# Verification report — AMR-022-7062 (Hayman–Lingham Problem 7.62, A. Hinkkanen)

Verification date: 2026-10-03.

**Verdict.** The answer to the question as posed is no: such an ω₁-family of convergent series cannot be constructed
in ZFC alone. Reading condition (b) for positive non-increasing sequences, as the first sentence of the problem
suggests, a family exists if and only if cof(𝒩) = ℵ₁, where cof(𝒩) is the cofinality of the null ideal. It exists
under CH and in the Sacks model; it does not exist under MA + ¬CH or in the Cohen and random models. As printed,
condition (b) cannot be satisfied at all, not even under CH. The non-existence in ZFC is not new in substance: it
also follows from results of Filmus and Geschke (2010/2013) and of Vojtáš (1987). The new part is the Tukey
equivalence with the null ideal, which confirms a conjecture of Todorčević recorded by Filmus, and the resulting
exact characterizations. The note is unrefereed.

## Statement checked
- **Primary source.** W. K. Hayman and E. F. Lingham, *Research Problems in Function Theory*, fiftieth anniversary
  edition, Springer 2019, doi:10.1007/978-3-030-25165-9; arXiv:1809.07200 (TeX source read).
  - Problem 7.62 (A. Hinkkanen) recalls that countably many convergent series with positive decreasing terms admit
    one converging more slowly than each of them. It asks whether, without any assumption about CH, one can assign
    to every countable ordinal α a convergent series Σ x_{n,α} with 0 ≤ x_{n+1,α} ≤ x_{n,α} such that
    (a) α < β implies x_{n,α}/x_{n,β} → 0, and (b) every x with x_n > 0 and Σ x_n < ∞ satisfies x_n/x_{n,α} → 0
    for some α.
  - Update 7.62 reports no progress.
  - Problem 2.66 is the analogue for maximum moduli of entire functions; Update 2.66 reports unpublished
    independence results of Hinkkanen. The note does not treat Problem 2.66.
- **Corpus record.** ulamai/UnsolvedMath, AMR-022-7062 (status `open`). Its statement matches the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (b) as printed: all positive summable x; members non-increasing | no family exists, in ZFC and in particular under CH (spike sequence y_{4^k} = k^−2) | Remark 7.1 |
| (b′): positive non-increasing x (the evident intent) | a family exists iff cof(𝒩) = ℵ₁; neither provable nor refutable in ZFC + ¬CH | Corollary 1.2 |
| (b) as printed, members not required to be monotone | a family exists iff cof(𝒩) = ℵ₁ | Remark 7.2 |
| (a) and (b′) both comparing remainders instead of terms | a family exists iff 𝔡 = ℵ₁ (monotone version of Vojtáš 1987) | Remark 7.3 |

## Results in the paper
- **Lemma 3.1** (first sentence of the problem, with a non-increasing bound): every countable subset of D has a
  strict ≺-upper bound in D. Here D is the set of positive non-increasing summable sequences and x ≺ y means
  x_n/y_n → 0.
- **Lemmas 3.2–3.3.** Apart from its first member, a Hinkkanen family is a scale of length ω₁ in (D, ≺); it exists
  iff cof(D, ≺) = ℵ₁. In general,
  a scale exists iff add = cof, and its length then has cofinality cof.
- **Theorem 1.1.** (D, ≺) is Tukey equivalent to (ℓ¹, ≤*) and hence to the null ideal; so cof(D, ≺) = cof(𝒩) and
  add(D, ≺) = add(𝒩).
  - Lemma 4.1: identity and strict bounds of running minima.
  - Lemma 4.2: y^x_k = 2^−k + sup{x(n)2^−n : 2^n ≥ k} and Cauchy condensation z ↦ (2^n z_{2^n}).
  - Lemma 4.3 (Bartoszyński): 𝒩 ≡ 𝒞 ≡ ℓ¹. Part (i) is cited (Bartoszyński 1984; Handbook chapter, arXiv version,
    Lemma 3.13); part (ii) (arXiv version, Lemma 4.12) is re-proved in the note.
- **Corollary 1.2** (answer): a family exists iff cof(𝒩) = ℵ₁. It exists under CH and in the Sacks model
  (𝔠 = ℵ₂); it does not exist under MA + 𝔠 > ℵ₁ or in the Cohen and random models. In the random model
  𝔟 = 𝔡 = ℵ₁, so these do not suffice.
- **Corollary 1.3.** A scale in (D, ≺) of some length exists iff add(𝒩) = cof(𝒩).
- **Proposition 5.1 and Corollary 5.2.** Filmus's monotone codes are Tukey equivalent to (D, ≺), by Kraft rounding
  and Filmus's completion lemma (LMCS 2013, Lemma 3.4). Hence the codes are Tukey equivalent to 𝒩 (Todorčević's
  conjecture, recorded by Filmus), and a scale of codes exists iff add(𝒩) = cof(𝒩). This sharpens Filmus's
  Theorems 4.2 and 4.8.
- **Propositions 6.1 and 6.2.** 𝔡 ≤ cof(D, ≺) and cov(𝒩) ≤ cof(D, ≺), with direct proofs that do not use
  Bartoszyński's theorem. They suffice for the non-existence half of Corollary 1.2.

## Computations (exact; scripts and outputs in reproducibility/)
These test finite instances of the elementary inequalities only; the theorems rest on the written proofs.
- **Lead** (`lead/verify_note.py`, standard library only, a few seconds). 9,732 exact checks:
  - Lemma 3.1 for 26 sequences;
  - Lemmas 4.1, 4.2 and 4.3(ii);
  - Proposition 5.1 (rounding), Propositions 6.1 and 6.2 (with b(k) = 2^{k+5}, exact big integers);
  - Remarks 7.1 and 7.3.

  All pass. `lead/negative_control.py` tightens four bounds and fails, as expected (439 failures).
- **Finder** (`finder/check_constructions.py`). Exact checks of the same kind for Lemmas 3.1, 4.1, 4.2,
  Propositions 6.1, 6.2 and Remark 7.1, plus an alternative argument not used in the note. All pass; the output was
  regenerated twice, byte-identical.
- **Independent verification run** (`independent/`, AI-assisted, written separately from the finder's script).
  - Checks V1–V5 and V1b cover Lemmas 3.1, 4.1, 4.2, Proposition 6.2 and Remark 7.2. All pass.
  - A negative control fails, as expected (610 failures).
- **Second independent verification run** (`independent_run_2/`, AI-assisted, written from the text of the note).
  - `run2_checks.py` makes 785,958 exact checks in Parts A–K. They cover:
    - Lemma 2.1, by brute force on small finite relational systems;
    - Lemma 3.1, for 19 sequences with exact tails;
    - Lemmas 4.1, 4.2 and 4.3(ii);
    - Proposition 5.1, with Filmus's completion procedure run on finite data and canonical prefix codes;
    - Propositions 6.1 and 6.2 (b(k) = 2^{k+5});
    - Remarks 7.1 and 7.3;
    - Table 1 against the standard values and the ZFC inequalities of Cichoń's diagram.

    All pass.
  - `run2_negative_control.py`: eleven deliberately broken claims all fail, as expected.
  - All six programs above were re-run from a fresh extraction of the version-1 source package. Their outputs
    were byte-identical to the recorded ones.

## First independent verification run
An independent verification run (AI-assisted, 2026-10-03) re-proved every step, read the problem source and the
relevant parts of Bartoszyński's survey and Filmus's paper in their TeX sources, and re-ran the scripts.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (reading (b′) evidently intended; literal (b) unsatisfiable, stated as such) |
| Proofs | CONFIRMED (every step re-proved; no mathematical error) |
| Computations | CONFIRMED |
| Answer as posed | CONFIRMED (no in ZFC; exact characterization cof(𝒩) = ℵ₁) |
| Novelty | CONFIRMED_WITH_FIXES (prior work of Filmus, Geschke and Vojtáš must be credited) |
| Presentation | CONFIRMED_WITH_FIXES |

All required fixes were applied:
1. The note cites Filmus, LMCS 9(3:7) (2013), arXiv:1308.1600, and MathOverflow question 32437 (Filmus) with
   Geschke's accepted answer.
   - It proves that the codes are Tukey equivalent to (D, ≺) (Proposition 5.1).
   - It credits the "not provable in ZFC" half to Filmus and Geschke.
   - It presents Theorem 1.1 as confirming Todorčević's conjecture, and the characterizations as sharpening
     Filmus's independence theorem.
2. Vojtáš 1987 (CMUC 28) is cited for the remainder order, which also gives 𝔡 ≤ cof(D, ≺) (Remark 7.3). Vojtáš
   1988 (Toposym VI) is cited for the non-monotone eventual-domination analogue (Remark 7.2).
3. Bartoszyński's survey is cited for the Tukey lemmas 𝒩 ≡ 𝒞 and 𝒞 ≡ ℓ¹ and the morphism lemma (Lemmas 3.13,
   4.12 and 2.5 of the arXiv version), not for the dual theorem of its Section 4. The proof of 𝒞 ≡ ℓ¹ is included.
4. The speculative remark on Problem 2.66 was removed. The claim that 𝔠 is unconstrained was removed; only the
   Sacks model (𝔠 = ℵ₂) is mentioned.
5. The reading (b′) is stated explicitly. The optional corollary on scales (a scale exists iff add(𝒩) = cof(𝒩))
   was added.

## Second independent verification run
A second independent verification run (AI-assisted, 2026-10-03) was carried out on the version revised after the first
run, before the corrections below were made.
- **Proofs.** It re-derived every proof.
- **Sources read.** It read the following again in their arXiv sources:
  - Problem 7.62, Update 7.62, Problem 2.66 and Update 2.66;
  - the cited results of Bartoszyński's survey. It computed the numbering of the survey's theorem environments from
    the TeX source: Lemma 2.5 is the morphism lemma, Lemma 3.13 is "N ≡ C" and Lemma 4.12 is "C ≡ ℓ¹";
  - the cited definitions and results of Filmus's paper (arXiv:1308.1600v2 is the LMCS version): Definitions 2.3,
    2.4 and 4.1, Lemma 2.5, Theorem 3.1, Lemma 3.4, Theorems 4.2 and 4.8, and the Discussion in Section 5.
- **Other sources.** It read Vojtáš 1987 in the journal scan and MathOverflow question 32437 with Geschke's answer.
- **References.** It checked all DOIs via Crossref.
- **Computations.** It wrote its own exact-arithmetic checks with a negative control, and re-ran all earlier programs
  (see above).

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED |
| Proofs | CONFIRMED (no mathematical error) |
| Computations | CONFIRMED (own checks pass; recorded outputs reproduced exactly) |
| Values in Table 1 | CONFIRMED (standard values for CH, the Sacks, Cohen and random models and MA + 𝔠 > ℵ₁) |
| Credit (Filmus, Geschke, Vojtáš 1987 and 1988, Kholshchevnikova, Bartoszyński) | CONFIRMED |
| Novelty | CONFIRMED (no earlier answer to Problem 7.62 and no resolution of Todorčević's conjecture found) |
| Presentation | CONFIRMED_WITH_FIXES |

Required fixes, all applied:
1. The remainder order of Remark 7.3 now has its own symbol ◁_r. Previously ◁ also denoted the order on codes.
2. Remark 7.3 no longer attributes the remainder comparison to du Bois-Reymond; that attribution was not supported
   by the sources read, and Vojtáš 1987 credits the ordering to Fichtenholz. The remark now also states that its
   two constructions are morphisms in both directions, so (D, ◁_r) is Tukey equivalent to (ω^ω, ≤*).
3. The introduction now says that results of the kind of the problem's first sentence "go back to"
   du Bois-Reymond and Hadamard, instead of attributing the decreasing-terms statement to them.
4. The abstract now reads "a scale of non-increasing convergent series, of some ordinal length, exists if and only
   if add(𝒩) = cof(𝒩)"; the earlier "of any length" could be misread.
5. The Verification paragraph records this run, and the search statement was updated.
6. The release documents now say that the theorems are proved in the note and rest on the written proofs.
7. Two optional clarifications were added:
   - the first member of a Hinkkanen family may lie outside D (introduction);
   - Corollary 5.2 also holds in the classical sense, for the directed preorder ι(c) = O(ι(d)).

## Relation to the literature, novelty and scope
- **Prior work.**
  - Filmus 2013 (codes; CH gives a scale; no scale after adding ℵ₂ Cohen generic codes; Todorčević's conjecture).
  - Geschke's 2010 MathOverflow answer (both consistency results for codes).
  - Vojtáš 1987 (remainder order: 𝔟 and 𝔡) and Vojtáš 1988 (eventual domination: cof(𝒩)).
  - Kholshchevnikova 1983 (remainder order).
  - Bartoszyński 1984 and his Handbook chapter (𝒩 ≡ 𝒞 ≡ ℓ¹).

  None of these mentions Hinkkanen's problem. The negative ZFC answer follows from Filmus's Theorem 4.8 together
  with Proposition 5.1, or from Vojtáš's remainder result; the note says so.
- **New, as far as could be determined.** The Tukey equivalence (D, ≺) ≡ 𝒩 (equivalently, Todorčević's conjecture
  for codes), add(D, ≺) = add(𝒩), and the exact characterizations cof(𝒩) = ℵ₁ and add(𝒩) = cof(𝒩). The deep
  ingredient (Bartoszyński) is classical and the new steps are short, so the result may be known to experts.
- **Searches (October 2026).** arXiv, Crossref, zbMATH, OpenAlex, Semantic Scholar, MathOverflow (Stack
  Exchange API), and four general web searches. No later work on Problem 7.62 and no resolution of Todorčević's
  conjecture was found.
- **Caveats.**
  - MathSciNet and Google Scholar were not searched.
  - Earlier printed versions of the problem list were not consulted.
  - Kholshchevnikova 1983, Vojtáš 1988 and Vojtáš 1993 were seen only through their zbMATH reviews; Vojtáš 1987
    was read in a scan of the journal.

  This negative search is not a proof of priority.
- **Scope.** The note answers Problem 7.62 in the reading (b′) and records the literal and alternative readings. It
  does not treat Problem 2.66 (entire functions).

## 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/