# Verification report — OWR-15219-013 (Brändén: majorization of the zeros of mixed characteristic polynomials, Conjectures 2 and 5)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete negative answer** to both conjectures of the record, by one explicit
counterexample with exact proofs.
- **Conjecture 2 is false.** P = (x1+x2)(x1+x3)(x2+x3)/8 lies in D_{3,3} and is symmetric;
  chi_P = (t-1)(4t^2-8t+1)/4, lambda(P) = (1+√3/2, 1, 1-√3/2), chi_{P_inf} = t^3 - 3t^2 + 2t - 2/9, and
  lambda_3(P) < lambda_3(P_inf) because 9√3 < 16. So lambda_1 + lambda_2 of P exceeds that of P_inf
  (2.8660254 > 2.8614085) (Theorem 1.1).
- **Conjecture 5 is false**, with r = 0 and every s > 0, for the flow exp(s(T-I)), under which the source's Lemma 3
  holds (Proposition 5.1): the orbit is exp(s(T-I))P = P_{A(q)}, q = e^{-s/3}, A(q) = ((1-q)/3)I + ((2+q)/6)(J-I),
  chi_q = t^3 - 3t^2 + (2+q^2/4)t - 2/9 - q^2/12 + q^3/18 has discriminant (2+q)^3(16-24q+18q^2-7q^3)/48 > 0, and
  chi_q(1-√3/2) = -(1-q)[4q^2 + (16-9√3)(1+q)]/72 < 0, so lambda_3(Psi_s P) > lambda_3(P) (Theorem 1.2(a)-(c)). The
  zeros are real by the discriminant alone; neither Lemma 3 nor stability of Psi_s P is used.
- **Under the sign as printed**, exp(s(I-T)), the same P refutes Conjecture 5 at k = 1 for every s in (0, s0],
  s0 = 3 ln u0 ≈ 0.7578; for every s > 0 the polynomial Psi_s P is not stable (it vanishes at a point of the
  positive orthant), so Lemma 3 fails as printed, and for s > s0 even chi has non-real zeros (Theorem 1.2(d)).
- **Not new:** Conjecture 2 was reportedly refuted in a 2017 KTH degree project (Samuel Zackrisson, DiVA
  diva2:1106332; full text not read), according to its abstract through Wanless's counterexamples to the
  Holens-Đoković conjecture, and according to a search-engine excerpt also by an explicit example with three
  weights, which we could not see. The note's example satisfies the Holens-Đoković inequalities, so it does not
  arise from Wanless's counterexamples; no priority is claimed for it or for any counterexample to Conjecture 2.
  No printed refutation of Conjecture 5 was found.
- **Not claimed:** which (m, d) admit counterexamples; which pairs 0 < r < s fail (some hold).

The note is unrefereed. Site result type: COMPLETE_COUNTEREXAMPLE.

## Statement checked
- **Primary source.** P. Brändén, "The van der Waerden and Kadison-Singer conjectures with symmetries", extended
  abstract in Oberwolfach Report 14/2017, *Real Algebraic Geometry With a View Toward Moment Problems and
  Optimization*, Oberwolfach Rep. 14 (2017), no. 1, 771-862 (published 2018 according to Crossref), Brändén's
  abstract pp. 825-826, doi:10.4171/OWR/2017/14.
  - Pages 825-826 read in the publisher's PDF of the full report (PDF pages 55-56 of 92), rendered as images for
    this note; the first verification run compared three text extractions of the same pages, the second a text
    extraction, and the third read the pages again in the PDF.
  - Definitions as printed: doubly stochastic = homogeneous of degree d with dP/dx_i(1) = d/m for all i; stable =
    no zeros with all Im x_i > 0; D_{m,d} = all doubly stochastic stable polynomials of degree d in R[x_1..x_m]
    (no multiaffinity or other condition); chi_P = (1-d/dx_1)...(1-d/dx_m)P at x_1 = ... = x_m = t (product over
    all m variables); lambda(P) its zeros in decreasing order; P_inf = (x_1+...+x_m)^d/m^d; "beta majorizes alpha,
    written alpha ⪯ beta" iff the partial sums of alpha are <= those of beta, with equal totals.
  - Conjecture 2: P in D_{m,d} implies lambda(P) ⪯ lambda(P_inf). Conjecture 5: P in D_{m,d} ∩ Sym_m and
    0 <= r <= s imply lambda(Psi_r(P)) ⪯ lambda(Psi_s(P)), where T(P) = (1/(md))(x_1+...+x_m)(d/dx_1+...+d/dx_m)P
    and the flow is printed as Psi_s = exp(s(I - T)). Lemma 3 (Psi_s preserves D_{m,d}, Psi_s(P) -> P_inf),
    Theorem 4 ("monotone decreasing"; "in particular lambda_1(P) <= lambda_1(P_inf)").
- **Corpus record** OWR-15219-013 (UnsolvedMath, snapshot d371ee9): label open; its statement reproduces both
  conjectures with the printed sign and leaves the definitions to the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Conjecture 2 with the printed definitions (D_{m,d} not multiaffine) | false: P in D_{3,3} ∩ Sym_3, failure at k = 2 | Theorem 1.1 |
| Direction of ⪯ reversed | also fails for P, at k = 1 (1.866 < 2.097); the printed direction is the only one consistent with Conjecture 1 and the source's display (1) | Theorem 1.1, Section 1.1 |
| D_{m,d} read as multiaffine polynomials | Conjecture 2 trivial (chi = (t-1)^d) and P_inf outside the class; not the source's reading | Remark 2.3 |
| Conjecture 5 with the convergent flow exp(s(T-I)) (Lemma 3 true) | false for r = 0 and every s > 0; also for r = 3 ln(100/99), s = 3 ln 2 | Theorem 1.2(c), Remark 4.4 |
| Conjecture 5 with the printed flow exp(s(I-T)) (Lemma 3 false) | false at k = 1 for every s in (0, s0]; Psi_s P is not stable for any s > 0, and for s > s0 lambda is not real | Theorem 1.2(d) |
| r = 0 allowed | yes ("0 <= r <= s"); Psi_0 = I | Section 1.1 |
| "monotone decreasing" in Theorem 4 | with the convergent flow it must mean non-decreasing; for P, lambda_1(Psi_s P) is strictly increasing | Remark 4.5 |

## Results in the paper
- **Lemma 2.1** (coefficients): chi_Q = sum_k (-1)^k E_k(Q) t^{d-k}, E_k = sum over |S| = k of d^S Q(1); for doubly
  stochastic Q, chi_Q is monic with zero sum d.
- **Lemma 2.2** (products of linear forms): P_M in D_{n,n} for M nonnegative doubly stochastic, E_k(P_M) =
  sigma_k(M) (sums of permanental minors); chi_{P_inf} = (-1)^n n! n^{-n} L_n(nt) (Laguerre), proved for all n.
- **Remark 2.3** (multiaffine reading).
- **Theorem 1.1** (Conjecture 2 fails) with a half-page proof.
- **Proposition 3.1**: Conjecture 2 for P_M plus Schur-concavity of e_k/e_{k-1} (Marcus-Lopes) gives the
  Holens-Đoković inequalities k n sigma_k >= (n-k+1)^2 sigma_{k-1}; so every Wanless counterexample refutes
  Conjecture 2 (presented as a reconstruction of the likely route of the thesis).
- **Remark 3.2**: A(1) = (J-I)/2 satisfies Holens-Đoković (9 >= 9, 27/2 >= 12, 9/4 >= 9/4), Tverberg-Friedland
  (3 >= 3, 9/4 >= 2, 1/4 >= 2/9) and the van der Waerden-Gurvits bound.
- **Remark 3.3** (computation only): the cycle products P_n = prod_i (x_i + x_{i+1})/2 violate Conjecture 2 for
  3 <= n <= 12 (exact Sturm/interval certificates); the observation lambda_n(P_n) = 1 - cos(pi/(2n)) is numerical.
- **Lemmas 4.1, 4.2**: eigenvectors P_inf, B0, C0 of T (eigenvalues 1, 1/3, 0), P = P_inf + B0/36 - C0/216, the
  orbit = P_{A(q)}, chi_{P_{A(q)}} = chi_q.
- **Theorem 1.2** (Conjecture 5 fails), parts (a)-(d).
- **Remark 4.3**: on the orbit, Conjecture 2 fails exactly for q* < q <= 1, q* = (3 - 9 mu_3)/2 ≈ 0.876338
  (alpha < alpha* = (9 mu_3 - 1)/6 ≈ 0.0412206).
- **Remark 4.4**: the failing pair r = 3 ln(100/99), s = 3 ln 2 (rational intervals); a holding pair r = 3 ln 2,
  s = 3 ln 10; lambda_3 is not monotone along the orbit: it is largest at q = 4/7, where lambda_3 = 1/7 (the earlier
  text said "about 0.55", from a coarse grid).
- **Remark 4.5**: lambda_1(Psi_s P) strictly increasing in s (implicit differentiation).
- **Proposition 5.1** (Lemma 3 for exp(s(T-I)); not needed for Theorems 1.1, 1.2): spectrum of T in
  {0, 1/d, ..., 1} with 1 simple; double stochasticity and the value at 1 preserved; stability preserved (an
  argument of Lieb-Sokal type and Hurwitz's theorem); limit Q(1) P_inf.

## Computations (sanity checks; programs and outputs in reproducibility/)
- `check_note.py` (standard library; exact arithmetic in Q and Q(√3)): 95 checks, 0 failures, about 1 s. It covers
  every finite and exact statement of Sections 2-4, the spectrum of T on all cubic forms, Proposition 5.1(a), (b)
  for eleven small (m, d), and Remark 3.3 for 3 <= n <= 12. Identities that are polynomial in q of degree at most
  3 (or 6 for the discriminant) are checked at 10 (or 13) values of q, which proves them.
- The programs with which the results were first obtained (`original/`, sympy and mpmath) and the programs of the
  three verification runs (`verification_run_A/`, `verification_run_B/`, `independent_run_2/`) are included with
  their recorded outputs. `independent_run_2/r2_verify.py` (sympy, mpmath, numpy, scipy; 101 checks, 0 failures,
  about 20 s) was written by the third run from the source definitions and the statements of the note.
- A complete re-run from a copy of the package on 2026-10-20 reproduced every recorded output byte for byte
  (`RERUN_LOG.txt`), except for the version line of `check_note.py` under another Python version and the final
  line `exit 0` that the shell added to the recorded output of `original/stress.py`.

## Independent verification runs
The results were first obtained, with proofs, by the programs in `original/`. Three independent verification runs,
all AI-assisted, followed, each with its own programs, on 2026-10-20. Runs A and B examined the first written
version of the result; the third run examined the final text of the note (see below).

| Item | Run A | Run B |
|---|---|---|
| Statement against the source (definitions, no multiaffinity, direction of ⪯, printed sign, r = 0 allowed) | CONFIRMED (three text extractions of the PDF pages) | CONFIRMED (text extraction) |
| Theorem 1.1 | PROVED (re-derived) and CONFIRMED by computation (chi by the definition and by the MSS determinantal formula) | PROVED and CONFIRMED (literal operator, rational intervals) |
| Lemmas 4.1, 4.2, Theorem 1.2(a)-(c) | PROVED; 10 x 10 matrix of T, diagonalisation, exp at s = 3 ln 2 | PROVED; exp(±s(T-I)) by the Jordan form; Q(√3) certificates; product form of the orbit found (now Lemma 4.2) |
| Theorem 1.2(d) (printed sign) | PROVED | PROVED (discriminant root count; u = 6/5, 3/2) |
| Proposition 5.1 | PROVED | PROVED as read |
| Proposition 3.1, Remark 3.2 | PROVED | PROVED / CONFIRMED |
| Remarks 3.3, 4.3, 4.4, Laguerre identity, Remark 2.3 | — | found and checked (n = 3..12 exact; alpha* exact; pair r > 0 exact; Laguerre n = 2..8) |
| Literature | thesis existence checked; content not readable | full literature check; Gurvits-Leake read; Brändén 2018 searched |
| Re-run of the first programs | identical outputs | identical outputs (stress: extra shell line) |

Neither run found a mathematical error. Required corrections (all made; see the note): Conjecture 2 is described as
"reportedly refuted" in the thesis (abstract only, full text not read), not as refuted in print; the thesis is cited
with the author's first name and with the checks a reader with access should make; the literature search is
described without call counts; the description of Gurvits-Leake, Theorem 7.1, was corrected (a majorization
statement of Schur-Horn type, not a largest-root theorem); the Oberwolfach report is cited consistently; and the
statement that lambda(Psi_s P) is defined by the discriminant alone, so that the refutation holds whichever sign
was intended, is made once at Theorem 1.2. Added when the note was written, checked by `check_note.py` but not
examined by the runs A and B: the factorised discriminant, the complete proofs in Remarks 4.3 and 4.5, the proof of
the Laguerre identity for all n (all examined by the third run).

### Third run (on the final text; programs in `reproducibility/independent_run_2/`)
| Item | Result |
|---|---|
| Statement against the source | Brändén's pp. 825-826 re-read in the publisher's PDF; every definition and statement quoted in Section 1 matches (no multiaffinity; chi over all m variables; P_inf; direction of ⪯; T; printed flow exp(s(I-T)); Lemma 3; Theorem 4 "monotone decreasing"; Conjectures 2 and 5 with r = 0 allowed) |
| Proofs | all theorems, propositions, lemmas and remarks re-derived line by line, including the parts added when the note was written (factorised discriminant, Remarks 4.3 and 4.5, Laguerre identity, Proposition 5.1) |
| Own program `r2_verify.py` | 101 checks, 0 failures: chi from the operator definition (symbolic q), the matrix of T on all ten cubic forms and exp(±s(T-I))P symbolically in s, discriminant, Q(sqrt3) identities, u0, s0, q*, alpha*, intervals of Remark 4.4, cycle family n = 3..12, Laguerre n = 2..8, Proposition 5.1(a), (b); numerical tests of Propositions 3.1 and 5.1(c) |
| Package | re-run from an extracted copy of the archive; all outputs reproduced (`independent_run_2/package_rerun_log.txt`) |
| Literature | Crossref for all DOIs; zbMATH review of Wanless (HD form); Gurvits-Leake Theorem 7.1 re-read (arXiv v4); arXiv, zbMATH, Semantic Scholar searches; one web search |
| Mathematical errors | none |

Required corrections of the third run (all made): (1) the novelty claim for the counterexample to Conjecture 2 was
withdrawn, after a new search-engine excerpt of the thesis reporting an explicit counterexample there; (2) the works
citing the report are now partly enumerated (Semantic Scholar); (3) Remark 4.4: the maximum of lambda_3 on the
orbit is at q = 4/7 (lambda_3 = 1/7), not at "about 0.55"; (4) Theorem 1.2(d): under the printed sign Psi_s P is not
stable for any s > 0 (its first factor vanishes at ((2+u)/(u-1), 1, 1), while a stable form Q of degree d satisfies
Q(ix) = i^d Q(x) ≠ 0 for positive x); (5) Section 6: check_note.py checks Proposition 5.1(b), (d) for three pairs
(m, d), not eleven; (6) the author order of Ravichandran-Leake (Math. Ann. 2020) follows the journal; (7) this
record of the run in the note and the package; (8) a layout point in Theorem 1.2.

## Relation to the literature, novelty and scope
- **Credits.** Question, definitions, Lemma 3, Theorem 4 and both conjectures: Brändén (OWR 14/2017, pp. 825-826).
  Mixed characteristic polynomials and the largest-root bound: Marcus, Spielman and Srivastava, Ann. of Math. 182
  (2015) 327-350, doi:10.4007/annals.2015.182.1.8. Van der Waerden type bound: Gurvits, Electron. J. Combin. 15
  (2008) R66, doi:10.37236/790. Holens-Đoković counterexamples: Wanless, Linear Algebra Appl. 286 (1999) 273-285,
  doi:10.1016/S0024-3795(98)10177-5. Concavity of e_k/e_{k-1}: Marcus and Lopes, Canad. J. Math. 9 (1957)
  305-312, doi:10.4153/CJM-1957-037-9. Tverberg's conjecture: Friedland, Linear Multilinear Algebra 11 (1982)
  107-120, doi:10.1080/03081088208817436. Stability argument: Lieb and Sokal, Comm. Math. Phys. 80 (1981) 153-179,
  doi:10.1007/BF01213009, and Hurwitz's theorem.
- **Prior work on Conjecture 2.** Samuel Zackrisson, *Coefficients and zeros of mixed characteristic polynomials*,
  degree project in mathematics, KTH Royal Institute of Technology, Stockholm, 2017, DiVA diva2:1106332. According
  to its abstract (seen only as quoted in search-engine results), it shows via Wanless's counterexample that the
  zeros of associated Laguerre polynomials need not majorize those of other mixed characteristic polynomials, which
  is Conjecture 2 in the square case. A second excerpt of the full text, seen by the third verification run only in
  search-engine results, attributes this conjecture to Marcus, in personal communication with Brändén, and says that
  it is disproved by an explicit choice of three weights w1, w2, w3, with a computation in an appendix; so the thesis
  apparently contains an explicit counterexample of its own, possibly as small as the note's. The full text could
  not be read: the DiVA site refused or timed out every connection from our machine (when the result was first
  obtained, in all three verification runs, and once more for this note); this was not worked around. **A reader
  with access should check** the theorem or example number, the example with the weights w1, w2, w3 and whether it
  is symmetric or 3 x 3, whether the thesis names Brändén's conjectures, and whether it contains anything on
  Conjecture 5 or the flow Psi_s. If it does, the corresponding part of the note is not new.
- **Conjecture 5.** No printed refutation found. Searched (2026-10-20): arXiv API (mixed characteristic
  polynomials; majorization with characteristic, stable, hyperbolic or real-rooted polynomials and permanents;
  doubly stochastic stable polynomials; Holens; Brändén's papers), zbMATH (Brändén since 2017; mixed characteristic
  polynomials; the Wanless review; the OWR title; Zackrisson), Crossref, Semantic Scholar, web search. Semantic
  Scholar lists six works citing the whole OWR report (2016-2021: three on sums of squares, one on camera
  arrangements, a book review listed twice); by their titles none concerns mixed characteristic polynomials; texts
  not read. OpenAlex refused the list of citing works (rate limit; it counts 2, Crossref 1). arXiv:2608.29595 (a
  2026 collection of counterexamples found with AI assistance), returned by the web search, treats other problems of
  Borcea and Brändén, not these conjectures (full text searched). Brändén,
  arXiv:1809.03255 (2018): no majorization statement (full text searched). Leake-Ravichandran, Math. Ann. 377
  (2020) 511-541 (arXiv:1609.04195): largest-root/paving bounds. Gurvits-Leake, STOC 2021, pp. 847-858,
  doi:10.1145/3406325.3451105 (arXiv:2007.08390 v4): Theorem 7.1, credited to Brändén through personal
  correspondence, is a majorization statement (for p in HStab_n(1), the roots of t -> p(t1 - x) are majorized by
  x), a Schur-Horn-type statement for hyperbolic polynomials; it does not compare chi_P with chi_{P_inf} and settles
  neither conjecture. A search that finds nothing is not a proof of novelty.
- **Scope.** One counterexample in D_{3,3} refutes both conjectures. No claim is made about which (m, d) admit
  counterexamples or about all pairs 0 < r < s. No priority is claimed for the refutation of Conjecture 2 or for any
  counterexample to it; the note's own claim is the refutation of Conjecture 5 (both signs of the flow) and the
  proof of Lemma 3 for exp(s(T-I)).

## 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/