# Verification report — OWR-1323-008 (Petersson's question on operator ranges of hypercyclic vectors)

Verification date: 2026-09-30 (revised after the second referee report).

**Verdict.** Partial answer with a mixed outcome. The note is unrefereed.
- First clause (does HC(T) ∪ {0} contain the range of an injective operator S?): **yes** for every hypercyclic
  operator on a separable Banach space (folklore), and for every hypercyclic operator on ω = K^N (not found in the
  literature). For weakly mixing (= hereditarily hypercyclic) operators on Fréchet spaces with a continuous norm, the
  source already answers it. The source states this without a reference; it follows from Bonet–Martínez-Giménez–Peris
  (2004), as recalled by Petersson (2007), and the paper uses it without proof.
- This last case contains the corpus formulation only if "bounded operator" in the corpus record is read as a normed
  setting. In the source's setting, a separable Fréchet space, case (O2) below remains open even for hereditarily
  hypercyclic T.
- Second clause, universal reading (is every linearly independent n-tuple in Im S hypercyclic for T ⊕ ⋯ ⊕ T, for every
  such S?): **no**, by an elementary argument, in every setting where the first clause is known.
- Second clause, existential reading (for some such S): holds if and only if T is weakly mixing, when X has a
  continuous norm or X = ω.
- **Not settled:** the first clause for non-weakly-mixing hypercyclic operators on non-normable Fréchet spaces with a
  continuous norm (O1), and on Fréchet spaces without a continuous norm other than ω (O2). For hereditarily
  hypercyclic T only (O2) remains.

## Statement checked
- **Primary source.** H. Petersson, "Hypercyclic structures in product spaces", in Mini-Workshop: Hypercyclicity and
  Linear Chaos, Oberwolfach Report 37/2006, pp. 2259–2261, doi:10.4171/OWR/2006/37. The question is on p. 2260.
  - The report PDF was fetched anonymously and read in full. Verifier 2 (round B) and referee 2 fetched it again;
    their copies have the same SHA-256.
  - Context in the source: by Bès–Peris (the source's reference [1]), the Hypercyclicity Criterion, hereditary
    hypercyclicity, weak mixing and syndetic hypercyclicity are equivalent. If X has a continuous norm, they are also
    equivalent to (v) L_T is SOT-hypercyclic and (vi) some S has (Sx_1, …, Sx_n) hypercyclic for ⊕ⁿT whenever {x_i}
    is linearly independent. No reference is attached to this second sentence.
  - The question asks whether, for any (hereditarily) hypercyclic T, HC(T) contains "the range Im S of a one-to-one
    operator" S, and/or whether, for such S, every linearly independent n-tuple in Im S is hypercyclic for ⊕ⁿT.
- **Corpus record.** ulamai/UnsolvedMath, OWR-1323-008 (status `open`), "Linear Subspaces of Hereditarily Hypercyclic
  Vectors". It states the question for hereditarily hypercyclic T and a bounded injective S, with "If so" in place of
  "and/or".

## Readings
| Reading | Answer | Witness / reference |
|---|---|---|
| Clause 1 literally, Im S ⊆ HC(T) | trivially false | 0 ∈ Im S, 0 ∉ HC(T); read as Im S \ {0} ⊆ HC(T) |
| "if T is such a mapping" (clause 2) | ambiguous, harmless | read as referring to S, or to an operator T for which clause 1 holds, with S from clause 1; both readings give (Q2)/(Q2′) |
| (Q1) for every hypercyclic T on a separable Banach space | yes | Thm 1.2: Su = Σ φ_k(u) T^k x₀ / k! (folklore) |
| (Q1) for every hypercyclic T on ω | yes | Thm 1.1 (a)⇒(c) |
| (Q1) for weakly mixing T on a Fréchet space with a continuous norm | yes (source; input used without proof) | source, (v)/(vi), with Lemma 2.5; stated in the source without a reference; it follows from Bonet–Martínez-Giménez–Peris 2004, Thms 3.1 and 3.5, as recalled in Petersson 2007, Prop. 2, with Bès–Peris 1999. Contains the corpus version only if "bounded operator" is read as a normed setting |
| (Q1) for non-weakly-mixing T on a non-normable Fréchet space with a continuous norm | not settled | (O1); such T exist on H(Ω) (Bayart–Matheron 2007) |
| (Q1) on Fréchet spaces without a continuous norm, not isomorphic to ω | not settled | (O2); the only open case for hereditarily hypercyclic T |
| (Q2) universal reading | no, in all settings where (Q1) is known | Thm 1.2 (pair (x₀, Tx₀)); Thm 1.3 (rank-one perturbation) |
| (Q2′) existential reading | iff T weakly mixing (X with continuous norm, or X = ω); iff Hypercyclicity Criterion on Banach spaces | Cor. 1.4(iii) |

## Results in the paper
- **Theorem 1.1 (ω).** For T ∈ L(ω) the following are equivalent: (a) T is hypercyclic; (b) p(T′) is injective for
  every nonzero polynomial p; (c) there is an injective S ∈ L(ω) such that (Su_1, …, Su_n) is hypercyclic for T^{⊕n}
  for all linearly independent u_1, …, u_n.
  - Then Im S is closed, so it is a hypercyclic subspace, and S is an SOT-hypercyclic vector for L_T.
  - (a)⇔(b) is due to Herzog–Lemmert (K = C; stated there as torsion-freeness of the dual of ω as a C[λ]-module,
    according to the zbMATH review) and to Shkarin (Existence theorems in linear chaos, Theorems 1.7–1.8;
    Demonstratio Math. 2011, Corollary 1.2). Only (a)⇒(c) is claimed.
  - Proof: Lemma 4.1 (a special case of Shkarin's Lemma 1.5, with a short proof via free K[t]-modules; the module
    viewpoint goes back to Herzog–Lemmert) and an explicit construction S = B^t, where B: φ → φ is prescribed on the
    independent family A^{n_s} g_i. Step 3 uses the mechanism of Shkarin's proof of his Theorem 1.1 (2011).
- **Theorem 1.2 (Banach, folklore).** Su = Σ φ_k(u) T^k x₀ / k! is nuclear, injective, with dense range, and
  Im S \ {0} ⊆ HC(T). Its range contains x₀ and Tx₀, and the pair (x₀, Tx₀) is never hypercyclic for T ⊕ T, since its
  orbit lies in the graph of T.
- **Theorem 1.3 (rank-one perturbation).** If S₀ has property (vi) for n ≤ 2, then
  S′ = S₀ + (TS₀a − S₀e) ⊗ ψ is admissible and its range contains x = S₀a and Tx.
- **Corollary 1.4.** The answers listed above. The continuous-norm input is used without proof in (i)–(iii).
- **Remark 4.2.** For T ∈ L(ω) and its iterates, the hypothesis of Menet's Theorem 4.8 (IEOT 2013), the existence of a
  hypercyclic subspace, and hypercyclicity are equivalent. This answers Menet's question in his Remark 4.10 for single
  operators on ω. The hypercyclic-subspace part of Theorem 1.1 (but not property (vi)) also follows from Shkarin's
  Lemma 1.5 combined with Menet's Theorem 4.8. Menet's general non-existence criterion (his Theorem 3.6, applied to
  universal series in his Theorem 4.2) never applies to the iterates of a hypercyclic operator on ω.
- **Remark 4.3.** Petersson, Hypercyclicity in omega, PAMS 135 (2007), read in full. Its Theorem 2: if a sequence of
  operators on ω has a hypercyclic subspace H, then some injective S ∈ L(ω) has Im S ⊆ H (a short triangular
  construction). So for a single operator on ω, (Q1) is equivalent to the existence of a hypercyclic subspace, and
  Theorem 1.1 shows that both always hold. The paper treats sequences of operators and assumes a hypercyclic subspace.
  It does not state that every hypercyclic operator on ω has one, and it does not consider property (vi). Its only
  discussion of L_T is its account of Chan, Chan–Taylor and Bonet–Martínez-Giménez–Peris.
- **Remark 4.4 (L_T).** As recalled in Petersson 2007, Prop. 2, Bonet–Martínez-Giménez–Peris 2004 give a compact
  SOT-hypercyclic vector of L_T on separable Fréchet spaces with a continuous norm, under the Universality Criterion,
  and this does not remain true without a continuous norm (their Example 3.2). On ω no injective operator is compact.
  Whether Martínez-Giménez–Peris 2003, Theorem 2.3 (L_T on locally convex spaces) applies to ω was not checked; if it
  does, property (vi) on ω follows from it with Bayart–Matheron, and (a)⇒(c) of Theorem 1.1 is not new.

## Computations (scripts and outputs in reproducibility/)
All proofs are by hand; no statement depends on a computation. The checks below are sanity checks of the finite
bookkeeping. They use exact rational arithmetic, except one floating-point check that is labelled as such.
- **Finder** (`claimant/`).
  - `omega_construction.py` (standard library, exact rationals, about 6 s) implements the construction of
    Theorem 1.1 for three torsion-free examples: the backward shift, (I + B)², and multiplication by t on
    K[t, 1/(t−1)] (a non-free module).
    - It runs 112 requirements (88 for pairs).
    - In each example, 784/784 end-to-end coordinate checks agree (7 test vectors × 112), and all 80 prescribed 2×2
      patterns are attained.
  - `identities.py` checks the rank-one identities of Theorem 1.3 (I1, I4) on 200 random exact instances. Its checks
    I2 (S e_j = T^j x₀/j!) and I3 (graph relation) are tautological as written (I2 compares a finite sum with its own
    definition; I3 compares T y with T y) and test nothing.
- **Independent referee 1** (`referee/indep_check.py`, written from the text, different algorithm: one global row
  reduction, an explicit row-finite matrix for S, primal iteration of T).
  - It covers the backward shift, an upper staircase matrix with non-constant coefficients, and multiplication by t on
    K[t, 1/t].
  - In each example, 250/250 coordinate identities hold. Exact target hitting uses one fixed independent triple and six
    random targets: 18 = 3 × 6 exact hits per example.
  - The degree bound of Lemma 4.1 is sharp for the shift. Two torsion examples fail for every n ≤ 60, as they must.
  - The identities of Theorem 1.3 hold on these instances.
  - The script seeds some choices with Python's `hash()` of a string, which is randomised per process. Its statistics
    lines vary between runs, and every run passes. The recorded output uses `PYTHONHASHSEED=0`.
- **Independent referee 2** (`referee2/r2_omega_check.py`, written from the paper before reading the other scripts;
  about 10 s). Seven examples (five torsion-free, among them multiplication by t on Q[t, 1/(t²+1)], and two torsion
  controls).
  - Row i of T^k equals A^k g_i: 1,456/1,456.
  - Lemma 4.1: every (N, l) eventually good; the bound of the proof is sharp for F and F²; both torsion controls fail
    for every n ≤ 60.
  - Menet's hypothesis against the translation in Remark 4.2: 5,040/5,040 cases agree.
  - Construction of Theorem 1.1 (66 requirements per example, shuffled): primal iteration on vectors that are not
    finitely supported matches the dual formula (786/786 coordinates); 90/90 exact target hits for random independent
    tuples.
  - Identities of Theorems 1.2 and 1.3 (50/50, 150/150); in floating point, the factorisation of Lemma 3.1 for
    f(z) = e^z − 1 (errors below 1e−15).
- **Reruns (2026-09-30).** All four scripts reproduce their recorded outputs, apart from timing lines (and, for
  `indep_check.py` without a fixed `PYTHONHASHSEED`, the statistics lines).

## Independent adversarial audit
Two independent verifiers (round B, 2026-09-29) checked every proof by hand, tried to refute each theorem, and
re-fetched the source anonymously. Verifier 1 also reran the finder's scripts and wrote an independent implementation
(`referee/indep_check.py`). Their verdicts:

| Item | Verifier 1 | Verifier 2 |
|---|---|---|
| Classification | PAPER_CANDIDATE | PAPER_CANDIDATE |
| Mathematics correct | yes (no error found) | yes (no error found) |
| Answers the question as intended | yes | yes |
| Readings (Im S \ {0}; S for T; "(hereditarily)" scope) | confirmed | confirmed (source PDF has the same SHA-256) |
| Suggested corpus status | solved, mixed answer (clause 1 yes, clause 2 no); not a bare "disproved" | universal clause 2 false; clause 1 yes on Banach spaces and ω, and in the source's setting; open elsewhere |
| Scripts | rerun, reproduced; independent implementation agrees | not reported |
| Novelty | only the ω results possibly new | only the ω theorem substantive; confirm with literature |

The verifiers' Theorems A, B, C are Theorems 1.2, 1.1, 1.3 of the paper.

Round-B required fixes and how they were applied:
1. **Petersson, Hypercyclicity in omega, PAMS 135 (2007) 1145–1149, doi:10.1090/S0002-9939-06-08584-4.** Cited and
   compared (Introduction, Remark 4.3). The earlier claim that none of Petersson's papers addresses the question was
   withdrawn. In round B the full text could not be obtained (the publisher served a bot check, which was not
   bypassed; the repository file was not retrievable). It was read in full in the revision after referee 2 (below).
2. **ω prior-art search.** The arXiv queries `abs:omega`/`all:omega` were ineffective, because abstracts write
   `$\omega$`.
   - The search was redone in zbMATH, and the papers citing Menet 2013 and Petersson 2007 (via Semantic Scholar) were
     inspected.
   - This found Shkarin, Mixing operators on spaces with weak topology, Demonstratio Math. 44 (2011) 143–150. Its
     Lemma 1.5 contains the note's Lemma 4.1, and its Corollary 1.2 gives (a)⇔(b).
   - The paper says that the hypercyclic-subspace statement follows quickly from Shkarin + Menet and may be known to
     specialists.
3. **Salas, RACSAM 105 (2011) 379–388, open questions.** Not accessible (bot check; closed access). The paper states
   that we do not know whether Theorem 1.1 answers one of them.
4. **Status for the corpus.** Reported as a mixed, partial answer. The paper says explicitly that for hereditarily
   hypercyclic T on spaces with a continuous norm the first clause is in the source, and it keeps (O1)/(O2) as
   unsettled cases.
5. **Minor.** The reason for the invertible minor (Step 3 of the proof of Theorem 1.1) was added. Lemma 4.1 is stated
   "for every n > d", with the degree measured in a fixed basis of the free module.
6. **Attribution and modesty.**
   - (a)⇔(b) is credited to Herzog–Lemmert and to Shkarin (Demonstratio Math. 2011, Corollary 1.2).
   - Verifier 2 pointed to Shkarin's survey, Existence theorems in linear chaos (J. Gen. Lie Theory Appl. 9 (2015),
     no. S1, Art. 009, doi:10.4172/1736-4337.S1-009; arXiv:0810.1192). Its Theorems 1.7–1.8 contain (a)⇔(b) and
     "hypercyclic ⇒ mixing" on ω, and its Corollary 1.10 shows that ω is the only separable infinite-dimensional
     Fréchet space without a hypercyclic non-mixing operator. We read these statements and the proofs of Theorems 1.7
     and 1.8 in the arXiv version, and the survey is now cited (revision after referee 2).
   - Bayart–Matheron (every hypercyclic operator on ω satisfies the Hypercyclicity Criterion) is cited.
   - Theorem 1.2 is labelled folklore, and the refutation of (Q2) is labelled elementary.
   - The paper is framed around Theorem 1.1.
   - Verifier 2 also suggested confirming the novelty of Theorem 1.1 with a specialist. No one was contacted; the
     novelty statement is kept modest ("not found in the literature", "may be known to specialists").

### Referee 2 (2026-09-30)
A second independent adversarial referee re-fetched the source, checked every proof line by line, wrote its own code
before reading the other scripts (`referee2/r2_omega_check.py`), reran all scripts, and searched arXiv, zbMATH,
Crossref, OpenAlex (metadata), Semantic Scholar and the web (one search). All its requests were anonymous. Its
verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity | Faithful; two small overstatements (fixes 7, 8) |
| Proofs | Correct; no gaps found; the one outside input (continuous-norm case) was misattributed (fix 4) |
| Computations | Reproduced, and confirmed by independent code; two descriptions inaccurate (fix 9) |
| Novelty | No prior publication of the same result found; the hypercyclic-subspace part follows quickly from Shkarin's Lemma 1.5 and Menet's Theorem 4.8; gaps in credits and search (fixes 1, 2, 5, 6) |
| Presentation | House style and release package correct; abstract and attributions to be tightened (fixes 3, 4, 6, 10) |
| Fatal | No |

Required fixes and how they were applied:
1. **Read Petersson 2007.** Done. The AMS site still served a bot check to automated requests, which was not
   bypassed, and the Chalmers file link still returns a server error. The publisher's PDF was read in full from its
   copy in the Internet Archive (5 pages). Findings, now in the Introduction, Remark 4.3, Remark 4.4, (O2) and "Scope
   and priority":
   - It treats sequences of operators on ω (single operators through their powers). Its Theorem 2 assumes a
     hypercyclic subspace H and gives an injective S with Im S ⊆ H.
   - It does not state the unconditional statement (every hypercyclic operator on ω has a hypercyclic subspace), and
     it does not consider property (vi).
   - Its only discussion of L_T is its account of Chan, Chan–Taylor and Bonet–Martínez-Giménez–Peris. It recalls that
     their criterion needs a continuous norm.
   - It asks whether its Theorem 2 holds on every separable Fréchet space; this is now mentioned under (O2).
2. **Shkarin's survey** (Existence theorems in linear chaos, 2015) is cited next to [Shk11] in the Introduction and in
   the proof of Corollary 1.4(iii). The sentence "which we did not check" was replaced (round-B fix 6 above).
3. **Abstract.**
   - (a) It now says that the second question is negative in the universal reading (for every such S), and that the
     existential reading holds exactly for weakly mixing operators, on ω and on spaces with a continuous norm.
   - (b) It now says that the hypercyclic-subspace consequence also follows quickly from Shkarin's Lemma 1.5 combined
     with Menet's Theorem 4.8.
   - (c) The phrase "removes … the hypothesis of a hypercyclic subspace from a theorem of Petersson (2007)" was dropped.
4. **Continuous-norm input.** The proof of Corollary 1.4(i) now says that the source states this without a reference
   (its [1] = Bès–Peris is attached to the preceding equivalences). It cites Bonet–Martínez-Giménez–Peris 2004,
   Theorems 3.1 and 3.5, as recalled in Petersson 2007, Proposition 2, together with Bès–Peris. Remark 4.4 adds Chan
   1999 and Chan–Taylor 2001, Corollary 6, for Hilbert and Banach spaces. The paper says that these papers were not
   read and that the input is used without proof in Corollary 1.4(i)–(iii). The same correction was made in this
   report and in RESULT.md.
5. **L_T literature.** Chan 1999, Chan–Taylor 2001, Martínez-Giménez–Peris 2003 and Bonet–Martínez-Giménez–Peris 2004
   are now cited in the Introduction, in the new Remark 4.4 and in "Scope and priority".
   - BMP04: according to Petersson 2007, its L_T theorems assume a continuous norm, and its Example 3.2 shows that they
     do not remain true without one. So BMP04 does not give property (vi) on ω. We have not read BMP04.
   - MGP03, Theorem 2.3: **not checked.** ScienceDirect answered automated requests with 403/bot pages, the CORE copy
     is behind a bot check, and no archived or repository copy was found. The paper says plainly that this was not
     checked, and that if its hypotheses allow ω, then property (vi) on ω follows from it with Bayart–Matheron and
     (a)⇒(c) of Theorem 1.1 is not new.
   - The (O2) example is now cited directly to BMP04 (as recalled in Menet 2013, Section 1, and Petersson 2007).
6. **Attributions.**
   - (a) Salas 2011: hypercyclicity (chaos if K = C) of upper staircase matrices, and a common hypercyclic subspace for
     countably many strictly upper triangular upper staircase matrices (zbMATH review; Bulancea–Salas for the first
     part).
   - (b) Bès–Conejero: single P(B_w) as described by Menet; countable families of non-constant polynomials of a fixed
     backward shift as described by Petersson 2007 and Bulancea–Salas (arXiv:2406.03650).
   - (c) Herzog–Lemmert's module formulation (condition (b)) and the module viewpoint of Lemma 4.1 are credited;
     Shkarin's remark that his proof idea is close to theirs is mentioned; Step 3 is noted to use the mechanism of
     Shkarin's proof of his Theorem 1.1.
   - (d) De La Rosa–Read gave the first non-weakly-mixing example, on a specially constructed Banach space;
     Bayart–Matheron gave the examples on ℓ^p, c₀ and separable Hilbert space.
   - (e) Remark 4.2 now names Menet's Theorem 3.6 as the general non-existence criterion and Theorem 4.2 as its
     application to universal series.
7. **Second clause.** "This must refer to S" (paper) and "misprint" (Readings table) were replaced by the neutral
   reading: the phrase can refer to S, or to T satisfying the first clause; both readings give (Q2)/(Q2′).
8. **Corpus scope.** The paper now says that for hereditarily (= weakly mixing) hypercyclic T the only case of (Q1)
   settled neither in the note nor in the source is (O2). This report qualifies "contains the corpus formulation".
9. **Descriptions of the checks** (paper, `reproducibility/README.md`, this report): `indep_check.py` uses one fixed
   independent triple and six random targets (18 = 3 × 6 hits); the checks I2 and I3 of `identities.py` are described
   as tautological, and I3 is no longer called a check of the graph obstruction.
10. **Scope and priority.** It now says that only the hypercyclic-subspace statement follows quickly from Shkarin 2011
    and Menet 2013; the property (vi) statement does not follow directly from them.
11. **Release.** Rebuilt with `tectonic --keep-logs` (0 overfull boxes; 2 underfull boxes in one bibliography entry,
    a long DOI), and all 12 pages rendered and inspected. `release/` (paper.pdf, main.tex, references.bib),
    `source.zip`, the three `zenodo/` files and the checksums in `ZENODO_METADATA.md` were refreshed, with the
    description in sync with the new abstract. Referee 2's script and output were added as `reproducibility/referee2/`.

Optional suggestions of referee 2: the title no longer leaves "Petersson" alone on the second line; the duplicated
"April 2026" was dropped from CSRV26; a zbMATH link was added for De La Rosa–Read; the sentence on the ineffective
arXiv queries was moved from the paper to this report. The dataset label still renders as "[ula26]": with a `key`
field, the BibTeX of the build tool renders the label as a bare year.

## Relation to the literature, novelty and scope
- **Read in full:** the source (OWR 37/2006); Petersson, PAMS 135 (2007) (publisher's PDF via the Internet Archive);
  Menet, IEOT 77 (2013) (arXiv:1302.6447); Shkarin, Demonstratio Math. 44 (2011) (arXiv:1209.0979).
- **Read in part:** Shkarin, Existence theorems in linear chaos (arXiv:0810.1192: Section 1, proofs of Theorems 1.7 and
  1.8); Bulancea–Salas (arXiv:2406.03650, Section 3). Also inspected: Menet, JOT 73 (2015) (arXiv:1312.5876);
  Bès–Menet, JMAA 432 (2015) (arXiv:1409.0995); Carvalho Silva–Ribeiro–Varão (arXiv:2606.18252).
- **Known only from reviews or secondary accounts:** Salas 2011; Bès–Conejero 2006; Herzog–Lemmert 1993;
  Bayart–Matheron 2007; Bès–Peris 1999; De La Rosa–Read 2009; Chan 1999; Chan–Taylor 2001; Martínez-Giménez–Peris 2003;
  Bonet–Martínez-Giménez–Peris 2004.
- **Searches (September 2026).** zbMATH, Crossref, arXiv, Semantic Scholar, and three web searches by the authors'
  side (one more by referee 2). OpenAlex was rate-limited at first. Full-text lookups also used the Internet Archive,
  CORE and the UPV repository (RiuNet). All requests were anonymous and are logged in the problem folder.
  - The first arXiv queries `abs:omega`/`all:omega` were ineffective, because abstracts write `$\omega$`; the search
    was redone in zbMATH.
  - We found no source stating that every hypercyclic operator on ω has a hypercyclic subspace or an operator with
    property (vi).
  - The hypercyclic-subspace statement follows quickly from Shkarin's Lemma 1.5 and Menet's Theorem 4.8, so it may be
    known to specialists. The property (vi) statement does not follow directly from these two results. Whether
    Martínez-Giménez–Peris 2003, Theorem 2.3 gives it was not checked. This negative search is not a proof of priority.
- **Scope.** Positive answer to the first clause on separable Banach spaces (folklore) and on ω. Negative answer to the
  universal second clause wherever the first clause is known. Characterization of the existential reading on spaces
  with a continuous norm and on ω. The first clause is not settled in the cases (O1) and (O2); for hereditarily
  hypercyclic T only (O2) remains.

## 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/