# Verification report — OWR-1782-017 (Brenti–Welker, Conjecture 8 of Welker, Oberwolfach Report 1/2008: real-rooted integer polynomials satisfy a g-theorem)

Verification date: 2026-10-17.

**Verdict.** The note proves the record's second conjecture in full; the record as a whole is answered only in part.
- **Settled.** Conjecture 8 of V. Welker, "Conjectures on f-vectors" (Oberwolfach Report 1/2008, p. 70), attributed
  there to F. Brenti and V. Welker, and quoted as "Conjecture 2" in the record: every h(t) ∈ N[t] of degree d with
  h_0 = 1, h_i ≥ 1 and only real roots satisfies a g-theorem (strict increase to a first peak r, no increase
  afterwards, and g_{i+1} ≤ g_i^⟨i⟩ for 1 ≤ i ≤ r − 1, with the classical pseudo-power). No symmetry is assumed
  (Theorem B). More generally the conclusion holds for every integer sequence with h_0 = 1 and
  i h_i^2 ≥ (i+1) h_{i−1} h_{i+1} (Theorem A).
- **Not new.** The palindromic case (Brenti's Problem 2.9) was proved by Guo and Kang, arXiv:2609.00781 (September
  2026, unrefereed), by a different method; the note gives a second proof (Corollary 5.2). Newton's inequalities,
  Macaulay's theorem and the g-theorem are classical. The source's Theorem 9 (Kubitzke–Nevo) and the theorem of
  Bell–Skandera are special or weaker cases (Corollaries 5.3, 5.5). The contribution of the note is Lemma D, the
  lower bound a^⟨k⟩ ≥ (k! a)^{(k+1)/k}/(k+1)!, and the resulting proof of the general, non-palindromic case.
- **Open.** The record's first conjecture, which is also its title statement (Conjecture 6 of the source:
  real roots of the h-polynomial of sd(∂P) for every, not necessarily simplicial, polytope P), is not treated; it
  holds for simplicial polytopes (noted in the source) and, as far as we know, is open in general (Ferroni–Matherne–
  Vecchi, arXiv:2411.04070v3 of May 2026, Question 5.8, still list it as open). The
  Kruskal–Katona strengthening (Kubitzke–Welker 2012, Question 4.4 = Brenti 2024, Problem 2.10) remains open; it is
  false under the (ULC) hypothesis alone (Remark 6.3).
- **Suggested label for the record:** partially solved (Conjecture 2 proved; Conjecture 1, the title statement, open).

The note is unrefereed.

## Statement checked
- **Primary source.** V. Welker, "Conjectures on f-vectors", extended abstract in Oberwolfach Report No. 1/2008,
  *Combinatorics* (workshop of 6–12 January 2008, organised by Jeff Kahn, László Lovász and Hans Jürgen Prömel),
  report pp. 67–71 (PDF pp. 63–67), doi:10.4171/OWR/2008/01 (report pp. 5–78).
  - File: EMS Press, serial-article-files/46145, 628,633 bytes, 74 pages, sha256
    `17b422edb6960f77689ee9bd018c016c56553395d97c79525d5c362505f760a9`; downloaded independently three times on
    2026-10-17 (when the result was obtained, by verification run A and by run B) and again when the note was
    written, always with this hash. The organisers and dates are on the title page.
  - Report p. 70, read in the text layer and as a rendered page image (three times independently): the expansion
    n = C(a_i, i) + … + C(a_j, j) "for numbers n ≥ i ≥ 1", and n^⟨i⟩ printed as C(a_i − 1, i − 1) + … +
    C(a_j − 1, j − 1); the definition of "satisfies a g-theorem"; Theorem 7 (g-theorem of Billera–Lee and
    Stanley); Conjecture 8 attributed to [4] = Brenti–Welker, "Conjectures on simplicial complexes and real rooted
    polynomials, in preparation, 2008" (not found); evidence: small-d calculations and Theorem 9 ([10] =
    Kubitzke–Nevo).
- **Corpus record.** ulamai/UnsolvedMath, OWR-1782-017 (dataset version 1.6.0; upstream status
  `partially_solved`). Its statement has two parts: "Conjecture 1" (= the source's Conjecture 6), which is also the
  record's title, and "Conjecture 2" (= the source's Conjecture 8, with "the classical g-theorem conditions … the
  Macaulay binomial-expansion inequalities"). The paraphrase is faithful.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Classical pseudo-power n^⟨i⟩ = Σ C(a_l + 1, l + 1), expansion for all n, i ≥ 1, first peak r, no symmetry | proved | Theorems A, B |
| Pseudo-power as printed, Σ C(a_l − 1, l − 1) | not tenable: the source's Theorem 7 would fail for (1,4,6,4,1); Conjecture 8 would fail for (1+2t)(1+4t) | Remark 2.2 |
| Kruskal–Katona operator Σ C(a_l, l + 1) | not tenable: Theorem 7 would fail for the cyclic polytope C(8,5), h = (1,3,6,6,3,1) | Remark 2.2 |
| Expansion only "for n ≥ i", missing pseudo-powers counted as failures | not tenable: Theorem 7 would fail for g = (1,5,3,2,1), h = (1,6,9,11,12,11,9,6,1) | Remark 2.2 |
| Expansion only "for n ≥ i", conditions with g_i < i dropped | not tenable: Theorem 7 would fail in the other direction for h = (1,4,5,10,5,4,1), whose g = (1,3,1,5) is not an M-sequence (Conjecture 8 under this reading is weaker and covered by Theorem B) | Remark 2.2 |
| "Classical g-theorem conditions" read as including symmetry h_i = h_{d−i} | trivially false (1 + 3t + 2t²); not the source's definition, where symmetry is separate (Theorem 7) | not treated |
| Differences up to ⌊d/2⌋ instead of the first peak | false for non-palindromic h with an early peak; not the source's definition | not treated |
| The record's Conjecture 1 (real roots for sd(∂P), P non-simplicial) | not treated | Scope and priority |

## Results in the paper
- **Theorem A.** Positive integers h_0 = 1, h_1, …, h_d with i h_i^2 ≥ (i+1) h_{i−1} h_{i+1} satisfy a g-theorem.
- **Theorem B.** Conjecture 8 of the source (= Conjecture 1.1 of the note) is true.
- **Proposition C.** For positive reals with (ULC) and h_{k−1} < h_k: (k+1)! g_{k+1} ≤ (k! g_k)^{(k+1)/k}
  (in fact strict). Steps (5)–(7); no integrality.
- **Lemma D.** a^⟨k⟩ ≥ (k! a)^{(k+1)/k}/(k+1)! for all integers a, k ≥ 1. Induction on k: facts (F1), (F2); the case
  n = k (a ≤ k, a^⟨k⟩ = a); for n ≥ k + 1 a chord bound, a one-variable minimum (9), and the inequality (10)
  proved by AM–GM and γ_k ≤ 3/4.
- **Lemmas 2.3, 2.4, 5.1.** Newton's inequalities give (ULC); shape of (ULC) sequences for d ≥ 1 (strict increase
  to r, at most one plateau, strict decrease); a real-rooted polynomial with nonnegative coefficients and h(0) = 1
  has all coefficients up to its degree positive.
- **Remark 2.2.** The printed operator is a slip under every reading of the source's definition (table above); it is
  the operator κ_i of the equivalent lower form κ_{i+1}(g_{i+1}) ≤ g_i of Macaulay's condition (Mu–Welker,
  arXiv:2503.24076v2, Section 2 and Theorem 2.1), applied in the source to g_i instead of g_{i+1}.
- **Corollary 5.2** (palindromic case = Guo–Kang, Theorem 1.1), **Corollary 5.3** (the source's Theorem 9, via
  Brenti–Welker's Theorems 2.2 and 3.1), **Remark 5.4** (Kubitzke–Nevo's Corollary 1.3 follows only together with
  their Corollary 4.7 on the peak), **Corollary 5.5** (Bell–Skandera, via Macaulay's theorem and A[y]).
- **Remarks 6.1–6.3.** Lemma D is asymptotically sharp for every k (ratio 1 + (k+1)/(2n) + O(n^{−2}) at a = C(n,k);
  1 + 3/(4n) in the middle of a block for k = 2; in the computed ranges (a ≤ 20,000, k ≤ 40) the smallest ratio is
  at k = 1, a heuristic observation, not proved in general); Proposition C is
  asymptotically sharp for every k (h_i = L^i/i!: ratio 1 − (k+1)/(2L²) + O(L^{−3})); the Kruskal–Katona version is
  not reached and is false under (ULC) alone ((1,4,8)).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The program of the note** (`check_note.py`, standard library, exact arithmetic, about 12 s; 32 checks, all
  PASS): uniqueness of binomial representations by exhaustive search (i ≤ 6, n ≤ 300); Remark 2.2 including
  Theorem 7 of the source on 135 h-vectors of cyclic and cross-polytopes (0 / 91 / 91 violations for the
  classical / printed / Kruskal–Katona operators); Lemma D for all k ≤ 30, a ≤ 2500, on 67,992 block points and 2000
  random pairs with a < 10^45, with (F1), (F2) and all constants and identities, (9) and (10) exactly; Proposition C
  with (5)–(7) exactly on 6000 random rational (ULC) sequences; Theorem A on all 370,612 integer (ULC) sequences
  with h_i ≤ 30, 2 ≤ d ≤ 10; Theorem B and Corollaries 5.2, 5.3, 5.5 on all real-rooted polynomials in boxes of degree
  2, 3, 4 (3008 + 7538 + 542, exact Sturm sequences), on (1+t)^n (n ≤ 200), Eulerian and Narayana polynomials
  (n ≤ 22), 3000 random products of real-rooted factors and 400 h-vectors of barycentric subdivisions; Remarks
  6.1–6.3 (the asymptotics numerically).
- **Programs with which the results were first obtained** (`original/`): Lemma D on 2,379,243 pairs; Conjecture 8 on
  all 11,089 real-rooted polynomials in boxes of degree 2–7 and on 20,000 products of linear factors; Theorem A on
  370,612 (ULC) sequences; the real inequalities numerically. The boxes of degree 5–7 contain only (1+t)^5, so the
  conditions with k ≥ 2 are exercised mainly by the (ULC) enumerations and the products (the recorded output
  `conj_check_part3.out` contains two artefact lines of its `part3` option; see `original/outputs/NOTES.txt`).
- **Verification run A** (`verification_run_A/`): Lemma D exactly on 990,415 pairs; Proposition C, (5), (6) on
  300,000 random rational (ULC) sequences (1.77 million indices); Theorem A on all 34,721,012 integer (ULC)
  sequences with h_i ≤ 55, 2 ≤ d ≤ 12; Conjecture 8 on 23,575 certified real-rooted polynomials of degree up to 120
  (classical families, products, 3000 polynomials with irrational roots; own Sturm sequences, cross-checked with
  sympy); every identity of Sections 3–4 symbolically or exactly.
- **Verification run B** (`verification_run_B/`): Remark 2.2 (three readings against Theorem 7 on 135 polytopes;
  pseudo-power against an independent monomial count); Lemma D exactly on 1,201,509 pairs (a < 10^60 included);
  Proposition C over the rationals (L up to 10^12); Theorem A on **all** integer (ULC) sequences with entries ≤ 10^5
  by a dynamic programme over consecutive pairs (2.27 × 10^10 cases), validated against direct enumeration;
  real-rootedness by Hermite's criterion (recount 11,089; 8000 products); 4011 polynomials with coefficients up to
  about 10^134; deliberately wrong inputs fail.
- **Verification run 2** (`independent_run_2/`, own library and programs, 38 checks, all PASS): Lemma D exactly on
  627,794 pairs (a < 10^80 included), with (F1), (F2) and (10) exactly for k ≤ 60, n ≤ k+400; Proposition C in its
  strict form over the rationals; Theorem A on all 7,038,047 integer (ULC) sequences with h_i ≤ 45, 1 ≤ d ≤ 12;
  real-rootedness decided by Descartes' rule of signs with bisection (boxes recounted: 3008, 7538, 542; 3500
  products); Corollary 5.3, Remark 5.4 and Brenti–Welker's formula on 974 simplicial complexes, with sd(Δ) counted
  from chains of faces; Remark 2.2 (including the lower form of Macaulay's condition on 1,669,016 triples);
  Remarks 6.1–6.3 numerically and symbolically.
- **Re-runs.** On 2026-10-17, when the note was written, every program of the package was run again from a copy of
  the package. All outputs are byte-identical to the recorded ones (for run A after removing the final `exit 0`
  line written by its wrapper). Verification run 2 ran every program again from a copy extracted from the archive
  `source.zip` and obtained the same result; it also ran its own programs again from the final archive (outputs
  reproduced apart from the lines reporting running times). Command lines and times are in
  `reproducibility/README.md` and `reproducibility/RERUN_LOG.txt`; the longest program takes about 6.5 minutes.

## Independent verification runs
The result was first obtained with the proofs and the programs in `original/`. Two independent verification runs,
both AI-assisted, followed on 2026-10-17; each re-derived the proofs line by line, read the source (with page 70 as
an image) and wrote its own programs. Run A concentrated on the statement and the proofs, run B on independent
computation and the literature.

| Item | Run A | Run B |
|---|---|---|
| Statement against the source; printed (−1,−1) slip; classical reading the only tenable one | CONFIRMED | CONFIRMED (three readings tested against Theorem 7) |
| Newton ⇒ (ULC); shape lemma | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| Proposition C | CONFIRMED (re-derived; exact checks) | CONFIRMED (re-derived; exact checks over Q) |
| Lemma D (all cases, Steps 1–4) | CONFIRMED (re-derived; exact checks) | CONFIRMED (re-derived; exact checks) |
| Theorems A and B | CONFIRMED | CONFIRMED (exhaustive DP to 10^5) |
| Palindromic case, source's Theorem 9, Bell–Skandera as consequences | CONFIRMED; Kubitzke–Nevo Cor. 1.3 needs their peak location | CONFIRMED; same qualification |
| Remarks of the first write-up | sharpness remark wrong (fixed) | sharpness remark wrong (fixed) |
| Bibliography | organisers wrong (fixed); Mu–Welker Thm 3.1 imprecise (fixed) | organisers wrong (fixed); rest correct (Crossref) |
| Novelty | no prior proof of the general case found | no prior proof of the general case found; Bell–Skandera 2007 and the Mohammadi–Welker survey unread |
| Record | PARTIAL (Conjecture 1 untreated) | PARTIAL |

Neither run found a mathematical error.

**Corrections required by runs A and B**, all applied in the note:
1. (A1, B1) The organisers of the workshop are J. Kahn, L. Lovász and H. J. Prömel: Section 1.1, bibliography entry
   of the source.
2. (A2) The binomial representation and the pseudo-power are stated for all n, i ≥ 1 with 0^⟨i⟩ = 0, and the case
   1 ≤ n ≤ i is made explicit (n^⟨i⟩ = n), as used in Lemma D: Section 2.1, Lemma 2.1. The source's restriction
   "n ≥ i" is discussed as a second imprecision with the example g = (1,5,3,2,1): Remark 2.2.
3. (A3, B2) Sharpness: Lemma D is asymptotically sharp for every k (at a = C(n,k) the ratio is (n+1)/P^{1/k}
   ≈ 1 + (k+1)/(2n); 1 + 3/(4n) in the middle of a block for k = 2); within a fixed range of a the smallest ratio
   occurs at k = 1; the original program's "min ratio" is a k-th power: Remark 6.1, `original/outputs/NOTES.txt`.
   Proposition C is asymptotically sharp for every k: Remark 6.2.
4. (A4, B3) "Contains Kubitzke–Nevo's Corollary 1.3" is replaced by: Theorem B gives the source's Theorem 9
   (first-peak form, Corollary 5.3), and together with Kubitzke–Nevo's Corollary 4.7 (peak in the middle) their
   Corollary 1.3: Remark 5.4, Section 1.3.
5. (A5) Mu–Welker's Theorem 3.1 is stated for ultra log-concave polynomials of degree d with x_d ≥ d, the
   real-rooted case being their Corollary 3.2(i); their binomial-expansion argument is named as the closest known
   technique: Section 1.3.
6. (A6, B4) Status: the record is partially solved; the note and this report name the proved conjecture explicitly
   (title, abstract, Section 1.1, Scope and priority, Verdict above).
7. (A7) The limited degree coverage of the exhaustive real-rooted test is stated (Section 7 of the note,
   `reproducibility/README.md`), and the artefact lines of `conj_check_part3.out` are explained in
   `original/outputs/NOTES.txt` (the recorded file is unchanged).

**Changes made when the note was written, after runs A and B** (examined by verification run 2, below; the finite
and numerical parts are also tested by `check_note.py`):
- Step 2 of the proof of Lemma D is presented with the substitution β = k s^{k−1}, under which
  (k+1)(β/k)^{k/(k−1)} − cβ = (k+1)s^k − cks^{k−1}; the minimum −c^k((k−1)/(k+1))^{k−1} is the one checked by both
  runs (check S3.5).
- Lemma 5.1 (positivity of the coefficients of a real-rooted polynomial with nonnegative coefficients and h(0) = 1)
  is stated and used in Corollaries 5.2, 5.3 and 5.5.
- The proof of Corollary 5.2 locates the first peak at ⌊d/2⌋ through the symmetric set of maximum positions (both
  runs had confirmed the location).
- The proof of Corollary 5.3 uses Brenti–Welker's Theorem 2.2 (nonnegative integer h-vector of sd(Δ)) and
  Theorem 3.1 (real roots), and treats trailing zeros (check S6.4).
- Remark 5.4 gives the short argument deriving Kubitzke–Nevo's Corollary 1.3 from Corollary 5.3 and their
  Corollary 4.7 (check S6.4).
- The expansions in Remarks 6.1 (middle of a block, k = 2) and 6.2 (constant (k+1)/2) are new in the text
  (checks S8.1, S8.3).

## Verification run 2 (of the written note)
After the note was written, a further independent verification run, also AI-assisted, examined its text
(2026-10-17), first the parts that runs A and B had not seen: Step 2 of the proof of Lemma D, Lemma 5.1, the proofs
of Corollaries 5.2 and 5.3, Remark 5.4 and the expansions of Remarks 6.1 and 6.2. It read the source again (same
file; page 70 rendered and the definition cropped), the arXiv versions of all cited papers that the note says were
read (version histories checked), and, for the status of the record's first conjecture, Ferroni–Matherne–Vecchi
(arXiv:2411.04070v3) and the introduction of Brändén–Leite (arXiv:2412.06595v3); it re-derived every proof line by
line, wrote its own programs before running those of the package (`reproducibility/independent_run_2/`, with a
README), re-ran the whole package from an extracted copy of the archive, and repeated the literature search.

| Area | Verdict of run 2 |
|---|---|
| Statement fidelity | CONFIRMED: the note's Conjecture 1.1 is Conjecture 8 of the source; the definitions, Theorems 4, 7, 9 and Conjecture 6 are quoted correctly; the printed operator (cropped image of p. 70) is a slip. Two completions of Remark 2.2 required: the reading in which conditions with g_i < i are dropped also breaks Theorem 7 (h = (1,4,5,10,5,4,1)); the printed operator is the operator κ_i of the equivalent lower form κ_{i+1}(g_{i+1}) ≤ g_i of Macaulay's condition, applied to the other side. The status of the record's first conjecture must be stated with a source |
| Proofs | CONFIRMED: all proofs re-derived, including Step 2 of Lemma D in the variable s (minimum −c^k((k−1)/(k+1))^{k−1}, equal to the β-form checked by runs A and B), Lemma 5.1, the argmax proof of Corollary 5.2, Corollary 5.3 (Brenti–Welker's Theorems 2.2 and 3.1, reversal, trailing zeros) and Remark 5.4 (with Kubitzke–Nevo's Corollary 4.7; the case r < ⌊d/2⌋ occurs, e.g. for a 3-simplex). No error. Lemma 2.4 needs d ≥ 1; the sentence of Remark 6.1 on the smallest ratio at k = 1 is a heuristic and is now stated for the computed range |
| Cited results | CONFIRMED in the versions read: [Wel08]; GK26 v1 (Theorems 1.1, 1.4); KN09 v1 (Corollaries 1.3, 4.7, Theorem 3.1); BW08 v1 (Theorems 2.2, 3.1); KW12 v1 (Proposition 4.3, Question 4.4); Bre24 v1 (Problems 2.9, 2.10); MW26 v2 (Theorem 3.1, Corollaries 3.2(i), 3.3); Brändén v1 (Lemma 1.1). The 13 DOIs resolve in Crossref with matching data |
| Computations | CONFIRMED: own programs, 38 checks, 0 failures (below); the re-run of every program of the package from the extracted archive reproduced every recorded output byte for byte (run A's files compared without their final `exit 0` line). Two ranges of degrees in the text are stated exactly now (2 ≤ d ≤ 10 for the 370,612 sequences, 2 ≤ d ≤ 12 for run A's 34,721,012) |
| Novelty and credit | CONFIRMED: no earlier proof of the general case and nothing equivalent to Lemma D found; credits complete; the novelty claim is no stronger than the evidence |
| Presentation | CONFIRMED with small corrections (status sentence in the abstract; two sentences of Section 1.1); house style, AI paragraph, references and all rendered pages in order |

**Own programs of run 2** (`reproducibility/independent_run_2/`, Python standard library except one SymPy script):
- Lemma D exactly on 627,794 pairs (160,000 with k ≤ 40, a ≤ 4000; 464,794 block points; 3000 random pairs with
  a < 10^80), with (F1), (F2) at every pair; the pseudo-power against an independent count of lex-segment monomials;
  the case n = k for k ≤ 200; Step 4 and (10) exactly for 2 ≤ k ≤ 60, k+1 ≤ n ≤ k+400; Steps 1–3 and (8) at 80
  digits; (10) fails at k = n = 2 and its smallest ratio is 1.0337 at (2,3).
- Proposition C in its strict form over the rationals (90,722 indices, near-extremal L up to 10^9); Lemma 2.4 and
  Theorem A on all 7,038,047 integer (ULC) sequences with h_0 = 1, h_i ≤ 45, 1 ≤ d ≤ 12.
- Real-rootedness decided exactly by Descartes' rule of signs with bisection: the boxes of degree 2, 3, 4 recounted
  (3008, 7538, 542), Theorem B and Corollary 5.5 on them and on 2500 products of real-rooted factors (and 300
  non-real-rooted controls rejected), Corollary 5.2 on 1000 palindromic products, Lemma 5.1, classical families.
- Corollary 5.3 and Remark 5.4 on 974 simplicial complexes (622 with h^Δ ≥ 0), with the f-vector of sd(Δ) counted
  from chains of faces, which also confirms Brenti–Welker's formula (Theorem 2.2) on all 974.
- Remark 2.2 (0 / 91 / 91 violations on the 135 polytopes; the lower form of Macaulay's condition on 1,669,016
  triples), Remarks 6.1–6.3 (constants numerically up to n, L = 10^12 and symbolically with SymPy; the smallest ratio
  at k = 1 for every range a ≤ A ≤ 20,000, k ≤ 40).

**Corrections required by run 2, all applied:**
1. Status of the record's first conjecture: "as far as we know" open, with a source (Ferroni–Matherne–Vecchi,
   arXiv:2411.04070v3, Question 5.8) and the simplicial case noted in the source (abstract, Section 1.1).
2. Remark 2.2: the reading in which the conditions with g_i < i are dropped (Theorem 7 fails for
   h = (1,4,5,10,5,4,1)); the printed operator identified as the operator of the lower form of Macaulay's condition
   (Mu–Welker, Section 2 and Theorem 2.1).
3. Lemma 2.4 assumes d ≥ 1.
4. Remark 6.1: the statement on the smallest ratio at k = 1 is restricted to the computed range and its explanation
   marked as heuristic.
5. Section 7, Verification paragraph and this report: 2 ≤ d ≤ 10 for the 370,612 (ULC) sequences, 2 ≤ d ≤ 12 for
   run A's 34,721,012.
6. Section 1.1: the g-theorem sentence says that the differences are nonnegative; r is defined as the largest index
   with h_0 < ⋯ < h_r.
7. This run is recorded in the Verification paragraph, in Section 7 and "Scope and priority", here, and in the
   package (`independent_run_2/`, README.md, RERUN_LOG.txt).

## Relation to the literature, novelty and scope
- **Searches (17 October 2026).** Queries to the arXiv API and arXiv texts (Guo–Kang 2609.00781; Kubitzke–Nevo
  0712.1560; Kubitzke–Welker 1108.2852; Brenti 2410.09897; Mu–Welker 2503.24076; Brenti–Welker math/0606356; Brändén
  1410.6601; further papers on real-rooted h-polynomials and subdivisions), Crossref (all DOIs of the bibliography),
  OpenAlex and Semantic Scholar (works citing Brenti–Welker 2008, Bell–Skandera 2007, Kubitzke–Welker 2012, Brenti
  2024 and Guo–Kang 2026), zbMATH Open, and six web searches. Several services answered with rate limits.
  Verification run 2 repeated the search: zbMATH Open (16 queries, among them all its 55 records of 2026 on
  real-rooted polynomials, screened by title), Crossref (13 DOI look-ups, 3 searches), the arXiv pages of the cited
  papers and of four papers on the record's first conjecture, and a seventh web search; its four queries to the
  arXiv API and four to OpenAlex were refused because of rate limits (no API key was requested).
  - No proof of Conjecture 8 for non-palindromic h was found, nor Lemma D.
  - The palindromic case: Brenti listed it as open in 2024 (Problem 2.9, checked for d ≤ 9, h_i ≤ 100); Guo and
    Kang proved it in September 2026 through the γ-vector (Gal), Bell–Skandera's theorem and ballot complexes; their
    method needs symmetry and makes no statement about the general case.
  - The Kruskal–Katona version (Kubitzke–Welker Question 4.4 = Brenti Problem 2.10) is open.
  - The record's first conjecture (= Brenti–Welker's question for all polytopes, [BW08, Question 1]) is described as
    open by Ferroni–Matherne–Vecchi (arXiv:2411.04070v3, 5 May 2026, Conjecture 1.8 and Question 5.8) and as a
    conjecture, proved for simplicial and (by Athanasiadis) cubical polytopes, by Brändén–Leite (arXiv:2412.06595v3,
    Adv. Math. 487 (2026) 110760, introduction).
- **What was read.** The source item completely (report PDF; page 70 also as an image); Guo–Kang completely;
  Kubitzke–Nevo: introduction, Section 3, Corollary 4.7; Kubitzke–Welker: Section 4.1; Brenti 2024: Section 2;
  Mu–Welker (arXiv v2 of 14 October 2026): Sections 1–3.1; Brenti–Welker 2008: introduction and the statements of
  Sections 2–3; Brändén: Section 1 (Lemma 1.1, Newton's inequalities); Ferroni–Matherne–Vecchi (v3): introduction
  and Section 5.3; Brändén–Leite (v3): introduction (not cited in the note).
- **Caveats.** Bell–Skandera (Discrete Math. 307 (2007)) was not read: the publisher's site refused automated access
  and no other copy was found; its theorem is quoted from Kubitzke–Welker, Guo–Kang and Mu–Welker, which agree on its
  content but number it differently (Theorem 3.6 vs. Theorem 4.2). The Mohammadi–Welker survey (LNM 2176, 2017) was
  not read. The paper "in preparation" to which the source attributes the conjecture was not found. The classical
  sources (Macaulay 1927, Bruns–Herzog, Billera–Lee 1981, Stanley 1980) were not consulted; they are cited for
  classical results. Journal versions of the arXiv papers were not compared. This negative search is not a proof of
  priority, and no priority is claimed.
- **Scope.** The note proves the record's Conjecture 2 (= the source's Conjecture 8) under the classical reading,
  and its (ULC) generalisation. The record's Conjecture 1 (its title statement) is not treated; as far as we know it
  is open.

## 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/