# Verification report — OWR-14299911-029 (Chepoi's corner-peeling conjecture for subcomplexes of Z^3)

Verification date: 2026-09-30.

**Verdict.** The conjecture is false, both for partial subgraphs (the setting of the source) and for
induced subgraphs of Z^3.
- G62 is a partial subgraph with 62 vertices in [0,3]^3. H73 is an induced subgraph with 73 vertices in
  [0,4]^3.
- In both, X(G) and every piece of every section are collapsible, so (1) and (2) hold. X(G) has no corner,
  so no corner peeling exists.
- A subdivision construction shows that the partial and the induced versions are equivalent. It turns
  G62 into a 231-vertex induced counterexample.
- All claims used in the disproof are exact. Solver results about small boxes and least sizes are
  reported separately as uncertified.
- The note is unrefereed.

## Statement checked
- **Primary source.** V. Chepoi (from discussions with J. Chalopin and M. Kokkou), Problem 15, "Corners in
  subcomplexes of Z3". It appears in *Median Geometry and Applications*, Oberwolfach Report 8/2026,
  Oberwolfach Rep. 23 (2026), no. 1, pp. 487–540, pp. 534–535, doi:10.4171/OWR/2026/8. The published PDF
  was read; two verifiers' anonymous downloads were byte-identical to the local copy.
  - G is a "finite connected (partial) subgraph of Z3".
  - A square or 3-cube is a cell of X(G) iff all its vertices **and** edges are in G.
  - A corner is a vertex in a unique maximal cell.
  - A corner peeling orders V(G) so that v_i is a corner of G_i = G[v_1..v_i]; the source notes that
    G_n = G.
  - Pieces are the components of the sections X(G) ∩ {x_i = α}, α ∈ R.
  - Conjecture: (1) X(G) simply connected and (2) every piece simply connected imply a corner peeling.
  - The source states, without proof, that the conjecture holds under an extra isometry condition (3), and
    that (3) may be imposed in one direction only.
- **Corpus record.** ulamai/UnsolvedMath OWR-14299911-029, status `open`.
  - In the local `data/problems.jsonl`, the statement fields (`for_research`, `original`, `upstream`) are
    truncated after the definition of a corner, before the conjecture, and `clean` is null.
  - The second verifier reports that the HF `clean_statement` is complete.
  - The author's anonymous attempts to re-read the HF record failed (timeout, HTTP 502).

## Readings
| Reading | Refuted? | Witness |
|---|---|---|
| The source's setting: finite connected (partial) subgraph; (1) and (2) for all real α | yes | G62 (62 vertices, [0,3]^3); also H73 |
| Induced subgraphs of Z^3 only | yes | H73 (73 vertices, [0,4]^3); G62^(2) (231 vertices, [0,6]^3) |
| (2) only for integer and half-integer pieces | equivalent (Lemma 2.2) | the same |
| G_i read as the subgraph of Z^3 induced by v_1..v_i (not the source's reading, since the source says G_n = G) | yes | H73: it is induced, so both readings coincide and v_n must be a corner of H73 |
| (2) required only in two of the three directions | yes (weaker hypothesis) | T12 (12 vertices, non-induced) and its subdivision T12^(2) (39 vertices, induced) |
| The source's theorem with the extra condition (3) in one direction | not refuted | both examples violate (3) in every direction, as they must |

## Results in the paper
- **Theorem 1.2.**
  - G62 is generated by 19 edge-orbit representatives under Γ = ⟨σ, ι⟩ ≅ Z/6, with σ(x,y,z) = (z,x,y)
    and ι(p) = (3,3,3) − p. It has 62 vertices, 114 edges, 54 squares and one cube.
  - H73 is the induced subgraph on the Γ'-orbits of 13 points, with ι'(p) = (4,4,4) − p. It has 73
    vertices, 132 edges, 60 squares and no cube.
  - For both, X(G) and all pieces are collapsible, and there is no corner.
  - Hand proofs:
    - no corners: Tables 1–2 and the corner criterion (Lemma 2.1);
    - collapsibility of all pieces: the x-section pictures (Figures 1–2) and Lemma 2.3 (a connected grid
      complex with χ = 1 is collapsible);
    - contractibility of X(G): the sweeping lemma (Lemma 2.5, slab retractions and gluing).
  - The collapsibility of X(G) itself is shown by explicit collapse certificates.
- **G62 is non-induced.** It has 24 missing lattice-adjacent pairs, in the Γ-orbits of 002–102, 011–021,
  012–022 and 013–023. Its induced closure has 12 corners. This is admissible by the source's "(partial)"
  wording and its vertex-and-edge rule for cells.
- **Theorem 1.3 (subdivision).** G ↦ G^(2), the induced subgraph of Z^3 on {2b + 1_D : b + [0,1]^D ∈ X(G)}.
  - |X(G^(2))| = 2|X(G)|, cell by cell.
  - The pieces of G^(2) are the dilates 2P of the pieces P of G.
  - A point coding a d-cell c lies in 2^d·m(c) maximal cells. Hence corners(G^(2)) = 2·corners(G).
- **Corollary 1.4.** The partial and induced versions are equivalent. G62^(2) is an induced
  counterexample.
- **Lemma 2.6 and Corollary 2.7 (corner deletion).**
  - X(G) collapses onto X(G − v), and (1), (2) and connectivity pass to G − v.
  - So the conjecture (in either class) holds iff every G with at least two vertices satisfying (1) and (2)
    has a corner.
- **Lemma 6.1.** If every half-integer piece has H_1 = 0, then H_2 = H_3 = 0 (Alexander duality). So
  under (2), (1) is equivalent to contractibility.
- **Proposition 6.2.** A counterexample in a box B yields a cornerless G* in B whose sections all have
  H_1 = 0 and with χ = 1. This is the reduction used by the solver searches.
- **Remarks.**
  - Condition (3) fails for both examples in every direction: G62 at α = 1/2 and 5/2, H73 at α = 1/2 and
    7/2.
  - The trough T12 shows that (2) cannot be required in two directions only, also for induced graphs.

## Computations (exact; scripts and outputs in reproducibility/)
- **Lead** (`lead/verify_paper.py`, standard library only, about 1 s). It checks:
  - G62 and H73 rebuilt from the orbit representatives, equal to the finder's and the referee's data files;
  - all numbers of Props. 3.1 and 4.1, Tables 1–2 and invariance;
  - corners by the raw definition and by Lemma 2.1;
  - d∘d = 0 and Betti numbers (1,0,0,0) over GF(2) and GF(1000003);
  - all 21 (G62), 27 (H73) and 39 (G62^(2)) sections, each a single collapsible piece;
  - condition (3);
  - the same for min62, G62^(2), T12 and T12^(2): T12 fails exactly at y ∈ {0, 1/2}, and T12^(2) at
    y ∈ {0, 1/2, 1, 3/2}.

  Certificates:
  - X(G62), X(H73) and X(G62^(2)) collapse to a vertex in 115, 132 and 458 elementary collapses.
  - Collapse sequences are given for every piece.
  - `lead/check_certificates.py`, which shares no code with the generator, replays all of them: VALID.
  - `lead/random_tests.py` checks Lemma 2.1, Lemma 2.6 and Theorem 1.3 on 3000 random subgraphs, with 6913
    corner deletions and 0 failures. A negative control with corrupted subdivisions was detected.
- **Finder** (`claimant/`).
  - Two independent exact verifiers, `verify.py` with `cubecx.py` and `verify2.py`, and `check_cond3.py`.
    The author reran `run_checks.sh` in the release copy: all outputs are byte-identical.
  - G62 was found by HiGHS with ⟨σ,ι⟩-symmetry imposed in the 4×4×4 box. min62 came from the
    vertex-minimising MILP.
- **First independent referee** (`referee/first_referee/indep.py`, from scratch).
  - Checks: Betti numbers over Q, GF(2) and GF(3); greedy collapse to a point; all pieces; corners;
    condition (3).
  - The author reran it on G62, min62, H73 and T12 (`indep_output.txt`).
  - This referee first pointed out that G62 is non-induced.
- **Second independent referee** (`referee/second_referee/`, written before reading the finder's code).
  - Z-homology by Smith normal form, explicit collapses, and the raw corner definition.
  - Rebuilt G62 from the orbit representatives.
  - Found H73 with an own CaDiCaL encoding (Γ'-invariant, induced, 5×5×5).
  - Proved the subdivision lemma and tested it on 300 random instances.
  - Independent SAT/MILP reruns of the solver claims.
- **Solver results (uncertified; no DRAT, no exact MILP certificates; none used in the disproof).**

  | box | class | result | runs |
  |---|---|---|---|
  | 3×3×3 | all | none | HiGHS and SCIP 9.2.4 (finder); complete CaDiCaL model with lex-leader symmetry breaking, and HiGHS (second referee); the CaDiCaL model rerun by the author |
  | 2×2×n (n ≤ 6), 2×3×3 | all | none | CaDiCaL (finder, second referee) |
  | 2×3×4 | all | none | CaDiCaL (second referee) |
  | 2×4×4, 3×3×4 | invariant under the box's point reflection | none | HiGHS (finder) |
  | 4×4×4 | Γ-invariant | least \|V\| = 62 | HiGHS (finder); CaDiCaL \|V\| ≤ 61 infeasible (second referee, rerun by the author) |
  | 4×4×4 | Γ-invariant, induced | none | CaDiCaL (second referee, rerun by the author) |
  | 5×5×5 | Γ'-invariant, induced | least \|V\| = 73 | CaDiCaL (second referee, \|V\| ≤ 72 rerun by the author) |

  - Wording: "no counterexample fits in a 3×3×3 box". The earlier phrases "smallest bounding box larger
    than 3×3×3" and "more than 3×3×3" were withdrawn.
  - The 62- and 73-vertex optima hold only among the symmetric configurations.
  - Unexamined without symmetry: 2×2×n (n ≥ 7), 2×3×n (n ≥ 5), 2×4×4, 3×3×4, 3×4×4 and larger.
  - The prism program `prism_dp.py` is TESTED-level only.
  - The true minimum size is unknown: at most 62 (partial) and at most 73 (induced).

## Independent adversarial audit
Verdicts (2026-09-29 and 2026-09-30):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (both verifiers read the source PDF; byte-identical to the local copy) |
| Correctness of G62 | CONFIRMED (the finder's two verifiers, both referees' from-scratch checkers, and the author's checker with an independent certificate checker; hand proof via symmetry) |
| Answer as posed | CONFIRMED (negative; the only literal feature used is non-inducedness, which the source explicitly allows) |
| Induced version | FALSE as well (H73; subdivision lemma), found by the second verifier |
| Novelty | CONFIRMED as far as can be checked (no priority claim) |
| Presentation | CONFIRMED_WITH_FIXES |

All required fixes were applied, in the paper and in `RESULT.md`:
1. G62, min62, the trough and the bent trough are stated to be partial (non-induced) subgraphs. The 24
   missing pairs and the 12 corners of the induced closure are given, and admissibility is justified by
   the "(partial)" wording and the vertex-and-edge rule.
2. The induced version is not listed as open. The induced counterexample H73 was added, with its
   13-row corner table and its x-section pictures for α ∈ {0, 1/2, 1, 3/2, 2}. The subdivision lemma was
   added with a proof: partial ⇔ induced, and G62 becomes a 231-vertex induced example.
3. All MILP/SAT statements are qualified as uncertified solver results, with the solvers named. The
   bounding-box wording was corrected, the unexamined boxes are listed, the symmetric optima are
   qualified, and the true minimum is stated to be unknown.
4. Collapsibility of X(G) and of every piece is certified by explicit elementary-collapse sequences, so
   (1) and (2) no longer rest on the graph-of-spaces argument. The paper also gives a hand proof, via
   Lemma 2.3 and the sweeping Lemma 2.5.
5. The sharpness remark was updated: the trough is non-induced, but its induced subdivision (39 vertices)
   is cornerless and contractible, with only y-pieces failing.
6. The HF status is labelled as our own unpublished result, and the literature check is cited. The
   truncated `problems.jsonl` statement is noted.
7. The literature check was completed (see below). OpenAlex was rate-limited again.

## Relation to the literature, novelty and scope
- **Searches (September 29–30, 2026; all anonymous).**
  - arXiv API author queries (Chepoi, Chalopin, Kokkou; newest first) and keyword queries (corner
    peeling(s), corner AND peeling, corners AND cube complexes, and related phrases);
  - the full text of arXiv 2602.12894, cited by the source, which does not mention corners;
  - zbMATH (corner peeling, corners cube complex, Chepoi 2023–2026), Crossref and V. Chepoi's publication
    page;
  - four web searches (one per agent);
  - OpenAlex: HTTP 429.

  Nothing states, proves or refutes the conjecture outside the OWR report.
- **Related work credited.**
  - Chalopin–Chepoi–Moran–Warmuth, JCSS 127 (2022): the 12-dimensional cornerless example recalled by
    the source.
  - Chalopin–Chepoi, JCTB 169 (2024).
  - Knauer–Marc, Europ. J. Combin. 112 (2023).
  - Chalopin–Chepoi–Kokkou, arXiv:2602.12894.
  - Chalopin–Kokkou, arXiv:2511.19208.
- **Caveats.**
  - The source's positive theorem under (3) is stated without proof; we only checked that our examples
    are consistent with it.
  - Unpublished work of the proposers cannot be ruled out.
  - This negative search is not a proof of priority.
- **Scope.**
  - The note refutes the conjecture as stated, for partial and for induced subgraphs, with fully exact
    proofs and certificates.
  - Open: the least size and bounding box of a counterexample, and whether (1) and (2) imply
    collapsibility of X(G).
  - Suggested corpus status: solved (disproved; our own unpublished result).

## 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/