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Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions

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Pith's one-line read It is consistent that the first derived limit of A is nonzero at every continuum size that already forces the higher ones nonzero.

desk verdict Solid forcing paper that finally gets lim^{1} A eq 0 into the same models as the higher lim^{k} nonvanishing under prescribed b = d. read the letter →

arxiv 2607.03995 v1 pith:FBIFMBUZ submitted 2026-07-04 math.LO math.AT

Nathaniel Bannister , Justin Tatch Moore This is my paper · ORCID
classification math.LOmath.AT MSC 03E3503E1718G1055N07
keywords derivedlimitsinversesystemAcoherentfamiliesnonlinearHechleriterationboundingnumberdominatingfinite-supportiterationsKnasterposets
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

A particular inverse system of abelian groups A, introduced to study strong homology, has derived limits whose vanishing or nonvanishing is controlled by set-theoretic hypotheses. Earlier work produced models in which the higher derived limits lim^k A are nonzero for all k between 2 and n while the continuum is ω_n, but left the first derived limit open. This paper supplies a nonlinear iteration that forces lim^1 A nonzero while arranging that the continuum is any prescribed well-founded partial order of uncountable cofinality. Combining that iteration with the combinatorial principles already known to produce the higher nonvanishings yields models in which every lim^k A for 1 ≤ k ≤ n is nonzero and the continuum is exactly ω_n, and likewise a model in which all of them are nonzero and the continuum is ω_{ω+2}. The same methods also show that many finite-support iterations of Knaster posets force lim^1 A to vanish, giving a clean contrast between the two regimes.

What carries the argument

The nonlinear iteration P_D of Definition 2.1: conditions carry finite stems and finite partial trivializations of a coherent family indexed by a well-founded poset D of uncountable branching number; the generic adds a cofinal copy of D inside ω^ω together with a nontrivial 1-coherent family on that copy, forcing lim^1 A eq 0.

What would settle it

If, after forcing with the product of P_{ω_n} and the relevant Cohen posets, either the 1-coherent family becomes trivial or the weak-diamond/square sequences needed by the earlier lemmas fail, then the simultaneous nonvanishing claim for 1 ≤ k ≤ n is false.

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Extended reading notes

Core claim

Relative to ZFC it is consistent that b = d = ω_n and lim^k A eq 0 for every 1 ≤ k ≤ n, and that b = d = ω_{ω+2} and lim^k A eq 0 for all k ≥ 1. Both statements are obtained by taking a product of a new nonlinear Hechler-style iteration (which adds a nontrivial 1-coherent family indexed by a cofinal copy of a prescribed well-founded poset) with the Cohen posets that force the weak-diamond and square principles needed for the higher derived limits.

Load-bearing premise

The product forcing of Section 3 must preserve both the nontrivial 1-coherent family and the combinatorial principles that force the higher lim^k to be nonzero; if either fails to transfer, simultaneous nonvanishing for k ≥ 2 collapses.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper constructs forcings that force lim^{1} A eq 0 for the Mardešić–Prasolov inverse system A, and shows these are compatible with earlier nonvanishing constructions for higher derived limits. The main new device is a nonlinear Hechler-style iteration P_D (Definition 2.1) indexed by a well-founded poset D with uncountable bounding number; Lemmas 2.2–2.5 establish that P_D is ccc, adds a cofinal copy of D inside (\omega^\omega, <*), and produces a nontrivial 1-coherent family, hence lim^{1} A eq 0. Products of this forcing with Easton-style Cohen posets (Section 3) yield the consistency of b = d = \omega_n together with lim^k A eq 0 for all 1 ≤ k ≤ n, and of b = d = ω_{ω+2} together with lim^k A eq 0 for every k ≥ 1, extending the Casarosa–Lambie-Hanson results that began at k = 2. Sections 4–5 develop a trivialization poset T_Φ and a linear iteration that forces MA(σ-linked) + lim^{1} A eq 0 while keeping the continuum arbitrary. Sections 6–7 adapt Kamo’s arguments to prove that lim^{1} A = 0 after any finite-support iteration of nontrivial Knaster posets of length of cofinality > ℵ_{1}.

Significance. The work closes a natural gap left by Casarosa–Lambie-Hanson: simultaneous nonvanishing of lim^k A can now begin at k = 1 rather than k = 2, and can hold for all positive k while b = d = ω_{ω+2}. The nonlinear iteration of Definition 2.1 is a clean, self-contained contribution that also produces prescribed cofinal suborders of (ω^ω, <*). The freezing/trivialization analysis of Section 4 and the Knaster-iteration vanishing theorem (Theorem 1.4) give a robust picture of when lim^{1} A vanishes under ccc forcing. All arguments are elementary forcing constructions relative only to ZFC; no large-cardinal hypotheses are required. The results therefore advance the program of determining the possible patterns of vanishing and nonvanishing for the derived limits of A.

minor comments (5)
  1. [Theorem 1.3 / Abstract] In the statement of Theorem 1.3 (and the corresponding abstract claim) the product is written V_{1≤k≤n} lim^k A eq 0; the intended meaning is the simultaneous conjunction, but the notation is nonstandard and could be replaced by an ordinary ∧ or by an explicit quantifier.
  2. [Definition 2.1] Definition 2.1, clause (4) of the order: the equality Φ_{b,p}(n,m) = Φ_{c,p}(n,m) is required only when both stems dominate m; a short parenthetical remark that this is the coherence condition for the eventual family would help the reader.
  3. [Lemma 2.5] Lemma 2.5: the density argument that produces disagreement with an arbitrary name Ψ is correct, but the choice of k = 1 + max s_{b,r}(n) is slightly opaque; a sentence explaining that this places (n,k) outside the graphs of the lower stems would clarify the construction.
  4. [Section 3] Section 3: the appeal to Easton’s lemma is standard, yet a one-line reminder that the Cohen factors C_k add no new reals over the P_{ω_n} extension would make the preservation of the nontrivial 1-coherent family completely explicit.
  5. [References] Several bibliographic entries (e.g., [1], [2], [7], [10]) are still listed as arXiv preprints; if any have appeared, the published references should be updated before final publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new forcings and their products are self-contained constructions that independently verify the hypotheses of prior black-box results.

full rationale

The paper constructs a nonlinear Hechler-style iteration P_D (Definition 2.1) and proves directly that it is ccc (Lemma 2.2), adds a cofinal copy of D in (ω^ω, <*) (Lemmas 2.3–2.4), and produces a nontrivial 1-coherent family (Lemma 2.5). These arguments are internal and do not presuppose lim^{1} A eq 0. The simultaneous nonvanishing theorems (3.1–3.2) are obtained by product with Easton-style Cohen posets; Easton’s lemma isolates the reals (preserving the family and the values of b and d), while the required weak-diamond and square principles are verified by the same combinatorial calculation already present in Casarosa–Lambie-Hanson. Those prior lemmas are used strictly as black boxes whose hypotheses are checked independently. The linear-iteration construction of §5 and the Knaster-iteration vanishing theorem of §6 are likewise self-contained adaptations of Kunen and Kamo, respectively; they do not redefine the target statement or smuggle it in via self-citation. No fitted parameters, definitional equivalences, or load-bearing uniqueness theorems appear. The derivation chain is therefore free of circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper works entirely inside ZFC plus standard forcing technology. No free parameters are fitted. The only background axioms are ordinary ZFC, the existence of the inverse system A (taken from Mardešić–Prasolov), and the combinatorial principles (weak diamond, square) already shown consistent by Casarosa–Lambie-Hanson. The new posets are invented entities but are fully defined and have no independent physical or empirical content.

assumptions (3)
  • standard math ZFC is consistent (used for all relative consistency statements).
    Standard metatheoretic assumption for forcing constructions; invoked throughout Sections 2–6.
  • domain assumption The inverse system A of Mardešić–Prasolov is well-defined and its derived limits control strong homology additivity and condensed derived functors.
    Taken as given from [24] and [11]; the paper never redefines A.
  • domain assumption The weak-diamond and square principles used in Casarosa–Lambie-Hanson Lemmas 5.7 and 5.10 remain valid after product with the new forcing P_D (via Easton's lemma).
    Invoked without re-proof in the proofs of Theorems 3.1 and 3.2; the paper asserts the verification is identical to the earlier one.
invented entities (2)
  • Nonlinear iteration P_a (Definition 2.1)
    purpose: Simultaneously force a nontrivial 1-coherent family and a prescribed well-founded cofinal suborder of (ω^ω,<∗).
    Fully defined by finite conditions with stems, Φ-maps and names for dominating reals; no external evidence required beyond the forcing relation.
  • Trivialization poset T_Φ (Definition 4.1)
    purpose: Analog of Kunen's gap-splitting poset that trivializes or freezes a 1-coherent family.
    Defined explicitly; used to prove ccc characterizations and indestructibility lemmas.

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Pith. "Pith review of Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions." pith.science (2026). https://pith.science/paper/FBIFMBUZ

@misc{pith2026260703995,
  author       = {Pith},
  title        = {Pith review of: Merging $\lim^1 \mathbfA \ne 0$ with other nonvanishing constructions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBIFMBUZ}},
  note         = {Machine review of arXiv:2607.03995}
}
abstract

We develop methods for forcing $\lim^1 \mathbf{A} \ne 0$, where $\mathbf{A}$ is a particular inverse system of abelian groups introduced by Marde\v{s}i\'c and Prasolov in their computation of certain strong homology groups. These methods allow us to extend previous nonvanishing results of Casarosa and Lambie-Hanson for $\lim^k \mathbf{A}$ for $k \geq 2$. Specifically we show that, for a given $n$, it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_n$ and $\lim^k \mathbf{A} \ne 0$ whenever $1 \leq k \leq n$ (previously established with $2 \leq k \leq n$). We also show it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_{\omega+2}$ and $\lim^k \mathbf{A} \ne 0$ for all $k \geq 1$ (previously established with $k \geq 2$). We also adapt proofs of Kamo to show that $\lim^1 \mathbf{A} = 0$ holds in many finite support iterated forcing extensions.

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