# Verification report — OWR-15428-003 (Seigal's conjecture on the Gram locus of real binary tensors)

Verification date: 2026-10-02.

**Verdict.** Seigal's Conjecture 1.5 is false for every n ≥ 4. For a real tensor of format 2×…×2 (n factors) let
d_i = det(M_i M_i^T) for its i-th flattening M_i; the Gram locus G(B) is the set of tuples (d_1, …, d_n) of tensors
in the Frobenius unit ball. The conjecture says G(B) = {d ∈ [0,1/4]^n : Q1(d) ≥ Q2(d)} for n ≥ 4. The point
d' = (9/100, 9/100, 9/100, 6/25) satisfies Q1 = 107811·10^−8 > 582·Q2 but is not the Gram tuple of any real or
complex tensor in the unit ball; zero padding gives counterexamples for every n ≥ 5, and the counterexamples form a
set with nonempty interior. The refutation is a short corollary of published theorems (Domanov–Stegeman–De
Lathauwer 2017, Theorems 4–5 with I = 2; Higuchi–Sudbery–Szulc 2003 for complex tensors), which the note credits.
The note also proves that for no n ≥ 3 is G(B) the nonnegativity set of a single polynomial inside the cube (for
n = 2 it is), gives the correct n-ary analogue of Seigal's Theorem 1.4, and corrects a misprinted inequality in
Theorem 1.4. The note is unrefereed.

## Statement checked
- **Primary source.** A. Seigal, "Gram determinants of real binary tensors", Linear Algebra Appl. 544 (2018)
  350–369, doi:10.1016/j.laa.2018.01.019; arXiv:1612.04420 (v1 is the only version).
  - Read: the arXiv LaTeX source, the accepted manuscript on PubMed Central (PMC6051437), and the version of record
    in the publisher-typeset PDF that the author hosts on her web page (the publisher's site was not accessible
    anonymously). The statements used are identical in all three: Definition 1.2 (B is the closed Frobenius unit
    ball), Q2 with the factor 1/2, Theorem 1.4 with "≤ 3/16", Conjecture 1.5, and the proof of Theorem 1.4 asking that
    the conic polynomials "be positive".
  - Conjecture 1.5: for n ≥ 4 the Gram locus is given by Q1 ≥ Q2 and 0 ≤ d_i ≤ 1/4.
  - zbMATH review Zbl 1390.15083 restates Theorem 1.4 (with "≤ 3/16") and the conjecture without comment.
  - Seigal's PhD thesis (UC Berkeley 2019, eScholarship qt9jv5j0f4), Chapter 4: Theorem 4.8 = Theorem 1.4 (same
    "≤ 3/16"), Conjecture 4.9 = Conjecture 1.5 unchanged; the random test is described as a test of the condition
    Q1 ≥ Q2 on Gram tuples of random tensors; Domanov et al. are cited only for necessary inequalities between the
    largest higher-order singular values.
- **Oberwolfach Report 20/2017** (Algebraic Statistics), doi:10.4171/OWR/2017/20, pp. 1226–1227, abstract by Seigal:
  the general form of the Gram locus is "concisely expressed as the non-negativity of a single polynomial", with a
  reference to Conjecture 1.5.
- **Corpus record.** ulamai/UnsolvedMath v1.6.0, OWR-15428-003 (record_id 30003480), status `partially_solved`; the
  statement asks whether the Gram locus of real binary tensors of fixed size has a semialgebraic description by the
  nonnegativity of a single polynomial.

## Readings
| Reading | Refuted? | Witness |
|---|---|---|
| Conjecture 1.5 as printed: G(B) = {d ∈ [0,1/4]^n : Q1 ≥ Q2}, n ≥ 4 | yes, every n ≥ 4 | d' (n = 4); (d', 0, …, 0) (n ≥ 5) |
| the same with any constant c > 0 instead of the factor 1/2 in Q2 | yes | (x, x, x, 3x − x^2) for small x > 0 (padded with zeros for n ≥ 5) |
| G(B) = the component of the complement of {Q1 = Q2} containing (1/4 − ε, …, 1/4 − ε) (Seigal's Section 3) | yes | d' lies in that component (explicit path, Remark 3.6) |
| unit sphere instead of unit ball | yes | G(B) = G(∂B) (Theorem 1.1) |
| complex instead of real tensors | yes | the same locus (Remark 2.4(a)) |
| "G(B) is the nonnegativity set of some single polynomial in d inside the cube" (the record's question) | answered no for every n ≥ 3; yes for n = 2 (−(d1 − d2)^2) | Theorem 1.4 |
| the same, together with Seigal's linear inequalities d_i ≤ Σ_{j≠i} d_j | still no, n ≥ 3 | Theorem 1.4 (the proof works inside the interior U of her polytope) |
| a single polynomial inequality in the coordinates λ_i = f(d_i) (not polynomial in d) | possible: Q1(λ) ≥ 0 | Remark 2.4(b) |
| the inclusion that Seigal's random test probes, G(B) ⊆ {Q1 ≥ Q2} | not settled for n ≥ 4 (consistent with Monte Carlo, n = 4, 5, 6); false for n = 3 | Problem 3.8; GHZ point (1/4, 1/4, 1/4) |

## Results in the paper
- **Theorem 1.1** (after Domanov–Stegeman–De Lathauwer and Higuchi–Sudbery–Szulc). G(B) = G(∂B) =
  {d ∈ [0,1/4]^n : Σ_{j≠i} sqrt(1 − 4d_j) − sqrt(1 − 4d_i) ≤ n − 2 for all i}; every point is attained by a unit
  tensor with nonnegative entries. Short proofs are included: Lemma 2.1 (polygon inequality μ_i ≤ Σ_{j≠i} μ_j for
  the smaller eigenvalues of the Gram matrices; DSL Theorem 4 with I = 2), Lemma 2.2 (explicit even-weight tensors;
  the I = 2 case of the construction for DSL Theorem 5), Lemma 2.3 (from the sphere to the ball, the one step not
  stated in the sources).
- **Corollary 1.2.** Conjecture 1.5 fails for every n ≥ 4; the failure set has nonempty interior. Explicit points:
  d' (Prop. 3.1: Q1(d') = (33/100)^3(3/100), Q2(d') = (1/2)(57/100)^2(15/100)^6, f(d') = (1/10, 1/10, 1/10, 2/5),
  ρ_4(d') = 11/5 > 2); (d', 0, …, 0) for n ≥ 5 (Lemma 3.2 and the comparison
  Q1 = Q1(d')(51/100)^(n−4) > (1/2)q^(n−3) ≥ (1/2)q^(2^(n−4)) = Q2, q = 2Q2(d'), because Q1(d') > q/2 and 51/100 > q);
  the family (x, x, x, 3x − x^2), 0 < x ≤ 1/12 (Prop. 3.3, with the valid chain
  3r1 − 2 > r4 ⇔ 3 − 6x − x^2 > 3r1 ⇔ (3 − 6x − x^2)^2 − 9(1 − 4x) = 30x^2 + 12x^3 + x^4 > 0, legitimate for
  0 < x < 5/36). Remark 3.4 gives the second example d* = (3/50, 3/50, 3/50, 171/1000) with the comparison
  Q1(d*)·0.351^(n−4) > (1/2)(q*)^(n−3), which holds because Q1(d*) > q*/2 and 0.351 > q*.
- **Theorem 1.3.** For n ≥ 3, G(B) = {N_n ≤ 0} ∪ {N_n ≥ 0, M_1 ≥ 0, …, M_n ≥ 0} inside the cube, with explicit integer
  polynomials N_n (degree 2^(n−1)) and M_i (degree 2^(n−2)). For n = 3, N_3 = −512(Q1 − Q2) and
  M_k = 16((d_i − d_j)^2 + (d_i + d_j)/2 − 3/16) (Prop. 4.1, hand proof), so Theorem 1.4 holds with "≥ 3/16".
- **Theorem 1.4.** For n ≥ 3 there is no polynomial F with G(B) ∩ U = {d ∈ U : F(d) ≥ 0}, U the interior of
  Seigal's polytope; for n = 2, G(B) = {−(d1 − d2)^2 ≥ 0}. The proof: each facet form divides F after the
  substitution d_j = (1 − r_j^2)/4; the sign changes r_j ↦ −r_j force all 2^n forms n − 2 − Σ ε_j r_j to divide with
  one odd multiplicity; the form n − 2 − Σ r_j then changes sign in the interior of the locus.
- **Remark 4.2 (misprint).** As printed (arXiv, accepted manuscript, version of record, zbMATH review, thesis
  Theorem 4.8), Theorem 1.4 excludes the GHZ point (1/4, 1/4, 1/4) ∈ G(B) and includes (1/5, 1/100, 1/100), which is
  outside the convex hull of G(B). With "≥ 3/16" it is correct.
- Remarks 3.5 and 3.7 report Monte Carlo estimates (labelled as computations): {Q1 ≥ Q2} \ G(B) fills about 13.6 %,
  7.6 % and 3.0 % of the cube for n = 4, 5, 6.

## Computations (scripts and outputs in reproducibility/)
- **Lead** (`lead/verify_paper.py`, standard library only, a few seconds; 59 checks, all PASS). Exact arithmetic from
  the definitions; products over sign vectors of sums of square roots are evaluated by an exact recursion, never by a
  closed form. It checks d', d*, the padded points (n up to 40), the family (polynomial identities, bounds, and the
  closed form (3) derived symbolically from the definition), the path of Remark 3.6, Lemma 2.2 for 230 random rational
  λ ∈ P_n (n = 2..8), Theorem 1.3 at 4,750 exact points (n = 3..6, including facets and L_+ = 0), Prop. 4.1 as
  polynomial identities, the expansion of N_4 (424 monomials; byte-identical to the finder's), the misprint
  (GHZ point; grid of 2197 points: printed version wrong at 772, corrected at 0), and the points used in the proof of
  Theorem 1.4 for n = 3..10. Lemmas 2.1 and 2.3 are tested in floating point. `lead/montecarlo.py` gives the Monte
  Carlo estimates.
- **Finder** (`finder/`, exact polynomial library over Fractions plus numpy/scipy): exact certificates for d', d* and
  the padded points; exact realisation of random λ ∈ P_n; the n = 3 identities; Theorem 1.3 at 1499 exact points for
  n = 4 and float checks for n = 5, 6; Monte Carlo volumes and the family. All outputs reproduce exactly on rerun.
- **First independent verification run** (`independent/`, AI-assisted, code written separately from the finder's): Q2 from
  its definition in multiquadratic number fields; a rigorous scale argument excluding d' without the ball-to-sphere
  lemma; numerical minimisation of the distance from d' to G(unit ball) (0.02333, equal to the distance to h(P_4));
  Prop. 4.1 by exact evaluation on a 5×5×5 grid; Theorem 1.3 at 7,000 exact points; Lemma 2.2 by linear programming
  and by numerical inversion of the Gram map; vertices of P_4; the path of Remark 3.6 (minimum of Q1 − Q2 along it
  1.08·10^−3, sampled).

- **Second independent verification run** (`independent_run_2/`, AI-assisted, code written from the definitions
  without reading or importing the earlier programs): Q2, N_n and M_i expanded as polynomials from their linear
  factors (formula (3), Lemma 3.2 for m = 3, Prop. 3.3 and Prop. 4.1 checked as polynomial identities); Q2 at d', d*
  and the padded points (n ≤ 8) in the formal algebra Q[g]/(g_j² − d_j); rigorous interval boxes of counterexamples
  around d' (radius 10^−3) and (d', 0, …, 0) (n = 5..7); Theorem 1.3 at 6,845 exact points and 3,000 rational d;
  Remark 4.2 (772 / 0 on the grid); Lemma 2.2 for 420 rational λ; floating-point tests of Lemmas 2.1 and 2.3,
  attainment by least squares, the distances from d' and d* to G(unit ball), and the Monte Carlo figures. It also
  reran every earlier program from an extracted copy of the source archive: all outputs byte-identical.

## Independent adversarial verification
Verdicts of the first independent verification run (2026-10-02), in its own terms:
- Correct: yes. Every load-bearing step was re-derived line by line and re-checked with separate code. One justification
  (the polygon failure along the family (x, x, x, 3x − x^2)) was invalid, although its conclusion is true.
- Answers the question as intended: yes. The printed definitions are used as they stand; no misprint is exploited;
  the failure has positive volume and is robust to the factor 1/2, to ball versus sphere, and to the "component"
  reading.
- Prior art: the exact locus is published in singular-value coordinates (DSL 2017, Theorems 4–5 with I = 2; HSS 2003);
  the refutation of Conjecture 1.5 is stated nowhere; Theorems 1.3 and 1.4 and the erratum were not found.
- Significance: a status correction backed by a short corrective note with full credit to DSL and HSS.

Required fixes and where they were applied:
1. Family (x, x, x, 3x − x^2): the invalid justification is replaced by the valid chain (Prop. 3.3); the comment in
   `finder/regions_n4plus.py` now gives the factorisation that makes the script's test equivalent.
2. Seigal's thesis: restates the conjecture unchanged (Conjecture 4.9) and Theorem 1.4 as Theorem 4.8, and cites DSL
   only for necessary inequalities (Introduction; this report).
3. Versions: PMC6051437 is the accepted manuscript; the version of record was read (publisher-typeset copy on the
   author's web page) and agrees (paper, scope paragraph; this report).
4. Citing works: Chen–Grochow–Qiao–Tang–Zhang (arXiv:2306.03135) cites Seigal only for orthogonal equivalence; it is
   counted among "three works that cite [Sei18] for other purposes" and not cited (this report lists it).
5. Credit: Theorem 1.1 is attributed to DSL (Theorems 4–5, I = 2) and HSS; the refutation is presented as a corollary;
   only the explicit refutation, Theorems 1.3 and 1.4 and the erratum are claimed, as "not found in the literature".
6. Completeness: the case n = 2 is in Theorem 1.4; the version with Seigal's linear inequalities is excluded by
   Theorem 1.4; the padding comparison is spelled out (proof of Corollary 1.2 for d', Remark 3.4 for d* with
   Q1^(4)(d*)·0.351^(n−4) > (1/2)q*^(n−3)).
7. Housekeeping: `finder/nary_theorem14.py` rewrites `N4_poly.txt` next to itself; the regenerated copy lives in
   `finder/`, the original in the problem folder is untouched, and all copies are byte-identical.

## Second independent verification run (2026-10-02)
An AI-assisted verification run, carried out after the fixes above, checked the paper and the package again from
scratch. It re-read the sources anonymously (arXiv:1612.04420 LaTeX source; the version of record, rendered at the
page with Theorem 1.4 and Conjecture 1.5; PMC6051437; the zbMATH review; the thesis; Oberwolfach Report 20/2017,
p. 1226; the arXiv versions of Domanov–Stegeman–De Lathauwer, Higuchi–Sudbery–Szulc and Krämer), re-derived every
proof line by line, wrote its own programs (`reproducibility/independent_run_2/`) and reran all earlier programs.

| Area | Verdict |
|---|---|
| Statement fidelity (OWR p. 1226; arXiv v1; accepted manuscript; version of record; zbMATH; thesis) | CONFIRMED |
| Corpus record and intended question | CONFIRMED (answered negatively for every n ≥ 3; n = 2 trivial) |
| Counterexamples and families (exact) | CONFIRMED |
| Theorem 1.1, Lemmas 2.1–2.3 and their attribution | CONFIRMED (attribution wording sharpened, fix 4 below) |
| Theorem 1.3, Proposition 4.1 | CONFIRMED (identities checked as polynomial identities) |
| Theorem 1.4 (no single polynomial; also with Seigal's linear inequalities; n = 2) | CONFIRMED (rigorous) |
| Remark 3.6, interior / positive volume | CONFIRMED (rigorous interval box around d') |
| Erratum to Seigal's Theorem 1.4 | CONFIRMED against the printed text and the GHZ tuple |
| Monte Carlo figures | CONFIRMED as computations (13.66 %, 7.56 %, 3.00 %; 1.11 % for n = 3) |
| Reproducibility of the release package | CONFIRMED (all 12 earlier outputs byte-identical) |
| Novelty and credit | CONFIRMED as far as can be checked; no priority claim |
| Presentation | CONFIRMED_WITH_FIXES |

No mathematical error was found. Required fixes, all applied:
1. Categories: math.SP (Seigal's own primary category) replaced by math.AG and math.RA in the paper's metadata and the
   Zenodo keywords.
2. The Verification paragraph of the paper records this second run.
3. This report: the wording of the first run's verdict corrected (the proofs were "re-derived line by line"); the
   lead program's count corrected to 59 checks; this section added.
4. Attribution: the introduction says that the theorems of Domanov–Stegeman–De Lathauwer hold for real and complex
   tensors (used with I = 2); the abstract says that the Gram locus follows from the published theorems plus an
   elementary step from the unit sphere to the unit ball (Lemma 2.3), as the Scope paragraph already stated.
5. Reproducibility: the README maps the labels of the earlier write-up used in the finder's and first run's scripts
   (Theorem A/B/C, Lemmas 1–4, Corollary 1) to the paper's numbering; stale theorem numbers in a docstring of
   `lead/montecarlo.py` and a comment of `lead/verify_paper.py` corrected (outputs unchanged).
6. The second run's code and outputs added in `reproducibility/independent_run_2/` (with a README); paper, source
   archive and Zenodo files rebuilt and their checksums recomputed.

## Relation to the literature, novelty and scope
- **Searches (2 October 2026).** Crossref (all DOIs of the bibliography), the arXiv API (all arXiv identifiers; title
  and abstract searches for "Gram locus", "binary tensors" with "Gram", "higher-order singular values" with
  "feasible"), zbMATH (Seigal, Domanov et al., Higuchi et al.), OpenAlex (works citing the LAA paper: 6; works citing
  Domanov et al.: 4), Semantic Scholar (works citing arXiv:1612.04420: 5), and one web search in each of the three
  passes (finding, independent verification, writing).
  - Works citing Seigal's paper: Domanov–Stegeman–De Lathauwer (SIMAX 2017), Krämer (SIMAX 2019), Seigal's thesis
    (2019), Fan–Nie–Zhou, "Completely positive binary tensors" (arXiv 2018, Math. Oper. Res. 2019; cites it for the
    term "binary tensor"), Chen–Grochow–Qiao–Tang–Zhang (arXiv:2306.03135; orthogonal equivalence), and a 2022
    chemistry paper (unrelated).
  - Domanov et al. remark, after their Theorem 5, that their result was independently proved for real 2×…×2 tensors
    in Seigal's preprint; Krämer (Section 1.4) repeats this. Neither notes the conflict with Conjecture 1.5.
  - No source states that Conjecture 1.5 is false, or contains Theorems 1.3 or 1.4 of the note or the correction of
    Theorem 1.4.
  - The second independent run repeated the searches (Crossref for all DOIs of the bibliography; OpenAlex works
    citing the LAA paper and Domanov et al.; Semantic Scholar citations of arXiv:1612.04420; the arXiv API for
    "Gram locus", "binary tensors" with Gram, "polygon inequalities" with tensor, "multilinear singular values" with
    feasible; one web search) with the same result.
- **Known results used.** Domanov–Stegeman–De Lathauwer, SIMAX 38(4) (2017) 1434–1453, Theorems 4–5; Higuchi–Sudbery–
  Szulc, PRL 90 (2003) 107902; see also Bravyi, QIC 4 (2004) 12–26. The polygon inequalities are standard in quantum
  information; that literature was not searched systematically, so the observation may have been made informally.
  This negative search is not a proof of priority.
- **Scope.** The note refutes Conjecture 1.5 as stated and in the natural variants above, and answers the record's
  question negatively for every n ≥ 3 in Gram-determinant coordinates (positively for n = 2). It leaves open whether
  G(B) ⊆ {Q1 ≥ Q2} for n ≥ 4 (Problem 3.8).

## Public release and license
Subject classification used for the release: math.AG, math.RA.
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/