# Verification report — OWR-13750332-001 (star transforms without Type 2 singular directions; Conflitti's conjecture)

Verification date: 2026-09-30. Revised on 2026-09-30 after a second independent verification run (AI-assisted).

**Verdict.** Both open items of the record are answered affirmatively, with complete proofs by hand. (a) For every odd
m and all nonzero real weights c_1, …, c_m there are unit rays γ_1, …, γ_m such that
P(ψ) = Σ_j c_j Π_{i≠j} ⟨ψ, γ_i⟩ has no zero on the unit circle: the star transform has no singular direction of
Type 2 (Ambartsoumian–Latifi Conjecture 1, the extension of their Theorem 6(b)). The rays of any such star are
pairwise non-parallel, in particular distinct. (b) For every even r and every n, the real zero set of the elementary
symmetric polynomial e_r in R^n contains no linear subspace of dimension r (Conflitti's Conjecture 8); the largest
dimension is min(n, r − 1). The new content of (b) is r ≥ 4 and n ≥ r + 2. Two independent verification runs found
no mathematical error. The note is unrefereed.

## Statement checked
- **Primary source.** G. Ambartsoumian (joint work with M. J. Latifi), "Injectivity and stability of the inversion of
  the star transform", Oberwolfach Report 21/2023, pp. 1128–1131, doi:10.4171/OWR/2023/21.
  - Read in the EMS Press PDF. On p. 1131 the extension of Theorem 6(b) to m = 2k + 1 > 3 is described as not yet
    proven, and Conflitti's conjecture is quoted with the extreme case e_{m−1} (m odd) proved there (Theorem 7).
  - Condition (6) and Definition 5 (p. 1130) print the Type 2 condition on a direction ψ_i as
    Σ_j c_j Π_{i≠j} ⟨ψ_i, γ_j⟩ = 0. This is the same index misprint as in AL21 (13), and the paper's footnote 1 now
    records both.
- **Ambartsoumian–Latifi**, J. Geom. Anal. 31 (2021) 11270–11291, doi:10.1007/s12220-021-00680-7; read in the
  preprint version arXiv:2005.01918v3.
  - Conjecture 1 is in Section 5.3; the Type 2 set is eq. (13).
  - In (13) the product factor is printed as ⟨ψ, γ_j⟩. The definition (23) of P_2 and the proof of Theorem 6(b) show
    that ⟨ψ, γ_i⟩ is meant; the paper uses this reading and says so in a footnote.
  - The sentence introducing Definition 2 takes the directions γ_i distinct and the weights c_i nonzero. Theorem 1.1
    states that the rays of a star without Type 2 directions are pairwise non-parallel, in particular distinct
    (proof in Remark 2.9), so its stars fit this definition.
- **Conflitti**, "Zeros of real symmetric polynomials", Appl. Math. E-Notes 6 (2006) 219–224. Read in the journal PDF.
  Conjecture 8, and the heuristic stated after it, are quoted correctly.
- **Corpus record.** ulamai/UnsolvedMath, record OWR-13750332-001 (record_id 30005519), status `open`.
  - In version 1.6.0 (commit c6e7f41, 25 August 2026), the version cited in the paper, the statement fields
    (`original`, `upstream` and `for_research`) quote the whole OWR passage of p. 1131, with both questions. The
    title is the first sentence of that passage, on the extension of Theorem 6(b) to odd m > 3. `statement.clean` is
    null. The literature note (checked 2026-08-22) concerns only the second question, Conflitti's conjecture.
  - In the dataset revision of 28 September 2026 (commit 372682f, still the dataset head on 30 September 2026), the
    record is titled "Real Subspaces in Zeros of Elementary Symmetric Polynomials". It has a cleaned statement of
    Conflitti's conjecture only (statement status `corrected_verified`). Its original statement still quotes both
    questions, and its status and literature note are unchanged.
  - The first version of this report quoted a cleaned statement that is not in the v1.6.0 record, and is not
    verbatim in revision 372682f either. This bullet replaces it. The paper (Section 1) now describes both versions.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Type 2 set = zeros on S^1 of Σ_j c_j Π_{i≠j} ⟨ψ, γ_i⟩ (AL21 (23), proof of Thm 6(b); the same misprint in AL21 (13) and OWR (6)); extension of Thm 6(b) to all odd m | yes: Type-2-free stars exist for every odd m and all nonzero weights; their rays are pairwise non-parallel | Thm 1.1 |
| Conflitti's Conjecture 8: e_r^{-1}(0) ⊆ R^n has no r-dimensional real subspace, r even, all n | true | Thm 1.3 |
| the same with "dimension ≥ r" | true (a larger subspace contains an r-dimensional one); maximum is min(n, r − 1) | Cor 1.4 |
| (r − 1)-dimensional subspaces for even r ≥ 4 | only coordinate subspaces | Thm 1.5 |
| Conflitti's heuristic after Conjecture 8 (a subspace containing a vector with ≥ r nonzero coordinates is a line) | false for every even r ≥ 4 (Chirvasitu's type-(b) subspaces; not new) | Rem 3.4 |
| odd r | the analogous statement fails once n ≥ 2r (Aron–Gonzalo–Zagorodnyuk) | Sec. 1 |

## Results in the paper
- **Theorem 1.1** (all odd m, all nonzero weights; proved by hand). Proof: reversing a ray absorbs the sign of its
  weight (Lemma 2.1); nearly parallel rays reduce the problem to a good list, i.e. positions b_j with
  g(t) = Σ_j a_j Π_{i≠j}(t − b_i) free of real zeros and Σ a_j ≠ 0 (Lemma 2.3); three poles at −1, b*, 1 with
  b* = D(2W_2 − S)/S² give B² − 4AC = S²(b − b*)² − 16AW_1W_2W_3/S² (Lemma 2.4); inserting a small copy of a good
  list for one entry keeps goodness (Lemma 2.5); greedy nesting reaches every multiset of magnitudes (Lemma 2.6).
  A parallel pair of rays forces a Type 2 direction (Remark 2.9), so the rays are pairwise non-parallel.
  Example 2.8 works the construction out for the zero-sum weights (−3, −3, 2, 2, 2), with a Gram-matrix certificate
  that can be checked by hand.
- **Theorem 1.3** (Conflitti's conjecture, all even r, all n). Lemma 3.1 (classical; Descartes' rule of signs):
  e_{r−1}(y) = e_r(y) = 0 forces at most r − 2 nonzero coordinates. Lemma 3.3: on a subspace V ⊆ e_r^{-1}(0) of
  dimension ≥ 2, some V ∩ R^J (|J| ≤ r − 2) has codimension ≤ 1, because e_{r−1} is an odd form.
- **Corollary 1.4.** Maximum dimension min(n, r − 1).
- **Theorem 1.5** (rigidity). For even r ≥ 4 every (r − 1)-dimensional subspace in e_r^{-1}(0) is a coordinate
  subspace. The cases n = r − 1 and n = r are trivial and n = r + 1 is covered by Chirvasitu, so the new content is
  n ≥ r + 2 (Remark 1.6).
- **Lemma 3.2.** e_{n−1} is irreducible for n ≥ 3, so the case n = r + 1 of Theorem 1.3 is elementary.
- **Remark 3.4.** Conflitti's heuristic fails for every even r ≥ 4; simplest counterexamples are Chirvasitu's
  type-(b) subspaces, e.g. span{(1,−1,0,0,0), (0,0,1,1,−1/2)} for r = 4.

## Proved for all parameters vs. certified for listed instances
- **Proved by hand, for all parameters:** Theorem 1.1 (every odd m, all nonzero real weights), Theorem 1.3 and
  Corollary 1.4 (all even r, all n), Theorem 1.5 (all even r ≥ 4, all n), Lemmas 2.1–2.6 and 3.1–3.3, Remark 3.4,
  Example 2.8 (its definiteness certificate is a 3 × 3 Gram matrix with three positive minors).
- **Certified by exact computation, listed instances only:** explicit Type-2-free stars for 82 weight vectors with
  m ≤ 15 (finder), for 20 further weight vectors with m ≤ 21 (re-implementation in `verifier/`), and for 62 weight
  vectors with m ≤ 17 (own implementation of the second independent verification run). These illustrate
  Theorem 1.1; no statement depends on them.
- **Numerical only (not claims):** conditioning values (min|P|/max|P|, kappa), and Grassmannian least-squares searches.

## Computations (exact unless marked; scripts and outputs in reproducibility/)
- **Finder** (`claimant/`).
  - `star_construction.py`: construction and exact certification (Sturm and Descartes bisection, plus P(1,0) ≠ 0) of
    82 stars; `check_certificates.py` re-verifies all 82 from the JSON alone (82/82).
  - `identities.py`: identities of Lemma 2.4 symbolically; Lemmas 2.1, 2.3, 2.5 on random instances.
  - `conflitti.py`: Lemma 3.1 on all 137,256 vectors in {−3..3}^n, n ≤ 6; an alternative family of counterexamples
    to the heuristic; numerical Grassmannian searches (evidence only).
  - `optimize_conditioning.py`: numerically optimised stars (floating point), each certified exactly.
  - Re-run for the first release: `identities.py`, `exactpoly.py` and `check_certificates.py` reproduce the saved
    outputs.
- **Re-check written for the release** (`verifier/`, AI-assisted, written from the paper's text; standard library only).
  - `verify_stars.py`: all 82 stars re-certified by Sturm and, separately, by an exact Taylor-bisection certificate
    on the two charts; Example 2.8 recomputed step by step.
  - `construct_independent.py`: re-implementation with random admissible middle positions, insertion at the other
    outer entry, δ, θ ∈ {3^{−j}}; 20 new weight vectors (m = 17, 19, 21 equal weights; zero-sum weights for
    k = 8, 9, 10; 1..17 and alternating; ratios up to 10^10; 8 random), all certified.
  - `check_identities.py`: Lemma 2.4 identities proved by a grid polynomial identity test; Lemma 2.5 and 2.1 checks.
  - `check_conflitti.py`: Lemma 3.1 exhaustive test; subspaces of Remark 3.4 in e_r^{-1}(0) for r ≤ 12 (grid proof);
    the divisibility step of Lemma 3.2 for n ≤ 7.
- **Second independent verification run** (`independent_run_2/`, AI-assisted). Its code was written from the text of
  the paper before the finder's scripts were read. It uses exact rational arithmetic, except for two numerical
  least-squares scripts.
  - `r1poly.py`: exact toolkit (Sturm; Descartes-rule bisection on the square-free part); self-test on 300 random
    polynomials with known real roots.
  - `ref1_lemmas.py`: Lemma 2.4 symbolically, and its equivalence on 2,302 random rational (W, b); Lemmas 2.1, 2.3
    and 2.5 exactly on 200 random instances each; Remark 2.7 on 300 random good lists and 300 even-length lists;
    Remark 2.9 on 200 stars with a parallel pair; the sign claim for the regular directions in Example 2.8.
  - `ref1_example.py`: every number of Example 2.8 recomputed from scratch.
  - `ref1_star.py`: an independent implementation of the whole construction. It uses b* or a random admissible middle
    position, alternately; δ, θ ∈ {5^{−j}}; and, for every third weight vector, an o(θ) perturbation of the rays
    (Lemma 2.3 with non-linear φ_j). Every final star is certified from (c, γ) alone: exact unit vectors, no
    parallel pair, P(1,0) ≠ 0, and Sturm, plus Descartes bisection for m ≤ 13. 62 weight vectors, m = 3..17, nine
    of them zero-sum: 0 failures.
  - `ref1_replay59.py`, `ref1_theta_scan.py`: definiteness is not monotone in θ. For weight vector #59 (m = 7) it
    holds for θ = 1/5, …, 1/12, fails for θ = 1/15, …, 1/30, and holds again for every tested θ from 1/40 down to
    10^{−12}. This is allowed by Lemma 2.3, which asserts definiteness only below a threshold. For very skewed
    weights (10^0, …, 10^24, m = 9) the threshold is about 10^{−28}. Remark 2.9 now notes the non-monotonicity.
  - `ref1_extra.py`: the exact value of min|P|/max|P| at the located extrema of the finder's m = 15 equal-weight
    star is 1.47·10^{−13} (the paper says ≈ 10^{−13}, a numerical value). The half-plane condition of
    Zhao–Schotland–Markel holds by an exact test for 82/82 finder stars and 62/62 run-2 stars.
    - The saved output has two further lines, (3) and (4): no parallel pair among the 3,507 ray pairs of the finder
      stars, and a summary of the 62 run-2 certificates.
    - Neither line is printed by the preserved script. `recheck_extra_34.py`, added for this revision (standard
      library only, own Sturm code), recomputes both, with the same results.
  - `ref1_conflitti.py`: Lemma 3.1 exhaustively on 652,828 integer vectors ({−4..4}^n for n ≤ 5, {−3..3}^6,
    {−2..2}^7 and {−2..2}^8; exact int64 arithmetic) and on 4,000 structured rational vectors: 0 violations.
    It also checks:
    - the subspaces of Remark 3.4 for r = 4..12 and n = r+1..r+3, by exact evaluation at 60 random rational points
      each;
    - 40 random Chirvasitu type-(b) subspaces;
    - Grassmannian least-squares searches, as evidence only (numpy and scipy).
  - `ref1_rigidity.py`: numerical evidence for Theorem 1.5, with controls (numpy and scipy).
  - **Reproduction of the release.** The second run extracted the release `source.zip` and re-ran every script of
    `claimant/` and `verifier/`.
    - All reproduced their saved outputs apart from timings.
    - `star_construction.py` rebuilt `certificates_star.json` byte for byte.
    - The claims of the paper's Verification paragraph were checked against the code and found accurate.
    - One documentation error was found: the numpy dependencies in `reproducibility/README.md`, now corrected.
  - **Re-run for this revision.** The files of `independent_run_2/` were re-run in a copy of the release layout.
    Every script reproduced its saved output apart from timings; for `ref1_extra.py`, this means lines (1) and (2).
    `ref1_star.py` rebuilt `ref1_star_certificates.json` identically apart from its `seconds` field.
  - The copies differ from the run's working files only in labels, one relative path and a shortened traceback path;
    `reproducibility/README.md` lists the changes.

## Independent verification and required fixes
First independent verification run (2026-09-30; its scratch scripts were not preserved):

| Item | Verdict |
|---|---|
| Source fidelity (OWR 21/2023, AL21 v3, Conflitti 2006) | CONFIRMED |
| Proofs (Lemmas A1–A5 = paper Lemmas 2.1 and 2.3–2.6; Theorem B = Theorem 1.3; B2 = Theorem 1.5) | CONFIRMED, checked step by step |
| Computations | CONFIRMED (certificate file rebuilt byte for byte; 82/82; an exact positivity certifier; 21 further weight vectors up to m = 21 with its own nesting code) |
| Answer as intended (no misprint exploitation) | CONFIRMED |
| Novelty | no earlier proof found |
| Classification | PAPER_CANDIDATE; suggested HF status "solved" |

All four required fixes were applied:
1. The case r = n − 1 of Conflitti's conjecture is stated to be elementary (e_{n−1} irreducible for n ≥ 3; Lemma 3.2);
   the new content is n ≥ r + 2 (and r ≥ 4, since r = 2 is Conflitti's). Remark 1.6, abstract, Scope paragraph.
2. The heuristic counterexample (former Proposition B3) is downgraded to Remark 3.4, citing Chirvasitu 2025,
   Thm 0.2(1)(b), whose type-(b) subspaces give simpler counterexamples in R^{r+1}.
3. Classical references for Lemma 3.1: Todhunter, *Theory of Equations* (1885), Chap. V, Arts. 68–72; Burnside and
   Panton, *Theory of Equations* (3rd ed., 1892), Art. 21 and Chap. IX, Ex. 18 (both read in scans).
4. Ambartsoumian's monograph (World Scientific 2023, doi:10.1142/12424), Chapter 5 "Star Transform" (pp. 143–163,
   doi:10.1142/9789811242441_0005): the publisher's page returned a bot check, which was not bypassed; Crossref and
   OpenAlex give no abstract. The paper's Scope paragraph says so.

Second independent verification run (2026-09-30; AI-assisted, adversarial; code and outputs in
`reproducibility/independent_run_2/`):

| Item | Verdict |
|---|---|
| Statement fidelity (OWR 21/2023, AL21 v3, Conflitti 2006, the corpus record) | PASS; three wording fixes needed (corpus-record description, the OWR misprint, distinct rays) |
| Proofs | PASS; every step checked, no mathematical error and no gap |
| Computations | PASS; its own exact code confirms every claim tested, and every script of the release reproduces its saved output |
| Novelty and credit | PASS; no earlier proof of either conjecture found; one credit fix needed (Zhao–Schotland–Markel 2014) |
| Presentation | PASS; a few factual details to correct |

All six required fixes were applied:
1. **Corpus record.** The paper (Section 1) no longer calls Conflitti's conjecture "the current statement" of the
   record. It says that in v1.6.0 the statement quotes both questions, the title is taken from the first, and the
   literature note concerns the second; and that the revision of 28 September 2026 has a new title and a cleaned
   statement of the second question only. The "Corpus record" bullet above replaces the earlier one.
2. **OWR misprint.** Footnote 1 now also records the misprint in OWR (6) and Definition 5 (p. 1130).
3. **Distinct rays.** Theorem 1.1 now states that the rays are pairwise non-parallel, in particular distinct, as in
   AL21 Definition 2. The proof refers to Remark 2.9.
4. **Web searches.** The Scope paragraph now separates the searches: arXiv, OpenAlex, zbMATH and Crossref for both
   questions, two web searches for Conflitti's conjecture, and one for the star-transform conjecture. The last one
   was run for this revision (see below).
5. **Zhao–Schotland–Markel 2014 credited** (Inverse Problems 30 (2014) 105001, doi:10.1088/0266-5611/30/10/105001;
   Section 8.2 and eq. (51) of the accepted version arXiv:1401.7655v2).
   - Their stability function f(θ) = Σ_k s_k / cos(θ − θ_k) is w.
   - With a sign argument they showed that it always has zeros for an even number of rays. For an odd number they
     showed that it has zeros whenever the vectors s_k û_k lie in one closed half-plane.
   - The paper now says so next to the even-m result of AL21, with m ≥ 3 for the odd case, since one ray has no
     Type 2 direction.
   - Remark 2.7 explains how the constructed stars meet the half-plane condition.
6. **README dependencies.** `claimant/identities.py` (through its import of `star_construction`),
   `optimize_conditioning.py` and `triage/test_sturm.py` also need numpy; `reproducibility/README.md` now says so.

Optional suggestions applied:
- Remark 3.4 uses n instead of m for the number of variables.
- The constants of Lemma 2.5 are renamed κ_1, Λ_1, κ_2, Λ_2, so that they do not clash with the weights c_i.
- The dataset is cited with the dataset's own recommended author and title, so the label is [Uns26], not [ula26].
- Remark 2.9 notes that goodness and definiteness need not be monotone in δ and θ.
- Remark 1.6 says that the cases n = r − 1 and n = r of Theorem 1.5 are trivial.
- Before Lemma 3.1, the classical statement now reads "two consecutive vanishing coefficients below the leading
  one".
- In Example 2.8, the contrast is made for the regular directions with mixed-sign weights, not for "the regular
  star", because AL21 Definition 5 builds unit weights into the word "regular".

Not applied:
- The remark that Conflitti states Conjecture 8 over subfields K ⊆ R. The real case implies the K case, and the
  record and OWR state the real case.
- Citing Shpilka (JCSS 2002, affine projections of symmetric polynomials). Its content was not checked.

## Relation to the literature, novelty and scope
- **Searches (September 2026).** All requests were anonymous and are logged.
  - arXiv API (finder and second run):
    - star transform(s) in title or abstract;
    - the authors Ambartsoumian, Latifi Jebelli, Zamindar, Terzioglu, Kuchment, Conflitti and Chirvasitu;
    - "elementary symmetric" with subspace, zero set, zero locus, hyperbolic or Fano.
  - zbMATH (star transform; citing documents), Crossref (all DOIs of the bibliography; bibliographic searches for
    both questions), and OpenAlex. OpenAlex lists 3 works citing Conflitti 2006, 5 citing AL21 and 2 citing
    OWR 21/2023, none on these questions.
  - Web searches:
    - two for Conflitti's conjecture (finder; second run; 9 links in the second, none a proof);
    - one for the star-transform conjecture, run for this revision. It returned 10 links: AL21, AAL25 (twice), the
      V-line 2-tensor paper arXiv:2306.13245 (twice), a ResearchGate profile, a SIAM V-line paper and an unrelated
      tomography item. None is a proof.
  - arXiv:2306.13245 (Ambartsoumian–Mishra–Zamindar) treats the star transform on symmetric 2-tensor fields. Its
    text defines singular directions for that transform and does not mention Conjecture 1.
  - No proof of either conjecture was found.
- **Related work, credited in the paper.**
  - Ambartsoumian–Latifi 2021 (m = 3; e_{n−1} for odd n).
  - Zhao–Schotland–Markel 2014: stability criterion; zeros of w for even m, and for odd m ≥ 3 when the c_iγ_i lie in
    a closed half-plane.
  - Conflitti 2006 (r = 2); Aron–Gonzalo–Zagorodnyuk 2000 (odd degree).
  - Chirvasitu, arXiv:2507.19163: Fano schemes of e_{n−1}. It covers n = r + 1 of Theorems 1.3 and 1.5 and refutes
    the heuristic.
  - Ambartsoumian–Auel–Latifi Jebelli, arXiv:2507.22138: symmetric star transforms; it does not address either
    conjecture.
- **Caveats.**
  - The monograph chapter was not seen.
  - The one web search attempted in the paper-writing round (for a classical reference for Lemma 3.1) returned
    nothing because of a tool limit and was not retried; the classical references were found by direct lookup of
    archive.org scans.
  - The second run reached the anonymous OpenAlex daily limit; the citing-work counts above are from the finder's
    saved responses of the same day.
  - The construction of Theorem 1.1 proves existence; it gives no quantitative stability bound, and its rays can be
    nearly parallel.
  - This negative search is not a proof of priority.

## 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/