.
  - The abstract reports that the property held "in all examples that we computed" and in the diagonal case.
- **Companion paper.** T. Kahle, K. Kubjas, M. Kummer, Z. Rosen, SIAM J. Appl. Algebra Geom. 1 (2017) 200–221, doi:10.1137/16M1074102; arXiv:1605.01678v2 was read.
  - Section 3 adds one standing assumption: E meets every maximal slice. Together these form the standing assumptions (1) of the paper.
  - The paper does not contain the convexity question.
- **Prior work that implies the answer.** K. Kubjas, Z. Rosen, J. Algebr. Stat. 8 (2017) 1–21, doi:10.18409/jas.v8i1.50. We read arXiv:1407.3254v2 and the journal PDF, which renumbers the results:

  | journal | arXiv v2 |
  |---|---|
  | Theorem 4 | Theorem 2.12 |
  | Example 4 | Example 2.13 |
  | Theorem 7 | Theorem 4.1 |
  | Example 9 | Example 4.2 |
- **Corpus records.**
  - ulamai/UnsolvedMath, OWR-15428-015, status `partially_solved`. Its literature note says that convexity holds in computed examples and in the diagonal case, and that no proof or counterexample is known.
  - The companion record OWR-15428-014 states both closing questions of the abstract: the degree of the boundary hypersurface H and the convexity question.

## Readings
| Reading | Answer | Witness / reason |
|---|---|---|
| N_E = R^E_{≥0} \ C_E, for all patterns E (literal) | no | any pattern with a pinned element (Prop. 3.1); e.g. 3×3, E = {11,12,21,33} |
| the same, restricted to points of the simplex Δ_E = {x ≥ 0, Σx ≤ 1} (probabilistic points; this also covers [0,1]^E) | no | Examples 7.1 and 7.2: both endpoints positive with coordinate sum < 1, and the midpoint is an interior point of C_E |
| the same, under KKKR's standing assumptions (1): #E = Σ(d_j − 1), every maximal slice met, C_E full-dimensional | no, except for the formats 2 × m and m × 2 | Theorem 1.5 and Proposition 5.1 (pinned admissible patterns exist in every other format) |
| the same, under only the two conditions stated in the Oberwolfach abstract: #E = Σ(d_j − 1) and C_E full-dimensional | no, except for the format 2 × 2 | Example 7.4: in 2 × m, m ≥ 3, E = {11,12,21} ∪ {(1,k): 4 ≤ k ≤ m}; x12 = x21 = 1/10, x11 = 1/200 and 159/200 are not completable, the midpoint 2/5 is (exact). In other formats, Proposition 5.1. In 2 × 2 no pattern of size 2 has a pinned element |
| the same, for patterns without finitely completable unobserved entries | no | Remark 7.3: 12 of the 228 pinned admissible 2×2×3 patterns, e.g. {111,122,212,223} (exact computation) |

## Results in the paper
- **Theorem 1.3.** Let n ≥ 2, d_j ≥ 2 and E ≠ ∅.
  - (a) If no element of E is pinned, then C_E = {G_E ≤ 0} for an explicit concave function G_E (Theorem 4.2). So N_E and N_E ∩ Δ_E are convex, and C_E has nonempty interior.
  - (b) If some element is pinned, N_E is not convex (Prop. 3.1). If moreover E meets every maximal slice and E ≠ D, then N_E ∩ Δ_E is not convex (Props. 3.2 and 3.3).
  - The hypothesis d_j ≥ 2 is needed (now also stated in the abstract): in the format 3 × 1 with E = D no entry is pinned, but N_E = {Σx ≠ 1} is not convex.
- **Corollary 1.4 (matrices).** N_S is convex iff every component of the bipartite graph G(S) with an edge is a star.
- **Theorem 1.5 (KKKR's standing assumptions).** An admissible pattern has no pinned element iff it is a corner pattern E*(s), or (n = 2, some d_i = 2) it meets every slice of the other coordinate once.
  - For n ≥ 3, and for matrices with d_1, d_2 ≥ 3, this gives exactly ∏ d_j patterns.
  - Proposition 5.1 gives explicit admissible pinned patterns in all other formats.
- **Proposition 6.1 (corner patterns).** x ∈ C_{E*(s)} iff inf_{A>0} A ∏_j (1 + X_j/A) ≤ 1, where X_j is the observed mass on the j-th axis. If at least two X_j are positive, this holds iff ∏_j(A + X_j) = A^{n−1} has a root in (0, 1], and then every completion comes from the root A = θ^s by explicit formulas. (Without this hypothesis completions with θ^s = 0 exist, e.g. n = 2, X_1 = 1, X_2 = 0.) By Descartes' rule of signs there are at most two completions (Remark 6.2(4)).
- **Proposition 6.3.** For corner patterns, the boundary in the open orthant lies on {Disc_A(P_X) = 0}. This discriminant has degree exactly n(n−1). The irreducible boundary polynomial f of KKKR Thm 3.13 divides it, so deg f ≤ n(n−1), with equality for 2×2×2. The dimension step in the proof of (c) now uses that the positive part of the boundary is the radial graph {G_E = 0} over the open simplex of directions, as the second referee suggested.
- **Examples 7.1–7.2.** Explicit rational counterexamples in 3×3 and 2×2×2, credited to the criteria of Kubjas–Rosen.
- **Example 7.4.** The 2 × m pattern above, which satisfies the two conditions of the Oberwolfach abstract but not the slice condition; due to the second referee.

## Computations (scripts and outputs in reproducibility/)
- **Finder** (`claimant/`).
  - `witnesses_exact.py` (exact): Examples 7.1 and 7.2, a 3×3×2 variant, and the witness of Prop. 3.1.
  - `census.py`: exhaustive over 17 formats. The classification of Theorem 1.5 is confirmed. Full-dimensionality is certified by a Jacobian determinant that is nonzero modulo a 61-bit prime. Examples: 2×2×2 has 32 admissible patterns (8 corner, 24 pinned); in 3×3×3, 141,723 of 153,828 are certified full-dimensional and 27 of them are unpinned.
  - `families_check.py`: Prop. 5.1 in 20 formats.
  - `kkkr_boundary_check.py`: the discriminant equals the KKKR Example 3.16 polynomial term by term, in integer arithmetic.
  - `numeric_tests.py`: numerical evidence only.
- **Audit** (`audit/`, exact, standard library).
  - `audit_checks.py` (a few seconds) checks the numbers of Examples 7.1–7.2, Prop. 6.1 in 7 formats, and the consistency equations of Prop. 5.1.
  - Its own exact full-dimensionality test reproduces the census counts of pinned full-dimensional patterns: 24, 228, 1240 and 4032 in 2×2×2, 2×2×3, 2^4 and 2×3×3.
  - It gives the counts of Remark 7.3: 0, 12, 64 and 72.
  - `revision2_checks.py` (under a second, written for this revision) checks Example 7.4 for 3 ≤ m ≤ 8 (size, missing slice, pinned entry, nonsingular Jacobian, the completion at the midpoint, B > 1 at both endpoints, coordinate sums), the 2 × 2 and 3 × 1 remarks, the midpoint 9/32 of Example 7.2, the degenerate example after Prop. 6.1 and the sign-change count behind Remark 6.2(4). All 75 checks pass.
- **Independent verifiers** (`referee/`, their own code, rerun on 2026-09-29).
  - An exact criterion for Example 7.1.
  - A float-rank census in 8 formats, matching `census.py`.
  - An own transcription of the KKKR Ex. 3.16 polynomial, equal to the discriminant at 200 random rational points.
  - Prop. 6.1 for 2×2×2 against a least-squares oracle: 150 agree, 0 disagree.
  - Random-line tests: no violation for 9 unpinned patterns. Violations are found for the pinned controls: in the full run 1/12 lines for {11,12,21,33}, 1/12 for {111,112,222}, and 12/12 for a 4-entry 2×2×2 pattern.
- **Second referee** (`referee2/`, written from the text of the paper before the finder's scripts were read).
  - `r2_census.py` (standard library): full census in 20 formats with its own parametrisation and Jacobian certificates modulo 2^61 − 1. In every format the unpinned patterns are exactly the corner and type (ii) patterns. It reproduces 2×2×2: 32 patterns (8 corner, 24 pinned); 3×3×3: 153,828 / 141,723 certified / 27 unpinned; 2×3×4: 51,468 / 48,624 / 24; 2^5: 159,936 / 149,056 / 32; 2 × m: 2^m − 2 type (ii) patterns; and the Remark 7.3 counts 0, 12, 64, 72.
  - `r2_examples.py`: exact checks of Examples 7.1 and 7.2, Prop. 3.1 for n ≤ 11, Prop. 6.1 at 40 random rational completions and Prop. 5.1 in 12 formats, with a numerical oracle for the examples.
  - `r2_disc.py`: Disc_A of the 2×2×2 corner cubic equals the referee's own transcription of the KKKR Ex. 3.16 polynomial term by term; degree n(n−1) and the leading form for n = 2..6.
  - `r2_families_fulldim.py`: Prop. 5.1 families in 18 formats and the Remark 7.3 example are certified full-dimensional.
  - `r2_convex_numeric.py` (numerical only): 163 sign agreements, 0 disagreements and 0 concavity violations for 7 unpinned patterns. A first run with an absolute-residual oracle had 2 disagreements at points with a coordinate near 1e-5 and G_E about −0.5; its log is kept.
  - `r2_noslice_census.py`, `r2_noslice_witness.py`: the scope check behind Example 7.4.
- **Reruns.** `witnesses_exact.py`, `kkkr_boundary_check.py` and `families_check.py` reproduce their recorded outputs byte for byte. `census.py` reproduces its output up to timings. After the second report, all `referee2/` and `audit/` scripts and `witnesses_exact.py` were rerun from a copy of `reproducibility/` and reproduce their recorded outputs (up to printed timings; `r2_noslice_witness_output.txt` keeps the last 8 lines of the run, see the README).

## Independent adversarial audit
Two independent verifications (2026-09-29) were run on the finder's dossier. Verdicts:

| Item | Verifier 1 | Verifier 2 |
|---|---|---|
| Classification | PAPER_CANDIDATE | HF_CORRECTION (counterexample alone); characterisation "could carry a short note" |
| Statement fidelity | CONFIRMED | CONFIRMED |
| Proofs | CONFIRMED (every step checked by hand) | CONFIRMED (minor gaps: Claim 5 edge case; sketchy empty-interior case) |
| Answers the question as intended | yes | yes |
| Novelty | characterisation and classification new as far as found; bare answer implicit in KR | counterexample follows from KR Example 4; not new mathematics by itself |

Required fixes and how they were handled:
1. Cite KR Example 9, 3(b) (arXiv 4.2(3b)) and 3(a), and state that the bare negative answer is implicit in KR: **applied** (introduction, Example 7.2, Scope and priority).
2. Credit KKKR Example 3.17 for the corner cubic and the Sturm/Ex. 3.16 relation, and present the boundary check as a re-confirmation: **applied**. KR Example 9, 3(c) is also credited for the cubic.
3. Note a "misprint" (missing radical) in KR Example 2.13: **checked and not applied**. In the rendered arXiv v2 PDF and in the journal PDF (Example 4) the radical extends over the whole sum. The apparent misprint is an artefact of plain-text extraction.
4. Harmonise theorem names between scripts and write-up: **applied** as a concordance table (`reproducibility/README.md`, `RESULT.md`). The finder's scripts are kept unchanged as original artefacts.
5. Keep the record update a status correction with modest novelty claims: **applied**.
6. Explain that the counterexamples have a finitely completable unobserved entry: **applied** (Remark 7.3). The remark adds, by exact computation, that pinned admissible patterns without such entries exist.
7. Patch Claim 5 when coordinate 1 has no R-parameter: **applied** (proof of Thm 4.2(ii)).
8. Tighten or drop the empty-interior case: **tightened**. Prop. 3.3 is a short reflection argument that needs no pinning.
9. Fixes addressed to `triage_part2.md` (outside the problem folder) were not applied to that file. The same statements are made in the paper and in `RESULT.md`.

### Second referee (2026-09-29)
An independent adversarial referee then checked the released version (11 pages) line by line, wrote its own code before reading the finder's scripts, reran the finder's and author's scripts from `source.zip`, fetched the Oberwolfach report anonymously and repeated the literature search. Verdicts:

| Item | Verdict |
|---|---|
| Mathematics | correct; no error affecting any theorem. One step in the proof of Prop. 6.3(c) under-justified (conclusion true) |
| Fatal problem | none. The bare negative answer is implicit in KR 2017, which the paper says openly |
| Statement fidelity | CONFIRMED, with the scope qualification of fix 1 |
| Proofs (Lemmas 2.1–2.4, Props. 3.1–3.3, Lemma 4.1, Thm 4.2, Remark 4.3, Cor. 1.4, Thm 1.5, Props. 5.1, 6.1, Remark 6.2, Prop. 6.3, Examples 7.1–7.2, Remark 7.3) | correct; Lemma 2.4, Prop. 3.3, Prop. 6.3(c) and Remark 7.3, previously checked by the author only, are now independently checked |
| Computations | every computational claim CONFIRMED, with separate code (`referee2/`) |
| Novelty | the characterisation, the classification and the general-n corner results were not found in the literature; the bare answer is not new, and the paper says so |
| House style, build, release package | compliant; clean build; hashes correct |
| Recommendation | accept after minor revisions |

All nine required fixes were applied:
1. **Scope.** "Always convex only in 2 × m" is now qualified, in the abstract, Section 1, the Scope paragraph and the Readings table above, as holding under KKKR's standing assumptions (1). Under the two conditions of the Oberwolfach abstract alone it holds only in 2 × 2. The 2 × 3 example {11,12,21} (x12 = x21 = 1/10, x11 = 1/200 and 159/200, midpoint 2/5) was added as Example 7.4, extended to 2 × m by padding with zero entries (checked exactly for m ≤ 8 in `audit/revision2_checks.py`). Section 1 now says which conditions of (1) the Oberwolfach abstract states.
2. **Abstract, trivial dimensions.** The abstract now says "format d_1 × ⋯ × d_n with n ≥ 2 and all d_j ≥ 2" and "2 × m and m × 2".
3. **Prop. 6.3(c).** The step "near a point of B_+ the set Z coincides with V(f)" was replaced. B_+ = {G_E = 0} ∩ (0,∞)^E is the radial graph u ↦ e^{−G_E(u)} u over the open (#E − 1)-dimensional set of directions (Thm 4.2(ii)–(iv)); the semialgebraic map x ↦ x/Σx maps it onto that set, so dim B_+ = #E − 1 and dim V(f) = #E − 1. The proof states that f is irreducible in R[x], as BCR Thm 4.5.1 requires.
4. **Example 7.2.** Added: 9/32 is the midpoint of 1/100 and 221/400, so N_E ∩ Δ_E is not convex.
5. **Remark 7.3.** "Finitely completable" is attributed to KKKR Section 2 (Props. 2.8–2.9, Ex. 2.11), and the rational-span definition is identified as the closure in the linear matroid of exponent vectors.
6. **Record OWR-15428-014.** Section 6, Section 8 and the [UM] bibliography note now say that 014 states both questions; the convexity half is answered by this note, the degree half remains open apart from deg f ≤ n(n−1) for corner patterns. The record-update text below says the same.
7. **Verification paragraph.** Item 3 now lists Remark 7.3 among the parts written after the first two verifications. A new item 4 records the second referee's checks of Lemma 2.4, Prop. 3.3, Prop. 6.3(c) and Remark 7.3, and its separate code. The referee's scripts are shipped in `reproducibility/referee2/`, and `source.zip` and the Zenodo hashes were regenerated.
8. **Prop. 6.1.** "Every completion arises in this way" is now stated under the hypothesis that at least two X_j are positive, and the proof gives the degenerate example n = 2, X_1 = 1, X_2 = 0 with θ^s = 0.
9. **Wording.** "The cells next to s along one axis" became "the cells that differ from s in exactly one coordinate". The reproducibility README now says to run the commands from `reproducibility/` and explains that the working directory does not matter.

The optional suggestions were also taken up in part: Remark 6.2(4) notes that, by Descartes' rule of signs, a corner-pattern point with at least two positive X_j has at most two completions, and compares this with Cai–Recke–Yahl. Kubjas–Metsälampi (Linear Algebra Appl., 2026) is listed below among the checked citing works; it is not cited in the paper.

Parts written after the first two verifications and checked by the second referee:
- the reflection argument of Prop. 3.3;
- the constant-rank proof of Lemma 2.4;
- the real-algebraic divisibility argument of Prop. 6.3(c);
- Remark 7.3.

Parts written after the second referee report, and checked by the author only (with `audit/revision2_checks.py` where computational):
- the rewritten dimension step of Prop. 6.3(c), which follows the referee's suggestion;
- the text of Example 7.4 and its extension to 2 × m, m ≥ 4;
- the degenerate example after Prop. 6.1;
- Remark 6.2(4).

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - OpenAlex: all 10 works citing KKKR and all 4 citing KR were checked. They include Cai–Recke–Yahl (Algebr. Stat. 15 (2024)), which mentions convexity only for moment polytopes, and Kubjas–Metsälampi (Linear Algebra Appl., 2026), on nonnegative-rank completion without the probability constraint.
  - arXiv API queries on completability, convexity, the independence model and "completable region".
  - zbMATH ("rank-one tensor completion").
  - Web searches: one by the finder, one per verifier, one at the paper stage and one by the second referee. They returned only KKKR, KR, a 2018 Aalto master's thesis that a verifier checked, and unrelated algorithmic papers.
  - The second referee repeated the arXiv, OpenAlex, zbMATH and Crossref searches independently (all requests anonymous and logged).
  - No work answers or revisits the question.
- **Priority.**
  - The negative answer follows in a few lines from KR Examples 4 and 9, 3(b). We found it written down nowhere, and it may be folklore.
  - The 2×2×2 corner cubic and its relation to the degree-6 boundary polynomial are due to KR (Example 9, 3(c)) and KKKR (Examples 3.16–3.17).
  - The pinned-element characterisation, the classification under KKKR's standing assumptions, and the general-n corner formula with the degree bound were not found in the literature.
  - Example 7.4 is due to the second referee.
  - This negative search is not a proof of priority.
- **Scope.**
  - The note settles the convexity question as posed, both for N_E and inside the simplex, and both in general and under the standing assumptions (1). Under the two conditions of the Oberwolfach abstract alone, "always convex" holds only in 2 × 2.
  - Corpus record OWR-15428-014 states both closing questions. Its convexity half is answered by this note. Its degree half (KKKR Problem 3.15) remains open; the note gives only deg f ≤ n(n−1) for corner patterns.
  - Suggested update for OWR-15428-015: status solved, answered negatively. Suggested note: "the negative answer is implicit in Kubjas–Rosen 2017; N_E is convex iff no observed entry is pinned (n ≥ 2, all d_j ≥ 2); under KKKR's standing assumptions 'always convex' holds only in the formats 2 × m and m × 2, and under the two conditions of the Oberwolfach abstract alone only in 2 × 2".
  - Suggested note for OWR-15428-014: "the convexity half is answered negatively (see OWR-15428-015); the degree half remains open, with deg f ≤ n(n−1) for corner patterns". Its status should stay open or partially solved.

## 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/