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

Verification date: 2026-09-30 (revised the same day after the second referee report; see "Second referee
report" below).

**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.
- A smaller partial counterexample G49, with 49 vertices in [0,3]^3, was found by the third independent
  referee (Remark 6.3). It is checked by computer only, by four programs. So the least size of a
  counterexample is at most 49; for induced subgraphs the bound stays 73.
- 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) |
| G required to be an isometric subgraph of Z^3 (not required by the source) | not addressed | G62, H73 and G49 are not isometric; none of the 21 pieces of X(G62) and 9 of the 27 pieces of X(H73) are isometric in the plane grid |
| The source's theorem with the extra condition (3) in one direction | not refuted | G62, H73 and G49 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.
  - Both examples are invariant under a cyclic group of order 6. The full symmetry group in the box has
    order 6 for G62 and order 12 for H73 (all coordinate permutations, with and without ι').
  - 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.
- **Remark 3.2.** The finder's vertex-minimising MILP returned min62, the image of G62 under the reflection
  (x,y,z) ↦ (x,z,y). It is congruent to G62, not a second example.
- **Lemma 6.1.** If every half-integer piece has H_1 = 0, then H_2 = H_3 = 0.
  - Alexander duality gives rank H_2 = the number of bounded complementary components, and there are none.
  - H_2 is torsion-free, because ∂_3 is injective modulo every n (top-face argument of Lemma 2.3(a)).
  - 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.
- **Remark 6.3 (G49).**
  - V49 consists of the 49 points of {0,1,2,3}^3 outside {0,1}^3 and outside {2,3}^3 ∖ {222}.
  - G49 is the induced graph on V49 (99 edges) minus the 12 edges in the ⟨σ⟩-orbits of 012-022, 022-023,
    031-131 and 113-213.
  - It is connected, σ-invariant (symmetry group ⟨σ⟩ of order 3), with f-vector (49, 87, 39, 0) and no
    corner.
  - X(G49) collapses in 87 steps, and all 21 sections are single collapsible pieces.
  - Its subdivision is an induced counterexample with 175 vertices, larger than H73.
  - It was found by the third referee's exploratory SAT search (σ-invariant, |V| ≤ 58), together with a
    disjoint hexagon. A 59-vertex σ-invariant counterexample G59, with one cube, was also found. These
    searches are uncertified; a run with |V| ≤ 48 did not finish.
- **Remarks.**
  - Condition (3) fails for all examples in every direction: G62 and G49 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.
  - Isometry: neither G62 nor H73 (nor G49) is an isometric subgraph of Z^3, and most of their pieces are
    not isometric in the plane grid. The conjecture does not require isometry; the isometric variant is
    not addressed.

## Computations (exact; scripts and outputs in reproducibility/)
Here the **finder** is the search, with its programs in `reproducibility/claimant/`, that produced G62;
the paper marks its runs (F).
- **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};
  - min62 is the image of G62 under (x,y,z) ↦ (x,z,y); exactly 6 of the 48 box symmetries map G62 onto
    it;
  - G49, rebuilt from its description and equal to the third referee's file, and G59: all of the above;
  - the symmetry groups in the box (orders 6, 12 and 3 for G62, H73 and G49) and the isometry statements.

  Certificates:
  - X(G62), X(H73), X(G62^(2)), X(G49) and X(G59) collapse to a vertex in 115, 132, 458, 87 and 109
    elementary collapses.
  - Collapse sequences are given for every piece.
  - `lead/check_certificates.py`, which shares no code with the generator, replays all of them, and the
    third referee's own certificates for G49 and G59: all VALID. The only pieces without a collapse
    sequence are the y-pieces of T12 and T12^(2), which are hexagons, as expected.
  - `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; it is the mirror image of G62 (see Remark 3.2).
- **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.
- **Third independent referee** (`referee2/`, written from the paper's text before reading any other
  program; author of the second referee report).
  - Own library `ref2lib.py`: every finite claim about G62, H73, G62^(2), T12 and T12^(2), including
    integer homology by Smith normal form; all nine figure panels; random tests of Lemmas 2.1, 2.3, 2.5,
    2.6, Theorem 1.3 and Lemma 6.1 on 3000 graphs; replay of all lead certificates (0 failures).
  - Found that min62 is the mirror image of G62, and that the symmetry group of H73 has order 12.
  - Third, eager SAT encoding (all region constraints for all edge-connected sets from the start; validated
    against brute force; lex-leader symmetry breaking validated to keep one solution per orbit). It
    reproduced every row of Table 3 (CaDiCaL 1.9.5; for 3×3×3 also Kissat 4.0.4; uncertified).
  - Exploratory searches found G49 and G59, checked by ref2lib, by the first referee's `indep.py` and by
    the lead's `check_certificates.py`. The author then checked both with `lead/verify_paper.py`.
- **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 all 420 region clauses from the start and lex-leader symmetry breaking over the 48 box symmetries, and complete HiGHS model without symmetry breaking (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) |

  - Region constraints. Most runs add them lazily. The finder's programs impose those for small regions
    from the start, which in the 3×3×3 box covers all regions. The second referee's complete 3×3×3 models
    impose all 420 region clauses from the start. The 3×3×3 CaDiCaL result also depends on the lex-leader
    symmetry breaking, which was tested separately.
  - CaDiCaL versions: 1.5.3 in the second referee's scripts, 1.9.5 (the PySAT default of
    `sat_search.py`) in the finder's. The paper cites the CaDiCaL 2.0 paper for the solver and states
    these versions.
  - 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 49 (partial; G49, Remark 6.3) 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.

## Second referee report (2026-09-30)
An independent adversarial referee reviewed the paper after the fixes above. The paper was not edited during
the review. The referee's code and outputs are in `reproducibility/referee2/`; the paper calls this referee
the third independent referee.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED |
| Proofs | CORRECT; one minor gap: Lemma 6.1 did not show that H_2 is torsion-free (the lemma is not used in the disproof) |
| Computational claims | CONFIRMED by independent code, with one misleading remark: the "second" 62-vertex graph of Remark 3.2 is the mirror image of G62 |
| Novelty | Not found in the literature; the modest priority wording is appropriate |
| Presentation / house style / release | ACCEPT WITH MINOR FIXES |
| Fatal | No |

New in that review: smaller counterexamples exist, G59 and G49 (see Remark 6.3). Nothing in the paper was
false because of them, but the bound "at most 62" was superseded.

All ten required fixes were applied:
1. **Remark 3.2.** min62 is now stated to be the image of G62 under (x,y,z) ↦ (x,z,y), congruent to G62
   and not a new example. The same correction was made in Verification items 1–2, `reproducibility/README.md`,
   this report and RESULT.md §2. `lead/verify_paper.py` now checks the congruence (its old check "min62
   differs from G62" is kept as "differs as an edge set").
2. **Lemma 6.1.** One argument was added: over Z/n, n ≥ 2, the top-face argument shows that ∂_3 is
   injective, so H_2 is torsion-free. Together with rank 0 this gives H_2 = 0.
3. **Abstract and Zenodo description.** "Both examples have a symmetry group of order 6" was replaced by
   "Both examples are invariant under a cyclic group of order 6". §4 now says that H73 is invariant under
   all coordinate permutations, so its symmetry group in the box has order 12.
4. **Isometry.** "Scope and priority" now says:
   - neither G62 nor H73 is an isometric subgraph of Z^3;
   - none of the 21 pieces of X(G62), and 9 of the 27 pieces of X(H73), are isometric in the plane grid;
   - the conjecture as stated does not require isometry, and the isometric variant is not addressed.
5. **Computations paragraph.** The blanket statement "imposed lazily" was corrected:
   - most runs add the region constraints lazily;
   - the finder's programs impose small regions from the start (all regions in 3×3×3);
   - the second referee's complete 3×3×3 models (CaDiCaL and HiGHS) contain all 420 region clauses from
     the start;
   - the CaDiCaL model also uses lex-leader symmetry breaking under the 48 box symmetries, on which its
     result depends; the finder's HiGHS and SCIP runs and the referee's HiGHS run use none.
6. **"The finder".** The jargon was replaced by "the search programs with which G62 was found, marked (F)".
   This was done in the Computations paragraph, the Table 3 caption, the Verification items and the
   transfer-matrix sentence.
7. **CaDiCaL citation.** The paper now writes "CaDiCaL [BFF+24] in the versions bundled with PySAT:
   version 1.5.3 in the programs (R), and version 1.9.5, the default of the SAT search script, in those of
   (F)". While applying this fix the author found that the finder's `sat_search.py` calls CaDiCaL 1.9.5 by
   default, not 1.5.3.
8. **Certificate output.** `lead/check_certificates_output.txt` no longer ends with "NOT ALL CERTIFICATES
   VALID (see above)".
   - The checker now knows the expected non-collapsible hexagon pieces of T12 and T12sub. It reports them
     as expected and ends with "ALL CERTIFICATES VALID (the only pieces without a collapse sequence are
     the expected hexagon y-pieces of T12 and T12sub, which are not collapsible)".
   - Any other failure still ends with "CERTIFICATE CHECK FAILED"; this was tested with corrupted
     certificates.
   - The README entry was reworded to match.
9. **Rebuild.**
   - The PDF was rebuilt (14 pages; no errors, warnings, overfull or underfull boxes), and every page was
     rendered and inspected.
   - Floats were repositioned, so that Table 1 and Figure 1 no longer interrupt Proposition 3.1 or the
     definition of G62, and Table 3 no longer interrupts the Computations paragraph mid-page.
   - source.zip and the zenodo/ copies were regenerated, and sha256, size and md5 were recomputed.
10. **Size bound.** G49 was added as Remark 6.3, with:
    - its description and the checks of the Verification paragraph;
    - collapse certificates by the author (`lead/certificates/G49_*`) and by the referee (`referee2/G49_*_ref2.txt`);
    - credit to the third referee's exploratory search.

    "At most 62" became "at most 49" in §6, Question (i) and RESULT.md; the induced bound stays 73. The
    Introduction and the abstract mention G49.

Optional suggestions, also applied:
- Chepoi–Maat, arXiv:2607.04014, is cited as related work on corner peelings. Its §9.3 treats corner
  peelings of ample sets in Cartesian products; it does not consider subcomplexes of Z^3.
- The search description was updated: OpenAlex answered on 2026-09-30 with nothing relevant, and there
  were five web searches.
- The third encoding's reproduction of Table 3 is recorded in Verification item 5.
- The floats were repositioned.
- The abstract says "the disproof can be checked by hand".

## 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;
  - five web searches (one per agent, including the third referee);
  - OpenAlex: HTTP 429 in the first attempts; on 2026-09-30 "corner peeling" returned 54 hits, none
    relevant (third referee).

  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.
  - Chepoi–Maat, Ample sets in Cartesian products, arXiv:2607.04014 (named by the third referee; corner
    peelings of ample sets in Cartesian products, not subcomplexes of Z^3).
- **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 (known: at most 49 vertices, at most 73 for
    induced subgraphs), whether (1) and (2) imply collapsibility of X(G), and the variant for isometric
    subgraphs.
  - Suggested corpus status: solved (disproved; our own unpublished result).

## Final wording change (2026-09-30, after the final readiness check)
- The abstract and Section 1 no longer say that "an independent referee" found the 49-vertex example. They
  now say that it was found in an exploratory solver search during an independent verification of this note.
- The Verification paragraph now states that the referees of items 3–5 are independent, AI-assisted
  verification runs made for this note, not peer reviews.
- No mathematical statement, proof, data file, program or certificate changed. The PDF was rebuilt with
  tectonic (14 pages, no warnings, no overfull or underfull boxes), and source.zip, the zenodo/ copies and
  the checksums were regenerated.

## 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/