# Verification report — OWR-12697710-006 (Reznick: is the set of forms with a sum-of-squares odd power a closed convex cone?)

Verification date: 2026-10-11.

**Verdict.** The note gives **explicit, exactly certified counterexamples to convexity in small dimension**. The
negative answer to the question itself is **not** due to the note: it is due to a prior preprint.
- **The question.** For even m and k ≥ 0, is Σ_{n,m}(2k+1) = { f : f^(2k+1) is a sum of squares } a closed convex
  cone? It is a closed cone (known). The open part was convexity.
- **Prior answer (not ours).** A. Kriebel, "Nonconvexity of fixed odd-power sum-of-squares cones", preprint,
  Zenodo, version 1.0, 6 October 2026, doi:10.5281/zenodo.23191300: the set is not convex for
  (n, m, 2k+1) = (3·10^62, 6, 3), and for every odd exponent ≥ 3 the set of sextics is not convex in some dimension;
  proved by averaging many copies of one seed form in separate variables. The examples of the note were found
  after this preprint had been read, and they use its principle with two copies instead of 10^62.
- **Proved in the note.** Theorem 1.1: an explicit pair of sextics in 5 variables, p = M_c(x,y,z), q = M_c(u,v,z),
  c = 41/16, with p^3, q^3 sums of squares and (p+q)^3 not. Theorem 1.2: the same for an explicit pair of quartics
  in 7 variables, built from Q_c with c = 13/4. Both theorems are **computer-assisted**: exact rational certificates
  (Gram matrices; a linear functional with positive definite moment matrix and negative value), verified by a
  short program in exact arithmetic. Corollary 1.3: Σ_{n,m}(3) is not convex for n ≥ 5, even m ≥ 6 and for n ≥ 7,
  even m ≥ 4. Theorem 1.4: for every even m ≥ 4 and odd exponent ℓ ≥ 3, non-convexity in all large dimensions
  (Kriebel's many-copies argument; the quartic seed is added). Proposition 1.5: M_c^3 is a sum of squares for all
  c ≤ 41/16 and Q_c^3 for all |c| ≤ 13/4; the same two functionals work on intervals of parameters. Remark 1.6:
  the sum p+q of either theorem is an explicit form whose cube is not a sum of squares and whose fifth power is.
- **Not new.** The answer "no"; the principle of copies in separate variables with a functional that is negative
  on a power of their sum.
- **Open.** Exponent 3: n = 3 and n = 4 (even m ≥ 6), and quartics in 4, 5, 6 variables. Exponents ≥ 5: every
  dimension below the non-explicit bound. Numerical searches found no example there; this is a test, not a proof.

The note is unrefereed.

## Statement checked
- **Primary source.** B. Reznick, "The Odd Powers of the Motzkin Polynomial, etc.", in: Real Algebraic Geometry
  with a View toward Koopman Operator Methods, Oberwolfach Reports 20 (2023), no. 1 (Report No. 14/2023,
  pp. 741–811), pp. 778–781, doi:10.4171/OWR/2023/14.
  - Read completely, as rendered pages (pp. 778–781), in the file of the publisher (734,622 bytes, SHA-256
    `5a9bdc9f9daa12ef35185ceae208c0f2f75e191875827f78b7b6ad0f5cdbf240`); at the writing and by all three
    verification runs.
  - p. 778: forms, psd, sos, the cones P_{n,m} and Σ_{n,m} (notation of Choi and Lam); Hilbert's theorem on the
    cases in which they differ (n ≥ 3 and m ≥ 6, or n ≥ 4 and m ≥ 4); the forms M, S, Q, R; the Newton polytope
    fact with the reference to Reznick's paper of 1978.
  - p. 779: M_c = x^4y^2 + x^2y^4 + z^6 − c x^2y^2z^2; M_c^3 is sos if c^3 ≤ 15/13, and c can be increased to
    about 1.1336 (stated without proof); the definition Σ_{n,m}(2k+1) = { f ∈ P_{n,m} | f^(2k+1) ∈ Σ_{n,(2k+1)m} };
    the question, in one sentence, which is the corpus record; closedness is called easy, convexity difficult,
    and the author guesses that the answer is negative; a positive answer would mean that p^(2k+1), q^(2k+1) sos
    implies (p+q)^(2k+1) sos.
  - p. 780: the theorem that p^(2k+1) and q sos imply (p+q)^(2k+1) sos, stated with the identity for the exponent 3
    as a hint of the proof; the added remark, without proof, that p^3, q^3 sos imply (p+q)^5 sos, and the
    conjecture for mixed exponents (proved as Theorem 5.3 of the paper below); the question whether a form p with
    p and p^3 not sos can have p^5 sos, with the guess "yes" and the statement that the author has no example.
- **The paper of the authors of the question.** G. Blekherman, K. Kozhasov, B. Reznick, "On odd powers of
  nonnegative polynomials that are not sums of squares": arXiv:2407.21779v1 (31 July 2024; the only version) and
  Forum of Mathematics, Sigma 14 (2026), e65, doi:10.1017/fms.2026.10221 (received 15 December 2024, revised
  19 August 2025, accepted 30 March 2026, published online 27 April 2026).
  - Definition (1.6) is the same set. The introduction proves closedness and says that it is unclear whether the
    set is convex for k > 0 (p. 4 in both versions). Section 6 says that the authors do not know whether, for a
    fixed odd exponent, the closed cone is convex, and that they expect the answer not to depend on the
    parameters (p. 22 of the arXiv version, p. 23 of the journal version). So the question is still stated as
    open in the journal version.
  - Theorem 5.1 (arXiv: 42): P_1^k and P_2 sos imply (P_1+P_2)^k sos. Theorem 5.3 (arXiv: 44):
    Σ_{n,d}(k) + Σ_{n,d}(k') ⊆ Σ_{n,d}(k+k'−1); hence the union over all odd exponents is a convex cone.
    Theorem 6.3 (arXiv: 47): the set of a with M_a^(2k+1) sos is (−∞, c_k]. Section 6: (15/13)^(1/3) ≈ 1.04886 ≤ c_1
    by an explicit identity; experimental values c_1 ≈ 2.56548 and c_2 ≈ 2.88905, attributed to a private
    communication of P. Parrilo (p. 23 of the arXiv version, p. 24 of the journal version).
  - Read at the writing: the introduction, Section 5 and Section 6 of the arXiv version; the introduction, the
    statements of Theorems 5.1 and 5.3 and Section 6 of the journal version. Sections 2 to 4 were not read.
    Run 2 read again, as rendered pages, pp. 3, 4, 22, 23 of the arXiv version and pp. 4, 23, 24 of the journal
    version, and the statements of Theorems 42, 44, 5.1 and 5.3; all page and theorem numbers of the note are
    correct.
- **Corpus record.** ulamai/UnsolvedMath, OWR-12697710-006 (dataset version 1.6.0; upstream status `open`). Its
  statement is the sentence of p. 779 with the running head of the page attached ("… Real Algebraic Geometry
  with a View toward Koopman Operator Methods 779") and without the definition; the source decides.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Is Σ_{n,m}(2k+1) a closed cone? | yes (known: Blekherman–Kozhasov–Reznick, introduction) | Section 1.1 |
| Is it convex for all n, m, k? | no. First shown in Kriebel's preprint (n = 3·10^62, m = 6, exponent 3; every odd exponent ≥ 3 in some dimension); explicit pairs in 5 and 7 variables in the note | Theorems 1.1, 1.2 |
| For which (n, m) is Σ_{n,m}(3) convex? | convex in Hilbert's cases (n ≤ 2; m = 2; (n, m) = (3, 4)); not convex for n ≥ 5, m ≥ 6 and n ≥ 7, m ≥ 4; open for n = 3, 4 (m ≥ 6) and n = 4, 5, 6 (m = 4) | Corollary 1.3, Section 1.4 |
| Exponents ≥ 5 | not convex for n ≥ n_0(m, ℓ), not explicit; open below | Theorem 1.4 |
| Is the condition f ∈ P_{n,m} in the definition needed? | no, it is automatic | Section 1.1 |
| The union over all k | convex (known, Theorem 5.3 of Blekherman–Kozhasov–Reznick); not contradicted: (p+q)^5 is sos in both examples | Remark 1.6 |
| One summand sos (the weaker statement of the source, p. 780) | true (known) | Lemma 2.4 |
| Second question of the source (p. 780): can p^5 be sos if p and p^3 are psd and not sos? | yes: the sum p+q of Theorem 1.1 or 1.2 is an explicit example. The existence of such forms follows already from Kriebel's preprint and Theorem 5.3 of Blekherman–Kozhasov–Reznick, without an explicit form | Remark 1.6 |
| Polynomials instead of forms | the same question with one variable less (Remark 1.5 of Blekherman–Kozhasov–Reznick); not treated separately | — |

## Results in the paper
- **Lemma 2.1 (Newton polytope lemma, in half-space form), Lemma 2.2 (duality), Lemma 2.3 (extra variables;
  cancelling a squared variable), Lemma 2.4 (Reznick's identity for cubes).** Proved in the text.
- **Lemma 3.1.** In both cases New(p+q) is a simplex Δ, and the set B used for the moment matrix is exactly the set
  of lattice points of (3/2)Δ = ½ New((p+q)^3); a sum-of-squares representation of (p+q)^3 is supported on B.
  Proved in the text (and tested again by the program).
- **Proposition 3.2 (exact verification).** (A) Gram identity and positive definiteness: four blocks of sizes
  5, 5, 5, 4 (19 monomials), respectively 6, 6, 6, 6 (24 monomials). (B) The moment matrix on B, of size 91,
  respectively 99, is positive definite; it splits into 16 blocks (sizes 3,3,3,3,3,4,4,4,4,7,7,9,9,9,9,10,
  respectively nine of size 3, six of size 10, one of size 12). (C) Λ((p+q)^3) =
  −29836645080831/20480000000000000 ≈ −1.4569·10^-3, respectively −22652335069827/320000000000000 ≈ −7.0789·10^-2.
- **Theorems 1.1 and 1.2** follow from Lemma 3.1, Proposition 3.2 and Lemma 2.2.
- **Corollary 1.3**: multiply by an even power of the shared variable; Lemma 2.3.
- **Proposition 4.1 and Theorem 1.4**: the many-copies argument of Kriebel's preprint (its Lemmas 1 and 2), with
  complete proofs of the ingredients; the quartic seed Q_{13/4} gives all even degrees m ≥ 4.
- **Proposition 1.5.** (a) By Lemma 2.4: M_c = M_{41/16} + (41/16 − c)(xyz)^2; for the quartics,
  Q_c is obtained from Q_{13/4} + ε w^4 by rescaling w. (b) φ(t) = Λ((p_t+q_t)^3) is an explicit cubic, strictly
  decreasing, with root t_I ∈ (2.5516680, 2.5516681), respectively t_II ∈ (3.1196392, 3.1196393); for larger t
  the cube of the sum is not sos.
- **Remark 1.6.** (p+q)^5 is sos, by an explicit identity; so P = p+q is an explicit psd form such that P and P^3
  are not sos and P^5 is sos. This is proved (through Theorems 1.1 and 1.2, hence computer-assisted), not only
  tested. By Theorem 5.3 of Blekherman–Kozhasov–Reznick the sum of every pair violating convexity for the
  exponent 3 is such a form, so the existence follows already from the prior preprint; the remark says so.

## Computations (programs and outputs in reproducibility/)
Theorems 1.1 and 1.2, Corollary 1.3 and Proposition 1.5(b) depend on the exact computation described in
Proposition 3.2; Theorem 1.4 and Proposition 1.5(a) depend on its part (A) only. Nothing depends on floating-point
arithmetic.
- **The exact program of the note** (`writing_stage/verify_note.py`, standard library, 0.3 s; ends with
  "BOTH CASES: ALL CHECKS PASSED"): checks (A), (B1), (B2), (C) of Proposition 3.2 and its proof, the cubics of
  Proposition 1.5(b) with their monotonicity and root enclosures, and the identities of Proposition 1.5(a) at
  sample parameters. Positive definiteness is tested by exact LDL^T (all pivots positive) on the full moment
  matrices and on all blocks, and by leading principal minors (fraction-free elimination) on all blocks. Smallest
  pivots: 2.20·10^-4 and 8.35·10^-3 (Gram), 3.75·10^-7 and 1.25·10^-5 (moment matrices, in the lexicographic order
  of B). It rejects eight kinds of altered certificates. After verification run 2 it also checks explicitly that
  all products of three monomials of p+q are of the form X^(a+b) with a, b in B and have their exponents among the
  keys of the table, so that Λ is defined on (p_t+q_t)^3 for every t.
- **Six further exact verifiers of the same files**: `original/scripts/verify_certificate.py` (standard library)
  and `verify_independent.py` (SymPy, python-flint) of the first results; `vA_verify.py` (standard library) and
  `vA_sympy_scratch.py` (SymPy) of verification run A; `verify_exact_B.py` (integers and python-flint) of
  verification run B; `independent_run_2/r2_verify.py` (standard library, integers only; the full matrices in two
  orders of B) of verification run 2. All obtain the two fractions of (C).
- **A second proof of positive definiteness, of a different kind** (`independent_run_2/r2_gershgorin.py`, run 2):
  for a floating-point Cholesky factor L of M − δI, taken as a matrix of rational numbers, the exact residual
  M − L L^T is strictly diagonally dominant with positive diagonal; hence M is positive definite. Done for the
  eight Gram blocks and for both full moment matrices; no exact elimination on M is used.
- **How the certificates were found** (`original/scripts/make_certificate.py`): floating-point semidefinite
  programs (Clarabel), rounding to denominator 10^13, exact projection of the Gram matrix. The construction is
  deterministic: it was repeated by verification run B and at the writing, with files identical byte for byte.
- **Numerical searches in the open cases** (tests only; `original/scripts/explore_*.py`, programs of
  verification run B): no example with (n, m) = (3, 6), (4, 6), (4, 4), (5, 4), (6, 4). Five points flagged by
  double precision solves (margins near −2·10^-6, reduced solver accuracy) were re-examined. The point with
  (n, m) = (4, 4) lies in the cone, and this is proved (run 2, `independent_run_2/r2_flagged_q4.py`): it is a form
  Q_{c'} with c' ≈ 0.99 in rescaled variables plus a sum of three squares of monomials, so its cube is a sum of
  squares by Proposition 1.5(a) and Lemma 2.4; its multiprecision margin is +3.3·10^-12
  (`writing_stage/gmp_q4_flagged.py`), which is the coefficient of a vertex monomial of the normalised cube and
  an upper bound for the margin. Three of the four points with (4, 6) have +6.6·10^-13 and twice 0 within the
  output resolution 10^-17 (verification run B); the fourth was not recomputed. These remain tests. Limits: only
  special families were searched; double precision cannot decide margins below about 10^-6, and the true margins
  of sums of boundary elements in three and four variables are mostly between 10^-14 and 10^-6; near the Motzkin
  form the boundary of Σ_{3,6}(3) lies within about 10^-4 of the boundary of the psd cone (in the direction of
  (x^2+y^2+z^2)^3).
- **Re-runs.** At the writing all exact programs were run again (`run_quick.sh`) and all 18 recorded runs of the
  programs of the first results were repeated (`rerun_claim.sh`). Verification run 2 extracted the archive and
  ran `run_quick.sh` again (0 failures or differences) and the multiprecision program for the point with
  (n, m) = (4, 4) (the recorded values); after its corrections the quick part, which now contains the programs of
  run 2, was run once more. Details in `reproducibility/RERUN_LOG.txt`.

## Independent verification runs
The results were first written down with the programs in `original/`. Three independent verification runs
(2026-10-11), all AI-assisted, followed; each wrote its own programs. Runs A and B were made in parallel on the
first written version of the results, and neither saw the other: run A examined the statement, the proofs and
the certificates; run B rebuilt the examples numerically, verified the certificates exactly, re-ran the first
programs, searched for smaller examples and searched the literature. Run 2, the second stage, was made on the
final text of the note, with the reports of runs A and B at hand; it examined first what had been added at the
writing, then the whole text, the program of the note and the package.

| Item | Run A | Run B | Run 2 (final text) |
|---|---|---|---|
| Statement, definitions, what is known, against the sources | CONFIRMED | CONFIRMED | CONFIRMED_WITH_FIXES (sources read again; wording, corrections 1 to 5 below) |
| Certificate (5, 6, 3): Theorem 1.1 | CONFIRMED (lemmas proved again; own exact verifier; second toolchain; controls; numerical search for a representation failed) | CONFIRMED (numerical rebuild; own exact verifier on the full matrix, two exact methods) | CONFIRMED (own exact verifier in integers; second proof of positive definiteness; controls) |
| Certificate (7, 4, 3): Theorem 1.2 | CONFIRMED (the same) | CONFIRMED (the same) | CONFIRMED (the same) |
| The set B is the half Newton polytope | CONFIRMED (proved; counted; computed in two ways) | CONFIRMED (proved; computed from barycentric coordinates and by linear programming) | CONFIRMED (Lemma 3.1 of the final text line by line; the printed formulas checked) |
| Lemmas 2.1 to 2.4 of the final text | proved again (first version) | proved again (first version) | CONFIRMED line by line |
| Proposition 3.2 and its proof | — | — | CONFIRMED_WITH_FIXES (a clause on the domain of Λ, correction 6; the order of B for the pivots, correction 7) |
| The program `verify_note.py` | — | — | CONFIRMED: read line by line; a passing run proves Proposition 3.2; two explicit checks added |
| Corollary 1.3; Theorem 1.4 (many copies) | CONFIRMED (each step proved again) | CONFIRMED | CONFIRMED (Proposition 4.1 in the form with disjoint copies; "even degree" to be written, correction 8; the count tested by enumeration) |
| Proposition 1.5(a) for M_c | CONFIRMED | CONFIRMED | CONFIRMED |
| Proposition 1.5(a) for Q_c (added at the writing) | — | — | CONFIRMED (the identity holds identically in c) |
| Proposition 1.5(b), the cubics | found by this run (case I; computed also for case II) | not its part | CONFIRMED for both cases (indeterminate t; Sturm; enclosures; interval for case II) |
| Remark 1.6 | identity for (p+q)^5 checked | — | CONFIRMED_WITH_FIXES: proved, not only tested; the question of p. 780 and what follows from the prior preprint to be stated (correction 9) |
| Flagged point with (n, m) = (4, 4) | — | flagged; artefact by rescaling (double precision) | multiprecision value reproduced; proved to lie in the cone (correction 10) |
| Programs of the first results | main verifier re-run, identical output; SymPy verifier read, not run | all 18 runs repeated: CONFIRMED_WITH_FIXES (one missing check; undocumented arguments) | exact programs re-run from the archive: the recorded outputs are reproduced |
| Description of the prior preprint | CONFIRMED_WITH_FIXES (three wording fixes); its data recomputed exactly | CONFIRMED; its data recomputed exactly | every sentence of the note compared with the preprint: exact; two additions required (correction 5); its data recomputed exactly |
| Smaller examples | not its part | none found (tests only) | not repeated |
| Novelty | no other source found | negative answer KNOWN (Kriebel); explicit small pairs, quartics, the bound 41/16: not found elsewhere | the same; no further source |
| Presentation, references | — | — | CONFIRMED_WITH_FIXES (abstract, corrections 1 and 2; references, correction 4; all DOIs resolve to the works named) |

No run found a mathematical error.

**Corrections required by runs A and B**, all applied:
1. (Run A) The first version said of the prior preprint that it contains "no formal proof". The preprint contains
   complete proofs; the note describes its theorem and its two lemmas as proved, and does not comment on how the
   preprint was produced.
2. (Run A) The first version called the new examples "independent". They were found after the preprint had been
   read, and their mechanism is the preprint's, with two copies instead of 10^62: said in the abstract, in
   Section 1.2 and in "Scope and priority".
3. (Runs A and B) The entries of the moment table of the preprint that a plain text extraction prints as "224" are
   typeset as 2^24 (read from the rendered page; the font sizes confirm it). The uncertainty expressed in the first
   version is removed; the header of `original/scripts/check_prior_note.py` was corrected; the run with the literal
   value 224 is kept as a control (it fails).
4. (Run A) The key `case` is listed among the keys of the certificate files (README of the package).
5. (Run B) Citing works: OpenAlex lists one work citing the paper of Blekherman, Kozhasov and Reznick (Kriebel's
   preprint); Semantic Scholar lists two others (Baldi et al., "Stubborn polynomials", arXiv:2602.01191, and
   Dressler, Kuhlmann and Schick, arXiv:2305.14848, on a different cone), neither of which concerns the question;
   Crossref lists none. Said in "Scope and priority".
6. (Run B) The preprint is cited with its version and the access date; the Zenodo record (one version, 1.0) showed
   a modification date of 10 October 2026; the file read has the checksum listed in the record.
7. (Runs A and B) The Oberwolfach abstract's "about 1.1336" is mentioned, and the values 1.04886 and 2.56548 are
   cited from both versions of the paper (after Proposition 1.5).
8. (Run B) `verify_independent.py` did not check that the monomials of p+q lie in the simplex whose vertices it
   takes as input. The check was added, together with a symmetry check of the Gram blocks (suggested by run A);
   with a wrong list of vertices the changed program now fails. The command-line arguments of the three
   `explore_hyperplane_VT*` runs are recorded in the README.
9. (Run B) Numerical thresholds are quoted as ≈ 2.56548 and ≈ 3.30151, not with more digits.
10. (Run B) The prior preprint is credited in the abstract; the open cases are stated, with the caveat that double
    precision arithmetic cannot resolve the thin layer.
Suggestions of run A that were followed: the interval of Proposition 1.5(b) (derived again exactly); Lemma 2.1
is presented as the Newton polytope lemma.

**Corrections required by run 2**, all applied:
1. Abstract: the sentence on the first negative answer names the prior preprint ("due to that preprint, and not to
   this note").
2. Abstract: the open cases in three and four variables are all even degrees m ≥ 6, not only sextics (as in
   Section 1.4 and in "Scope and priority").
3. Section 1.1: the Oberwolfach abstract states the theorem on one sos summand with a hint of the proof; the note
   said "proves". The exponents of Theorem 5.3 are written ℓ, ℓ'; the abstract has the case ℓ = ℓ' = 3 without
   proof and the general case as a conjecture.
4. References: five older papers (Stengle; Berg–Christensen–Jensen; Choi–Dai–Lam–Reznick; Reznick 1995;
   Scheiderer) were cited only in a list of papers that were not read. They are now cited where they belong (two
   sentences of history in Section 1.1, taken from the abstract and from the introduction of the paper; the
   attribution of the seed in Section 1.2), and "What was read" says that what is said about them is taken from
   these sources.
5. Section 1.2: Lemmas 1 and 2 of the prior preprint are stated for forms of any even degree, and the preprint
   notes that the conclusion holds for every non-negative form that is not sos and has an sos ℓ-th power; it
   attributes its seed to Berg, Christensen and Jensen. Both facts are now said.
6. Proofs of Proposition 3.2 and of Proposition 1.5(b): Λ is defined on the span of the monomials X^γ, γ ∈ B+B;
   that all products of three monomials of p+q have their exponents in T (so that Λ((p+q)^3), Λ(S^3), Λ(S^2R),
   Λ(SR^2), Λ(R^3) are defined) was true and checked by the program for (p+q)^3, but not said. It is now said, and
   `verify_note.py` checks it explicitly for all such products; its recorded output was renewed.
7. Proof of Proposition 3.2: the smallest pivots of M_Λ are those of the lexicographic order of B (in the
   reversed order they are 2.63·10^-7 and 1.21·10^-5).
8. Proposition 4.1 is stated for a form of even degree (the proof uses d = ℓ m_0/2).
9. Remark 1.6: the second question of p. 780 is described as a question to which the author guesses the answer
   and has no example; it is said that the sum of every pair violating convexity for the exponent 3 is such a
   form, so that the existence follows already from the prior preprint, where no form is made explicit.
10. Section 6.1: the flagged point with (n, m) = (4, 4) is proved to lie in the cone (Proposition 1.5(a) and
    Lemma 2.4); the multiprecision value is kept with its meaning; an impersonal wording replaces "our numerical
    runs"; the first sentence of the subsection marks the one statement that is not a test.
11. Section 6: the verifier of run 2 and its second proof of positive definiteness are named; runs A and B are
    called by their names instead of "first" and "second".
12. Paragraph "Verification": rewritten to the final state.
13. "Scope and priority" and "Searches": the recomputation and the searches of run 2 (nine web searches in all;
    DataCite; OpenAlex lists no work citing the prior preprint).
14. Package: this report; `reproducibility/independent_run_2/`; README, `RERUN_LOG.txt` and `run_quick.sh` updated.
15. Zenodo metadata: description renewed from the final abstract; the relation "cites" to the DOI of the prior
    preprint is noted there.

**Changes made when the note was written, after runs A and B.** All of them were examined by run 2.
- `verify_note.py` was written, with knowledge of the other verifiers. Run 2 read it line by line: a passing run
  proves Proposition 3.2.
- Proposition 1.5(a) for the quartics Q_c with |c| < 13/4 was new: the first version and runs A and B had left
  these values as tested only. The argument (add ε w^4, apply Lemma 2.4, rescale w) is three lines; the identities
  are checked by `verify_note.py` at c = 3, 1, 1/2. With it the interval for case II in Proposition 1.5(b) was new.
  Confirmed by run 2 (the identity holds identically in c).
- Proposition 1.5(b): the cubic is derived from the expansion (S − tR)^3 instead of interpolation, and its
  monotonicity (negative leading coefficient and negative discriminant of the derivative) replaces the numerical
  isolation of the roots.
- Proposition 4.1 is written for copies in disjoint sets of variables, as in the preprint; the first version used
  copies sharing one variable. The functional with value 1 on the constant is obtained by adding a small multiple
  of a Gaussian integral, as in the preprint.
- Remark 1.6: the comparison with the second question of the Oberwolfach abstract (p. 780) was added; the
  identity for (p+q)^5 had been checked by run A.
- The multiprecision value for the flagged point with (n, m) = (4, 4), and a re-run of `recheck_q4.py`, of which
  no output had been recorded.
- The two checks in `verify_independent.py` (correction 8).

**Statements added by run 2**, which no other run has examined: the proof that the flagged point with
(n, m) = (4, 4) lies in the cone (the exact check is `independent_run_2/r2_flagged_q4.py`), and the sentence of
Remark 1.6 on what follows from the prior preprint (a two-line deduction from Theorem 5.3).

## Relation to the literature, novelty and scope
- **Prior answer.** Kriebel's preprint (Zenodo record 23191301, concept DOI 10.5281/zenodo.23191300; version 1.0,
  the only version; publication date 6 October 2026; file `odd_power_sos_nonconvexity.pdf`, 92,284 bytes, MD5
  `a125402b725f8c77ebcef71dc7f17b3a` as listed by Zenodo on 11 October 2026). Read completely (6 pages) at the
  writing and by all three runs. Its Theorem 1: with N = 10^62 and
  p_i = x_i^4y_i^2 + x_i^2y_i^4 + z_i^6 − x_i^2y_i^2z_i^2 on disjoint triples of variables, every p_i lies in the set
  for (3N, 6, 3) and the average of the p_i does not; and for every odd q ≥ 3 some finite n(q) gives non-convexity
  for sextics. It states that no conclusion is drawn for three variables or for the smallest dimension. Its
  Lemmas 1 and 2 are stated for forms of any even degree, and it notes that the conclusion holds for every
  non-negative form that is not sos and has an sos q-th power. Its explicit data (the identity for the cube of the
  seed; the moment matrix of size 220, positive definite; a_1 = −1, a_2 = 11292·10^53, a_3 = 35039520·10^116; the
  sign at N = 10^62) were recomputed exactly by four programs of the package (`check_prior_note.py`,
  `vA_prior_note.py`, `check_kriebel.py`, `r2_prior_preprint.py`). The verification archive attached to the
  preprint was not read. Run 2 compared every sentence of the note about the preprint with the preprint: all are
  exact.
- **What the note adds.** Explicit pairs in 5 variables (sextics) and 7 variables (quartics), hence Corollary 1.3;
  the quartic case of Theorem 1.4; Proposition 1.5, in particular c_1 ≥ 41/16 = 2.5625 (proved in the literature:
  ≥ 1.04886; experimental: 2.56548; Oberwolfach abstract: "about 1.1336" without proof).
- **Searches (11 October 2026).** arXiv API, Crossref, OpenAlex, Semantic Scholar, zbMATH, Zenodo, DataCite, and
  nine web searches (two with the first results, two by run A, three by run B, one at the writing, one by run 2).
  Run 2: eleven queries to the arXiv API, ten to Zenodo (the only record on the question is the prior preprint),
  OpenAlex (one work citing the paper of Blekherman, Kozhasov and Reznick, namely the prior preprint; no work
  citing the prior preprint), Crossref, zbMATH (three queries: the paper, Zbl 1586.14068, and "Stubborn
  polynomials"). No source was found with an example in small dimension, for three variables, or for quartics,
  and none with a proved bound for c_1 beyond 1.04886. A search that finds nothing is not a proof of novelty; no
  priority is claimed.
- **What was not read.** The verification archive of the prior preprint; Sections 2 to 4 of the paper of
  Blekherman, Kozhasov and Reznick; of Baldi et al. (arXiv:2602.01191) everything except the abstract and the
  statement of Theorem 2.2 (its text was searched for statements on a fixed exponent: none); the older papers of
  Hilbert (1888), Choi and Lam (1977), Reznick (1978, 1995), Stengle (1979), Berg, Christensen and Jensen (1979)
  and Choi, Dai, Lam and Reznick (1982); of Scheiderer (2012) only the abstract was read. Nothing in the note
  depends on them; what the note says about them is taken from the Oberwolfach abstract, from the introduction of
  the paper and from the prior preprint. All bibliographic data were checked on 11 October 2026 with Crossref, the
  arXiv API and the Zenodo API, and again by run 2 with Crossref (ten DOIs) and DataCite (the DOI of the prior
  preprint) (for Reznick 1978 Crossref gives no pages, and for Choi–Dai–Lam–Reznick no authors; these are taken
  from the reference lists of the Oberwolfach abstract and of the paper). Run 2 read the papers in the files
  that the earlier stages had fetched on the same day (two copies of each, byte-identical) and checked the
  present state of the records: Zenodo lists one version of the prior preprint and the same MD5 for its PDF; the
  arXiv API lists one version of each of the two arXiv papers.
- **Scope.** Exponent 3 in dimension ≥ 5 (even degree ≥ 6) and ≥ 7 (even degree ≥ 4); other odd exponents only in
  large, non-explicit dimension. Ternary forms, which are the case of the Motzkin form itself, are not decided.

## 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/