# Verification report — OWR-1787-006 (Costakis: polynomial multiples of MacLane's universal functions)

Verification date: 2026-10-19.

**Verdict.** The note gives a **complete negative answer** to the question; every statement is proved as stated.
Three independent AI-assisted verification runs checked the results: two on the first written version and a
third on the final text of the note (see below).
- **Theorem A.** There is an explicit f_A ∈ HC(D) such that (z f_A)^(n)(0) = n a_{n−1} is 0 or of modulus ≥ 2 for
  every n ≥ 0; hence no derivative of z f_A lies in the open set {h : |h(0) − 1| < 1}, and z f_A ∉ HC(D). More
  generally c z^d f_A ∉ HC(D) for all c ≠ 0, d ≥ 1. This already answers Question 5 of the source negatively
  (p(z) = z).
- **Theorem B.** There is one f_B ∈ HC(D) such that p f_B ∉ HC(D) for every non-constant polynomial p: for n ≥ n_*(p),
  (p f_B)^(n)(0) is 0 or of modulus ≥ S + 2 (S = Σ|c_i|), and finitely many derivatives do not affect density.
- **Consequence.** Theorem 6 of the source (a residual set G ⊂ HC(D) with pf ∈ HC(D) for all f ∈ G and all non-zero
  p) cannot be extended from a residual set to all of HC(D).
- **Credit and limits.** The question and Theorem 6 are Costakis's. The functions are built by the classical block
  construction of MacLane-type universal functions (Taylor data of a dense sequence of polynomials in sparse
  blocks); the formula (pf)^(n)(0) = Σ_i c_i n(n−1)···(n−i+1) f^(n−i)(0) is the Leibniz rule at the origin, and the
  only new observation is that it can be controlled on such functions (with ramps of large coefficients for
  Theorem B). The argument is elementary; no printed answer was
  found, but a folklore version may exist (the monographs of Bayart–Matheron 2009 and Grosse-Erdmann–Peris 2011 were
  not accessible to us). Not decided: whether (1 + z) f_A ∈ HC(D); the structure of {f ∈ HC(D) : zf ∈ HC(D)} beyond
  Theorem 6.

The note is unrefereed. Under the dataset's policy for unrefereed claims, the status of the record stays open
(or partially solved) with the evidence type unrefereed_claim.

## Statement checked
- **Primary source.** G. Costakis, "Which maps preserve universal functions?", in Mini-Workshop: Complex
  Approximation and Universality, Oberwolfach Rep. 5(1) (2008), Report 6/2008, pp. 297–346, the abstract on
  pp. 328–331, doi:10.4171/OWR/2008/06. Read in the open PDF on the EMS Press site (50 pages; the abstract on PDF
  pages 32–35), by the first version, by both verification runs, again at the writing and again by the third
  verification run (same file, sha256 9bd27d84…03a5; pages 328–331 also rendered as images).
  - Definition 4 (MacLane), p. 330: an entire f is universal with respect to derivatives if {f^(n) : n = 0, 1, 2,
    …} is dense in H(C); the set is HC(D), the hypercyclic vectors of D on H(C).
  - Theorem 5, p. 330: f + e^{az} ∈ HC(D) for f ∈ HC(D), |a| ≤ 1; some g ∈ HC(D) has g + e^{az} ∉ HC(D) for all
    |a| > 1 (the proof of the second part uses a member of HC(D) of optimal growth, due to Grosse-Erdmann 1990
    and Shkarin 1993).
  - Theorem 6, p. 330: a residual G ⊂ H(C), G ⊂ HC(D), with pf ∈ HC(D) for every non-zero polynomial p and every
    f ∈ G; the proof is not given and uses a result of the preprint "Common Cesàro hypercyclic vectors" (published
    as Studia Math. 201 (2010) 203–226).
  - Question 5, p. 330 (paraphrased): is pf ∈ HC(D) for every f ∈ HC(D) and every non-constant polynomial p?
  - Questions 1–3 (p. 329) concern Nestoridis's class U(D, 0) of universal Taylor series (multiplication by z − a,
    |a| ≥ 1, and by non-constant polynomials; derivatives); Question 4 asks for an inner function in U_1(D, 0). They
    form record OWR-1787-005, not this record. Theorem 3 (p. 329): (z − a)f stays in U_1(D, 0) for a in the closed
    disc and in U(D, 0) for a in the open disc.
- **Corpus record.** ulamai/UnsolvedMath, OWR-1787-006 (dataset version 1.8.0; status `open`; literature assessment:
  only a residual subclass of f is known to have the property). The statement is Question 5 together with
  Definition 4; it is a faithful restatement.
- **Conventions.** H(C) carries the topology of uniform convergence on compact sets (the source names it for H(D)
  and uses it for H(C) in the proof sketch of Theorem 5). The orbit includes n = 0; finitely many orbit points do
  not matter (Lemma 2.2), so this convention cannot change the answer.

## Readings
| Reading | Answer | Where |
|---|---|---|
| "for every f ∈ HC(D) and every non-constant p" (the source's reading) | no: (f_A, z), and f_B for all p | Theorems A, B |
| "for every f ∈ HC(D) there is some non-constant p with pf ∈ HC(D)" | no: f_B defeats every p | Theorem B |
| "there is f ∈ HC(D) with pf ∈ HC(D) for all p" (existential) | yes, residually many: Theorem 6 of the source | Theorem 1.1 |
| constant p ≠ 0 | trivially yes, (cf)^(n) = c f^(n) | Section 1.1 |
| p restricted to roots in the unit disc (as in the source's Theorem 3 for universal Taylor series) | still no: p(z) = z | Theorem A |
| pf read as p(D)f | trivially yes (p(D) commutes with D and has dense range); excluded by the source's wording ("multiplication") | Section 1.1 |

## Results in the paper
- **Lemma 2.1** (Taylor data): f^(n)(z) = Σ_k a_{n+k} z^k/k!; (pf)^(n)(0) = Σ_i c_i n^(i) a_{n−i} (Cauchy product).
- **Lemma 2.2** (obstruction): if g^(n)(0) ∈ {0} ∪ {|w| ≥ 2|w_0|} for all n ≥ N, then g ∉ HC(D) (the open set
  {h : |h(0) − w_0| < |w_0|} minus the finitely many g^(n), n < N, is open and non-empty).
- **Lemma 2.3**: the polynomials with all coefficients in Q(i) \ {0} form a countable dense set P.
- **Construction A, Lemma 3.1, Proposition 3.2**: with rational M_j ≥ max(1, max|b_{j,k}|), δ_j ≤ min|b_{j,k}|,
  (A1) n_j δ_j ≥ 2 and (A2) N_j ≥ max(d_{j−1} + 1, j − 1), (d_j + 1) M_j (j − 1)^{N_j}/N_j! ≤ 2^{−j}; then
  sup_{|z|≤j} |f_A^(n_j) − g_j| ≤ 2^{−j}, f_A is entire and in HC(D). Explicit admissible n_j are given.
- **Theorem A** (proof in Section 3): c n^(d) a_{n−d} is 0 or ≥ (n_j + k + 1)|c||b_{j,k}| > 2|c| in modulus.
- **Remark 3.3**: n a_n ∈ {0} ∪ {|w| ≥ 2}, so f_A ∉ HC({nD^n}) and f_A lies outside the residual set
  ∩_b HC({n^b D^n}) of Costakis 2010, Theorem 1.7.
- **Construction B, (L1), (L2), Lemma 4.1**: ramps a_{n_j − l} = M_j (2j)^l (1 ≤ l ≤ j) before the blocks;
  (B1) n_j ≥ j(M_j/δ_j + 1), (B2) N_j = n_j − j − n_{j−1} ≥ d_{j−1} + j + 1 and
  (j + d_j + 1) M_j (2j)^j (j − 1)^{N_j}/N_j! ≤ 2^{−j}; at least j zeros before and j + 1 after each zone; f_B ∈ HC(D).
- **Proposition 4.2**: for p of degree d ≥ 1, j_0 = max(d, ⌈(2S + 4)/c_min⌉), n_* = n_{j_0} − j_0: for n ≥ n_*,
  s_n = 0 or |s_n| ≥ S + 2. Proof: the leftmost non-zero position m* of the window lies in a zone Z_j, j ≥ j_0, and
  all non-zero positions of the window lie in Z_j; Case R (m* in the ramp): |s_n| ≥ 2j_0 c_min − S ≥ 3S + 8; Case K
  (m* in the block): i* ≥ 1 because all zone entries are non-zero, and (B1) gives |s_n| > j M_j c_min/2 ≥ S + 2.
- **Theorem B** (proof in Section 4): Lemma 2.2 with w_0 = 1, N = n_*.
- **Remarks 5.1, 5.2**: why ramps are needed ((1 + z) f_A takes the values g_j(0) at the block starts of the size
  enumeration; whether (1 + z) f_A ∈ HC(D) is not decided); where each hypothesis is used (the size of the ramp
  entries in Case R only, their non-vanishing also in Case K — corrected after the third run).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program written with the note** (`check_note.py`, standard library, exact, about 14 s; 644 checks, 0 failed):
  Lemma 2.1(b) against repeated termwise differentiation (300 random instances, a wrong variant detected); (3),
  N! ≥ (N/e)^N and (j − 1)^N/N! ≤ 2^{−N} on finite ranges; for the first 120 members of two enumerations of P (the
  size enumeration of the first version and a mixed one with long blocks and small coefficients first): (A1), (A2),
  (B1), (B2), (L1), (L2), the explicit choices after (A2), (B2); Theorem A for z and six other monomials at every n
  up to the end of block 120, Remark 3.3; Proposition 4.2 for 13 polynomials with rational-modulus coefficients
  (exact j_0 from 3 to 58) at every n from n_* to the end of zone 120 (5882–7914 non-zero values each, all with
  |s_n| ≥ S + 2; at most 3 exceptions below n_*); Remark 5.1 (96 and 110 exceptional n, all block starts); tail sums;
  exact majorants of (4), (5). For the size enumeration n_120 = 23770 (A) and 54162 (B).
- **Program of the first version** (`original/verify_r48.py`, corrected copy): the same constructions for J = 60,
  60 with `--mix`, 120; it tests the first version's conclusion s_n = 0 or |s_n| ≥ 2 (not |s_n| ≥ S + 2), from the
  cruder threshold n_*′ ≥ n_* (rational bounds for S and c_min). Corrections made in the copy: the Remark 5.1 line now states what it counts (60, 59, 96 exceptional n, all
  block starts; 5, 21, 15 of them in the disc |w − 1| < 1); the final line "ALL EXACT CHECKS PASSED" is printed only
  if no check failed. Its outputs are otherwise identical to those recorded at the time.
- **Programs of the verification runs** (`verification_run_A/`, `verification_run_B/`, `independent_run_2/`), see
  below; the second run's tests of Proposition 4.2 check the weaker form s_n = 0 or |s_n| ≥ 2. Re-run at the
  writing from a copy of the package: `vA2_check.py`, `vb_lemma1.py`, `vb_wrong.py`, `vb_crosscheck_claim.py 60`,
  `vb_check.py 120 1 plain` — outputs identical up to timing lines.

## Independent verification runs
The results were first obtained with written proofs and the program in `original/`. Two independent verification
runs, both AI-assisted, examined this first written version (2026-10-18/19); each read the source in the EMS Press PDF, re-derived all proofs
line by line and wrote its own programs; neither read the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, quantifiers, readings, conventions against the source | CONFIRMED (pp. 328–331 re-extracted; readings incl. p(D)f, roots in the disc, n from 0 or 1) | CONFIRMED (PDF read; problem session pp. 343 ff. has no item on pf) |
| Lemma 2.1 (Leibniz at 0) | CONFIRMED (two proofs; SymPy) | CONFIRMED (SymPy, 476 instances; wrong variant detected) |
| Lemmas 2.2, 2.3; enumeration with repetition | CONFIRMED | CONFIRMED |
| Construction A, explicit n_j, Lemma 3.1, Proposition 3.2 | CONFIRMED (explicit formula checked exactly) | CONFIRMED (approximation bound also numerically, j ≤ 60) |
| Theorem A and the monomial case | CONFIRMED (150 adversarial blocks) | CONFIRMED (two enumerations, 120 blocks, six monomials) |
| Construction B, (L1), (L2), Lemma 4.1 | CONFIRMED | CONFIRMED |
| Proposition 4.2 (Claim), Theorem B | CONFIRMED (incl. i* = d in Case K; 43 + 35 polynomials on 130 zones, tightest layout, \|s_n\| ≥ S + 2; small coefficients; three mutations detected) | CONFIRMED (in the form s_n = 0 or \|s_n\| ≥ 2: 18 polynomials globally; local zone test, j_0 up to 1 000 002, ~16.4 million windows per seed; irrational coefficients; nine defective inputs detected) |
| Dependence of f_B on p | none (only n_* depends on p) | none |
| First-version program | re-run: byte-identical outputs; two cosmetic defects found | re-run: identical outputs; printed n_j reproduced independently; cruder threshold noted |
| Literature and bibliography | zbMATH, Crossref, IMPAN (Costakis 2010, Theorem 1.7 read), Gardiner 2014 (publisher PDF), Costakis–Tsirivas (arXiv 1511.06145), one web search | arXiv (17 queries), zbMATH (11, incl. cited-by), Semantic Scholar (citations), Crossref, publisher pages, four web searches |

**Corrections required by the runs, all made:**
run A — the counts of exceptional n for (1 + z) f_A in the first-version text (60 / 59 / 96, not "59–60"); the
program's label of these counts and its unconditional final line (corrected copy, outputs re-recorded); the
unverified reference to a theorem number in Bayart–Matheron dropped; Grosse-Erdmann 1990 and Shkarin 1993 described
as the source does (optimal growth); Question 4 of the source concerns U_1(D, 0) (record OWR-1787-005 covers
Questions 1–4 over U(D, 0) and U_1(D, 0)); Costakis, Studia Math. 201 (2010), Theorem 1.7 cited, with the remark that
f_A ∉ HC({nD^n}) (Remark 3.3). Run B — Shkarin, Israel J. Math. 180 (2010), final section (no non-trivial ideal in
HC(D) ∪ {0}, via f·f*) cited, with the reason why it does not answer the question (the multiplier is transcendental
and depends on f); Costakis 2010, Theorem 1.7; the folklore caveat with the two monographs "not accessible to us";
the cruder threshold n_*′ ≥ n_* of the first-version program stated (note, Section 6; README; program header).

**Third verification run (on the final text of the note, 2026-10-19).** A further AI-assisted run, independent of
the two above, examined the final text of the note and the package. What it checked:
- the sources again (OWR 6/2008, Gardiner 2014, Costakis 2010 and Shkarin 2010, arXiv v1; downloaded anew, checksums
  identical to the copies read before; the source pages also rendered as images), against every sentence of the
  note about them (Definition 4, Theorems 3, 5, 6, Questions 1–5; Gardiner p. 330; Costakis 2010 p. 205 and
  Theorem 1.7 on p. 207; Shkarin's Theorem 1.4 and final section);
- every proof of the note line by line, in particular the parts written or changed at the writing: the rational
  bounds M_j, δ_j; the bound |s_n| ≥ S + 2 of Proposition 4.2 with the factor bound m* + 1 > n_j in Case K (which
  needs only n_j ≥ j M_j/δ_j); the thresholds j_0, n_*; the independence of f_B from p; the role of Lemma 2.2
  (finitely many derivatives do not matter); Remark 3.3;
- its own program `independent_run_2/run2_check.py` (two seeds, 202 checks each, all passed): Lemma 2.1(b) as a
  symbolic identity (d ≤ 6, n ≤ 22); Constructions A and B for three enumerations (the size enumeration, 100
  adversarial members of P with degrees up to 14 and moduli 1e−6..1e6, and g_j = q_{v(j)+1}) with the least
  admissible n_j, the explicit choices after (A2), (B2) and two tight layouts; Theorem A and Remark 3.3;
  Proposition 4.2 with |s_n| ≥ S + 2 for 13 polynomials at every n ≥ n_*, every inequality of its proof checked
  (Step 1, (6), Case R ≥ 3S + 8, Case K); four polynomials with irrational |c_i| by interval arithmetic; single
  zones placed at n_j = ⌈j M_j/δ_j⌉ or slightly later, for j_0 up to 6002; (4) and (5) for all 1 ≤ j < i ≤ 120; n_120 = 23770 / 54162 and
  the numbers 96 / 15 of Remark 5.1 reproduced; three mutations (ramps removed: failures only in Case K with
  i* = 0; ramps of size 1e−6: failures only in Case R; (B1) dropped: failures only in Case K);
- the package programs again from the source archive (`check_note.py`, the first-version program in three
  configurations, `vA2_check.py`, `vA_check.py 40 60`, eight runs of the second run's programs including
  `vb_local.py 2`): all outputs identical to the recorded ones up to elapsed times;
- the literature search again (below), the bibliography (all DOIs via Crossref), house style and every page.

It found no error in a theorem, proposition, lemma or proof. Corrections it required, all made: Remark 5.2 said
that the ramps are used only in Case R, but their non-vanishing is also used in Case K (at the first d positions
of a block); the abstract presented the Leibniz formula itself as the new observation; the note did not say that
the first-version program and the second run test the weaker conclusion s_n = 0 or |s_n| ≥ 2; precision points:
Costakis "states" Theorem 6 (the proof is not in the report), Theorem 5(ii) gives one g for all |a| > 1, Theorem 3
covers U_1 for a in the closed disc, Bourdon's theorem on p(T)x credited (as recalled by Shkarin), the
enumeration behind the numbers of Remark 5.1 named, "compare" for MacLane and Birkhoff (not read), volume and pages
of Mortini–Pflug 2023 (zbMATH); the record of this run in the note, in this report and in the package; and the
README (the names used in the comments of the earlier programs; the weaker form tested by two programs).

## Relation to the literature, novelty and scope
- **Searches.** First version (2026-10-18): arXiv API (9 queries), zbMATH (7), Crossref (4), Semantic Scholar
  (citations of the report), three web searches; read: the report, Costakis's one-page pre-workshop abstract on the
  workshop page, Gardiner 2014 (arXiv 1301.2082), Shkarin 2010 (arXiv 1209.0984), Bernal-González–Conejero–
  Costakis–Seoane-Sepúlveda 2018 (arXiv 1710.10413), Chan–Hofstad–Walmsley 2020 (arXiv 1912.02371). Runs A and B:
  see the table. Writing stage (2026-10-19): Crossref for every DOI of the bibliography, zbMATH for Birkhoff 1929,
  two arXiv API queries for recent work (newest relevant items: arXiv 2602.19343, 2506.12763; neither concerns
  polynomial multiples); no web search. OpenAlex was not available to us at that time.
  Third verification run (2026-10-19): Crossref (all 13 DOIs of the bibliography; three keyword or bibliographic
  queries), OpenAlex (the records of the report and of Costakis 2010, with 3 and 10 citing works; the lists of
  citing works were not available to us), zbMATH Open (works citing the report: 2,
  Meyrath–Nieß 2011 and Mortini–Pflug 2023; works citing Costakis 2010: 6, none on products with polynomials; three
  keyword queries), arXiv API (9 queries), one web search; the survey of Aron and Markose, J. Korean Math. Soc. 41
  (2004), could not be fetched (zbMATH review read). Nothing new was found.
- **No printed answer to Question 5 was found.** Closest: Shkarin's remark on ideals (entire, f-dependent
  multiplier); Costakis 2010, Theorem 1.7 (residual weighted orbits, the tool behind Theorem 6); Gardiner 2014
  (the U(D, 0) analogue answered negatively); hypercyclic algebras (Shkarin 2010, Theorem 1.4: g ∘ f, not p·f).
  Works citing the report (zbMATH, Semantic Scholar): Meyrath–Nieß 2011, Mortini–Pflug 2023; also Mouze–Munnier
  2016; abstracts read, unrelated to pf. A search that finds nothing is not a proof of novelty.
- **Not accessible to us.** Bayart–Matheron, Dynamics of Linear Operators (2009), and Grosse-Erdmann–Peris, Linear
  Chaos (2011); the observation may appear there as an exercise. Not read: MacLane 1952, Birkhoff 1929,
  Grosse-Erdmann 1990, Shkarin 1993 (cited for background, the last two as in the source); the journal version of
  Shkarin 2010 (the arXiv version was read).
- **Scope.** Answered: Question 5, negatively (Theorem A), with one f_B for all non-constant p (Theorem B). New to our
  knowledge: only the observation that formula (1) can be controlled on block-built MacLane functions (with ramps
  for Theorem B). No priority is claimed.
- **Suggested label for the record.** OWR-1787-006 answered negatively (unrefereed note); under the dataset policy
  the status stays open (or partially solved) with evidence unrefereed_claim.

## 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/