# Verification report — OWR-14299905-036 (Schildkraut; Halberstam–Schildkraut: the word-search constants C_2(ABBB) and C_3(ABB))

Verification date: 2026-10-10.

**Verdict.** The note determines **two concrete constants** asked for in the record, with their extremal grids; the
record as a whole (Question 10 for all pairs; C_d(ABB) for all d > 2) is **not** settled. Scope:
- **Settled, computer-assisted with exact certificates.**
  - C_2(ABBB) = 8/5 (Theorem 1.1). This is Question 9.3 of Halberstam and Schildkraut and the last part of
    Question 11 of the Oberwolfach report.
  - C_3(ABB) = 6 (Theorem 1.6). This is the case d = 3 of the first part of Question 11 and of Question 9.2 of
    Halberstam and Schildkraut (ABB is 3-stackable).
  - Both values also hold in the normalisation of the report, for n × … × n arrays without wraparound
    (Corollaries 1.2(b) and 1.7).
  - The extremal grids are classified in both cases (Theorems 1.3 and 1.6(b)), with gaps 3/70 and 47137/294912 per
    window that is not of the extremal kind. So for the pairs (2, ABBB) and (3, ABB), and only for these, the
    supremum is attained by a periodic grid; this is what Question 10 of the report (Question 2.5 of Halberstam
    and Schildkraut) asks for all pairs.
- **Further results for ABBB in dimension 2.** The closure of the set of pairs (density of A, concentration) is the
  quadrilateral with the vertices (0,0), (1/5, 8/5), (3/8, 3/2), (1,0) (Theorem 1.4). Question 9.5 of Halberstam
  and Schildkraut has a positive answer for (ABBB, 2) and for (ABB, 3). Question 9.6 has a negative answer for
  (ABBB, 2) in the form in which it is stated (constant 1); a linear bound holds with the optimal constant 14/5
  for grids on two letters, and with the constant 9002/15 for all alphabets (Corollary 1.5).
- **Partial.** 18 ≤ C_4(ABB) ≤ 62/3, 54 ≤ C_5(ABB) ≤ 64, 24/5 ≤ C_3(ABBB) ≤ 11/2 (Corollary 1.8).
- **Open.** Question 10 in general; C_d(ABB) for d ≥ 4; Questions 9.5 and 9.6 for other pairs; the best constant
  in Corollary 1.5 for alphabets with more than two letters.
- **Not new.** The lower bounds (the five-queens grid; stacking) are constructions of Halberstam and Schildkraut.
  The technique of the upper bounds (linear programming over window statistics, with dual "potential"
  certificates) is standard.
- **What "computer-assisted" means here.** The reductions (Lemmas 2.1, 2.2, 2.4, 2.5, Proposition 2.3) and the
  deduction of all theorems and corollaries from finitely many inequalities are proved in the text. The finite
  inequalities are verified in exact integer arithmetic: 65,536 inequalities for each of the six certificates in
  dimension 2, and 2^27 = 134,217,728 for each of the two certificates in dimension 3. Each certificate was
  accepted by at least four different programs. Anyone can run the checks again in at most two seconds each
  (dimension 2, mostly pure Python) or in 13 to 26 seconds each with between 2 MB and 2.5 GB of memory
  (dimension 3; one further program, a walk through all patterns in the order of a Gray code, takes 83 and 169
  seconds).

The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Reports 23 (2026), no. 1, pp. 5–79, Report No. 1/2026 "Combinatorics"
  (organisers J. Fox, P. Keevash, W. Samotij), doi:10.4171/OWR/2026/1; problem session, contribution of
  C. Schildkraut, pp. 72–73.
  - Read in the publisher's open-access PDF (805,949 bytes, sha256
    `dcaab409041b3d91a88d4c46d97612c2120feff8e29ce62ad9dcce163dd35b76`); pages 72 and 73 were rendered and looked at.
  - The problem is stated for n × … × n arrays without wraparound, with the directions {−1,0,1}^d ∖ {0}; the
    maximal number of occurrences is written as C_d(w) n^d − O(n^{d−1}).
  - Question 10 asks whether, for all d and w, C_d(w) is always attained in the limit by some periodic
    construction. Question 11 asks for C_d(ABB) for d > 2 and for C_2(ABBB); the text adds that the value 8/5 is
    believed for the latter and describes the five-queens construction.
  - The two words are ordinary text in the report (Question 11, the neighbouring sentences, the captions of
    Figures 2 and 3).
- **Companion paper.** Z. Halberstam, C. Schildkraut, "Words with Repeated Letters in a Grid", arXiv:2511.19678,
  version 2 of 26 November 2025 (the latest version on 2026-10-10). Used for the definitions (toroidal grids of
  arbitrary shape and alphabet; an appearance is a pair of a cell and a direction; concentration; C_d(w) as a
  supremum; letter distribution; total variation distance with the factor 1/2) and for Questions 2.5, 9.2, 9.3,
  9.5, 9.6. All numbers refer to version 2.
- **Corpus record.** ulamai/UnsolvedMath, OWR-14299905-036 (dataset version 1.6.0; status `open`). Its statement
  consists of Questions 10 and 11 of the report, followed by text which belongs to the preceding problem of the
  report.
- **Read again by verification run 2 (final text).** Pages 72–73 of the report in the same file, with both
  figures (Figure 3 of the report is the grid with A at j ≡ 2i + 2 (mod 5)); the TeX source of version 2 of the
  companion paper, which was also compiled to obtain its numbering. Every numbered citation of the note agrees
  with the source: Sections 2.1, 2.5, 5.3, 6.2 and 9, Propositions 1.3, 2.6, 3.1, 3.2, 5.2, Lemmas 2.2, 3.9, 5.4,
  5.6, Theorems 1.4(c) and 2.1, Questions 2.5, 9.2, 9.3, 9.5, 9.6, Figure 5, and the bound 59526/35459 of
  Appendix A.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Question 11, second part / Question 9.3: the value of C_2(ABBB), toroidal grids of every shape and alphabet | 8/5 | Theorem 1.1 |
| The same for n × n arrays without wraparound (normalisation of the report) | (8/5)(n−6)(n−10) ≤ M(n) ≤ (8/5) n² (lower bound for n ≥ 6, upper bound for all n) | Corollary 1.2(b) |
| Question 11, first part / Question 9.2: the value of C_d(ABB) for d > 2 | d = 3: 6; d = 4: between 18 and 62/3; d = 5: between 54 and 64; open for d ≥ 4 | Theorem 1.6, Corollary 1.8 |
| The same for n × n × n arrays | 6(n−4)²(n−6) ≤ M_3(n) ≤ 6n³ for n ≥ 6 | Corollary 1.7 |
| Question 10 / Question 2.5: is the supremum attained by a periodic grid, for all d and w | yes for (2, ABBB) and (3, ABB), with the complete list of the extremal grids; open in general | Theorems 1.3, 1.6(b) |
| Question 9.5: do all extremal grids have the same letter distribution | yes for (ABBB, 2): (1/5, 4/5); yes for (ABB, 3): (1/3, 2/3) | Theorems 1.3, 1.6(b), Corollary 1.5(a) |
| Question 9.6 as stated (total variation distance ≤ δ whenever the concentration is ≥ C_d(w)(1 − δ); δ quantified universally as in the one-dimensional Proposition 3.2 of the companion paper) | no for (ABBB, 2): the distribution would have to be (1/5, 4/5), and there are two-letter grids with δ → 0 and distance (112/67) δ | Corollary 1.5(b), Proposition 5.3 |
| Weaker forms of Question 9.6 for (ABBB, 2) | distance ≤ (14/5) δ for grids on two letters, optimal; distance ≤ (9002/15) δ for all alphabets, not optimised | Corollary 1.5(c), Theorem 1.4(c) |
| Counting each line once instead of pairs (cell, direction); only rows and columns; non-contiguous occurrences | not the question; not treated | Section 1.1 |

## Results in the paper
- **Lemmas 2.1, 2.2, 2.4, 2.5, Proposition 2.3.** Only the positions of the letter A matter; the number of
  (binary) appearances is the sum over all window positions of a weighted count Ψ_0 of the arrows of the window
  that read the word; potentials (telescoping differences of a function on two overlapping blocks; polynomials
  whose coefficients sum to zero on every congruence class of monomials) have sum zero over all window positions;
  hence a pattern-wise bound Ψ_0 + potential ≤ M gives C_d ≤ M, with the equality case.
- **Theorem 1.1, Corollary 1.2.** C_2(ABBB) = 8/5 from Proposition 3.1 (certificate with q = 20); every word that
  contains XYYY or YYYX has C_2 ≤ 8/5; the array version.
- **Theorem 1.3.** From Proposition 4.1 (certificate with exactly ten tight patterns, gap 3/70) and Lemma 4.2:
  the extremal grids are the grids with A on j ≡ s i + t (mod 5), s ∈ {2, 3}, on tori with both sides divisible
  by 5.
- **Theorem 1.4, Corollary 1.5.** c ≤ 8α, c ≤ (12 − 4α)/7, c ≤ (12/5)(1 − α) (Proposition 5.1); the closure of
  the set of pairs is the quadrilateral Q (Lemma 5.2); the family of Proposition 5.3 with δ → 0.
- **Theorem 1.6, Corollary 1.7.** C_3(ABB) = 6 from Proposition 6.1 (two certificates: polynomials of degree 4 in
  the 27 cell variables); the extremal grids are the grids with A on a·y ≡ c (mod 3), a ∈ {−1,0,1}³ ∖ {0}, with
  3 | n_i whenever a_i ≠ 0; gap 47137/294912.
- **Corollary 1.8.** By restriction to sublattices (Lemma 7.2) and explicit rational combinations.
- **Section 8** describes how the certificates were found (linear programs, rounding); every number there is
  labelled "computed, not certified" unless an exact counterpart is named.

## Computations (programs, certificates and outputs in reproducibility/)
- **Certificates** (`certificates/`): `cert_ABBB_q20.txt`, `cert_ABBB_q280.txt`, `cert_ABBB_density_q280.txt`,
  `cert_ABB_3D.json` (the note); `mycert_ABBB.json`, `mycert_ABBB_density.json`,
  `mycert_ABBB_density_piece3.json`, `mycert_ABB_3D.json` (verification run B). SHA-256 values are in
  `reproducibility/README.md`.
- **Checks of the certificates** (exact integer or rational arithmetic; all accepted):

  | Statement | Certificates | Programs that accepted it |
  |---|---|---|
  | (6): Ψ_0 + D_g + D_g^t ≤ 8/5 on 65,536 patterns | q = 20 (568 tight patterns); q = 280 (ten tight patterns, gap 3/70); run B, q = 2520 (ten tight patterns, gap 53/1260) | `check_gcert.py`, `check_gcert_numpy.py` (note); `vA_check2d.py` (run A); `verify2d.py` (run B); `check_window_cert.py` (writing stage); `r2_check2d.py` (run 2) |
  | (7): Ψ_0 + (4/7) n_A/16 + D_g + D_g^t ≤ 12/7 | q = 280 (206 tight patterns); run B, q = 2520 (206 tight patterns) | `check_density.py`, `check_gcert_numpy.py --density` (note); `vA_check2d.py`; `verify2d.py`; `check_window_cert.py`; `r2_check2d.py` |
  | (8): Ψ_0 + (12/5) n_A/16 + D_g + D_g^t ≤ 12/5 | run B, q = 2520 (65 tight patterns) | `vA_check2d.py`; `verify2d.py`; `check_window_cert.py`; `r2_check2d.py` |
  | Proposition 6.1 on 2^27 patterns: class sums zero; Ψ_0 + Q ≤ 6; exactly the 39 hyperplane windows tight | note (gap 1079/90720); run B (gap 47137/294912) | `check_abb3d.py` (note); `verify3d.py` (run B); `vA_3d_zeta.c` and `vA_3d_gray.c` (run A; equal checksums over all 2^27 values: `c6847e70920edc41` and `61ea377e23c930ef` for the two certificates); `r2_prep3d.py` with `r2_check3d.c` (run 2; no table of 2^27 numbers; its values agree with a direct evaluation in fractions at 1,694 patterns and with the exact sum over all patterns) |

- **Tests of the reductions.** The identity "sum over all window positions of (Ψ_0 + potential) = number of
  binary appearances" was tested with the actual certificates on random grids, including grids with sides
  shorter than the window and with a third letter: 400 + 10 grids (note), 240 + 22 grids (run A),
  112 + 7 grids (run B). Corrupted certificates, wrong weights, wrong bounds and wrong index conventions are
  rejected (rejection tests of the note and of both runs).
- **Exhaustive searches from the definition** (no linear program involved): all grids on two letters on all tori
  with at most 25 cells (note, run B), at most 30 cells and 2×16, 4×8, 5×7, 6×6 (run A) and at most 30 cells
  (run 2) in dimension 2; with at most 24 cells (note, run B), 3×3×3 (run B), at most 28 cells and 2×3×5, 2×4×4,
  2×2×8 (run A) and at most 28 cells (run 2) in dimension 3.
  No grid exceeds 8/5, respectively 6. The value 8/5 occurs only on 5×5, for exactly the ten lattice grids; the
  value 6 occurs exactly on the shapes with a side divisible by 3, for (3/2)(3^k − 1) grids. Run 2 also tested,
  for every grid of its enumeration (6,199,895,886 grids on 58 tori; 1,805,645,902 grids on 62 tori), the three
  bounds of Theorem 1.4(a) and the gap inequalities of Theorems 1.3(b) and 1.6(c): no violation. Simulated
  annealing on larger tori (all three runs) found nothing above the bounds.
- **Writing stage.** `strip_family.py`: the grids of Proposition 5.3 counted from the definition for 20 pairs
  (L, N). `density_region.py`: the vertices of Q, Lemma 5.2 on the sides of Q and on 200 pairs of random grids,
  Theorem 1.4(a), and the inequalities in the proof of Corollary 1.5(c) on all 111,954 three-letter grids of ten
  small shapes and on 700 further grids. `density_bounds_exhaust.py`: the three bounds of Theorem 1.4(a) for all
  grids on two letters on the 46 tori with at most 25 cells, from the definition.
- **Verification run 2** (`independent_run_2/`; standard library and C; written from the statements of the
  note). Besides the checks of the certificates and the enumeration named above: the grids of Theorems 1.1, 1.3,
  1.4, 1.6 and all 190 grids of Proposition 5.3 with N ≤ 400, counted from the definition; Lemma 4.2 and
  Lemma 6.2(b) by enumeration; Lemma 5.2 on 400 pairs of random grids (298 of them with a part of fewer than
  seven columns); the statements in the proof of Corollary 1.5(c), cell by cell, on 116,590 grids with three or
  four letters; the identity (5) with all eight certificates on 150 grids; Lemma 7.1, the inequalities (12) on
  170 grids, the combinations in the proof of Corollary 1.8 and the linear programs of Remark 7.3, solved exactly
  (62/3, 64, 596/3, 11/2); the example at the end of Section 8; the constructions of Corollaries 1.2(b) and 1.7.
  Every program ends with `RESULT PASS`. Its maxima agree with those recorded by run A on all 120 tori and with
  those recorded by run B (maxima and numbers of maximisers) on all 97 tori which run B enumerated.
- **Re-runs (10 October 2026).** `sh run_all.sh PYTHON --searches` was run from an extracted copy of the archive,
  one process at a time. All 72 steps ended without error; the checksums of the two C programs agree for both
  certificates of dimension 3; both lists of the 39 tight patterns agree with the recorded list; and all 73 output
  files agree with `expected_outputs/` (running times removed). The outputs also agree with those recorded
  earlier by the original computation and by the two verification runs (40 comparisons). The six largest
  enumerations of run A (2×16, 4×8, 5×7, 6×6, 2×4×4, 2×2×8), which are not part of the script, were run again,
  one process per shape (84 minutes in total), and gave the recorded maxima and numbers of maximisers. In the
  course of verification run 2 the script was run once more with all its parts from an extracted copy of the
  archive (72 steps without error; equal checksums; 73 output files identical to `expected_outputs/`), and then
  the script of run 2, `independent_run_2/run_run2.sh --searches` (23 steps; 21 output files identical to the
  recorded ones). Logs, running times and memory: `reproducibility/RERUN_LOG.txt` and
  `reproducibility/README.md`.

## Independent verification runs
The results were first obtained on 2026-10-10 with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed on the same day. Each wrote its own programs before reading those of the note.
Runs A and B examined the first written version of the results, and neither saw the report of the other: run A
examined the statements and the proofs; run B recomputed the results without the certificates of the note and
made the literature search. Run 2 (the second round of verification; in the note: the third verification run)
examined the final text of the note; it had the reports of runs A and B.

| Item | Run A | Run B | Run 2 (final text) |
|---|---|---|---|
| Statement, conventions, questions against the report and the companion paper | CONFIRMED_WITH_FIXES (two remarks of the first written version about the sources; the array normalisation for d = 3) | CONFIRMED_WITH_FIXES (the same first remark; the array normalisation for d = 3) | CONFIRMED (report pages and figures and the source of the companion paper read again; every numbered citation compared; one wording: "in this sense") |
| Lemmas 2.1, 2.2, 2.4, 2.5, Proposition 2.3 | CONFIRMED (re-derived) | Lemma 2.5 re-derived | CONFIRMED (re-derived; the identity (5) tested with all eight certificates on 150 grids, also with sides 1, 2, 3 and a third letter) |
| Theorem 1.1 (C_2(ABBB) = 8/5) | CONFIRMED (lemmas re-derived; both certificates accepted by its checker) | CONFIRMED (own linear program: 1.6000 on the 4 × 4 window, floating point; own exact certificate; certificates of the note accepted) | CONFIRMED (all three certificates accepted by its own checker: 568, 10, 10 patterns with equality) |
| Corollary 1.2 | CONFIRMED (ranges of n corrected) | CONFIRMED | CONFIRMED |
| Theorem 1.3 (extremal grids, gap 3/70) | CONFIRMED | CONFIRMED | CONFIRMED; the direct proof of Lemma 4.2(b) CONFIRMED_WITH_FIXES (one sentence added) |
| Theorem 1.4(a), bounds 8α and (12 − 4α)/7; the 4 × 4 grid | CONFIRMED | CONFIRMED (own certificate) | CONFIRMED (both certificates: 206 patterns with equality; the grid recounted) |
| Theorem 1.4(a), bound (12/5)(1 − α) | not its part | found and certified by run B | CONFIRMED (certificate accepted: 65 patterns with equality) |
| Theorem 1.4(b) (the closure is the quadrilateral) with Lemma 5.2 (mixing) | not in the first written version | not in the first written version | CONFIRMED (both inclusions; boundary terms of the lemma, also for parts with fewer than seven columns); notation of the lemma made unambiguous |
| Proposition 5.3 (strip family) and Theorem 1.4(c) | family supplied by run A, 20 grids counted | CONFIRMED (two-letter bound) | CONFIRMED for all admissible L, N (argument re-derived; all 190 grids with N ≤ 400 counted from the definition) |
| Corollary 1.5(a), (b) and the two-letter bound in (c) | CONFIRMED (the negative answer is an answer to the question as literally stated; family with δ → 0 supplied) | CONFIRMED | CONFIRMED (Question 9.6 read again in the source; quantifiers correct) |
| Corollary 1.5(c), bound (9002/15) δ for all alphabets | not in the first written version | not in the first written version | CONFIRMED (derivation repeated; the statements of the proof tested cell by cell on 116,590 grids) |
| Theorem 1.6 (C_3(ABB) = 6, extremal grids, gap) | CONFIRMED (Lemma 2.5 re-derived; two programs with equal checksums) | CONFIRMED_WITH_FIXES (own certificate with the larger gap; array normalisation required) | CONFIRMED (both certificates accepted by its own program: class sums zero, 39 patterns with equality, gaps 1079/90720 and 47137/294912; the direct proof of Lemma 6.2(b); the count (3/2)(3^k − 1)) |
| Corollary 1.7 | proved by run A (it was missing) | proved by run B (it was missing) | CONFIRMED |
| Corollary 1.8, Lemma 7.2, Remark 7.3 | CONFIRMED (explicit rational combinations, re-derived) | CONFIRMED (exact rational optima of the linear programs) | CONFIRMED (the present proof of Lemma 7.2; the combinations; the four linear programs solved exactly with primal and dual solutions) |
| Exhaustive and random searches | CONFIRMED | CONFIRMED (identical maxima in all 97 shapes) | CONFIRMED (58 + 62 tori; identical maxima on all common tori) |
| Novelty and credit | not its part | CONFIRMED_WITH_FIXES (no earlier proof found; credits and wording) | CONFIRMED_WITH_FIXES (searches repeated, no earlier proof found; two descriptions of earlier work corrected) |

No run found a wrong theorem, proposition or lemma, or a gap.

**Corrections required by runs A and B**, all applied in the note (details: audit file `fixes_applied.md`, not
part of this package; summary here):
1. (Runs A and B) The two words of Question 11 are ordinary text in the report and are present in the corpus
   record; the record is garbled only by appended text of the preceding problem. The note says exactly this
   (Section 1.1).
2. (Run A) A remark on counting lines without orientation was wrong as written; it was deleted.
3. (Runs A and B) The array normalisation for ABB in dimension 3 was added: Corollary 1.7.
4. (Run A, recommended) Question 9.6: the quantifier on δ is stated; what fails is the constant 1; a family with
   δ → 0 is given: Corollary 1.5, Proposition 5.3.
5. (Run A) Corollary 1.2(b): the lower bound is stated for n ≥ 6 and the upper bound for all n.
6. (Run B) The literature paragraph says that Semantic Scholar lists no later work citing the three papers and
   that the citation data of OpenAlex are incomplete for them.
7. (Run B) All values without an exact certificate are labelled "computed, not certified" (Section 8); several
   exploratory values of the first written version were left out.
8. (Run B, optional) Theorem 1.6(c) uses the larger gap of the second certificate; Theorem 1.4(a) contains the
   third bound; the answer to Question 9.5 for (ABB, 3) is stated.
9. (Run B) Credits as in "Scope and priority".

**Added when the note was written, after runs A and B** (all of it examined by run 2, see below):
- Theorem 1.4(b) (the closure of the set of pairs is Q) with Lemma 5.2 (mixing), which generalises the locality
  argument of run A; tested by `writing_stage/density_region.py`.
- Proposition 5.3 for all multiples L, N of 20 with 20 ≤ L ≤ N − 20 (run A had counted 20 cases and given the
  locality argument); recounted by `writing_stage/strip_family.py`.
- The bound for arbitrary alphabets in Corollary 1.5(c) (constant 9002/15) with its proof; tested by
  `writing_stage/density_region.py`.
- Direct proofs of Lemma 4.2(b) and Lemma 6.2(b), which the runs had verified with programs; the proof of
  Lemma 7.2 in its present form.
- The two certificates of run B that are used in Theorems 1.4(a) and 1.6(c) were checked again with the programs
  of the note and of run A (table above).

**Run 2: what it examined, and the result.**
- The additions first. Lemma 5.2: the appearances starting in a column depend on seven columns, so at most six
  columns of each part are affected by the two interfaces (all of them, if the part has fewer than seven); the
  error is at most 12 · 8m, hence 96/N; the admissible ratios L/N are all rationals of (0, 1); convexity of the
  closure follows. Theorem 1.4(b): the four half-planes cut out exactly the quadrilateral, and its vertices are
  pairs of grids. Proposition 5.3: the locality argument holds for all admissible L, N, and one count fixes the
  constant 54. Corollary 1.5(c): every cell with a third letter lies in a window that is not a lattice window or
  on a binary appearance that is not an appearance; 16 · 70/3 = 1120/3, (1120/3)(8/5) = 1792/3,
  14/5 + 1792/3 = 9002/15. Lemma 4.2(b), Lemma 6.2(b), Lemma 7.2. Result: correct.
- Then all other proofs, line by line (Lemmas 2.1 to 2.5 and Proposition 2.3, also for tori with sides shorter
  than the window and for larger alphabets; Theorems 1.1, 1.3, 1.4, 1.6 with the gluing arguments, the
  divisibility conditions and the letters other than A; Corollaries 1.2, 1.5, 1.7, 1.8). Result: correct.
- The sources (see "Statement checked") and the literature (see below).
- Its programs reproduce every finite statement of the note (see "Computations"), and the package's own script
  reproduces all recorded outputs.
- It required twelve corrections of the text and three of the package; all were made, and none changes the
  content of a statement:
  1. the Verification paragraph of the note describes the final state;
  2. Section 1.2: Alon and Kravitz proved d-stackability for every word with distinct letters (the note had
     said "many"); their theorem is cited as restated in Theorem 2.1 of the companion paper;
  3. Section 1.4: the weights in Proposition 5.2 of the companion paper are real numbers; they are non-negative
     in the linear program of its Section 5.3 (the note had called them non-negative throughout);
  4. the value 1.6000 of run B's linear program is labelled "computed, not certified" in the Verification
     paragraph;
  5. proof of Proposition 5.3: the counts of run A are quoted precisely (20 grids), and the 190 counts of run 2
     are added;
  6. Table 1, caption: "are determined in this note" instead of "are new";
  7. Lemma 5.2: the concentrations of the two grids are written c^(1), c^(2) (c_2 clashed with c_2(ABBB, ·)), and
     the periodic extension is described precisely;
  8. proof of Lemma 4.2(b): every lattice window has at least three entries 1 (needed for "the ten windows are
     distinct");
  9. after the proof of Theorem 1.4(a): "another proof" instead of "a third proof";
  10. Section 1.1: "In this sense Question 10 of the report is Question 2.5";
  11. Sections 1.4 and 9: the programs of run 2, the numbers of programs per certificate, the ranges of the
      enumerations, the running times and the re-runs;
  12. "Scope and priority": the searches repeated by run 2;
  13. this report;
  14. the folder `reproducibility/independent_run_2/`, and the files `README.md` and `RERUN_LOG.txt` of the
      package;
  15. the archive and the deposit metadata (hashes).

## Relation to the literature, novelty and scope
- **Searches (10 October 2026).** arXiv: the versions of 2511.19678 (v1 of 24 November 2025, v2 of 26 November
  2025, no later version), the listings of both authors, and queries for the title words, for word searches in
  grids, queens, stackable words, ABBB, Patchell–Spiro and Alon–Kravitz; OpenAlex and Semantic Scholar (citing
  works); Crossref; zbMATH Open; DataCite and Zenodo (queries for the constants, the title, the authors and the
  record identifier); the publisher's page of the report; six web searches. Run 2 repeated the queries to these
  databases later on the same day (arXiv: the versions, a request for a version 3, which does not exist, and the
  listings of both authors, the newest entry being of 14 September 2026) and made one further web search.
  - No proof of C_2(ABBB) = 8/5 or of C_3(ABB) = 6, no classification of the extremal grids and no later version
    of the companion paper announcing one were found.
  - Semantic Scholar lists no later work citing the companion paper, Alon–Kravitz or Patchell–Spiro. OpenAlex
    lists none either, but its citation data are incomplete for these papers.
  - MathSciNet and Google Scholar were not consulted.
- **What was read.** Pages 72–73 of the report (with the figures); of the companion paper, version 2: Sections 1,
  2, 5.1–5.4, 6.1 and 9, the passages of Section 6.2 on ABBBB and BABBB, the statements of Propositions 3.1, 3.2,
  8.1 and of Lemma 3.9, and the passage of Appendix A with the bound 59526/35459. Its proofs were not checked,
  except that of Lemma 2.2, the only result of the companion paper that is used in the proofs of the note.
  Patchell–Spiro and Alon–Kravitz were not read; what the note says about them is taken from the companion paper
  and from the report. Of Stolee (2014) and Huang et al. (2016) only the abstracts were read; Razborov (2007) is
  cited as in Section 2.5 of the companion paper. All bibliographic data were checked with Crossref or the arXiv
  API, and once more by run 2 (eight DOIs, four arXiv identifiers). Run 2 read Sections 1, 2 and 9 of the
  companion paper in the TeX source, the proof of its Lemma 2.2, and the other cited statements.
- **Credits.** The problem goes back to Patchell and Spiro and to Alon and Kravitz. The setting, the bounds
  3^{d−1} C_1 ≤ C_d ≤ ((3^d − 1)/2) C_1 with the notion of stackability, the five-queens grid with the lower bound
  8/5 and the conjectured value, C_2(ABB) = 2, the bound for ABB on grids of shape (Z/3)^d, the local
  linear-programming method with the bound 59526/35459, and Questions 2.5, 9.2, 9.3, 9.5, 9.6 are due to
  Halberstam and Schildkraut; Questions 10 and 11 are from Schildkraut's contribution to the Oberwolfach problem
  session.
- **Caveats.** A search that finds nothing is not a proof of novelty. The sources are recent (2025–2026) and their
  authors are active; the note was written without knowledge of unpublished progress of the authors of the
  problem. No priority is claimed.
- **Scope.** The note determines C_2(ABBB) and C_3(ABB) and classifies the extremal grids in these two cases,
  which settles Question 10 of the report (Question 2.5 of the companion paper) for these two pairs only.
  Question 10 in general and C_d(ABB) for d ≥ 4 stay open (only bounds for d = 4, 5).

## 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/