{"id":"5c3e5217-2ffc-4ea5-b3d5-a7aadde91d6a","arxiv_id":"2607.03995","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"It is consistent that b=d=ω_n and lim^k A ≠ 0 for all 1≤k≤n, and that b=d=ω_{ω+2} with lim^k A ≠ 0 for every k≥1, by new forcings for lim^1 A ≠ 0 compatible with prior nonvanishing methods.","lead":"The paper constructs forcings that make the first derived limit of a key inverse system A nonzero, and merges them with earlier constructions so that lim^k A can fail for every k from 1 up to a prescribed n (or all k). This fills the remaining gap in known consistency results about when these derived limits vanish, with consequences for strong homology and condensed mathematics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the only external dependence—the transfer of weak diamond and square via Easton’s lemma and the cited combinatorial calculations—and rates the overall risk low. That dependence is standard and does not threaten the new material (the construction and analysis of P_D, the linear iteration of §5, or the Knaster vanishing argument). The paper’s internal density and Δ-system arguments are complete enough for a pure-forcing paper of this type. Consequently the ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":12176,"tokens_out":468,"duration_ms":4383,"concrete_test":"Independently recompute the weak-diamond and square sequences in the product model of §3 (following the exact template of Casarosa–Lambie-Hanson Lemmas 5.6 and 5.10) and confirm that the same combinatorial hypotheses hold after the product with P_{ω_n} (resp. P_{ω_{ω+2}}); if they do, the simultaneous nonvanishing for 1 ≤ k ≤ n is secured.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Theorems 1.3/3.1–3.2) rest on the product of the new ccc nonlinear Hechler-style iteration P_D (Definition 2.1) with the Easton-style Cohen posets C_k that supply the weak-diamond and square principles used by Casarosa–Lambie-Hanson. The paper’s own arguments that P_D is ccc, adds a cofinal copy of D, and produces a nontrivial 1-coherent family (Lemmas 2.2–2.5) are self-contained and standard. Easton’s lemma correctly isolates the reals (hence preserves nontriviality of the family and the values of b and d). The only external step is the claim that the combinatorial principles transfer “identically” to the product; that transfer is routine for these forcings and does not introduce a new soft spot. The vanishing results for Knaster iterations (Theorem 1.4 / §6) are likewise a clean adaptation of Kamo. No load-bearing gap appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs forcings that force lim^{1} A \neq 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 \neq 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 \neq 0 for all 1 ≤ k ≤ n, and of b = d = ω_{ω+2} together with lim^k A \neq 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 \neq 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}.","tokens_in":12391,"tokens_out":1094,"duration_ms":8395,"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.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem 1.3 (and the corresponding abstract claim) the product is written V_{1≤k≤n} lim^k A \neq 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.","section":"Theorem 1.3 / Abstract"},{"comment":"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.","section":"Definition 2.1"},{"comment":"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.","section":"Lemma 2.5"},{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"References"}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, technically clean contribution that fits comfortably in a set-theory or set-theoretic topology journal. The only external dependence is the black-box use of Casarosa–Lambie-Hanson’s combinatorial lemmas; those lemmas are correctly cited and their hypotheses are verified by routine Easton-product arguments. I see no reason to delay acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The main advance is the nonlinear Hechler-style iteration of Definition 2.1: it forces a nontrivial 1-coherent family while realizing any prescribed well-founded cofinal order D in (ω^ω, <*). Product with the Easton Cohen posets that supply the weak-diamond and square principles of Casarosa–Lambie-Hanson then yields the first models with b = d = ω_n and lim^{k} A \neq 0 for every 1 ≤ k ≤ n (and the analogous statement for all k when b = d = ω_{ω+2}). That closes the remaining open case in their program.\n\nThe construction is clean. Lemmas 2.2–2.5 give ccc, the order isomorphism, cofinality, and nontriviality of the coherent family in detail. Section 4 adapts Kunen’s gap analysis to produce the trivialization and freezing posets T_Φ; those lemmas are then used both for the MA(σ-linked) + lim^{1} A \neq 0 iteration of Section 5 and for the vanishing theorem under long finite-support Knaster iterations (Theorem 1.4). The latter is a transparent update of Kamo. Citations are accurate and the dependence on earlier nonvanishing results is properly black-boxed.\n\nThe only soft spot is the claim that the combinatorial principles transfer “identically” to the product; it is routine for these forcings and Easton’s lemma isolates the reals correctly, so it is not load-bearing. A few density arguments are compressed, but nothing appears broken.\n\nThis is for people already working on derived limits of A or on coherent families and gaps. It deserves a serious referee and should be accepted after ordinary polishing. I would cite the new forcings and the simultaneous nonvanishing statements.","headline":"Solid forcing paper that finally gets lim^{1} A \neq 0 into the same models as the higher lim^{k} nonvanishing under prescribed b = d.","tokens_in":13082,"tokens_out":487,"would_cite":true,"duration_ms":4596,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E35","03E17","18G10","55N07"],"pacs":[],"model":"grok-4.5","headline":"It is consistent that the first derived limit of A is nonzero at every continuum size that already forces the higher ones nonzero.","keywords":["derived limits","inverse system A","coherent families","nonlinear Hechler iteration","bounding number","dominating number","finite-support iterations","Knaster posets"],"falsifier":"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.","tokens_in":13018,"feed_emoji":"∞","tokens_out":836,"duration_ms":6496,"temperature":0.7,"pith_summary":"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.","feed_headline":"First derived limit of A can be nonzero at every continuum size","feed_subtitle":"New nonlinear forcing merges lim^{1}A \neq 0 with all higher nonvanishings while fixing b = d = ω_n","key_machinery":"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 \neq 0.","core_discovery":"Relative to ZFC it is consistent that b = d = ω_n and lim^k A \neq 0 for every 1 ≤ k ≤ n, and that b = d = ω_{ω+2} and lim^k A \neq 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.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Consistent: b=d=ω_n and lim^k A ≠0 for every 1≤k≤n","Nonlinear forcing merges lim^{1}A ≠0 with higher nonvanishings at b=d=ω_n","lim^{1}A nonzero with all lim^k A for k≤n when continuum is ω_n","Extends nonvanishing of lim^k A down to k=1 at b=d=ω_n","lim^k A ≠0 for all k≥1 consistent with b=d=ω_{ω+2}"],"cache_read_input_tokens":128,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Consistent: b=d=ω_n and lim^k A ≠0 for every 1≤k≤n","Nonlinear forcing merges lim^{1}A ≠0 with higher nonvanishings at b=d=ω_n","lim^{1}A nonzero with all lim^k A for k≤n when continuum is ω_n","Extends nonvanishing of lim^k A down to k=1 at b=d=ω_n","lim^k A ≠0 for all k≥1 consistent with b=d=ω_{ω+2}"]},"model":"grok-4.5","effort":"low","cost_usd":0.006604,"raw_usage":{"total_tokens":1738,"prompt_tokens":860,"num_sources_used":0,"completion_tokens":143,"cost_in_usd_ticks":66040000,"prompt_tokens_details":{"text_tokens":860,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":735,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":860,"tokens_out":143,"duration_ms":5416,"temperature":1.0,"reasoning_tokens":735,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T22:25:43.530414+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}