# Verification report — OWR-15428-014 (degree of the algebraic boundary in rank-one tensor completion)

Verification date: 2026-09-30 (revised the same day after a second-round referee report; final pre-release check the same day).

**Verdict.** This is a partial answer. The question asks for the degree of the boundary hypersurface H as a function of n, d_1, …, d_n and E. The note answers it for three families and for six small formats, proves a general upper bound, and proves a structural identity for saturated patterns (E is *saturated* if B Z^N = Z^E, i.e. the maximal minors of the incidence matrix B are coprime; earlier drafts and all scripts call this "unimodular"). The families are:
- all corner patterns, in every format;
- all admissible matrix patterns;
- all saturated patterns with W = 2.

The degree is not a function of (n, d) alone. In 2 × 2 × 2 the two orbits have degrees 4 and 6. A general formula remains open.
- A second-round independent referee found the proofs correct, reproduced every computation stated as proved with its own code, found no fatal problem, and recommended release as an unrefereed partial-results note after 13 required fixes. All 13 were applied (see below). None of them changes a theorem.
- The positive real forms of several structural lemmas are due to Cai–Recke–Yahl (2024); this is now credited precisely (fix 1).
- The note is unrefereed: the checks below are independent internal verifications, not journal peer review.

## Statement checked
- **Primary source.** Oberwolfach Report 20/2017, Workshop "Algebraic Statistics", DOI 10.4171/OWR/2017/20. The abstract is "Rank One Tensor Completion" by M. Kummer, joint work with T. Kahle, K. Kubjas and Z. Rosen, pp. 1270–1272 (its reference list is on p. 1272).
  - The PDF was fetched anonymously from EMS Press. It is byte-identical to the corpus copy; the second-round referee fetched it again and found the same file.
  - The authors "focus on the case where this set is full-dimensional and |E| = Σ(d_i − 1)". Apart from coordinate hyperplanes, the algebraic boundary is one irreducible hypersurface H.
  - The abstract states only these two conditions. The third condition, that E meets every maximal slice, is a standing assumption of KKKR Section 3 (arXiv v2, p. 11), used in the proof of their Theorem 3.13. The paper now says so in Section 1 (fix 13).
  - First closing question: "What is the degree of the hypersurface H?" The abstract adds that it should be approachable with toric or tropical geometry.
- **Paper.** Kahle–Kubjas–Kummer–Rosen, SIAM J. Appl. Algebra Geom. 1 (2017) 200–221, arXiv:1605.01678v2; the arXiv v2 numbering is used.
  - Standing assumptions (Section 3): |E| = Σ(d_j − 1), E meets every maximal slice, and the completable region has nonempty interior.
  - Theorem 3.13: H = V(f) with f irreducible, obtained by elimination from G_E + ⟨l_E⟩.
  - Problem 3.15 asks for deg f "as a function of n, d_1, …, d_n, and E".
  - Example 3.16 gives deg f = 6 for {211, 121, 112}. Example 3.4 has l_E = 2 − θ_1 − θ_2 − θ_3; the paper's ⟨w, θ⟩ is −3 l_E (fix 12).
- **Corpus record.** ulamai/UnsolvedMath OWR-15428-014, status "partially solved". It states both closing questions; the convexity question is answered in the companion note (Zenodo 10.5281/zenodo.23041938, registered and findable, version 1.0, per DataCite).
  - Suggested correction for the maintainer: the clean_statement should include the standing hypotheses (full-dimensional region, |E| = Σ(d_i − 1), every maximal slice met). Without them "the hypersurface H" is not the object asked about; for example, the diagonal patterns, whose boundary has degree n^(d−1) in the paper's notation (d^(n−1) in the notation of Kubjas–Rosen), belong to a different setting.

## Readings
| Reading | Answered? | Result |
|---|---|---|
| deg H as a function of (n, d_1, …, d_n) only | yes (negatively) | 2×2×2: two orbits, degrees 4 and 6; 2×2×2×2: degrees 6, 6, 6, 8, 8, 9, 11, 12, 12, 14, 18 |
| deg H as a function of (n, d, E), KKKR Problem 3.15 | partially | corner patterns n(n−1); matrices 2 + 2·#pinned; saturated W = 2 formula; bound m!/∏(d_j−1)!; complete tables for 2×2, 3×3, 3×4, 2×2×2, 2×2×3, 2×2×2×2 |
| setting without the standing hypotheses (e.g. diagonal patterns) | not treated | not the object of the question; the diagonal degrees are known (Kubjas–Rosen JAS Prop. 6 = arXiv Prop. 5.2; KKKR Section 4) and are now cited |

## Results in the paper
- **Lemma 2.1.** Under |E| = m and the slice condition, the region is full-dimensional iff rank B = m (B is the incidence matrix). This gives an exact admissibility test. The implication rank B = m ⇒ full-dimensional is also a case of Cai–Recke–Yahl, Corollary 4.7.
- **Lemma 2.2.** Critical points of the parametrization form Im B^T. KKKR's l_E is proportional to ⟨w, θ⟩ on the normalized parameter space. H is the closure of {x(λ) : x_e = |λ|^{−n} ∏_j L_{j,e_j}(λ)}. Over the positive reals, (a) and the parametrization in (c) are the independence-model case of Cai–Recke–Yahl, Proposition 4.4 and Corollary 4.5.
- **Theorem 1.1 (corner patterns).** f = c·Disc_A(∏_j(A + X_j) − A^{n−1}) and deg f = n(n−1) in every format.
  - Top form: ∏_{i<j}(X_i − X_j)^2. Lowest form: ±(n−1)^{n−1}(X_1⋯X_n)^{n−2}.
  - The companion note had f | Disc. The new part is irreducibility, hence equality.
- **Theorem 1.2 (saturated patterns).**
  - (a) The complex completions of a generic x are the W roots of one Laurent polynomial equation Φ_x(τ) = 1. A positive x is completable iff min_{τ>0} Φ_x ≤ 1. The statements about completions in Δ (at most two; Lemma 4.2) are contained in Cai–Recke–Yahl, Theorems 4.11, 4.13, 4.16 and Proposition 4.14, and the relation between two positive completions (Lemma 4.1) is their Proposition 4.8. The complex count is new as far as we know.
  - (b) Den·∏(1 − γ_i) = c f^q, Den = c′ f_low^q, and **q·deg f = W* + deg Den**. Equivalently, deg f − deg f_low = W*/q, the number of distinct nonzero critical values.
  - (c) The location of the factors of Den, in terms of the Laurent polynomials U_j^± (now defined in Section 1). In binary formats Den is a monomial.
  - Only this identity is proved. q = 1 and W* = W are proved only for corner patterns and for W = 2.
- **Theorem 1.3 (saturated, W = 2).** Completability iff √β_1 + √β_2 ≤ 1.
  - f = c x^M((1 − β_1 − β_2)^2 − 4β_1β_2) and deg f = 2 + Σ_e M_e.
- **Corollary 1.4 (matrices).** deg f = 2 + 2·#pinned.
  - For d_1 ≤ d_2 the number of pinned entries takes exactly the values 0, …, P = max(0, min(d_1 + d_2 − 5, 2d_1 − 4)). The largest degree 2 + 2P equals 2 for 2×2 (added in this revision, fix 8), 2(d_1 + d_2 − 4) if 3 ≤ d_1 ≤ d_2 ≤ d_1 + 1, and 4d_1 − 6 if d_2 ≥ d_1 + 1. For example, it is 6 for 3×5.
  - This is a corollary of Kubjas–Rosen's matrix criterion, plus irreducibility and a denominator computation.
- **Theorem 1.5.** For every admissible E, saturated or not, deg f ≤ m!/∏_j(d_j − 1)!.
  - This is the degree of the Segre variety. The proof uses a linear projection of a hyperplane section of the Segre variety and the refined Bézout theorem.
  - Equality holds for 2×2, for the 2×2×2 corner patterns, and for 2×2×3 {111, 122, 212, 223} (degree 12).
- **Proposition 4.3 / Corollary 4.4 (saturated E).** ord_{x_e} Den ≤ μ_e, with explicit Newton-polygon multiplicities μ_e. If Den is a monomial (e.g. binary formats), deg f ≤ W + Σ_e μ_e.
- **Proposition 7.1.** The degree of every orbit of admissible patterns is proved in 2×2, 3×3, 3×4, 2×2×2, 2×2×3 and 2×2×2×2, including the orbit {1111, 1122, 2212, 2221} that is not saturated (degree 12).
- **Computational observation and Conjectures 7.2–7.3.** In all 2271 saturated patterns of a modular census:
  - q = 1 and W* = W, i.e. deg f = W + deg f_low;
  - ord_{x_e} Den = μ_e.
  These are not proved. Conjecture 7.3 gives deg f = W + Σ_e μ_e for monomial Den only together with Conjecture 7.2 (fix 4).

## Computations (scripts and outputs in reproducibility/)
- **Finder** (`claimant/`).
  - τ-reduction library and census of 2343 patterns in 25 formats. The census is exact modular arithmetic at random lines and primes: correct with high probability, not a proof. 2–3 independent runs agreed for every pattern. For patterns that are not saturated the count along the curve x_e = y_e^c is c·deg f and is divided by c.
  - Explicit certificates for 16 patterns: an integer polynomial F with exact grid-vanishing proof, plus a rank lower bound.
  - Rank lower bounds for 15 patterns.
  - Exact checks of Theorem 1.1 for n ≤ 8.
  - Exact evaluation of Theorem 1.3 for the 290 saturated W = 2 orbits; no cancellation in 1602 checks.
- **Author** (`lead/`, exact arithmetic).
  - `exact_orbits.py` does the following:
    - exact orbit census with the rank criterion of Lemma 2.1. Orbit counts are 1, 1, 2, 2, 2, 4, 6, 7, 2, 9, 16, 39, 11, 30, 122, 229, with 1, 1, 5, 4 not saturated; representatives are identical to the census;
    - its own implementation of Theorem 1.3 for all 290 W = 2 orbits;
    - the tropical bound of Corollary 4.4, compared with the census degrees of the 132 orbits that `census_output.txt` lists individually. The bound equals the census degree for all 25 of these orbits, of any W, that satisfy the criterion of Theorem 1.2(c) (every non-constant w_j attains its maximum and its minimum only once). Over all exhaustive orbits the criterion holds for 67 orbits, 31 of them with W ≥ 3, so this comparison does not cover all of them (fix 6).
  - `matrix_max.py`: the range in Corollary 1.4, by brute force up to 5×5, 4×6, 3×7 and 2×8.
  - `lower_bounds.py`: the rank lower bounds of Table 1.
  - `revision2_checks.py` (new, standard library, under a second): the terminology facts (maximal minors ±2 and ±3 of the corner patterns of 2×2×2 and 2^4, gcd 1; elementary divisors 1, 1, 1, 2 of the pattern that is not saturated), the closed forms of Corollary 1.4 including 2×2, the inequalities in the proof of Lemma 5.2, the sizes of diagonal patterns, the Cai–Recke–Yahl correspondence (Σ_κ w(κ)θ^κ = ⟨w, θ⟩, slice sums, rank A = m + 1, rank A_E = m) and ⟨w, θ⟩ = −3 l_E. All checks pass.
- **Independent verifier 1** (`referee/`, written from scratch; it imports nothing from the finder's folder, apart from reruns of the finder's programs in a scratch copy).
  - Exact orbit enumeration with gcd-of-minors saturation.
  - Certificates (lower bound plus explicit F with proved vanishing) for all 18 orbits of 2×2, 3×3, 3×4, 2×2×2 and 2×2×3. This includes a 217-term F of degree 12.
  - Its own discriminants for n ≤ 4, with top and lowest forms, and a grid proof of D_n(ξ(a)) = 0 for n = 3, 4.
  - A certificate for the 2×3×3 corner pattern.
  - Theorem 1.3 re-derived for all 290 orbits.
  - A theory-free implicitisation census, with proved lower bounds up to degree 14.
  - Numerical counts of complex completions.
  - Reruns of the finder's programs, reproducing the outputs.
  - The author ran this verifier's certificate routine on the 2^4 orbit that is not saturated on 2026-09-30 (`cert_nonunimod_2222.py`). F has 78 terms and degree 12, its vanishing was proved, and the lower bound was proved.
- **Independent verifier 2** (`referee2/indep_implicit.py`, `run1-4.py`). An independent implicitisation from KKKR's Proposition 3.12 kernel, with rational points and full rank mod p. It proves lower bounds in 17 cases, among them 2^4 degrees 9, 11, 12, 12, 14 and 18. The author reran its scripts on 2026-09-30 and saved the outputs.
- **Second-round referee** (all other files in `referee2/`; written from the text of the paper before any script of the finder, the author or the verifiers was read; standard library, numpy and a small C rank routine).
  - `orbits_census.py`: all exact columns of Table 2 and the labelled counts of §7.2 (483 orbits, 11 not saturated, 290 saturated with W = 2, Segre bounds); Table 1's representatives and W. The Smith form of the 2^4 pattern that is not saturated is (1, 1, 1, 2).
  - `w2_degrees.py`: Theorem 1.3 for all 290 orbits and 1602 pairs (0 cancellations); 2 + 2·pin in every matrix orbit; the Theorem 1.3 polynomial vanishes at 3 random points of H mod 2^61 − 1 in every orbit.
  - `tropical.py`: μ_e and the bounds of Corollary 4.4 (9, 11, 12, 14, 18 in 2^4; 12 for 2×2×3 {111, 122, 212, 223}).
  - `lowerbounds.py` + `rankmodp.c`: all 22 non-matrix orbits of Table 1 have full column rank at degree N − 1 mod 2^31 − 1 (degree 18: 5985/5985).
  - `implicit_exact.py`: F reconstructed for all orbits of 2×2×2 and 2×2×3 and for the 2^4 orbit that is not saturated, with vanishing on H proved exactly in integer arithmetic (up to 117,649 grid points); 217 and 78 terms as stated. With the lower bounds this proves every degree of 2×2×2, 2×2×3 and the 2^4 orbit that is not saturated.
  - `corner_check.py`: D_3 equals the KKKR Example 3.16 sextic exactly (38 terms, same sign); degree, top and lowest forms and irreducibility of D_n on random integer lines for n = 2..7; multiple roots of h_ξ(a) for n ≤ 8; the 2×2×3 corner F equals D_3(x213, x123, x111 + x112). The polynomial of Remark 5.1 equals the reconstructed F (`remark_and_irrtest_output.txt`).
  - `matrix_range.py`: the range in Corollary 1.4 in 16 formats; it found that the closed maximum formula omitted 2×2 (fix 8).
  - `compare_census.py`: all 132 orbits listed individually in the census output match the referee's orbits; W, the 90 listed W = 2 degrees, the 14 listed corner degrees and the tropical bound agree.
  - `census_spot.py`: for the 72 listed orbits of 2×2×4, 2×3×3 and 2×2×5 with at most 6500 monomials, (L) proves deg f ≥ census degree and the corank at the census degree is 1; for the 56 of them with a theorem upper bound the census degree is proved.
  - `extra_224_deg13.py` (not used in the paper): deg f = 13 for 2×2×4 {111, 112, 123, 214, 221}, by Corollary 4.4 and (L).
  - `fibre_numeric.py` (numerical evidence only): exactly W complex completions in six saturated cases, and 4 = 2W in the case that is not saturated.
  - `unimodular_terminology_output.txt`: the 2×2×2 corner matrix has a maximal minor ±2 and the 2^4 corner matrix ±3 (fix 3).
  - Two paths were adapted for the release, as documented in the README: `lowerbounds.py` writes temporary files to the system temporary folder instead of the referee's scratch folder, and `compare_census.py` reads `../claimant/census_output.txt`.
- **Reruns after the second-round report** (from a scratch copy of `reproducibility/`, 2026-09-30): all scripts of the second-round referee, including `census_spot.py 6500` (13 min), and `lead/revision2_checks.py` were rerun with the commands of the README. They reproduce the recorded outputs up to printed timings, and the recorded JSON files byte for byte; this also tests the two adapted paths.

## Independent adversarial audit
Two independent verifiers, 2026-09-29/30:

| Item | Verifier 1 | Verifier 2 |
|---|---|---|
| Classification | PAPER_CANDIDATE | PAPER_CANDIDATE |
| Proofs correct | yes (all theorems re-derived; tropical inequality checked) | yes (Theorems 1.1, 1.2, 1.3 and 1.5 re-derived; numbered Theorems 1–4 in the version it read) |
| Answers the question as intended | yes, partially | partially (families, bound, structure; the general formula is open) |
| Suggested corpus status | partially solved (unchanged) | partially solved (unchanged) |

All required fixes were applied.

Verifier 1:
1. "q = 1, W* = W" is now an observation and a conjecture; only q·deg f = W* + deg Den is stated as proved.
2. The tangent-cone statement is corrected: Den = c f_low^q; if Den is constant, then 0 ∉ H.
3. The bound is reproved. The proof mentions that the hyperplane section can be reducible when some w_j is constant, and that deg H̄ = deg f.
4. The mod-p Jacobian test is replaced by the exact rank criterion. The orbit counts are now exact.
5. The matrix maximum is corrected to 2 + 2·max(0, min(d_1 + d_2 − 5, 2d_1 − 4)).
6. The matrix formula is presented as a corollary of Kubjas–Rosen. Novelty is phrased as "not found in the literature".
7. Certificates are given per orbit, including the verifier's 217-term polynomial.
8. The unirational parametrization is now Lemma 2.2(c).

Verifier 2:
1. Kubjas–Rosen is credited for Theorem 4, Example 4 and Example 9, parts 3(b) and 3(c). The JAS numbering was checked against the journal PDF; in arXiv v2 these are Theorem 2.12, Example 2.13 and Example 4.2.
2. The companion note is credited for f | Disc and deg f ≤ n(n−1).
3. The title and abstract are partial, and the observations are labelled as such.
4. The census is labelled as modular Monte Carlo.
5. The scope for patterns that are not saturated is stated in the abstract and introduction.
6. Literature log: Semantic Scholar citers re-fetched and OpenAlex rate limits recorded.
7. The Kubjas–Rosen theorem number was checked (see item 1).
8. A corpus-record note was added.

### Second-round referee (2026-09-30)
An independent referee then checked the revised version (15 pages) line by line, wrote its own code before reading any other party's script, read only the other parties' outputs (and one docstring) afterwards, fetched the Oberwolfach report and the literature anonymously (33 logged requests, one web search) and repeated the literature search. Verdicts:

| Aspect | Verdict |
|---|---|
| Statement fidelity | Correct. The first closing question of OWR 20/2017 (= KKKR Problem 3.15) is answered partially, and the paper is titled and abstracted as partial. |
| Proofs | Correct. Every proof was checked line by line; no mathematical error; a few small expository gaps (fixes 8, 9). |
| Computations | Reproduced. Every computational claim stated as proved was reproduced with independent code; the census claims (heuristic, labelled as such) are consistent wherever checked. |
| Novelty | No prior determination of deg H found. The overlap with Cai–Recke–Yahl on the structural lemmas (positive real part) was under-credited; the known degrees for diagonal patterns (outside the standing assumptions) were not mentioned. |
| Presentation / house style | Good; conforms to the template. The nonstandard word "unimodular" misled; a few small inaccuracies and two typographic collisions. |
| Release package | Complete and consistent; hashes correct; no personal data or absolute paths. The verification report needed small updates. |
| Fatal | No. |
| Recommendation | Fit for release as an unrefereed partial-results note once the required fixes are made; none changes a theorem. |

All 13 required fixes were applied:
1. **Cai–Recke–Yahl credited precisely** (§1, §2, §4, Scope and priority, this report). Their hypothesis |E| = rank A_E = rank A − 1 is |E| = rank B = m for the independence model (rank A = m + 1, A_E = B^T), so it holds for every admissible pattern; "for suitable patterns" was removed. Their ν is the cell-weight vector. Their Proposition 4.4 and Corollary 4.5 are cited as the positive real form of Lemma 2.2(a) and of the parametrization in (c) (remark after Lemma 2.2, which also explains the correspondence ν^T p = ⟨w, θ⟩); Proposition 4.8 as the positive real form of Lemma 4.1 (remark after it); Theorems 4.11, 4.13, 4.16 and Proposition 4.14 for the statements on completions in Δ in Lemma 4.2 and Theorem 1.2(a) (remark after Lemma 4.2). The paper claims as new only the complex fibre count, Theorem 1.2(b)–(c) and the degree results. The paper uses the numbering of arXiv:2312.15154v2 and says so. In addition, Corollary 4.7 of Cai–Recke–Yahl is credited for one direction of Lemma 2.1(b).
2. **Diagonal patterns.** A new paragraph in §1 cites Kubjas–Rosen JAS Proposition 6 (arXiv Proposition 5.2; degree n^(d−1) in the paper's notation) and KKKR Section 4 (the polynomial Q̃_{n,d} of degree n^(d−1)), and notes that a diagonal pattern satisfies the standing assumptions only in 2×2, where it is a corner pattern of degree 2. "The only value computed there…" and "We found no other work on the degree of H" now refer to the standing assumptions (1); Scope and priority likewise.
3. **Terminology.** The notion was renamed *saturated* throughout the paper (abstract, all sections, "n.s." in Table 2). The abstract defines it ("the maximal minors of its incidence matrix are coprime; this is weaker than unimodularity"). §1 states the equivalence with saturation of the lattice spanned by the columns of A_E = B^T (the condition of KKKR Corollary 2.10) and gives the ±2 and ±3 minors of the corner patterns of 2×2×2 and 2^4. §7.1 and the README say that the scripts and outputs still use "unimodular" and "n.u.".
4. **Conjecture 7.3** now reads "Together with Conjecture 7.2, this gives deg f = W + Σ_e μ_e whenever Den is a monomial". A sentence after the conjectures notes that Conjecture 7.3 and Theorem 1.2(b) alone give only deg f = (W* + Σ_e μ_e)/q.
5. **Abstract.** "Admissible" is defined in the abstract; the determination by certificates is stated "for all admissible patterns in the six formats 2×2, 3×3, 3×4, 2×2×2, 2×2×3 and 2×2×2×2", matching §1 and Proposition 7.1; the census statement is restricted to the saturated patterns of the census ("we do not prove this in general").
6. **Verification item 2.** It now says that the tropical bound equals the census degree for all 25 orbits, of any W, among the 132 orbits listed individually in the census output that satisfy the criterion of Theorem 1.2(c), and that over all exhaustive orbits the criterion holds for 31 orbits with W ≥ 3 alone. The same change was made in this report and in the README.
7. **§7.2.** For patterns that are not saturated, the number of points of H on the curve x_e = y_e^c is c·deg f, and the program divides it by c.
8. **Corollary 1.4.** The closed form of the maximum now includes "2 if d_1 = d_2 = 2".
9. **Lemma 5.2 and Corollary 1.4.** The justification is now "t + 3 ≤ 2p + 1" and "t + 3 ≤ p + p′ ≤ 2p′". The proof of Corollary 1.4 says that a single-edge tree has no internal edges and that Lemma 5.2 is applied with rows and columns exchanged when p_l > p′_l, so that the bound 2p_l − 2 still holds.
10. **Bibliography and records.** The OWR entry gives pp. 1270–1272 (references.bib and this report). The verifier table above no longer uses the stale "Theorems 1–4". The OpenAlex citer lists of 2026-09-29 are located and copied (see the literature section). Today's DOI lookups (10 and 4 citers) are recorded. The Cai–Recke–Yahl overlap and this referee's checks are added.
11. **Typography.** U_j^± are now defined in §1 and Theorem 1.2(c) uses them, which removes the collision on p. 3; "the closure of Σ ∖ U_0" is written in words, which removes the collision in the proof of Theorem 1.2(b). Four further inline sums with stacked subscripts (three in the proof of Theorem 1.3, one in the new remark after Lemma 2.2) were written in words as a precaution. The revised PDF (17 pages) builds with no warnings and no overfull or underfull boxes; all pages were rendered at scale 1.4 and inspected, and the lines with sub- and superscripts also at scale 3.
12. **Theorem 1.1, Step 1.** "(For n = 3 this is −3 l_E, a constant multiple of the form l_E = 2 − a_1 − a_2 − a_3 of [KKKR17, Example 3.4].)"
13. **Statement fidelity.** §1 now notes that the OWR abstract states only |E| = Σ(d_j − 1) and full-dimensionality, and that the slice condition is a standing assumption of KKKR Section 3, as discussed in the companion note.

Optional suggestions of the referee, not applied in this revision (they add results rather than correct the text): the rigorous value deg f = 13 for 2×2×4 {111, 112, 123, 214, 221} (recorded in `referee2/extra_224_deg13_output.txt`); the statement that the census is rigorously confirmed on the 56 orbits of `census_spot_output.txt` with a theorem upper bound; and a comparison of the fibre polynomial with KKKR Example 3.17.

Parts written after the second-round report, and checked by the author only (with `lead/revision2_checks.py` where computational):
- the remarks relating Lemmas 2.2, 4.1 and 4.2 to Cai–Recke–Yahl, and the paragraph on Cai–Recke–Yahl in §1;
- the terminology remark in §1 and the paragraph on diagonal patterns;
- the rewritten steps in the proofs of Lemma 5.2 and Corollary 1.4, which follow the referee's wording.

## Relation to the literature, novelty and scope
- **Sources read.** The OWR abstract; KKKR arXiv v2 (source); Kubjas–Rosen arXiv v2 and the JAS version; the full text of Cai–Recke–Yahl, arXiv:2312.15154v2 (Algebraic Statistics 15 (2024) 225–247), whose numbering the paper uses; the companion note.
- **Searches.** arXiv API (about 20 queries by the finder, the verifiers and the author, and 6 by the second-round referee), Crossref, zbMATH, OpenAlex, Semantic Scholar, and four web searches (one each by the finder, verifier 2, the author and the second-round referee, 2026-09-29/30). The web searches found only KKKR and algorithmic rank-one completion papers. All requests were anonymous and are logged in `queries.log` of the dossier.
  - Semantic Scholar citers of KKKR (14 works) and KR (7 works), fetched 2026-09-30. The arXiv abstracts of the six KKKR citers missing from the OpenAlex list (2604.23104, 2601.07658, 2504.00398, 2408.05431, 2404.08171, 2009.10533) are about general or noisy rank-one completion and algorithms, not about deg H.
  - OpenAlex citers of KKKR and KR: the lists of 2026-09-29 (10 and 4 works) were saved in the dossier of the companion note (`../OWR-15428-015/audit/referee2/lit/oa_cites_kkkr.json`, `oa_cites_kr.json`) and are now copied to this dossier as `audit/lit/oa_cites_kkkr_20260929_from_OWR-15428-015.json` and `audit/lit/oa_cites_kr_20260929_from_OWR-15428-015.json`. The OpenAlex citation queries made from this dossier during the revision (2026-09-30) returned HTTP 429, so `lit/oa_cites_kkkr.json`, `audit/lit/oa_cites_*_20260930.json` and `audit/referee2/lit/oa_cites_*.json` contain only error bodies; the DOI lookups of the second-round referee still gave cited_by_count 10 (KKKR) and 4 (KR). In the final pre-release check (2026-09-30) the citation filter succeeded (`audit/lit/oa_cites_kkkr_20260930_final.json`, `audit/lit/oa_cites_kr_20260930_final.json`) and returned exactly the same 10 and 4 works as on 2026-09-29.
  - The newest citers (Kubjas–Metsälampi, LAA 2026, nonnegative-rank matrix completion; Zhou–Nie–Peng–Zhou, SIMAX 2025, algorithmic) and the zbMATH/arXiv hits 2604.23104, 2511.06062 and 2009.10533 (algorithmic rank-one completion without probability constraints) do not concern deg H.
- **Finding.** Under the standing assumptions, no work determines deg H beyond KKKR Example 3.16 (degree 6) and the companion bound deg f ≤ n(n−1) for corners.
  - Outside the standing assumptions, the boundary degrees of the diagonal patterns are known: Kubjas–Rosen JAS Proposition 6 (arXiv Proposition 5.2) and KKKR Section 4, degree n^(d−1). They satisfy the standing assumptions only in 2×2 (degree 2, consistent with Theorem 1.1).
  - Cai–Recke–Yahl: their hypothesis covers all admissible patterns; the positive real forms of Lemma 2.2(a),(c), Lemma 4.1 and the statements on completions in Δ in Lemma 4.2 and Theorem 1.2(a) are theirs. They give a procedure for the algebraic boundary but no degree results, and they do not count complex completions.
- **Prior results used and credited.**
  - Kubjas–Rosen: the matrix criterion, the 3×3 {11, 12, 21, 33} criterion, the 2×2×2 criteria, and the diagonal degrees.
  - KKKR: the setting, Theorem 3.13, Propositions 3.8 and 3.12, Examples 3.4 and 3.16, Corollary 2.10 (saturation), and the diagonal case (Section 4).
  - Cai–Recke–Yahl: Proposition 4.4, Corollary 4.5, Corollary 4.7, Proposition 4.8, Theorems 4.11, 4.13, 4.16, Proposition 4.14.
  - The companion note: pinned entries, the classification, f | Disc, and the discussion of the slice condition.
- **Novelty.** Theorems 1.1, 1.3 and 1.5, Corollary 1.4 (beyond the Kubjas–Rosen criterion), Theorem 1.2(b)–(c), the complex count in Theorem 1.2(a), Proposition 4.3 and the certified tables were not found in the literature. This negative search is not a proof of priority.
- **Scope.** A partial answer to KKKR Problem 3.15 and to the first OWR question.
  - Open: the general formula; Conjectures 7.2 and 7.3; the non-monomial factors of Den; patterns that are not saturated beyond the general bound.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
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