# Verification report — OWR-12872-018 (Bousquet-Mélou, corners in square lattice loops: the growth rate of corner-weighted quadrant loops)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete proof of a scoped case**; every statement it makes is proved, and the
scope is as follows.
- **Settled by the note.** For every real a ≥ 0 the limits of Q_n(a)^{1/n} and L_n(a)^{1/n} exist and both
  equal G(a) = (2 + √(2+2a))²; hence ρ_Q(a) = ρ_L(a) = (2 + √(2+2a))^{-2}, i.e. Conjecture 11 of Albert and
  Bousquet-Mélou and the second question of Bousquet-Mélou's problem-session contribution hold on [0, ∞);
  moreover Q_n(a) ≤ G(a)^n for all n (Theorem 1.1), and G(a)^n e^{-C_a √(n log n)} ≤ Q_n(a) (Proposition 3.1).
- **Known before.** a = −1 and a = 1 (Albert and Bousquet-Mélou, Propositions 15 and 17); for a ≥ 0 the
  one-sided bound ρ_Q(a) ≥ (2 + √(2+2a))^{-2} (Proposition 17); the radius of unconfined and half-plane loops
  for all a ≥ −1 (Propositions 21, 22). The note re-proves the unconfined radius only for a ≥ 0.
- **Open.** −1 < a < 0. This range contains the values a = −S•(t_c) ∈ [−1/3, 0) on which the conjectural
  growth rate of permutations sortable by two stacks in parallel depends (Theorem 23 and Corollary 24 of
  Albert and Bousquet-Mélou), so the note does not make that growth rate unconditional. The numerical data
  for the open range are evidence only.
- **Novelty.** No proof of the case a ≥ 0 (other than a = 1) was found in the sources searched; for a ≥ 1
  the result seems to follow after a short argument from known results on weighted walks in cones, and a
  Markov-additive theorem of Ozawa nearly covers a ≥ 0 (Remark 4.1). No priority and no new method are
  claimed.

The note is unrefereed.

## Statement checked
- **Problem source.** M. Bousquet-Mélou, "Corners in square lattice loops", problem-session contribution,
  in: Enumerative Combinatorics, Oberwolfach Report 12/2014, Oberwolfach Rep. 11 (2014) 635–720,
  doi:10.4171/OWR/2014/12, p. 711. Read in a local text copy of the report. Definitions: square lattice
  loops with N, S, E, W steps, confined to the first (nonnegative) quadrant; Q_n(a) counts quadrant loops of
  length 2n by the number of NW and ES factors (example NEENSESWWW: one factor ES). The second conjecture: the
  exponential growth of the numbers Q_n(a), a ≥ −1, equivalently the radius of Q(a,u) = Σ u^n Q_n(a), is the
  same as for unconfined loops; the exact value is in arXiv:1312.4487.
- **Conjecture.** M. Albert, M. Bousquet-Mélou, "Permutations sortable by two stacks in parallel and quarter
  plane walks", Europ. J. Combin. 43 (2015) 131–164, doi:10.1016/j.ejc.2014.08.024, arXiv:1312.4487 (read in
  the arXiv source, version v3, whose shared theorem counter the note uses): Conjecture 11 gives
  ρ_Q(a) = (2 + √(2+2a))^{-2} for a ≥ −1/2 and −a/(2(a−1)²) for −1 ≤ a ≤ −1/2. Functional equation:
  Theorem 7, Eq. (1). (A reference to "Theorem 4" in the preparatory material was corrected to Theorem 7 by
  counting the environments of the v3 source.)
- **Corpus record.** ulamai/UnsolvedMath, OWR-12872-018 (snapshot d371ee9, line 11332; status `open`): "For
  every a ≥ −1, is the exponential growth rate of Q_n(a) equal to that of the corresponding unconfined
  loops?" This is the second question of the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Growth rate for a ≥ 0 (all weights nonnegative; limsup Q_n^{1/n} vs limsup L_n^{1/n}) | equal, both limits exist and equal (2+√(2+2a))² | Theorem 1.1 |
| a = −1 | equal (8), known | [AB, Props. 15, 17] |
| −1 < a < 0 (signed sums; growth = limsup |Q_n(a)|^{1/n}) | open; numerical evidence only | Section 5 |
| Other corner pairs (WN, ES), (WN, SE), (NW, SE) | same Q_n(a), L_n(a) by the Bernardi–Courtiel involution [AB, Prop. 14]; Theorem 1.1 holds for them | Section 1.1 |
| (a+1)-positivity [AB, Conj. 10], the derivative [AB, Conj. 12], exact asymptotics | not treated | — |

## Results in the paper
- **Lemma 2.1 (all walks).** W_{m+1} = S W_m + (a−1) T W_{m−1} (every real a); for a ≥ 0 the roots are real,
  0 ≤ W_m ≤ (m+1) Λ^m, and W_m(1,1) ≥ c_a λ(a)^m with c_a = 1 (a ≤ 1) and 2/√(2+2a) (a > 1).
- **Lemma 2.2** reversal–negation keeps c. **Lemma 2.3** L_{m+1} ≥ W_m(1,1)²/(m+1)² (insert NS, Cauchy–Schwarz).
  **Lemma 2.4** loops reaching x = −k−1 or y = −k−1 weigh at most max(1,a)(2n−1)(n+1)² e^{−2δ(k+1)} Λ_δ^{2n}
  (cut at the first passage, exponential tilt). **Lemma 2.5** the corner-free padding P = N^k E^{k+1} W,
  F = E W^{k+1} S^k. **Lemma 2.6** Q_{m+n} ≥ Q_m Q_n, Q_n ≥ 1.
- **Theorem 1.1** (a ≥ 0), proof in Section 2.1 (Fekete's lemma; k = ⌈εn⌉, δ = min(1, ελ/(2K))).
- **Proposition 3.1** G(a)^n e^{−C_a √(n log n)} ≤ Q_n(a) ≤ G(a)^n (k ≈ C√(n log n), δ = kλ/(2nK)).
- **Remark 3.2** the three essential uses of a ≥ 0 (and its use, for convenience only, in two elementary estimates). **Remark 4.1** close routes (six-step model and
  Denisov–Wachtel for a ≥ 1; Ozawa; Denisov–Zhang; Padhi). **Section 5** why the method cannot work for
  −1 ≤ a < −1/2 (the conjectured growth exceeds λ(a)² = lim W_{2n}(1,1)^{1/n}), Table 1, shuffle classes.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program written with the note** (`check_note.py`, standard library, deterministic, about 3 s; TOTAL
  failures: 0): parts A–J check Lemma 2.1 (recurrence on all words of length ≤ 7 at six exact points, exact
  lower bounds in Q(√2), Q(√3) for m ≤ 60), Lemma 2.2 and the diagonal symmetry, Lemma 2.3, Lemma 2.4 (exact
  B^x_n(k) = B^y_n(k) against the bound, n ≤ 9, k ≤ 3), Lemma 2.5 (9,651 loops), the polynomials Q_n (n ≤ 30)
  and L_n (n ≤ 12) with the closed forms at a = ±1, Lemma 2.6, Q_n ≤ G^n exactly, the recorded data
  (n ≤ 200), the δ-inequalities of Theorem 1.1 and Proposition 3.1, the constants and the identity of
  Section 5, the six-step identity (n ≤ 10), and the shuffle-class sums (the two negative values exactly).
- **Programs with which the result was first obtained** (`original/`): integer transfer matrix for
  Q_n(p/q), n ≤ 200, at a ∈ {−1, −3/4, −1/2, −1/4, 0, 1, 2}; ratio analysis (Table 1); brute-force lemma
  checks; shuffle-class polynomials; two exploratory random searches whose seeds were not recorded (their
  two examples are recomputed deterministically by `check_note.py`).
- **Programs of the verification runs** (`verification_run_A/`, `verification_run_B/`; and
  `independent_run_2/` for the further run on the text of the note, see below).
- **Re-run** (2026-10-20, `RERUN_LOG.txt`): every recorded output of the three groups of programs was
  reproduced byte for byte. The further verification run repeated this re-run from an extracted copy of the
  source archive: all outputs identical (for `check_note.py` up to the Python version line and the timings).

## Independent verification runs
Two independent verification runs, both AI-assisted, each with its own programs, examined the result
before the note was written; the note was then written from the result with the corrections they required.

| Item | Run A | Run B |
|---|---|---|
| Statement, definitions, corner pair, half-length convention, [AB] numbering (v3 source) | CONFIRMED | CONFIRMED |
| Lemma 2.1 (recurrence, discriminant, bounds, c_a) | CONFIRMED (re-derived; checked on all words m ≤ 7) | CONFIRMED (re-derived; symbolic check m ≤ 7, mutant detected) |
| Lemmas 2.2, 2.3 | CONFIRMED | CONFIRMED |
| Lemma 2.4 | CONFIRMED_WITH_FIXES (wording of the y-case) | CONFIRMED (suggestion: bound over all cut points) |
| Lemmas 2.5, 2.6 | CONFIRMED | CONFIRMED (mutants detected) |
| Theorem 1.1 (choice of δ, Fekete, limit) | CONFIRMED | CONFIRMED |
| Remark (2) of the result, now Proposition 3.1 | plausible, proof to be written | plausible, proof to be written |
| Exact data n ≤ 200 | own transfer matrix agrees for n ≤ 36 | all 7 × 200 values recomputed, identical |
| Open range −1 < a < 0 | not claimed; correct | evidence and both negative class sums reproduced |
| Prior work / novelty | not stale; add Denisov–Zhang, Padhi | not stale; close routes (six-step model + Denisov–Wachtel for a ≥ 1; Ozawa) to be cited |
| Bibliography | — | checked with Crossref; Kuszmaul is Europ. J. Combin. 60 (2017) 100–113 |

Corrections made in the note: the y-case of Lemma 2.4 now says that it uses only the symmetry of S and T under x ↔ y; the requested
parenthetical "the corner statistic itself is not symmetric" was not adopted, because the reflection in the
diagonal (N ↔ E, S ↔ W) maps NW to ES and ES to NW, so it keeps c (checked by `check_note.py`, parts B2 and
D2); the note mentions this symmetry as not needed. Lemma 2.4 now says that the bound holds for any cut
point. Proposition 3.1 is proved. Kuszmaul is cited "according to its abstract". Denisov–Zhang, Padhi, Ozawa,
Denisov–Wachtel, Garbit–Raschel and Duraj are cited in Remark 4.1. The [AB] number of the functional
equation is Theorem 7, Eq. (1).

### Further verification run on the text of the note (AI-assisted, 2026-10-20)
A further independent verification run, also AI-assisted, then checked the written note line by line (the
two runs above had examined the result before the note was written). Programs and outputs:
`reproducibility/independent_run_2/`.

| Item | Verdict |
|---|---|
| Statement against the problem text (p. 711) and the [AB] v3 source; shared theorem counter recounted (Theorem 7 = functional equation, Eq. (1); Conjectures 10–12; Propositions 14, 15, 17, 19–22; Theorem 23; Corollary 24; Sections 4, 5.1, 6.3) | CONFIRMED; every number cited from [AB] is right |
| Lemmas 2.1–2.6, Theorem 1.1, Proposition 3.1 (re-derived line by line: discriminant ≥ 8, roots 2 ± √(2+2a), c_a (a = 7: λ = 6, r = 1/3, c_a = 1/2), endpoint count (m+1)², the tilt bound, K (a = 0: 1.8392; a = 1: 2), δ, M ≤ (1+2ε)n+4, the √(n log n) choice with C²λ/(2K) = 6, the gap filling) | CONFIRMED, no gap |
| Own computations: brute force over all words (n ≤ 5) against own polynomials Q_n (n ≤ 20) and L_n (n ≤ 12); Q_1…Q_5 as printed; closed forms at a = ±1; Q_n(a) ≤ G(a)^n exactly for n ≤ 20 at a = 0, 1/2, 1, 2, 7; Lemma 2.4 exactly (B^x = B^y) for n ≤ 12, k ≤ 3; six-step identity n ≤ 10; both shuffle-class sums; all 7 × 200 recorded values and Table 1 with its own transfer matrix | all identical / all pass |
| Re-run of the archive from an extracted copy | identical outputs |
| Literature (arXiv, Semantic Scholar, OpenAlex, Crossref, zbMATH, one web search) and all DOIs (Crossref) | nothing new; no printed proof found; DOIs correct |

Its corrections, all made: Remark 3.2 now speaks of three essential uses of a ≥ 0 and mentions the
elementary estimates (discriminant, f and f′) where b ≥ −1 is also used; Section 5 now says that (P₁) is
listed in [AB] among properties that may hold, and that the second negative shuffle-class example is a pair of
bilateral Dyck paths whose vertical path is not a Dyck path (its class contains no quadrant loop); the
attributions to Elvey Price–Guttmann were made precise (Raschel's exponent formula is reported there as a
conjecture; "not rigorous" is the wording of [AB] for the radius prediction); "Scope and priority" now names
the close routes of Remark 4.1. No mathematical error was found.

## Relation to the literature, novelty and scope
- **Searches (20 October 2026).** The Semantic Scholar list of 22 works citing [AB] (titles; abstracts where
  relevant); zbMATH (2 citing works); the OpenAlex citing list (16 works; retrieved once, later requests got
  HTTP 429); arXiv and Crossref queries (two stacks in parallel; quarter-plane or quadrant loops; corners and
  quadrant walks; Markov-additive and Markov-modulated walks in cones); a few web searches. No printed answer
  to the question was found; [AB], Elvey Price–Guttmann (2017) and Elvey Price (2019, who writes that he
  could not prove the three conjectures of [AB]) treat the case as open.
- **Close prior work (none applies directly).** (a) For a ≥ 1, a^c = Σ over subsets of corners of
  (a−1)^{|subset|}; this reads [AB, Theorem 7, Eq. (1)] as the kernel equation of a six-step model with
  nonnegative weights, and Denisov–Wachtel (Ann. Probab. 43 (2015)) should give the case a ≥ 1 after a short
  argument (a sketch, not in print as far as found, not checked in detail); i.i.d. cone results
  (Denisov–Wachtel, Garbit–Raschel, Duraj) do not cover 0 ≤ a < 1 directly (negative diagonal weight).
  (b) Ozawa, Queueing Syst. 97 (2021), Section 2: the convergence parameter of irreducible nonnegative block
  tridiagonal matrices; its irreducibility condition (a5) fails for the natural encoding of quadrant walks.
  (c) Denisov–Zhang, J. Theoret. Probab. 38 (2025) (Markov chains on R^d; the position process here is not
  Markov) and Padhi, arXiv:2608.11290 (2026, unrefereed; run-modulated tandem walks with three fixed steps).
- **What was read.** The problem text (p. 711); [AB], Elvey Price–Guttmann and Elvey Price 2019 in their arXiv
  sources; Kuszmaul (abstract only; its scope, Dyck or bilateral Dyck paths, is not visible from the
  abstract); Ozawa (statement of the lemma and its conditions); Padhi (statement of the excursion theorem and
  its surroundings); Denisov–Zhang (abstract). Not read in full: Elvey Price's PhD thesis (2018, predates his
  2019 statement), Trotignon's PhD thesis (2019), Kenyon–Miller–Sheffield–Wilson (Phys. Rev. E 95 (2017)
  052146 and Ann. Probab. 47 (2019); abstracts; the arXiv source of the 2017 paper was read by run B: Peano
  curves of six-vertex configurations, no lattice walks with corner weights).
- **Caveats.** A search that finds nothing is not a proof of novelty. The tools are standard (Fekete's
  lemma, exponential tilting, Cauchy–Schwarz, concatenation and padding of walks); no method claim.
- **Scope.** a ∈ [0, ∞) proved; a = −1 known; −1 < a < 0 open, including [−1/3, 0).

## 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/