# Verification report — OWR-14299904-015 (Konvalinka: optimizing over nonzero Littlewood–Richardson coefficients; the quantum max d-cut of K_{n−k,k})

Verification date: 2026-10-15.

**Verdict.** The note answers the problem **completely**: for all n, k, d the maximum is determined exactly, with proofs.
- **Settled.** For d ≥ 2 and 1 ≤ k ≤ n−1, Δmax(n,k,d) = 2k(n−k) + 2·max{Ψ_A(f), Ψ_B(f,π,y)}, a maximum over an explicit
  list of at most (d−1)(1 + k(n−k)) integers built from balanced partitions (Theorem C); Δmax = 0 for d = 1 and
  4k(n−k) for d ≥ n. The reduction (Theorem A) and the structure of an optimum (Theorem B) are proved on the way.
- **Corollary.** Conjecture 7.7 of Klep, Štrekelj and Volčič (arXiv:2503.20942v5 = Quantum 10, 2203 (2026)) holds, with
  explicit optimal triples (P ⊎ Z, P, Z) or (P ⊎ Z ⊎ (π+y), P ⊎ (π), Z ⊎ (y)), P and Z balanced (Corollary D).
- **Form of the answer.** A maximum over a candidate list of polynomial size, each entry an explicit integer; no single
  closed formula is claimed (KSV remark that a straightforward formula is not to be expected).
- **Credit.** The problem is Konvalinka's (Oberwolfach Report 2/2026); the quantum max d-cut problem was introduced by
  Carlson, Jorquera, Kolla, Kordonowy and Wayland (arXiv:2309.10957). The identification with the quantum max d-cut of
  K_{n−k,k}, the identity η = N(N−1) − 2c (KSV (B.2), Lemma B.1), the cases d ≤ 3 and k ≤ 4, the merged maximum and
  Conjecture 7.7 are due to KSV; d = 2 is the Lieb–Mattis theorem; the dominance bound of Lemma 3.1(b) is classical.
  New, to our knowledge: Theorems A–C in general and the proof of Conjecture 7.7.

The note is unrefereed.

## Statement checked
- **Primary source.** G. Chapuy, S. Elizalde, I. Fischer (organizers), *Enumerative Combinatorics*, Oberwolfach Reports
  23 (2026), no. 1, 81–156 (Report No. 2/2026, workshop 11–16 January 2026), doi:10.4171/OWR/2026/2; open problem
  session (collected by G. Chapuy), Problem 10, "Optimizing over nonzero LR coefficients (problem of Matjaž
  Konvalinka)", communicated by Igor Klep, p. 150.
  - Read in the open file of the publisher (EMS Press, serial-article-files/53596, PDF page 70 = report page 150),
    fetched on 2026-10-15 (1,423,064 bytes) by the first computations, by both verification runs and at the writing.
  - Content: for a partition λ "of length d", η(λ) = n² + d(d−1)(2d−1)/6 − Σ_{i=1}^{d}(λ_i − (i−1))²;
    Δ(λ,µ,ν) = η(λ) − η(µ) − η(ν); given n, k, d with k ≤ n/2, maximize Δ over |λ| = n, |µ| = k, |ν| = n−k with
    c^λ_{µ,ν} ≠ 0. The item states no partial results.
- **Origin of the problem.** I. Klep, T. Štrekelj, J. Volčič, "Quantum Max d-Cut via qudit swap operators",
  arXiv:2503.20942v5 (2 September 2026) = Quantum 10, 2203 (2026), doi:10.22331/q-2026-09-03-2203: (6.4) (η_λ with
  n = |λ|; valid for at most d rows, last sentence of the proof of Prop. 6.7), (7.3)–(7.5) (Δ as the eigenvalues of
  the Hamiltonian of K_{n−k,k}), Conjecture 7.7, Prop. 7.8, Remark 7.9, Theorem 7.27, Corollaries 7.13, 7.28, 7.29,
  Theorem 7.2, Example 1.9, Appendix B ((B.2), Lemma B.1).
- **Corpus record.** ulamai/UnsolvedMath, OWR-14299904-015 (dataset version 1.6.0; upstream status `open`). Its
  statement is a garbled text extraction of the same paragraph; its literature note cites only Taşkın–Gümüş–Işık–Ulvi,
  arXiv:2208.06541 (bounds for nonzero LR coefficients).

## Readings
| Reading | Answer | Where |
|---|---|---|
| Partitions with at most d parts, padded with zeros; n in η(π) is \|π\| (the reading of KSV) | Theorem C for all n, k, d | Section 1.1, Theorem C |
| The given n in all three η | every Δ decreases by n² + 2k(n−k); same maximizers | Lemma 2.1 |
| λ with exactly d nonzero parts | the same maximum for d ≤ n (proof); no triple for d > n | Remark 6.2 |
| k ≤ n/2 | a normalization (symmetry in µ, ν); all results for 1 ≤ k ≤ n−1 | Section 1.1 |
| d = 1; d ≥ n | 0; 4k(n−k) | Theorem C |
| Maximum over merged triples λ = µ ⊎ ν only (KSV Theorem 7.27) | not the question; 2km + 2·max_f Ψ_A(f) | Remark 6.3 |

## Results in the paper
- **Lemma 2.1** (KSV (B.2), Lemma B.1): η(π) = |π|(|π|−1) − 2c(π); Δ = 2km − 2F, F = c(λ) − c(µ) − c(ν).
- **Lemma 3.1**: for µ ⊆ λ, the column filling r − µ'_s is an LR tableau of content ν* (conjugate of the sorted column
  lengths of λ/µ); every admissible ν is dominated by ν*; c(ν) < c(ν*) for ν ≠ ν*. (a), (b) classical (Remark 3.2).
  **Lemma 3.3**: c(λ) − c(µ) − c(ν*) = D(b) − Σ a_s b_s.
- **Theorem A** (Lemmas 4.1, 4.2, level sets): Δmax = 2km + 2·max Ψ(µ,x), Ψ(µ,x) = Σ_r Σ_{s=µ_r+1}^{µ_r+x_r}(µ'_s − µ_r);
  an optimal triple is (sort(µ+x), µ, ν*). Greedy rule for fixed µ (Remark 4.3).
- **Theorem B** (Lemmas 5.2–5.5: pairwise form, moving T-cells, at most one mixed row by a convexity argument,
  balancing): an optimum of shape A (merged, balanced) or B (one mixed row (π,y), balanced P-rows ≥ π, balanced
  T-rows ≥ π + y).
- **Theorem C**: the value; the list is a superset of the shapes of Theorem B and consists of lower bounds (Remark 6.1).
- **Corollary D**: optimal triples; KSV Conjecture 7.7.
- **Remark 6.2** (exactly d parts), **Remark 6.3** (Δ(µ⊎ν,µ,ν) = 2km + 2Σ min(µ_i,ν_j); Ψ_A(f) = (Δ(µ⊎ν,µ,ν) − 2k(n−k))/2),
  **Proposition 6.4** (shape B with g = 0 is never optimal; a criterion for merged optima; merged optima for d ≤ 3 and
  k ≤ 4, an independent proof of KSV Prop. 7.8).
- **Proposition 7.1**: d = 2 (Lieb–Mattis), d = 3, the star graph, merged optima for k ≤ 4 or d ≤ 3 (d ≥ 2), all derived
  from Theorem C. **Table 1**: the 24 cases with n ≤ 20 (k ≤ n/2, 2 ≤ d ≤ n) without a merged optimum, with optimal triples.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`, standard library, exact, about 8 s; TOTAL checks 495,166, TOTAL failures 0):
  (i) all LR coefficients with |λ| ≤ 12 by skew Kostka numbers and the inverse Kostka matrix (no LR tableaux;
  8,312 pairs, 20,237 nonzero coefficients; symmetry and dimension identity), Lemmas 2.1, 3.1, 3.3 on all pairs,
  brute-force Δmax with η as printed for all 572 triples 2 ≤ n ≤ 12, 1 ≤ k ≤ n−1, 1 ≤ d ≤ n = Theorem C, the two
  other readings, the merged maxima; (ii) Corollary D's triples for all 506 triples with 2 ≤ d ≤ n ≤ 12 (575
  triples) and the construction of Theorem A; (iii) Theorem C = Theorem A for all 8,555 triples n ≤ 30,
  1 ≤ k ≤ n−1, 2 ≤ d ≤ n, two implementations of the list (n ≤ 16), d ≥ n; (iv) Proposition 6.4 (n ≤ 24),
  Proposition 7.1 (n ≤ 40), examples, Table 1; (v) the numbers on the formulas of Jakab–Solymos–Zimborás;
  (vi) all 10,737 configurations for 2 ≤ d ≤ n ≤ 9.
- **Programs with which the results were first obtained** (`original/`): LR tableau enumeration |λ| ≤ 14 (25,301
  pairs; 483 triples n ≤ 14, k ≤ n/2, 1 ≤ d ≤ n); Theorem C vs exhaustive Theorem A, 13,959 triples n ≤ 44,
  k ≤ n/2, 2 ≤ d ≤ n; Corollary D's triples by LR tableaux, 708 triples n ≤ 17, k ≤ n/2, 2 ≤ d ≤ n−1; random lemma
  tests (331,555 checks); KSV's cases; symmetry; JSZ eq. (11).
- **Verification run A** (`verification_run_A/`): all LR coefficients |λ| ≤ 19 without tableaux (272,896 pairs,
  1,431,061 nonzero coefficients), Theorem C = brute force for all 2,280 triples n ≤ 19, 1 ≤ k ≤ n−1, 1 ≤ d ≤ n, and
  Corollary D's triples; Lemma-3.1 solver for all 8,555 triples n ≤ 30 (also "exactly d parts" and JSZ eq. (11));
  Theorem A = Theorem C for 84,540 triples n ≤ 80 and 7,000 random triples (n ≤ 500; large k with d ≤ 8); KSV's
  proved cases n ≤ 150 / 120; random lemma tests (about 430,000 checks); the JSZ formulas.
- **Verification run B** (`verification_run_B/`): LR tableau enumeration |λ| ≤ 22 (1,022,205 pairs; enumerator tested
  by symmetry and the dimension identity); Theorem A = brute force for all 3,542 triples n ≤ 22; Theorem A's
  construction (780 triples); Theorem C = Theorem A for 13,959 triples n ≤ 44 and = KSV Cor. 7.28 implemented from
  KSV's text (3,590 triples); lemma tests on 40,000 random configurations; Theorem B on all configurations n ≤ 9;
  deliberately wrong inputs detected; all 20 recorded outputs of `original/` reproduced byte for byte.
- **Re-runs at the writing** (`RERUN_LOG.txt`): 18 programs of the three folders re-run from a copy; all outputs
  reproduced (byte for byte or up to timing fields).
- **Third verification run** (`independent_run_2/`, on the final text): all LR coefficients |λ| ≤ 18 by the
  Brauer–Klimyk formula (no LR tableaux, no inverse Kostka matrix; 172,459 pairs, 808,203 nonzero coefficients) and the
  brute force for all 2,244 triples n ≤ 18, 1 ≤ k ≤ n−1, 1 ≤ d ≤ n+2 = Theorem C, with the other two readings,
  Remarks 6.2 and 6.3, Corollary D's triples (2,035) and Theorem A's construction (3,971); the largest eigenvalue of
  the Hamiltonian (2) computed directly on a weight space of (C^d)^{⊗n} for 138 triples n ≤ 10 (dimension up to
  113,400 for (10,5,5): 72), the whole spectrum compared with the Δ values in 81 small cases; Theorem C = Theorem A
  for 9,425 triples n ≤ 30, all k; Propositions 6.4 and 7.1 (n ≤ 40); all 212,612 configurations n ≤ 13 (Lemmas
  5.2–5.5, the convexity and endpoint claims inside the proof of Lemma 5.4, Theorem B); the JSZ numbers; six negative
  controls detected; 0 failures. It re-ran `check_note.py` and 47 recorded runs of the three earlier folders from an
  extracted copy of the archive: every output reproduced (`independent_run_2/r2_package_rerun_v1.txt`).

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed (2026-10-15); each wrote its own programs from the definitions. Runs A and B examined
the results before the note was written, and neither saw the report of the other: run A was the designated proof
reading; run B concentrated on independent computation, the literature and novelty. The third run examined the final
text (section "Third independent verification run" below).

| Item | Run A | Run B |
|---|---|---|
| Statement and readings against the source and KSV | CONFIRMED (source read; readings checked; fixed-n shift n² + 2k(n−k)) | CONFIRMED |
| Lemma 2.1, Lemmas 3.1, 3.3 | CONFIRMED (re-derived; checked on all pairs \|λ\| ≤ 19) | CONFIRMED (read; checked on all pairs \|λ\| ≤ 22) |
| Theorem A | CONFIRMED (re-derived) | CONFIRMED (= brute force n ≤ 22) |
| Lemmas 5.2–5.5, Theorem B | CONFIRMED (re-derived; E-rows to be mentioned) | CONFIRMED (read; random tests; exhaustive n ≤ 9) |
| Theorem C | CONFIRMED | CONFIRMED_WITH_FIXES (state the superset and lower-bound property) |
| Corollary D (KSV Conjecture 7.7) | CONFIRMED | CONFIRMED |
| Known cases (d = 2, d = 3, k ≤ 4, star, Remark 7.9, Example 1.9) | CONFIRMED | CONFIRMED |
| Statements about JSZ | CONFIRMED_WITH_FIXES (name the variant behind every count) | CONFIRMED_WITH_FIXES (precise sentence) |
| Bibliography and attributions | fixes (Lemma 2.1 is KSV (B.2); dominance bound classical; cite the record's literature source) | fixes (journal references; OWR volume; pointer to KSV's padding remark; ranges of d) |
| Novelty | no prior solution found | no prior solution found |

Neither run found an error in a proof or a gap; run A found a wrong factor ½ in a remark of the first version (correction 1 below).

**Corrections required by runs A and B**, all applied in the note (details in the audit file `fixes_applied.md`):
1. (A1) Ψ_A(f) = (Δ(µ⊎ν,µ,ν) − 2k(n−k))/2 (factor ½): Remark 6.3.
2. (A2, B1) The JSZ counts with the variant behind each: eq. (13) 24 of 1,185; eq. (16) as printed 956 (943 with ties
   rounded up); with the centre d·n_A/(n_A+n_B) 24; closed form (15) with (16) 956 / 249 (947 / 251 with ties
   rounded up); the second case of (15) 11,753 of 149,530 as printed, 0 with the sign changed; eq. (11) agrees:
   Section 1.3.
3. (A3) The count of sentences of the source item is not stated.
4. (A5, B3) Rows of type E in the first step of the proof of Lemma 5.4: they are now among the rows used there.
5. (A6) Lemma 2.1 credited to KSV (B.2), Lemma B.1; Lemma 3.1(a), (b) classical, with Azenhas's statement as quoted by
   JSZ (Remark 3.2); the record's literature source Taşkın–Gümüş–Işık–Ulvi cited (Section 1.3).
6. (B2) Theorem C states that the list is larger than the set of shapes of Theorem B and consists of lower bounds;
   Remark 6.1 gives the triples.
7. (B4) Björnberg–Rosengren–Ryan, Adv. Appl. Math. 151 (2023) 102572; Allerstorfer et al., Comm. Math. Phys. 407
   (2025), article 21, doi:10.1007/s00220-025-05483-3 (Crossref: online 2025-12-08, issue 1 of January 2026);
   OWR volume 23; the pointer to KSV's remark on rows of length zero (last sentence of the proof of Prop. 6.7).
8. (B5) The ranges of d covered by each computation: Section 8 of the note and `reproducibility/README.md`.
The internal remarks A4 and A7 concern files of the project and are not part of the note.

**Changes made when the note was written, after runs A and B.**
- Remark 6.2 (λ with exactly d parts: the same maximum for d ≤ n) was proved; run A had only computed it (n ≤ 30).
- Proposition 6.4 (shape B with g = 0 is never optimal; a criterion for merged optima; d ≤ 3 and k ≤ 4) and the
  derivations of Proposition 7.1 from Theorem C were added; the formula Δ(µ⊎ν,µ,ν) = 2km + 2Σ min(µ_i,ν_j) in
  Remark 6.3 was written out.
- Table 1 and Figure 1 were added; the numbering was changed (correspondence in `reproducibility/README.md`).
- Runs A and B did not examine these additions; `check_note.py` checks them (part iv), and the third run re-derived
  them first (next section).

## Third independent verification run (on the final text)
An AI-assisted run examined the written note and the package line by line, as if nothing had been checked, and the
four additions above first. It read the source (the same file, by SHA-256) and the quoted parts of KSV v5 (now also
the beginning of §1.5), JSZ v2 and BRR (abstract, introduction, the discussion of the complete bipartite graph)
again, re-derived every proof, wrote its own programs before reading `check_note.py` (results above, under
"Computations"), re-ran the package from the archive, and repeated the literature search (arXiv API, Crossref,
OpenAlex, Semantic Scholar, zbMATH, one web search).

| Item | Verdict of the third run |
|---|---|
| Statement and readings | CONFIRMED; the attribution sentence of Section 1.1 went beyond the source (fixed) |
| Lemmas 2.1, 3.1, 3.3, 4.1, 4.2, 5.2–5.5; Theorems A, B, C; Corollary D; Remark 6.1 | CONFIRMED (re-derived; tested) |
| Remark 6.2 | CONFIRMED (proof re-derived; brute force n ≤ 18; all configurations n ≤ 13) |
| Remark 6.3 | formula CONFIRMED; the deduction of the merged maximum was too terse (written out) |
| Proposition 6.4 (a)–(c) | CONFIRMED; it is a proof of KSV Prop. 7.8 (d ≥ 2) without their case analysis |
| Proposition 7.1 | (a)–(c) CONFIRMED; (d) lacked the hypothesis d ≥ 2 and was false for d = 1 (fixed) |
| Examples, Table 1, Figure 1, the JSZ numbers | CONFIRMED (recomputed) |
| Package | all recorded outputs reproduced; one docstring gave a wrong running time (fixed) |
| Novelty and credit | no earlier solution found; the introduction of the d-QMC problem by Carlson et al. was not credited (fixed) |

**Corrections required by the third run**, all applied: (1) Proposition 7.1(d) now assumes d ≥ 2 (for d = 1 the only
triple ((n),(k),(n−k)) is not merged); (2) Section 1.1 follows the source: "problem of Matjaž Konvalinka",
"communicated by Igor Klep", Problem 10 of the report on the open problem session; (3) Section 1.3, "Scope and
priority" and the bibliography credit Carlson, Jorquera, Kolla, Kordonowy and Wayland, arXiv:2309.10957, for the
quantum max d-cut problem (KSV §1.5); (4) Remark 6.3 now reads a merged triple as a configuration and derives the
merged maximum with Lemma 5.2 and Lemma 5.5; (5) the Verification paragraph no longer says that runs A and B
"confirmed all statements" (run A found a wrong factor ½ in a remark of the first version); (6) this run is recorded in
the note (Verification, Section 8, Scope and priority, What was read), here and in `reproducibility/`; (7) the docstring
of `check_note.py` gives its running time correctly (about ten seconds). Details in the audit file
`VERIFICATION_RUN_2.md`.

## Relation to the literature, novelty and scope
- **Searches (15 October 2026: when the results were first obtained, by runs A and B, in part again at the writing,
  and by the third run).** arXiv API (quantum max d-cut; quantum max cut on complete bipartite graphs and for qudits;
  Littlewood–Richardson with quantum max cut, swap operators, Hamiltonians, ground states; extendibility of Werner
  states; Lieb–Mattis for SU(d); interchange
  models on complete bipartite graphs; the authors Konvalinka, Klep, Štrekelj, Volčič), Crossref, zbMATH, the works
  citing KSV and JSZ in OpenAlex and Semantic Scholar, and five general web searches (the fifth by the third run,
  which repeated the arXiv, Crossref, OpenAlex, Semantic Scholar and zbMATH queries on the same day).
  - No proof of the general case or of KSV Conjecture 7.7 was found. The only work citing KSV is Štrekelj,
    arXiv:2512.03740 (complete tripartite graphs, d ≤ 3).
  - Jakab–Solymos–Zimborás, arXiv:2208.13743v2 (unpublished preprint), give a general formula; the note reports
    neutrally which of their displayed formulas agree with the exact values and which do not (Section 1.3).
  - Related: Björnberg–Rosengren–Ryan (thermodynamic limits on complete bipartite graphs); Allerstorfer et al.
    (clique case for Werner states); Taşkın–Gümüş–Işık–Ulvi (bounds for nonzero LR coefficients).
- **What was read.** The source item and front matter; KSV v5: Definition 1.5, Sections 1.4.3–1.4.4, the beginning
  of 1.5, 6.2, Theorem 7.2, Section 7.2, Appendix B (its case analysis not checked); JSZ v2: main text and Supplement
  B; abstract, main theorems and introduction of BRR; abstracts of Allerstorfer et al., Carlson et al., Štrekelj,
  Taşkın et al. Azenhas (1999) not read (its statement as quoted by JSZ). All bibliographic data were checked with Crossref and the arXiv API on 2026-10-15.
- **Caveat.** A search that finds nothing is not a proof of novelty; no priority is claimed.
- **Scope.** Complete answer in the stated reading, as an explicit maximum over a polynomial-size list; no closed
  formula for max Ψ_B.

## 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/