# Verification report — OWR-9790358-011 (Xu: Gaussian cubature of degree 2n−2 for the product Chebyshev weight in three variables)

Verification date: 2026-10-15.

**Verdict.** The note gives a **partial answer**; every statement it makes is proved, and the scope is as follows.
- **Settled.** For n = 4 (20 nodes, degree 6) no Gaussian cubature rule for the product Chebyshev weight of the
  second kind on [−1,1]³ has a node set invariant under a reflection in a coordinate plane (Theorem 6.2), a
  half-turn about a coordinate axis (Theorem 6.3), two coordinate reflections (Theorem 6.1) or x ↦ −x
  (Proposition 4.2, for every n); hence none is invariant under a non-trivial subgroup of the sign-change group
  Z₂³ (Corollary 6.4, Theorem 1.4). These statements hold for real and for complex nodes and weights. For n = 3 an
  explicit ten-node rule of degree 4 with nodes and weights in Q(√2, √3) is given (Proposition 5.1).
- **Computer-assisted.** Theorems 6.2 and 6.3: exact Nullstellensatz identities Σ q_k E_k = 1 with rational
  coefficients. Theorem 6.1, Proposition 4.2, Proposition 5.1 and the reduction (Theorem 1.2) have proofs without
  computation; their finite parts (the moments and Table 1 of Proposition 5.1, the values of G in the proof of
  Theorem 6.1, Lemmas 3.2–3.3 for small d and n) are also checked by the program of the note.
- **Not new mathematics.** Theorem 1.2 (a Gauss rule of degree 2n−2 exists iff the quadratic multi-Hankel system
  (⋆) has a real solution) is a reformulation of Xu's theory of Gaussian cubature of even degree (1994; in two
  variables Xu 2018, Theorem 12), going back to Morrow–Patterson (1978) and Schmid (1978) in two variables; Hankel
  structure of this kind appears in Xu 2018, eq. (21) (constant weight) and in Sauer–Xu (arXiv 2023, using a lemma
  of Xu 1994). The existence for n = 3 is stated in the source. No priority is claimed for these.
- **Open.** n = 4 for node sets without a symmetry in Z₂³ (the symmetry group may still be non-trivial, e.g.
  generated by (x,y,z) ↦ (−y,−x,z), (y,x,−z), (y,x,z) or the rotation (x,y,z) ↦ (y,z,x)); every n ≥ 5; the weight of
  the first kind. Numerical searches (n = 4, 5) found no rule; this is evidence only.

The note is unrefereed.

## Statement checked
- **Primary source.** Y. Xu, "Polynomial interpolation and cubature rules", abstract in Mini-Workshop:
  Interpolation, Approximation, and Algebra, Oberwolfach Reports 19 (2022) no. 1, report 7/2022, pp. 383–428
  (the abstract: pp. 415–417), doi:10.4171/OWR/2022/7 (Crossref: issue published 2023-03-10).
  - Read in the publisher's open PDF (EMS Press, fetched again on 2026-10-15: 781,751 bytes, sha256
    `850a7a2e520351a1a470843f2812bf2eaaa86c008a3e95bdbb82156a3403872d`); PDF pages 33–35 rendered and read.
  - p. 415: a cubature rule Σ λ_k f(x_k) has nodes x_k ∈ R^d and weights λ_k ∈ R; degree m means exactness on
    Π^d_m and failure for some polynomial of degree m+1; N ≥ dim Π^d_{n−1} for m = 2n−1 and m = 2n−2; Gaussian iff
    equality.
  - p. 416: for m = 2n−2 the nodes are common zeros of a basis of quasi-orthogonal polynomials of degree n;
    Gaussian rules do not exist for centrally symmetric w and Ω (in context: degree 2n−1); the existence for
    degree 2n−2 is characterised by a non-linear system in the coefficient matrices of the three-term relations;
    the product Chebyshev weight "of the second kind" is printed with the formula 1/(√(1−x₁²)√(1−x₂²)) of the first
    kind; the question (whether such a rule exists for the product Chebyshev weight in three variables); existence
    known only for n = 2 and n = 3; Möller's bound N ≥ dim Π²_{n−1} + ⌊n/2⌋ (two variables, degree 2n−1).
  - p. 417: references [1] Dunkl–Xu (2014) and [2] Xu (2018).
- **Corpus record.** ulamai/UnsolvedMath 1.6.0, record OWR-9790358-011 (record_id 30005013). Its statement text is
  a garbled extraction of the sentences of p. 416; the source decides. Record-level remark (for the notice only,
  not a defect of the note): the record's literature source labelled "De Marchi–Vianello–Xu, cube
  hyperinterpolation" resolves (Crossref) to M. Caliari, S. De Marchi, M. Vianello, "Hyperinterpolation in the
  cube", Comput. Math. Appl. 55 (2008) 2490–2497, doi:10.1016/j.camwa.2007.10.003.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Weight of the second kind (the name in the source; the two-variable theorem it refers to) | n = 3: explicit rule; n = 4: node sets with a symmetry in Z₂³ excluded; otherwise open | Prop. 5.1, Thms 6.1–6.3, Cor. 6.4 |
| Weight of the first kind (the printed formula) | not treated by the theorems (the Hankel form (⋆) does not hold); tested only: no 4-rule found (400 starts) | Section 7 |
| "Such rules exist for every n, as in two variables" | not settled; the theorems make the symmetric route impossible for n = 4 | Sections 6, 7 |
| "Such rules exist for some n ≥ 4" / "for no n ≥ 4" | not settled | — |
| Nodes in the cube, positive weights | weights are automatically positive (Lemma 2.1); the non-existence results hold a fortiori | Lemma 2.1 |
| Complex nodes and weights | the non-existence results hold (bilinear form; unisolvence from ∫ p p̄ W > 0) | Thm 1.2(c), Lemmas 2.1(a), 3.1(a) |
| Degree exactly 2n−2 | automatic: no rule with dim Π_{n−1} nodes is exact on Π_{2n−1} | Thm 1.2(b), Lemma 3.4 |

## Results in the paper
- **Theorem 1.2** (reduction, d ≥ 2, n ≥ 2): a Gauss rule of degree 2n−2 exists iff (⋆) has a real solution h on
  Λ_{2n−1}: Σ_β h(α+β)h(α'+β) − δ_{αα'} depends only on α+α'. Node sets ↔ real solutions (bijection); nodes = common
  zeros of q_α = P_α + Σ_β h(α+β)P_β; weights 1/K_{n−1}(x_k,x_k) > 0; degree exactly 2n−2; complex rules give complex
  solutions. Proof: Lemma 2.1 (K-orthogonality), Lemma 3.1 (Xu's matrix conditions (S), (C), both directions,
  with the block computation of the commutator), Lemmas 3.2–3.3 (multi-Hankel structure by a connectivity argument),
  Lemma 3.4 (h = 0 is not a solution).
- **Remark 3.5** (two variables): h = δ_{(n,n−1)} gives the Morrow–Patterson rules; nodes (cos iπ/(n+2), cos jπ/(n+1)),
  i + j odd; invariant under y ↦ −y for odd n and under x ↦ −x for even n. Footnote: the polynomials in Xu 2018,
  Theorem 5 are printed with a minus sign; with the nodes printed there the sign must be plus.
- **Lemma 4.1** (sign changes ⇒ support of h), **Proposition 4.2** (no centrally symmetric node set, any n ≥ 2, d ≥ 2).
- **Proposition 5.1** (n = 3): the D₄-invariant ten-node rule; proof by the ten D₄-invariants (Table 1); its
  solution h* = (1/√2)(δ₀₀₅ + δ₂₂₁ − δ₄₀₁ − δ₂₀₃ − δ₀₄₁ − δ₀₂₃). Remark 5.2: numerically a five-dimensional family of
  3-rules, generically without symmetry (evidence only).
- **Theorem 6.1** (two reflections; direct proof with three unknowns a, b, c), **Theorems 6.2, 6.3** (one
  reflection, 20 unknowns; half-turn, 16 unknowns; certificates), **Corollary 6.4**.
- **Section 7** (tested only): n = 4 unsymmetric, n = 5, first kind, lattice candidates; small residuals are
  escapes to infinity (residual ≈ 2.7/|h|).

## Computations (programs and outputs in reproducibility/)
- **The program of the note**, `check_note.py` (standard library, about 1 s; 27 checks, 0 failed): Proposition 5.1
  exactly in Q(√2,√3) (35 moments, 6 failing monomials of degree 5, signs, Table 1, weights 1/K₂, h* and its
  quasi-orthogonal polynomials); Lemmas 3.2–3.3 as linear algebra for (d,n) = (2,4), (3,3), (3,4) (ranks 114 and 75
  for (3,4)); the 135 equations for n = 4 by two routes (117 distinct up to sign, rank 75, same span as the 75
  conditions of (⋆)); the fifteen values of G in the proof of Theorem 6.1; the three certificates (support from
  Lemma 4.1; every stored equation compared with a generated one; Σ q_k E_k expanded exactly with the generated
  equations = 1); Remark 3.5 for n ≤ 8; negative controls (all rejected).
- **The certificates**: Theorem 6.1: 11 non-zero multipliers (integer coefficients ≤ 8, 151 terms, only
  equations E₁₂); Theorem 6.2: 63 multipliers, 1575 terms, numerators ≤ 6623, denominators ≤ 144; Theorem 6.3: 60
  non-zero multipliers, 394 terms, numerators ≤ 89, denominators ≤ 39; all multipliers of degree ≤ 2.
- **The programs with which the results were first obtained** (`original/`), with recorded outputs.
- **Re-runs of 15 October 2026** (`RERUN_LOG.txt`): the deterministic programs of `original/` and the quick
  programs of both runs reproduce the recorded outputs byte for byte; new outputs: Gröbner bases ([1] for the
  two-reflection class at n = 4 and for the half-turn class; exactly two real solutions ±h* for the n = 3 classes)
  and the drift of unconstrained Levenberg–Marquardt runs (residual × |h| constant, 2.61–3.03).
- **The third verification run** (`independent_run_2/`, see its README): an own generator from the definitions
  (A_{k,i} as integrals, (S) for a general Γ, all blocks of the commutators of J_i(Γ_h)), the identities (7) with its
  own equations, the direct proof of Theorem 6.1, Gröbner bases, Proposition 5.1 and Table 1 in an own exact field,
  the D₄ ansatz, Remark 3.5 and its footnote; and a re-run of the package from `source.zip`, which reproduced the
  recorded outputs (`outputs/rerun_package/`).

## Independent verification runs
The results were first obtained with the programs in `original/`, in a first written version. Three independent
verification runs, all AI-assisted, followed (2026-10-15); each wrote its own programs. Run A examined the
statement, the proofs and the certificates of the first written version; run B made its own computations,
searches, and the literature and bibliography check; neither saw the report of the other. The third run examined
the final text and the package (see the last subsection).

| Item | Run A | Run B | Writing stage |
|---|---|---|---|
| Statement and reading against the source | CORRECT (PDF read) | CORRECT (PDF read) | source read again; first/second kind noted neutrally |
| Lemma 2.1, Lemma 3.1 (both directions) | PROVED (re-derived; block commutator recomputed) | PROVED (re-derived) | proofs written out in full |
| Lemmas 3.2, 3.3 (multi-Hankel) | PROVED; dimensions checked (also: fails for the first kind) | PROVED; counts exact | checked as linear algebra by `check_note.py` |
| Lemma 4.1 and half-turn variant | PROVED | PROVED | stated for every sign change |
| Central symmetry | supplied the direct argument (h = 0 contradicts (⋆)) | direct argument via (C) with Γ = 0 | Proposition 4.2 with Lemma 3.4 |
| Complex nodes | argument supplied (bilinear form, ∫ p p̄ W) | argument supplied | Lemma 2.1(a), Thm 1.2(c) |
| Proposition 5.1 | CORRECT (own exact arithmetic; weights = 1/K₂) | CORRECT (own exact arithmetic; tamper test) | proof by the ten D₄-invariants |
| Theorems 6.1–6.3 | CORRECT: all 135 equations regenerated (0 mismatches); identities = 1 against own equations; Gröbner [1] over Q for 6.1 and 6.3, modulo 32003 for 6.2 | CORRECT: own parser and expander, equations matched to own generator, tamper test; Gröbner [1] over Q for 6.1 and 6.3 | direct proof of 6.1; all three identities expanded again by `check_note.py` |
| Morrow–Patterson remark | CORRECTED (reflection depends on the parity of n) | CORRECTED (same) | Remark 3.5 |
| Open part, numerical evidence | TESTED only; sphere searches n = 4, 5; drift ≈ 2.7/|h|; n = 3 family | TESTED only; node-space searches; escape to infinity | Section 7, labelled evidence |
| Novelty | no prior treatment found | no prior treatment found; record not stale | searches repeated (one web search) |

No run found a wrong lemma, proposition or theorem.

**Corrections required by run A (13) and run B (6), all applied** (details in the audit file of the writing stage):
1. (A1) Status of n ≥ 5: n = 5 numerical tests only (an exploratory search and run A's norm-limited search, no
   solution with |h| ≤ 40); n ≥ 6 not tested — Section 7, abstract.
2. (A2, B1) "Integer" certificates → rational coefficients, with sizes (1575 and 394 terms; numerators ≤ 6623,
   denominators ≤ 144; numerators ≤ 89, denominators ≤ 39) — Section 6.
3. (A3) Central symmetry: direct argument instead of Möller's bound; Möller's bound is stated in the source for
   d = 2 only; the p. 416 sentence on centrally symmetric weights concerns degree 2n−1 — Proposition 4.2,
   Lemma 3.4, Theorem 1.2(b), Section 4.
4. (A4, B2) Complex nodes justified (bilinear form; unisolvence by ∫ p p̄ W > 0) — Lemmas 2.1, 3.1, Theorem 1.2(c).
5. (A5) First kind: tested only; the two-variable rules found numerically have nodes outside the square — Sections 1,
   7.
6. (A6, B4) Symmetries not covered stated exactly (groups meeting Z₂³ trivially; examples) — Section 1.2,
   abstract, Section 6.
7. (A7, B3) Morrow–Patterson reflection: y ↦ −y for odd n, x ↦ −x for even n — Remark 3.5.
8. (A8) The n = 3 rules form a (numerically) five-dimensional family, generically unsymmetric; "only known
   mechanism" dropped — Remark 5.2.
9. (A9, B6) Small residuals are escapes to infinity (residual ≈ 2.7/|h|), not near-solutions — Section 7.
10. (A10) Xu 2018, Theorem 5 is introduced with Morrow–Patterson only; Xu 1994 is the source of Theorem 12; (21) is the
    constant-weight case — Sections 1.3, 3.
11. (A11, B5) zbMATH document ids replaced by Zbl numbers 0809.41028, 0812.65013, 0402.65013 (verified) — "Scope and
    priority", this report.
12. (A12) The complex-search sentence of the first version was dropped (no log exists).
13. (A13) The later mini-workshop report OWR 2023/43 was read: no update on the three-variable question —
    "Scope and priority".

**Changes made when the note was written.** A direct proof of Theorem 6.1 (three unknowns, six equalities from
(⋆)); a proof of Proposition 5.1 by the ten D₄-invariants (Table 1); the program `check_note.py`; Remark 3.5 proved
from Theorem 1.2 (including the footnote on the sign printed in Xu 2018, Theorem 5); renumbering (Lemmas 1–5 →
2.1, 3.1, 3.2, 3.3, 4.1; Proposition 1 → 5.1; Theorems 2–4 → 6.1–6.3).

### Third run: the final text (2026-10-15)
An independent AI-assisted run examined the final text and the package as if nothing had been checked, and the two
proofs added when the note was written first. It fetched and read the source (pp. 415–417), Theorems 2, 5 and 12 and
equations (19)–(21) of Xu 2018 (arXiv v1, p. 7 rendered: the printed nodes and polynomials of Theorem 5), the end of
Section 4 of Sauer–Xu (arXiv v1), the report OWR 2023/43 and the public summary of the epilogue of Xu's monograph
again; re-derived every proof line by line; and wrote its own programs before reading those of the package.

| Item | Third run |
|---|---|
| Statement, reading, first/second kind, page of the question (p. 416) | CORRECT |
| Lemmas 2.1, 3.1–3.4, Theorem 1.2 | PROVED (re-derived; the block computation of the commutator also verified symbolically: all blocks outside the last diagonal block vanish for Γ = Γ_h) |
| Remark 3.5 and footnote | CORRECT (n = 2..8 with 50 digits; the polynomials printed in Xu 2018, Thm 5, do not vanish at the printed nodes, those with + do) |
| Lemma 4.1, Proposition 4.2, reduction for d = 3 | PROVED |
| Proposition 5.1 (proof by the ten D₄-invariants) | PROVED (dimension of the invariant space = 10; all entries of Table 1 and M_k in an own exact field; λ_k = 1/K₂(x_k,x_k)) |
| h\*, uniqueness of the D₄ ansatz | CORRECT; the package's sympy computation treats one placement of the orbits, the other two are excluded by the moments of x²y² and (x²+y²)² (now said in the note) |
| Theorem 6.1 (direct proof) | PROVED (all table vectors and the fifteen values of G re-derived and computed; the six relations have Gröbner basis {1}) |
| Theorems 6.2, 6.3 | CORRECT: the identities (7) expand to 1 with its own equations (own generator, own parser); stored equations identical to its own; sizes of the multipliers as printed; Gröbner basis {1} over Q for 6.1 and 6.3 recomputed |
| Numbers of Sections 5 and 7 | compared with the recorded logs: all agree |
| Package | re-run from `source.zip`: `check_note.py` (27/27), the quick programs of `original/`, of run A (including the drift search of Section 7) and of run B reproduce their recorded outputs |
| Novelty | no prior publication found (arXiv, Crossref, OpenAlex, zbMATH, the Encyclopaedia of Cubature Formulas, one web search) |

It found no error in a lemma, proposition or theorem and no gap in the proofs. **Corrections required by the third
run (5), all applied:**
1. Abstract: the symmetries that are not covered are now described by the symmetry group (no non-trivial sign
   change; examples (x,y,z) ↦ (−y,−x,z), a rotation of order three, the trivial group); the numerical searches are
   stated for n = 4 and n = 5 only; the claim of novelty reads "the exclusion of coordinate reflections and half-turns
   for n = 4 and the explicit rule for n = 3" (Proposition 4.2 may be known).
2. Section 5: the uniqueness of the D₄ ansatz is justified for all placements of the two orbits of size four.
3. Section 7: the symmetric classes for n = 5 were searched for 1 ≤ R ≤ 20 (was "R ≤ 20").
4. Verification paragraph: the complex-node justification and the correction of Remark 3.5 are credited to both
   earlier runs, and run B's second argument for the central symmetry is mentioned; the third run is recorded there,
   in Sections 6 and 8 and in "Scope and priority" (four web searches).
5. Package: this report, the package README, `independent_run_2/` (programs, outputs, re-run, request log), rebuilt
   archive and deposit metadata.

## Relation to the literature, novelty and scope
- **Searches (15 October 2026: first computation, runs A and B, writing stage, third run).** arXiv API
  (Gaussian/Gauss cubature, Xu's work on cubature, cubature with Chebyshev weights in three variables,
  quasi-orthogonal polynomials, even degree, Chebyshev weight of the second kind), Crossref, OpenAlex (including
  works citing Xu 1994 (17), Xu 2018 (2) and the source (0)), zbMATH, Semantic Scholar (partly rate-limited), the
  Encyclopaedia of Cubature Formulas (its bounded regions carry only the constant weight), four web searches. No publication constructs or excludes a Gaussian
  rule of degree 2n−2 for this weight in three variables for some n ≥ 4; no earlier treatment of the symmetric case
  n = 4 was found. A search that finds nothing is not a proof of novelty.
- **Closest items.** Li–Sun–Xu 2009 (cube rules with about n³/4 nodes), De Marchi–Vianello–Xu 2009 (degree 2n+1,
  about n^d/2^{d−1} nodes), Xu 1994 (even-degree theory), Sauer–Xu 2023 (Hankel characterisation for odd degree),
  Xu's monograph (2025), whose epilogue discusses cubature rules of more than two variables as an open problem
  (publisher's public chapter summary; the chapter was not accessible), OWR 2023/43 (no statement on the question).
- **What was read.** The source pp. 415–417; Xu 2018 (arXiv version) Sections 2, 3 and 5.1; Sauer–Xu (arXiv v1),
  abstract and end of Section 4; OWR 2023/43 (searched in full, abstracts of Xu and Sauer read); abstracts of Li–Sun–Xu
  and De Marchi–Vianello–Xu; zbMATH entries of Xu 1994 (Zbl 0809.41028), Schmid–Xu 1994 (Zbl 0812.65013) and
  Morrow–Patterson 1978 (Zbl 0402.65013). Not read: Xu 1994, Morrow–Patterson 1978, Schmid 1978, Möller 1976,
  Schmid–Xu 1994 (what the note says about them is taken from the source, Xu 2018 and Sauer–Xu). All DOIs checked with
  Crossref.
- **Scope.** The question is not answered for any n ≥ 4. No priority is claimed for the reduction or for the existence
  of 3-rules. New, to our knowledge: the explicit rule of Proposition 5.1 and Theorems 6.1–6.3; Proposition 4.2 is a
  simple consequence of the theory and may be known.

## 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/