Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$

Nathaniel Bannister

REVIEW 3 major objections 6 minor 1 cited by

Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the higher derived limits of the inverse system $\mathbf{A}$ do not vanish under the hypotheses $d = \aleph_n$ and $w\diamondsuit(S_{k+1}^k)$ for $1 \le k < n$, and that in the Mitchell model the nonvanishing derived…

desk verdict Good core results on nonvanishing derived limits, but the paper's headline theorem rests on Lemma 5.5, whose proof as written has an unproven coding step. read the letter →

arxiv 2506.14183 v2 pith:JWH4E3XA submitted 2025-06-17 math.LO

Nathaniel Bannister This is my paper · ORCID

classification math.LO MSC 03E3503E0503E17

keywords higherderivedlimitscoherentfamiliesweakdiamondMitchellmodelMillerdominatingnumberinversesystemAsettheoryofthereals

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a general nonvanishing theorem for the higher derived limits of the standard inverse system of abelian groups $\mathbf{A}$ indexed by $\omega^\omega$. It shows that whenever the dominating number $d$ equals $\aleph_n$ and weak diamond principles $w\diamondsuit(S_{k+1}^k)$ hold for $1 \le k < n$, the $n$-th derived limit $\lim^n \mathbf{A}$ is nonzero. This refines earlier results by removing the hypothesis $b=d$ and by removing the full weak diamond at $\omega_1$. In the Mitchell model, the paper completes the picture: $\lim^n \mathbf{A} \ne 0$ exactly for $n = 0, 2$. A key corollary is that higher derived limits do not vanish in the Miller model, disproving a published conjecture.

What carries the argument

The central object is the inverse system $\mathbf{A}$ of abelian groups indexed by $\omega^\omega$ (with $\mathbf{A}_x = \bigoplus_{i<\omega} \mathbb{Z}_{x(i)}$ and projection maps), whose $n$-th derived limit vanishes exactly when every $n$-coherent family of functions is $n$-trivial. The paper's engine is a recursion on an internally approaching chain of elementary substructures $M_\alpha$, combined with the weak diamond principle $w\diamondsuit(S)$: at each stage, if the current family has a trivialization $\Psi$, the construction splits into two branches, one using $\Psi$ and one using $\Psi + \Theta$ where $\Theta$ is an $n$-coherent family nontrivial below $y \wedge g$, so that no single trivialization can cover both branches. The weak diamond then selects a branch $h$ through the binary tree along which every coded trivialization attempt is diagonalized away. The auxiliary notion ``trivial below $g$'' (with $g$ possibly the constant function $\omega$) lets the induction climb from $n$ to $n+1$.

What would settle it

A model of $d = \aleph_2$ with $w\diamondsuit(S_2^1)$ in which every 2-coherent family is trivial (so $\lim^2 \mathbf{A} = 0$) would falsify Theorem 5.1 at $n = 2$; equivalently, one could try to find a concrete recursion where the assumed uniform binary encoding of trivialization attempts provably does not exist, which would show that the diagonalization step of Lemma 5.5 fails.

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Extended reading notes

Core claim

The central discovery is that the nonvanishing of higher derived limits follows from the dominating number alone together with weak diamonds on the stationarity ladder $S_{k+1}^k$, without needing $b=d$ or $w\diamondsuit_{\omega_1}$. The proof constructs, for any attempt at a trivializing family, a binary-branching recursion indexed by ordinals; the weak diamond principle guarantees a branch along which every trivialization attempt fails, yielding a nontrivial $(n+1)$-coherent family below a carefully chosen function $g$. As a consequence, in the Mitchell model the derived limits $\lim^n \mathbf{A}$ are zero for $n \neq 0, 2$ and nonzero for $n = 0, 2$, and in models such as the Miller, Laver, Mathias, length-$\omega_2$ Cohen, and length-$\omega_2$ Hechler models, $\lim^2 \mathbf{A} \ne 0$, contradicting the conjecture that all higher derived limits vanish in the Miller model.

Load-bearing premise

The load-bearing premise is that every possible trivialization of the families built along the recursion can be uniformly encoded as a binary sequence so that the weak diamond principle can diagonalize against them, and the paper does not specify this encoding.

Editorial extensions

If this is right

Under the hypotheses $d = \aleph_n$ and $w\diamondsuit(S_{k+1}^k)$ for $1 \le k < n$, the $n$-th derived limit $\lim^n \mathbf{A}$ is nonzero, so every model with these hypotheses contains a nontrivial $n$-coherent family.
In particular, $\lim^2 \mathbf{A} \ne 0$ in the Miller, Mitchell, Laver, Mathias, length-$\omega_2$ Cohen, and length-$\omega_2$ Hechler models, since each satisfies the relevant diamond hypotheses after forcing.
The Mitchell model is completely analyzed: $\lim^n \mathbf{A} \ne 0$ if and only if $n = 0, 2$, so its derived-limit pattern matches the proper-forcing-axiom pattern.
The conjecture that all higher derived limits of $\mathbf{A}$ vanish in the Miller model is false: the second derived limit is provably nonzero there.
With $d = \aleph_n$, for any nonzero abelian groups $G_i$ indexed by $i < \omega_n$ there is a nontrivial $n$-coherent family valued in $\bigoplus_{i<\omega_n} G_i$, showing that small continuum forces an abundance of nontrivial coherent families.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof of the key lemma only uses weak diamond to choose one of two branches at stationarily many stages; if a coding-free version of that choice could be made, Theorem 5.1 might hold under weaker guessing principles such as club guessing, which the author does not claim.
Because the group-valued result replaces weak diamonds with the large cardinality of the value group, the paper suggests that the size of the target group is what creates nontrivial coherent families; one could try to transfer this to other inverse systems by enlarging value groups rather than adding diamonds.
The preservation results in Section 7 are stated for 1-coherent families under Cohen, Miller, $\sigma$-distributive, strongly proper, and $\omega^\omega$-bounding forcings; the same methods may extend to $n$-coherent families for $n \ge 2$ only if a version of ``trivial below $g$'' retains its inductive strength, which the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this paper has real, usable results: Theorem 3.1 (diamond on S_1^2 plus d = aleph_2 gives lim^2 A nonzero) drops b from Bergfalk's theorem, and the Mitchell model analysis in Section 6 (lim^n A nonzero iff n = 0, 2) is clean, assuming the Cohen preservation argument checks out. Second, the advertised result, Theorem 5.1 removing weak diamond from Casarosa-Lambie-Hanson, is not proven as written. The issue is Lemma 5.5's final paragraph. It says that for any reasonable encoding, a binary sequence of length alpha codes a trivialization of the family at stage alpha. That only makes sense if |A_alpha| <= |alpha|. The lemma only assumes |A_alpha| = aleph_n, so for alpha < omega_n this is false. In the intended application S = S_n^{n-1}, the ordinals in S do have size aleph_{n-1} and match |A_alpha|, so the lemma is likely repairable. But the statement needs an explicit hypothesis like |alpha| = |A_alpha| for alpha in S, and the proof needs to show the bijections can be chosen coherently. As written, this is a load-bearing gap, not a stylistic hiccup.

What is genuinely new: Section 4's corollaries for standard models (Miller, Laver, Mathias, Cohen, Hechler) follow from Theorem 3.1 and are worth having. Section 7's preservation results, especially Lemma 6.2 on Cohen forcing not trivializing nontrivial 1-coherent families, are clean and useful. The recursive construction in Section 3 is clearly presented. The self-citations are only for context; I see no circularity.

Soft spots beyond Lemma 5.5: Corollary 4.5 lists models without verifying their hypotheses, which is a minor presentational issue. The statement of Lemma 5.5 says S subset omega_{n+1} but the application uses S_n^{n-1} subset omega_n; the index shift should be stated explicitly.

Who is this for: set theorists working on higher derived limits, strong homology, and Cech cohomology. It deserves a serious referee, not a desk rejection, but the referee should insist on a complete proof of Lemma 5.5 before the main theorem is accepted. If the coding can be patched, the paper will be a solid contribution.

Referee Report

3 major / 6 minor

Summary. The paper studies higher derived limits lim^n A of the Mardešić–Prasolov inverse system A indexed by ω^ω. The main theorem (Theorem 5.1) states that if d = ℵ_n and w♢(S_{k+1}^k) holds for all 1 ≤ k < n, then lim^n A ≠ 0, which is presented as a common generalization of results of Bergfalk and of Casarosa–Lambie-Hanson, with the improvement that the weak diamond on ω_1 is no longer needed. The paper also proves lim^2 A ≠ 0 in several standard models (Mitchell, Miller, Laver, Mathias, Cohen, Hechler) and completes the computation of derived limits in the Mitchell model, showing lim^n A ≠ 0 iff n = 0, 2. A further theorem (Theorem 5.9) constructs nontrivial ⊕ G_i-valued n-coherent families under d = ℵ_n. Section 7 establishes preservation of nontrivial 1-coherent families under several classes of forcings.

Significance. If the main arguments are correct, the paper resolves an open question by disproving a conjecture of Bergfalk–Hrušák–Lambie-Hanson about the Miller model, and it provides the first computation of the full pattern of derived limits in the Mitchell model. The removal of w♢_{ω_1} from the hypothesis of the nonvanishing theorem is a genuine strengthening. The paper is careful in many of its constructions, and the preservation results in Section 7 are of independent interest. However, the proof of the central Lemma 5.5 contains a substantial gap concerning the coding of trivializations, and the corollary listing models does not supply the required verifications; these issues need to be addressed before the results can be considered established.

major comments (3)

[Section 5, Lemma 5.5, final paragraph] The proof uses the phrase 'For any reasonable choice of encoding attempts at trivialization' and then defines a function F on 2^{<ω_{n+1}} to apply w♢(S). This is not a proof: to apply w♢, one must specify a concrete coding by which a binary sequence b of length α (for α ∈ S) encodes a trivialization of the family Φ_s on A_α, and one must ensure that these codes are coherent under restrictions along the chain A_β (β < α). The lemma as stated allows |A_α| = ℵ_n for every α, while a sequence of length α can only code objects of size |α|; for α < ω_n this is impossible. In the actual application (Lemma 5.8) one has S = S_n^{n-1} and therefore |α| = ℵ_{n-1} = |A_α| for α ∈ S, but this is not part of the lemma and the proof does not describe the bookkeeping of bijections e_α: α → (A_α)^n × ω needed for a coherent coding. Without such a coding, the function F may not exist and the diagonalization argument in the last paragraph fails. Since Lemma 5.5 is the engine of Theorem 5.1, this is a load-bearing gap.
[Section 4, Corollary 4.5] The corollary asserts lim^2 A ≠ 0 in six models (Mitchell, Miller, Laver, Mathias, length-ω_2 Cohen, length-ω_2 Hechler) but contains no verification that the hypotheses of Corollary 4.2 (or any other theorem) hold in these models. For example, for the Miller model the paper does not show that the ground model satisfies ♢(S_1^2), that the relevant forcing has cardinality ℵ_2, or that it preserves the stationarity of all stationary subsets of S_1^2; indeed Miller forcing in L has size ℵ_1, so Corollary 4.2 does not apply directly. Since the Miller model is featured in the abstract as a disproof of a conjecture, the verification must be supplied or precise references given.
[Section 6, proof of Corollary 6.4] The proof of lim^1 A = 0 in the Mitchell model is quite terse. The reflection step 'there is a club C ⊆ κ such that whenever α ∈ C has uncountable cofinality, ˙Φ reflects to an M_α name ˙Φ_α' is stated without proof or precise definition of the structures M_α, and the subsequent argument relies on the assertion 'all reals added by Mitchell forcing are added on the Cohens coordinate', which is neither proved nor referenced. Since this completes the computation of derived limits in the Mitchell model (Theorem 6.1), the proof needs more detail.

minor comments (6)

[Section 5, Lemma 5.5] The line 'We claim that for some h ∈ 2^{ω_1}' should read h ∈ 2^{ω_{n+1}}; as written, h↾α for α up to ω_{n+1} is not defined.
[Section 5, Lemma 5.5] The third hypothesis uses Φ for the hypothesized n-coherent family, but Φ is already used for the constructed (n+1)-family; in the proof this family is called Θ. Rename in the statement for clarity.
[Section 5, proof of Lemma 5.5] In the successor step, the text defines the extensions on tuples in (A_{α+1})^{n+1} \ (A_α)^n; the complement should be (A_{α+1})^{n+1} \ (A_α)^{n+1} for the definition to match the arity of the families.
[Section 2] The labels 'F act 1' and 'F act 2' have extra spaces and should be formatted as 'Fact 1' and 'Fact 2'.
[References] Reference [11] gives the author as 'K. Kenneth'; the correct name is 'K. Kunen'.
[Section 6] In the first sentence, 'inacessible' should be 'inaccessible'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the main theorems are direct constructions under external hypotheses; the only flagged passage is a coding gap in Lemma 5.5, not a circularity.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 3.1 assumes diamond plus d = aleph_2 and constructs a nontrivial 2-coherent family by an elementary-chain recursion; the contradiction uses the diamond sequence only to locate a stage where a hypothetical global trivialization agrees with the local family. Theorem 5.1 is likewise a construction: Lemma 5.5 propagates nontriviality along an increasing chain using w-diamond, and Lemma 5.8 supplies the induction hypotheses from external cardinal assumptions. No fitted parameter is later renamed a prediction, and no theorem is defined in terms of its own conclusion. The cited external results (Bergfalk's theorem, Casarosa--Lambie-Hanson, Todorcevic's Theorem 3.3.13) are used as independent ingredients, while the author's self-citations [1] and [2] appear only for context and question motivation (e.g., 'see [2, Theorem 7.7] and [1, Theorem 1.2]'), not as load-bearing premises of the main proofs. The one passage deserving scrutiny is Lemma 5.5's final paragraph: 'For any reasonable choice of encoding attempts at trivialization...' This is an unproved and potentially substantial coding assertion about representing trivializations by branches of 2^{<omega_{n+1}}; if no coherent local coding exists, the w-diamond diagonalization does not go through. But that is a gap or incompleteness, not circularity: the existence of such a coding is not assumed by the statement of the conclusion, nor does the argument rename the conclusion as an input. Accordingly, no circular step is identified; the score of 2 reflects at most the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper relies on several cited theorems from the set-theoretic literature, most importantly the coherent-family characterization of derived limits and Goblot's vanishing theorem. No free parameters are fit to data, and no new mathematical entities are postulated; the theorems are proven within ZFC plus the stated hypotheses (d=ℵ_n, weak diamonds).

assumptions (5)

domain assumption Equivalence between vanishing of lim^n A and triviality of n-coherent families (Fact 1, Section 2).
Bridges the algebraic derived-limit question to the combinatorial coherent-family framework used throughout the paper; cited to [3] rather than proved.
standard math Goblot's vanishing theorem (Theorem 2.2): any H-valued n-coherent family indexed by a set of size < ℵ_n is trivial.
Used at every recursion step to extend families to larger index sets.
standard math Todorčević's theorem (Theorem 3.5) providing a non-extendable coherent sequence along a <*-chain.
Core input to Lemma 3.4, which builds the initial nontrivial 1-coherent family.
domain assumption Preservation of ♢(S) under forcings of size κ preserving stationary subsets of S (Fact 2).
Used to derive Corollary 4.2 and its model corollaries.
domain assumption In the Mitchell model, all new reals are added by the Cohen coordinate, and the forcing has κ-cc.
Essential for the proof that lim^1 A = 0 in the Mitchell model (Section 6).

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Cite this review

Pith. "Pith review of Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$." pith.science (2026). https://pith.science/paper/JWH4E3XA

@misc{pith2026250614183,
  author       = {Pith},
  title        = {Pith review of: Nonvanishing Higher Derived Limits without $w\diamondsuit_\omega_1$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWH4E3XA}},
  note         = {Machine review of arXiv:2506.14183}
}

abstract

We prove a common refinement of theorems of Bergfalk and of Casarosa and Lambie-Hanson, showing that under certain hypotheses, the higher derived limits of a certain inverse system of abelian groups $\mathbf{A}$ do not vanish. The refined theorem has a number of interesting corollaries, including the nonvanishing of the second derived limit of $\mathbf{A}$ in many of the common models of set theory of the reals and in the Mitchell model. In particular, we disprove a conjecture of Bergfalk, Hru\v{s}\'ak, and Lambie-Hanson that higher derived limits of $\mathbf{A}$ vanish in the Miller model.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Higher limits of wider systems
math.LO 2025-07 unverdicted novelty 8.0 of 10

Under GCH plus diamond principles, and in Gödel's constructible universe, the higher derived limits lim^n A_λ are nonzero for every cardinal λ where Goblot's vanishing theorem does not force them to zero.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

Bannister

N. Bannister. Additivity of derived limits in the Cohen model . To appear in Israel Journal of Mathematics

work page
[2]

Bannister, J

N. Bannister, J. Bergfalk, J. Tatch Moore, S. Todorˇ cevi´ c.A descriptive approach to higher derived limits. J. Eur. Math. Soc. (2024), published online first

work page 2024
[3]

Bergfalk

J. Bergfalk. Strong homology, derived limits, and set theory , Fund. Math. 236 (2017), no. 1, 17–28

work page 2017
[4]

Bergfalk, C

J. Bergfalk, C. Lambie-Hanson. Simultaneously vanishing higher derived limits, Forum Math, Pi 9 (2021), Paper no. e4, 31pp

work page 2021
[5]

Bergfalk, M

J. Bergfalk, M. Hruˇ s´ ak, and C. Lambie-Hanson. Simultaneously vanishing higher derived limits without large cardinals , Journal of Mathematical Logic 2023 23:01

work page 2023
[6]

Casarosa

M. Casarosa. Nonvanishing higher derived limits without scales . 2024, arXiv preprint: 2404.08983

arXiv 2024
[7]

Casarosa and C

M. Casarosa and C. Lambie-Hanson. Simultaneously nonvanishing higher derived limits . 2024, arXiv preprint:2411.15856

arXiv 2024
[8]

Clausen and P

D. Clausen and P. Scholze. Analytic stacks . https://youtu.be/YxSZ1mTIpaA?si= 8PvF- sTN6GSKkWaWy, 2023. Posted by Institut des Hautes Etudes Scientifiques (IH ´ES)

work page 2023

Show all 15 references

[9]

A. Dow, P. Simon, and J. Vaughan. Strong homology and the Proper Forcing Axiom . Proc. Amer. Math. Soc. 106.3 (1989), 821—828

1989
[10]

Mardeˇ si´ c and A.V

S. Mardeˇ si´ c and A.V. Prasolov.Strong homology is not additive . Trans. Amer. Math. Soc., 307(2):725–744, 1988

1988
[11]

K. Kenneth. Set Theory: An Introduction to Independence Proofs . Studies in logic and the foundations of mathematics, vol. 102. North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1980, xvi + 313 pp

1980
[12]

Todorˇ cevi´ c.The first derived limit and compactly Fσ sets

S. Todorˇ cevi´ c.The first derived limit and compactly Fσ sets. J. Math Soc. Japan, 50(4):831- 836, 1998

1998
[13]

Todorˇ cev ´ ıc.Walks On Ordinals and Their Characteristics

S. Todorˇ cev ´ ıc.Walks On Ordinals and Their Characteristics . Progress in Mathematics, vol
[14]

Velickovic, A

B. Velickovic, A. Vignati. Non-vanishing higher derived limits. Communications in Contem- porary Mathematics, World Scientific Publishing Company, 2023

2023
[263]

Birkh¨ auser Verlag, Basel, 2007, vi + 324 pp

2007

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Bannister

N. Bannister. Additivity of derived limits in the Cohen model . To appear in Israel Journal of Mathematics

work page

[2] [2]

Bannister, J

N. Bannister, J. Bergfalk, J. Tatch Moore, S. Todorˇ cevi´ c.A descriptive approach to higher derived limits. J. Eur. Math. Soc. (2024), published online first

work page 2024

[3] [3]

Bergfalk

J. Bergfalk. Strong homology, derived limits, and set theory , Fund. Math. 236 (2017), no. 1, 17–28

work page 2017

[4] [4]

Bergfalk, C

J. Bergfalk, C. Lambie-Hanson. Simultaneously vanishing higher derived limits, Forum Math, Pi 9 (2021), Paper no. e4, 31pp

work page 2021

[5] [5]

Bergfalk, M

J. Bergfalk, M. Hruˇ s´ ak, and C. Lambie-Hanson. Simultaneously vanishing higher derived limits without large cardinals , Journal of Mathematical Logic 2023 23:01

work page 2023

[6] [6]

Casarosa

M. Casarosa. Nonvanishing higher derived limits without scales . 2024, arXiv preprint: 2404.08983

arXiv 2024

[7] [7]

Casarosa and C

M. Casarosa and C. Lambie-Hanson. Simultaneously nonvanishing higher derived limits . 2024, arXiv preprint:2411.15856

arXiv 2024

[8] [8]

Clausen and P

D. Clausen and P. Scholze. Analytic stacks . https://youtu.be/YxSZ1mTIpaA?si= 8PvF- sTN6GSKkWaWy, 2023. Posted by Institut des Hautes Etudes Scientifiques (IH ´ES)

work page 2023