# Verification report — AMR-126-0025 (Benedetto–Ingram–Jones–Manes–Silverman–Tucker: piecewise linear recursions for deg(fⁿ))

Verification dates: 2026-10-17 (two runs on the first write-up) and 2026-10-18 (one run on the note as written).

**Verdict.** The note gives a **complete negative answer** to the question in the reading stated below; it rests on
one published theorem (Bell–Diller–Jonsson, Acta Math. 225 (2020)) and on standard Diophantine tools.
- **Reading.** Finitely many fixed sequences L^(1), …, L^(r), each satisfying a linear recursion with constant
  coefficients, with coefficients and initial values in any field of characteristic 0, and an arbitrary index function
  ι: N → {1, …, r}, with deg(fⁿ) = L^(ι(n))_n for all n ≥ 1. The different "switched recursion" reading, in which
  the recursions are applied to the degree sequence itself, deg(fⁿ) = Σ_j c_j^(ι(n)) deg(f^(n−j)) with coefficients
  changing with n, is **not addressed**.
- **Settled.** Theorem 1.1: if a positive-integer sequence (a_n) with lim a_n^(1/n) = δ agrees on the fibres of an
  index function with finitely many such sequences, then δ = |θ| for an algebraic number θ. Theorem 1.2: the
  dominant rational map f = g ∘ h_{1+2i} of P² of Bell, Diller and Jonsson (explicit forms over Z, degree 10) has
  transcendental dynamical degree (their Main Theorem), so its degree sequence has no such description; the answer
  to the question is **no**. Corollary 1.3: the same for every N ≥ 3 by the birational maps of Bell, Diller, Jonsson
  and Krieger (Proc. London Math. Soc. 128 (2024), Theorem 1.1); for N = 1 the answer is trivially yes.
  Proposition 1.4: degree sequences of monomial maps are piecewise linear recurrent (maximum of at most
  (N+1)^(N+1) integer recurrence sequences), which explains the remark of the source on the Hasselblatt–Propp
  example.
- **Credit.** The question is due to Benedetto, Ingram, Jones, Manes, Silverman and Tucker (2019). The
  transcendental dynamical degree is due to Bell, Diller and Jonsson (2020) and, for N ≥ 3, to Bell, Diller,
  Jonsson and Krieger (2024). The growth lemma rests on the subspace theorem through Evertse's inequality for sums of
  S-units (Evertse 1984; van der Poorten–Schlickewei 1982), used in the form stated by Bell, Diller and Jonsson,
  Theorem 3.5 (= Bell–Diller–Jonsson–Krieger Theorem 4.1 with T = {v_0} = Evertse–Győry Proposition 6.2.1).
  Earlier results exclude a single linear recursion for particular maps (Hasselblatt–Propp 2007, Bedford–Kim 2008,
  Lin–Wulcan 2014; Nguyen, Math. Ann. 396 (2026), Corollary B, for the generating series of the Bell–Diller–Jonsson
  maps).
- **What the note adds.** The short deduction that piecewise linear recurrence (with an index function) forces an
  algebraic growth rate (Theorem 1.1), which with Bell–Diller–Jonsson answers the question negatively. The novelty is
  modest; specialists may regard it as known; no priority is claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** R. Benedetto, P. Ingram, R. Jones, M. Manes, J. H. Silverman, T. J. Tucker, "Current trends
  and open problems in arithmetic dynamics", Bull. Amer. Math. Soc. (N.S.) 56 (2019), no. 4, 611–685,
  doi:10.1090/bull/1665; arXiv:1806.04980 (v1 13 June 2018, v2 2 July 2018; the arXiv listing also names
  L. DeMarco, whose author line is commented out in the TeX of v2).
  - Read: Section 14 "Dynamical Degrees of Rational Maps" in the TeX sources of v1 and v2 (fetched 2026-10-17).
    Theorem-like environments share one counter per section. In both versions: Example 14.1 (the Fibonacci degree
    map), Definition 14.2 (dynamical degree), Remark 14.3 (morphisms, algebraic stability), Theorem 14.4
    (Hasselblatt–Propp 2007, Bedford–Kim 2008), Example 14.5 (the map [YW, ZW, X², XW] of P³, degrees
    2, 3, 4, 6, 9, 12, 17, 25, …), **Question 14.6**. In v2 the question is followed by Remark 14.7 (Diller–Favre,
    Favre–Jonsson) and Conjecture 14.8 (the integrality conjecture of Bellon–Viallet); v1 has no such remark and
    numbers the conjecture 14.7.
  - The question (paraphrased): for a dominant rational map f: P^N ⇢ P^N, are there finitely many linear
    recursions with constant coefficients L^(1)_n, …, L^(r)_n and an index function ι: N → {1, …, r} such that
    deg(fⁿ) = L^(ι_n)_n for every n ≥ 1? The source states no expected answer. Conventions (first paragraph of Section 14): f is given by forms of
    the same degree with common factors removed; deg f is their common degree; dominant means no non-trivial
    polynomial relation among the forms; δ_f = lim deg(fⁿ)^(1/n).
  - Published item number: not verified (the AMS page could not be accessed).
    It is most likely Question 13.6; the note says so and does not assert it.
- **Corpus record.** ulamai/UnsolvedMath v1.8.0 (snapshot d371ee9), record AMR-126-0025, status `open`; title
  "Open Problems in Arithmetic Dynamics — Question 25" (running count of the extraction), source arXiv:1806.04980.
  Its statement is the TeX of Question 14.6.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Finitely many fixed LRS with constant coefficients and initial values in a field of characteristic 0 (Z, Q, Q̄, R, C, …), arbitrary index function, equality for all n ≥ 1 | **no** (one explicit map of P² suffices; every N ≥ 2) | Theorems 1.1, 1.2, Corollary 1.3 |
| Same, index function restricted (periodic, eventually periodic, computable) | no (a fortiori) | Theorem 1.1 uses only one infinite fibre |
| Same, recursions with a constant term, or valid only for n ≥ n′, or with vanishing last coefficients | no | Remark 4.1(2), Lemma 2.2 |
| Recursions over a field of characteristic p, equality in that field | trivial (yes: p constant sequences) | Section 1.1 |
| Switched recursions: deg(fⁿ) = Σ_j c_j^(ι(n)) deg(f^(n−j)) with finitely many coefficient vectors | **not addressed**; a transcendental growth rate is no obstruction there (a_n = 2^(n−⌊nρ⌋)3^⌊nρ⌋, ρ algebraic irrational, has lim a_n^(1/n) = 2(3/2)^ρ, transcendental by Gelfond–Schneider) | Section 1.1 |
| N = 1 | yes (deg fⁿ = dⁿ) | Corollary 1.3 |
| deg as degree of coprime forms vs. (f^(n*)H · H^(N−1)) | the same number | Sections 1.1, 5 |

## Results in the paper
- **Lemma 2.1 (specialization).** An LRS over a field F of characteristic 0 that takes integer values on a set S can
  be replaced by an LRS with algebraic coefficients and values that agrees with it on S: map the finitely generated
  Q-algebra A = Q[c, L_1..L_k] to Q̄ through a maximal ideal (Zariski's lemma, Atiyah–Macdonald Prop. 7.9). No
  Noetherian argument: the relations L_n = a_n are equalities in A and are preserved by the homomorphism.
- **Lemma 2.2 (structure).** Shift away vanishing last coefficients; exponential-polynomial representation
  (Evertse's lecture notes, Theorem 8.17); algebraicity of the polynomial coefficients by uniqueness.
- **Lemma 2.3 (residue classes).** Passing to q-th powers (q = lcm of the orders of the root-of-unity quotients)
  splits the sequence into finitely many non-degenerate power sums with algebraic data.
- **Theorem 3.1 (Evertse's inequality)**, quoted from Bell–Diller–Jonsson Theorem 3.5; **Remark 3.2** (provenance:
  Evertse 1984, Theorem 2, read in the rendered scan, is stated for integral vectors, every non-empty T ⊆ S and
  with the factor ‖x‖^(−ε), ‖x‖ the maximum of the conjugates of the coordinates; Theorem 3.1 follows from it by
  multiplying x with a suitable S-unit, which exists by Dirichlet's S-unit theorem, since both sides of Theorem 3.1
  are invariant under such multiplication). **Theorem 3.3** (Skolem–Mahler–Lech, Lech 1953, in the form of Evertse's
  notes, Theorem 8.18).
- **Lemma 3.4 (growth).** For a non-degenerate power sum v_t = Σ h_l(t)β_l^t with algebraic data and
  M = max|β_l|: |v_t| ≥ Mᵗ e^(−εt) for all large t (Skolem–Mahler–Lech excludes vanishing subsums for large t;
  Theorem 3.1 at the place of the inclusion K ⊂ C; product formula for the S-units β_l; polynomial bounds for
  H_S and Π_S Π_l |x_l|_v). For two conjugate roots and constant coefficients this is Bell–Diller–Jonsson–Krieger,
  Corollary 4.2.
- **Theorem 1.1** (Section 4): pigeonhole an infinite fibre; specialize; structure; residue class with infinitely
  many fibre indices; compare |v_t|^(1/t) → |θ_i|^q with a_{qt+s}^(1/t) → δ^q along the fibre; δ = |θ_i|,
  δ² = θ_i θ̄_i ∈ Q̄. Remark 4.1: δ need not be an algebraic integer by this argument; inhomogeneous and eventually
  valid recursions are covered.
- **Lemma 5.1, Lemma 5.2, Theorem 1.2** (Section 5): h_{1+2i} = [x0³x1² : x1³x2² : x0⁴x2]; f = g ∘ h by the forms
  (3), coprime (a common irreducible factor would divide P0 + P1 = 2u2) and of degree 10; g ∘ g = m·id; h dominant
  (det 5); (1+2i)ⁿ ∉ R since (−3+4i)/5 is not an algebraic integer; Bell–Diller–Jonsson: λ(f) transcendental; their
  degree is the source's degree; Theorem 1.1 gives the contradiction. The two conventions of Bell–Diller–Jonsson for
  h_ζ (introduction (y1^a y2^b, y1^(−b) y2^a) versus §1.3 (y1^a y2^(−b), y1^b y2^a)) are conjugate by the coordinate
  swap, which commutes with g.
- **Corollary 1.3**: N = 2 (Theorem 1.2), N ≥ 3 (Bell–Diller–Jonsson–Krieger Theorem 1.1; coefficients in
  {−1, 0, 1}, so over Q; their degree is the degree of the preimage of a general hyperplane = the source's degree),
  N = 1 (morphisms).
- **Proposition 1.4** (Section 6): deg h_B = Σ_i max_j(−e_{j,i}(B)); Cayley–Hamilton; sum of maxima = maximum over
  choice functions. The Hasselblatt–Propp map is h_A with A = [[−1,1,0],[−1,0,−1],[−1,0,0]]; its growth rate is the
  modulus 1.3562… of the complex roots of X³ + X² + X − 1, in accordance with Theorem 1.1.

## Computations (sanity and exactness checks; program and output in reproducibility/)
No theorem, lemma, proposition or corollary depends on a computation.
- **Program of the note** (`check_note.py`, Python 3 standard library, about 22 seconds, 38 checks, 0 failed):
  Part A, Lemma 5.2; Part B, the forms of Lemma 5.1 and (3), the coprimality identities, g ∘ g = m·id, the coordinate
  swap, a non-zero Jacobian (dominance sanity); Part C, formula (6) against direct iteration (Hasselblatt–Propp map
  n ≤ 40, 1,200 random cases), the piecewise representation of Proposition 1.4 for n ≤ 60, agreement with
  Bell–Diller–Jonsson's ψ; Part D, a certified enclosure λ ∈ [6.857557409279008598, 6.857557409279008599]
  (ζ = 1+2i) and λ = 13.449607681785413449… (ζ = −3+4i), matching the values printed by Bell–Diller–Jonsson, and
  e_1..e_10 from their Proposition 2.7; Part E, deg fⁿ = 10, 66, 454, 3114 (n ≤ 4) on three random lines modulo
  2^61 − 1; Part F, the example of the switched reading.
- **First program** (with which the result was first obtained; SymPy and mpmath): Hasselblatt–Propp degrees for
  n ≤ 10 by iteration and by formula (6); d_j = ψ(ζ^j) for j ≤ 200; λ = 6.857557409279 and 13.449607681785; the
  explicit forms of f; deg f = 10 and deg f² = 66 by an exact multivariate gcd (common factor x0¹⁶x1¹⁶x2²). Not part
  of the package; verification run B re-ran it with identical output.

## Independent verification runs
The result was first obtained with a written proof and the first program. Two independent verification runs, both
AI-assisted, followed (2026-10-17); each re-derived the proofs and wrote its own programs, and neither read the
other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, reading, numbering against the source (TeX of v1 and v2) | CONFIRMED; v1 numbering corrected (no Remark 14.7 in v1; conjecture = 14.7) | CONFIRMED (v2 TeX) |
| Lemma 2.1 (specialization, no Noetherianity needed) | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| Lemma 2.2 (structure; shift; algebraic coefficients) | CONFIRMED (re-derived; Evertse's notes Thm 8.17 read) | CONFIRMED (re-derived) |
| Lemma 2.3 (residue classes) | CONFIRMED | CONFIRMED |
| Theorem 3.1 as quoted | CONFIRMED against Bell–Diller–Jonsson Thm 3.5 and Bell–Diller–Jonsson–Krieger Thm 4.1; Evertse 1984 display not readable (text layer only); O_K → S-integer reduction supplied | CONFIRMED against Bell–Diller–Jonsson Thm 3.5 and Bell–Diller–Jonsson–Krieger Thm 4.1 |
| Lemma 3.4 (growth) | CONFIRMED (re-derived); Bell–Diller–Jonsson–Krieger Cor. 4.2 noted as the two-root case | CONFIRMED (re-derived); 9 test sequences consistent |
| Theorem 1.1 (Steps 1–5, variants) | CONFIRMED | CONFIRMED |
| Theorem 1.2 (Bell–Diller–Jonsson map, hypothesis, dominance, degree convention) | CONFIRMED; deg fⁿ = 10, 66, 454, 3114 (n ≤ 4) on random lines mod 2^61 − 1; the two conventions for h_ζ noted | CONFIRMED; degrees of the iterates up to n = 6 (ζ = 1+2i) and n = 5 (ζ = −3+4i) modulo primes near 2^20 equal to the recursion of Bell–Diller–Jonsson; λ to 20 digits |
| Corollary 1.3 (N ≥ 3, N = 1) | CONFIRMED | CONFIRMED |
| Proposition 1.4 (monomial maps) | CONFIRMED (re-derived; Hasselblatt–Propp degrees by iteration) | CONFIRMED (2,400 random comparisons) |
| Literature and novelty | no prior statement of the answer found among the sources read | no prior statement found (Semantic Scholar lists of works citing the source (80) and Bell–Diller–Jonsson (31), arXiv and zbMATH searches, web searches); one wrong statement about a later citation corrected |
| Bibliography | journal data of Bell–Diller–Jonsson and Bell–Diller–Jonsson–Krieger confirmed (Crossref) | title of the Evertse–Győry book corrected; DOIs confirmed (Crossref) |

No run found an error in the mathematics.

**Corrections required by the two runs**, all applied in the note:
1. (Run A) The arXiv numbering: Remark 14.7 and Conjecture 14.8 exist in v2 only; in v1 the conjecture is 14.7
   (Section 1.1).
2. (Run A) An explicit one-sentence statement of the reading in the abstract and the introduction (abstract,
   Section 1.1 "The reading", "Scope and priority").
3. (Run A) Provenance of Evertse's inequality: quoted from Bell–Diller–Jonsson Theorem 3.5 (= Bell–Diller–Jonsson–
   Krieger Theorem 4.1 = Evertse–Győry Proposition 6.2.1); the displayed formulas of Evertse 1984 were not read; the
   reduction from O_K-vectors to S-integer vectors is given (Theorem 3.1, Remark 3.2).
4. (Run A) The two conventions of Bell–Diller–Jonsson for h_ζ, conjugate by the coordinate swap, which commutes
   with g (Section 5).
5. (Run A, optional) Bell–Diller–Jonsson–Krieger Corollary 4.2 cited as the two-root case of the growth bound
   (Section 1.3, after Lemma 3.4).
6. (Run B) The published number of the question is not asserted (most likely 13.6, not verified); the statement that
   a later paper cites this question was wrong (the citation concerns another question of the source) and is not
   made; no source read cites Question 14.6 (Section 1.1, Verification paragraph).
7. (Run B) Title: Evertse–Győry, *Unit Equations in Diophantine Number Theory* (Cambridge University Press, 2015).
8. (Run B) Bibliographic data from Crossref: Bell–Diller–Jonsson, Acta Math. 225 (2020), no. 2, 193–225,
   doi:10.4310/ACTA.2020.v225.n2.a1; Bell–Diller–Jonsson–Krieger, Proc. London Math. Soc. 128 (2024), no. 1,
   e12573, doi:10.1112/plms.12573 (authors Bell, Diller, Jonsson, Krieger, as in arXiv:2107.04113v2).

**Added when the note was written.** The program `check_note.py` (with the certified enclosure of λ and the
exact checks of Lemma 5.1); the observation that the generating series of the degree sequence of h_{1+2i} already
has a natural boundary (Nguyen, Theorem D), so that natural-boundary results cannot decide the question; the
deduction of the Hasselblatt–Propp growth rate from the proof of Theorem 1.1 (Section 6). These additions were
examined by the third run (next section).

## Verification run on the note as written (2026-10-18)

A third AI-assisted run examined the written note as if nothing had been checked before (its programs and recorded
outputs are in `reproducibility/independent_run_2/`).
- **Sources, downloaded again and compared with every quoted statement:** the TeX of both arXiv versions of the
  source (environments of Section 14 counted: question 14.6 in both; remark 14.7 and conjecture 14.8 in v2, conjecture
  14.7 in v1; conventions, Definition 14.2 and the paragraph after it, Remark 14.3, Theorem 14.4 with its citations,
  Example 14.5 with the quotation, Remark 14.7); Bell–Diller–Jonsson (Main Theorem; the introduction's h and the
  condition ζⁿ ∉ R; char ≠ 2 only for g; §1.2 degree convention; §1.3 monomial maps, both conventions, ψ and
  |z| ≤ ψ(z) ≤ √5|z|; §2.1 g; Propositions 2.7 and 2.8 and Theorem 3.5 by counting the environments; the closing
  sentences of §3; the printed values of λ); Bell–Diller–Jonsson–Krieger (Theorem 1.1 and the paragraph after it,
  the degree convention of the introduction and of §2.1, Theorem 4.1, Corollary 4.2 with its proof); Nguyen
  (Theorem A, Corollary B, Theorem D(1), the degree convention; the eigenvalue hypothesis of Theorem D holds for
  h_{1+2i}); Evertse's lecture notes (§8.4, Theorems 8.17 and 8.18, page rendered); Evertse 1984 in the Numdam
  scan, rendered page by page, so that the normalization of the absolute values, the height ‖x‖ and the displayed
  inequality (14) of Theorem 2 could be read; the corpus record; Crossref for all 13 DOIs; the arXiv API for the
  six arXiv identifiers.
- **Proofs:** Lemmas 2.1–2.3, Theorem 3.1 as quoted, Remark 3.2, Theorem 3.3, Lemma 3.4, Theorem 1.1 (Steps 1–4),
  Remark 4.1, Lemmas 5.1 and 5.2, the two conventions, Theorems 1.2, Corollary 1.3, Proposition 1.4, the
  Hasselblatt–Propp example and the deduction of its growth rate, Section 7 (including the certification logic of
  Part D of `check_note.py` and the lower-bound logic of Part E) — all re-derived line by line. **No error found.**
  One provenance statement had to be made precise: Remark 3.2 described Evertse 1984 from a text layer; the
  displayed inequality carries ‖x‖^(−ε) rather than H_S(x)^(−ε), and the passage to Theorem 3.1 is by an S-unit
  scaling (now written out in the remark), not only by the integrality reduction.
- **Own programs:** `run2_checks.py` (48 checks, 0 failed; sympy and mpmath): the forms of h and f built from the
  affine definitions, coprimality over Q and modulo 3, 5, 7, 11, 13, 101 and the common factor modulo 2 (control),
  the identities of Lemma 5.1, g ∘ g = m·id, the coordinate swap, the Jacobian, deg f² = 66 by an exact trivariate
  gcd, formula (6) against direct composition (Hasselblatt–Propp map, n ≤ 60; 90 random matrices in dimensions 2,
  3, 4), the 256 sequences of Proposition 1.4, ψ for both conventions, d_j for j ≤ 400, an independent exact
  certification of the enclosures λ ∈ (6.857557409279008598, 6.857557409279008599) and
  λ ∈ (13.449607681785413449, 13.449607681785413450), BDJ's recursion, Lemma 5.2, the eigenvalues of the matrix of
  h_{1+2i}, the growth rate 1.3562… of the Hasselblatt–Propp degrees (deg(hⁿ)^(1/n) at n = 1000 and 2000), the
  example of Section 1.1, and a numerical sanity check of Lemma 3.4(b). `run2_degrees_modp.py` (numpy; FFT products
  with a modular self-test; vectorized Euclid): deg fⁿ on new random lines modulo 2^31 − 1: 10, 66, 454, 3114,
  21368, 146488 for n ≤ 6 (ζ = 1+2i; n = 6: raw degree 213,680, common factor of degree 67,192, about 2 minutes),
  and 16, 224, 2970, 39900 for n ≤ 4 (ζ = −3+4i), all equal to BDJ's recursion.
- **Package:** `check_note.py` re-run from an extracted copy of `source.zip`: 38 checks, 0 failed, output identical
  apart from the running time.
- **Literature (18 October 2026):** arXiv API (8 queries), zbMATH Open (4 queries), Semantic Scholar (31 works citing
  Bell–Diller–Jonsson, 80 citing the source; the same counts as the day before; abstracts of the 2026 entries read),
  OpenAlex (single-work lookups answered; the lists of citing works refused, rate limit), one web search, and the
  slides of Bell's BIRS 2024 talk on transcendental dynamical degrees (read; nothing on the question). The AMS page
  of the published source could not be accessed again; the published item number remains unverified.
  No printed statement of the answer found.
- **Presentation:** all pages rendered and read (11 before and 12 after the corrections); house style checked;
  bibliography checked against Crossref.
- **Corrections required and made:** the date; Remark 3.2 (above); "constant coefficients" for
  Bell–Diller–Jonsson–Krieger, Corollary 4.2 (Section 1.3 and after Lemma 3.4); the Verification, "Scope and
  priority" and "What was read" paragraphs; this report, the README of the package and the Zenodo metadata.

## Relation to the literature, novelty and scope
- **Searches (17 October 2026; repeated on 18 October 2026 by the third run, see above).** arXiv API abstract searches (dynamical degree with recurrence or
  recursion; degree sequence with rational map or recurrence or dynamical degree; transcendental dynamical degree;
  degree growth with linear recurrence; algebraic entropy with transcendental; Bellon–Viallet conjecture); zbMATH
  Open (keyword and title searches); Semantic Scholar lists of the works citing the source (80 records) and
  Bell–Diller–Jonsson (31 records), by title and abstract, several read in full text (Hu 2026, Nguyen 2026, Xie
  2024, Takenawa 2026, Weinreich); Crossref; four web searches. the lists of citing works in OpenAlex and the AMS page
  of the published source could not be accessed.
  - No printed answer to the question was found in these sources. A search that finds nothing is not a proof of
    novelty.
  - Bell–Diller–Jonsson cite the source only for the algebraicity question of dynamical degrees; their only recurrence
    statement concerns the monomial map h (Hasselblatt–Propp). Bell–Diller–Jonsson–Krieger remark that their series
    coefficients are maxima of several integer linear recurrences (the structure of Proposition 1.4), not the
    question. Nguyen (Corollary B) excludes a rational generating series (hence a single recursion) for the
    Bell–Diller–Jonsson maps, not finitely many recursions with an index function. Hu (2026) studies degree growth
    up to polynomial factors and cites the source for a different question. Silverman's survey (2026) recalls the
    transcendental examples and does not restate the question.
- **Caveats.** The transcendence theorems of Bell–Diller–Jonsson and Bell–Diller–Jonsson–Krieger are used as
  published; their proofs were not checked. Evertse's inequality is used in the form stated by Bell–Diller–Jonsson;
  the Evertse–Győry book was not read; Theorem 2 of Evertse 1984 was read in the scan by the third run, and
  Remark 3.2 of the note explains how the form used follows from it.
- **Scope.** The note answers Question 14.6 negatively in the reading stated above; the switched-recursion reading
  is not addressed. No priority is claimed.

## 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/