{"id":"c153d2ee-2029-41e4-b9ba-fdd3db740755","arxiv_id":"2507.03734","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove b(I) ≤ b_R(I) for every ideal I, characterize b_R(I) structurally, and show both numbers coincide for a range of definable ideals.","lead":"This set theory paper compares two ways of generalizing the bounding number b to ideals, and proves that one version is always at most the other. It also computes these numbers for several known ideals and answers an open question about null-ideal gaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central inequality b(I) ≤ b_R(I) is internally well-supported, and the only soft spot (the terse construction in Lemma 6.2) is standard and fillable.","rationale":"The reader identified the construction in Lemma 6.2 as the weakest assumption, and I agree that this is the least explicit part of the paper. However, the paper's central claim, b(I) ≤ b_R(I) for every ideal, is proved in Theorem 3.2 with a correct and self-contained argument modulo the cited characterization Theorem 3.1. I checked the algebra of the separation proof and the use of the P_I partition; both are sound. The consistency results rely on cited theorems from Canjar and the authors' earlier work, but no circularity is evident. The Lemma 6.2 gap is a matter of omitted standard details rather than a mathematical error: the required simultaneous choice of A_n and dense limit sequences is possible by a standard transfinite recursion using the fact that ω1 ≤ 2^ω and each U_n contains a Cantor set. Since the repair is routine and does not change any conclusion, the accept verdict should stand. I therefore report no significant objection to the central claim, while noting that the authors should expand Lemma 6.2 for clarity.","tokens_in":13928,"tokens_out":45769,"duration_ms":513746,"concrete_test":"Repair and verify Lemma 6.2 explicitly: choose pairwise disjoint Cantor sets C_n compactly contained in each basic open U_n, set A_n = Q∩C_n, and for each α < ω1 choose a dense sequence ⟨y_{n,α} : n < ω⟩ with y_{n,α} ∈ C_n, all y_{n,α} distinct across pairs (n,α); then define x_{n,α} as a sequence in A_n converging to y_{n,α}. Check that the proof's separation step goes through unchanged; if any choice fails, Theorem 6.3 would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 3.2, is proved by a direct use of Theorem 3.1: for λ < b(I), any candidate (ω, λ)-gap gives a family E of sequences of pairwise disjoint I-sets of size λ, and the characterization yields a partition C_n ∈ I that separates the gap. The inclusion B \\ C ⊆ ⋃_n (C_n ∩ ⋃_{i≤n} E^B_{A_i}) checks out because the C_n form a partition and an element of E^B_{A_m} outside C falls into the first C_j with j ≥ m. No circularity or sign error is apparent; the proof is self-contained modulo the published characterization. The only genuine soft spot in the paper is Lemma 6.2, where the construction of A_n and the limits y_{n,α} is terse: one needs a simultaneous choice of countably infinite A_n ⊆ Q∩U_n whose closures are nowhere dense Cantor sets and, for each α < ω1, a pairwise distinct sequence of limits y_{n,α} ∈ U_n that is dense in [0,1]. This is achievable by standard recursion (choose disjoint Cantor sets C_n compactly inside U_n, set A_n = Q∩C_n, and pick ω1 dense sequences with distinct coordinates), but the paper does not spell out the density or distinctness conditions. Since the gap is easily repairable and does not affect the main theorem, it does not constitute a load-bearing objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares two ideal generalizations of the classical bounding number b: the invariant b(I) introduced earlier by the authors, defined as the least size of an unbounded family in (D_I, ≥_I), and the Rothberger number b_R(I), defined as the least κ for which there exists an I-(ω,κ)-gap. The main result is Theorem 3.2, which proves b(I) ≤ b_R(I) for every ideal I, using a direct argument based on the authors' earlier characterization of b(I). The paper also proves a structural characterization of b_R(I) (Theorem 4.1) in the style of the b(I) characterization, derives consistency results showing that b(I) and b_R(I) can differ in various ways (Theorems 5.3 and 5.6), computes b(I)=b_R(I)=ω1 for every ideal between CONV and CTBL (Theorem 6.3), proves b_R(I)=b(I)=b for every P-ideal with the Baire property (Theorem 6.6), and answers a question of Kankaanpää by showing b_R(NULL) ≥ p (Theorem 7.1). The exposition is clear and the central arguments are generally rigorous, with the main proofs being self-contained modulo cited theorems from the literature.","tokens_in":14219,"tokens_out":12401,"duration_ms":129764,"significance":"The inequality b(I) ≤ b_R(I) is a clean and useful unifying result: it relates two previously separate ideal versions of the bounding number and yields immediate computations for several concrete ideals. The characterization in Theorem 4.1 is a valuable technical tool, and the consistency results provide a fairly complete picture of the possible relationships between b(I), b_R(I), b, d, and c. The paper also settles an open question of Kankaanpää about NULL-Rothberger gaps. The proofs are mostly detailed and check out; in particular, the proof of Theorem 3.2 is clever and correct, and the use of previously published characterizations is appropriate. The main weakness is an under-specified construction in Lemma 6.2, which supports Theorem 6.3; however, the gap is standard and readily repairable, and it does not affect the central inequality or the other main results.","major_comments":[{"comment":"The construction of the sets A_n and the sequences x_{n,α} is incomplete as written. First, 'A_n is a perfect nowhere dense set' cannot be interpreted literally for a subset of Q, since perfect subsets of [0,1] are uncountable; the authors presumably mean that the closure of A_n is perfect and nowhere dense, and this should be stated. Second, the proof requires that for each α < ω1 the set {y_{n,α} : n ∈ ω} be dense in [0,1] in order to conclude that the closure of B_α \\ C is [0,1]; but picking y_{n,α} ∈ U_n for a fixed enumeration of a base only ensures that each y_{n,α} lies in the corresponding U_n, which does not imply density. This can be fixed by enumerating the base with infinitely many repetitions and choosing the A_n and the limits so that for each basic open set U and each α there are infinitely many n with y_{n,α} ∈ U, but the manuscript does not say this. Third, the line 'B_α \\ C = [0,1]' is false as an equality of sets (B_α is countable), and should be replaced by the intended statement that the closure of B_α \\ C is [0,1], or equivalently that B_α \\ C ∉ CTBL. These points need to be addressed, since Lemma 6.2 underpins Theorem 6.3, one of the advertised main results.","section":"Lemma 6.2"}],"minor_comments":[{"comment":"The introductory sentence says 'As a collorary of Theorem 4.1', but the proof actually uses Theorem 3.2 and cited results on b_R(I); the reference to Theorem 4.1 appears inaccurate.","section":"Corollary 3.4"},{"comment":"The title contains the typo 'Rathberger' and should read 'Rothberger'.","section":"Section 6 title"},{"comment":"The phrase 'On can also show' should be 'One can also show'.","section":"Section 2"},{"comment":"The text reads 'I is a Borel ideals and and there is X'; this should be 'I is a Borel ideal and there is X'.","section":"Proposition 2.2(3)"},{"comment":"The displayed formula before Theorem 4.1 has a typographical error in the indexing: '{Eαn :n<ωα<κ}' should be '{E^α_n : n < ω, α < κ}'. A similar notational issue appears in the statement of Theorem 4.1 where the index sets are written in a compressed way.","section":"Section 4 preamble"},{"comment":"The notation A_n is used ambiguously for both the subset of Q and its closure; the authors should consistently distinguish between a set and its closure, for example by writing cl(A_n) when referring to the closure.","section":"Lemma 6.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution to the study of ideal versions of the bounding number. The central Theorem 3.2 is correct and well motivated, and the consistency results and computations are interesting. The main issue is the under-specified proof of Lemma 6.2, which affects Theorem 6.3; the gap is standard to repair, and I would be happy to accept after the authors fill in the details. I do not see any issue with the reliance on the authors' earlier characterization of b(I), as that is a published theorem with an independent proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing to know: this paper proves b(I) ≤ b_R(I) for every ideal I, gives a new characterization of b_R(I) mirroring the known one for b(I), and uses it to compute values for natural ideals. It also answers Kankaanpää's question about NULL-gaps. That is a solid, useful paper.\n\nWhat's new: the inequality is genuinely new, the characterization in Theorem 4.1 is a nice structural tool, and the consistency results in Theorem 5.6 separate the two numbers in a controlled way. Theorem 6.3 computes both numbers for ideals between CONV and CTBL, and Theorem 6.6 does the same for P-ideals with the Baire property. All of these are real additions.\n\nI checked the main proofs. Theorem 3.2 works; the inclusion at the end is correct. Theorem 4.1 checks out. The reliance on earlier published results, including the authors' own characterization of b(I), is fine — no circularity.\n\nThe one genuinely soft spot is Lemma 6.2. It says A_n is a perfect nowhere dense set, but A_n ⊆ Q cannot be perfect in [0,1]; the intended statement is that the closure of A_n is a perfect nowhere dense set. More importantly, the proof needs that for each α < ω1 the set {y_{n,α}: n ∈ ω} is dense in [0,1] to conclude B_α \\ C = [0,1]. The paper does not state or justify the simultaneous choice of these limits. It is a standard recursion, so the gap is easily repairable, but it should be written out. There are also a couple of minor typos, e.g., Lemma 6.5 states b(I, Fin, Fin) where it should be b_R(I, Fin, Fin).\n\nWho it's for: set theorists working on cardinal characteristics and ideals. It deserves a serious referee and, after a small revision addressing Lemma 6.2 and the typos, should be accepted.","headline":"A genuinely new inequality between two ideal versions of the bounding number, with clean proofs and one easily fixable gap in a technical lemma.","tokens_in":14774,"tokens_out":5149,"would_cite":true,"duration_ms":49493,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E17","03E05","03E35","03E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every ideal I, the Rothberger number b_R(I) is an upper bound for the alternative ideal bounding number b(I), and the two numbers coincide for several natural classes of ideals.","keywords":["bounding number","Rothberger gap","ideal on omega","cardinal characteristics","P-ideal","Baire property","CONV ideal","CTBL ideal"],"falsifier":"Try to find an ideal I with b_R(I)<b(I); the proof of Theorem 3.2 would then fail, so a concrete check is whether a Rothberger gap of size κ forces an unbounded family of size κ in (D_I,≥_I). For the CONV–CTBL equality, check the simultaneous-choice step in Lemma 6.2: if it cannot be justified, a model with b_R(CONV)>ω1 would refute Theorem 6.3.","tokens_in":13717,"feed_emoji":"♾️","tokens_out":8498,"duration_ms":89443,"temperature":0.7,"pith_summary":"The paper compares two ideal-based generalizations of the classical bounding number b. One, written b(I), measures the smallest unbounded family inside a restricted poset of functions that take each value only on an ideal-small set; the other, the Rothberger number b_R(I), measures the smallest size of an I-Rothberger gap, a pair of orthogonal families with no separating set. The paper's central result is that b(I) ≤ b_R(I) for every ideal I. It then computes both numbers for several classes: they both equal ω1 for every ideal between the convergence ideal CONV and the countable-set ideal CTBL, and they both equal b for every P-ideal with the Baire property. It also proves b_R(NULL) ≥ p, which consistently rules out a NULL-Rothberger gap of size add(NULL), answering a question from the literature.","feed_headline":"Rothberger number dominates the new ideal bounding number","feed_subtitle":"The paper proves b(I) ≤ b_R(I) for every ideal and computes exact values for several definable ideals.","key_machinery":"The central device is a pair of matching characterizations. Theorem 3.1 characterizes b(I) as the least size of a family of sequences of pairwise disjoint I-small sets that 'catches' every partition of ω into I-small sets. Theorem 4.1 characterizes b_R(I) the same way except that disjointness is required simultaneously across all sequences in the family. Because the simultaneous condition is stronger, any family witnessing b_R(I) witnesses b(I), yielding b(I)≤b_R(I). The later sections use Fubini products and direct sums to transfer values between ideals, and a density construction with perfect nowhere dense sets to build the ω1-sized gaps for the CONV–CTBL interval.","core_discovery":"The discovery is that the two seemingly different ideal versions of the bounding number are ordered: the poset-based number never exceeds the gap-based number. The proof works by showing that if λ < b(I), then no I-Rothberger gap of size λ can exist: given any candidate gap, the characterization of b(I) produces a separating set. The same comparison tool also yields exact evaluations: b(I)=b_R(I)=ω1 for all ideals with CONV⊆I⊆CTBL, b(I)=b_R(I)=b for P-ideals with the Baire property, and b_R(NULL)≥p, with consistency results showing b(I)<b_R(I) and d<b(I)=b_R(I) are both possible.","pith_inferences":["The inequality b(I)≤b_R(I) suggests a general pattern: the gap-based version of an ideal cardinal is the stronger one, so any lower bound proved for b_R(I) automatically applies to b(I); one could test whether the same ordering holds for other pairs of ideal invariants defined through gaps versus posets.","The construction behind Lemma 5.2 shows b_R(I⊗{∅})=b(I), so any ideal with a large b(I) can be converted into an ideal with equal Rothberger number; iterating this product might produce longer chains of distinct ideal bounding numbers.","The CONV–CTBL interval result raises the question whether all 'small' definable ideals have both numbers equal to ω1; a natural target would be the ideal of nowhere dense sets, where b_R is known to be add(M).","The paper leaves open whether a Borel ideal can have b(I)<b_R(I)≤c; if such an ideal exists, it would show the inequality is strict inside the definable realm, and the product/direct-sum toolbox here is a plausible route to build it."],"forward_implications":["For every ideal I, the Rothberger number b_R(I) is an upper bound on b(I); in particular any ideal with a small Rothberger gap has a small unbounded family in the restricted function poset.","For every ideal I with CONV⊆I⊆CTBL, both numbers are exactly ω1, so b(CONV)=b_R(CONV)=ω1 and b(CTBL)=b_R(CTBL)=ω1; this can be strictly below b.","For every P-ideal with the Baire property, b_R(I)=b(I)=b, extending the known equality for analytic P-ideals.","b_R(NULL)≥p, and it is consistent that no NULL-(ω,add(NULL))-gap exists, settling the open question.","It is consistent that b=b(I)<b_R(I)≤c, that b<b(I)<b_R(I)≤c, and that d<b(I)=b_R(I)≤c, so the two numbers can differ by a controlled amount."],"supporting_citations":[{"why":"Defines b(I), proves its basic properties including b(I)≥ω1 and the characterization used in Theorem 3.1, and supplies the equality b(I)=b(I⊗J).","marker":"[9]"},{"why":"Introduces I-Rothberger gaps and the number b_R(I), and gives values b_R(I)=ω1 for fragmented ideals used in Corollary 3.4.","marker":"[3]"},{"why":"Proved b_R(NWD)=add(M) and raised the question about NULL-Rothberger gaps answered in Theorem 7.1.","marker":"[13]"},{"why":"Provides p≤b(NULL) and the consistency of add(N)<p used in Theorem 7.1, plus the ideal with d<b(J)≤c used in Theorem 5.3.","marker":"[15]"},{"why":"Gives a maximal P-ideal with b(J)=cf(d) under d=c, used in Theorem 5.3(1).","marker":"[5]"},{"why":"Provides maximal ideals with prescribed b(J) in the Cohen model, used in Theorem 5.6.","marker":"[4]"},{"why":"Provides maximal ideals with prescribed b(J) in the Cohen model, used in Theorem 5.6.","marker":"[6]"},{"why":"Supplies the Baire-property criterion (if infinitely many intervals [m_k,m_{k+1}) are contained in A then A∉I) used in Lemma 6.4.","marker":"[20]"},{"why":"Provides the definition of p and the consistency of add(N)<p used in Theorem 7.1's 'in particular' conclusion.","marker":"[2]"}],"fun_headline_variants":["Two b's: one always smaller","b(I) ≤ b_R(I) for every ideal","Ideal bounding: poset never exceeds gap","Rothberger bound never below new ideal bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the ω1 result for ideals between CONV and CTBL depends on being able to choose, for every n and every countable ordinal α, a sequence inside the nth perfect set with the required limit point, and the paper does not spell out why all these choices can be made simultaneously.","fun_headline_variants_meta":{"raw":{"variants":["Two b's: one always smaller","b(I) ≤ b_R(I) for every ideal","Ideal bounding: poset never exceeds gap","Rothberger bound never below new ideal bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000601,"raw_usage":{"total_tokens":2676,"prompt_tokens":681,"completion_tokens":1995,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":297,"completion_tokens_details":{"reasoning_tokens":1936}},"tokens_in":297,"tokens_out":1995,"duration_ms":17646,"temperature":1.0,"reasoning_tokens":1936,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:06:29.641352+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Try to find an ideal I with b_R(I)<b(I); the proof of Theorem 3.2 would then fail, so a concrete check is whether a Rothberger gap of size κ forces an unbounded family of size κ in (D_I,≥_I). For the CONV–CTBL equality, check the simultaneous-choice step in Lemma 6.2: if it cannot be justified, a model with b_R(CONV)>ω1 would refute Theorem 6.3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines b(I), proves its basic properties including b(I)≥ω1 and the characterization used in Theorem 3.1, and supplies the equality b(I)=b(I⊗J)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces I-Rothberger gaps and the number b_R(I), and gives values b_R(I)=ω1 for fragmented ideals used in Corollary 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved b_R(NWD)=add(M) and raised the question about NULL-Rothberger gaps answered in Theorem 7.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides p≤b(NULL) and the consistency of add(N)<p used in Theorem 7.1, plus the ideal with d<b(J)≤c used in Theorem 5.3."},{"cited_title":"Michael Canjar, Cofinalities of countable ultraproducts: the existence theorem, Notre Dame J","cited_arxiv_id":null,"evidence_quote":"Gives a maximal P-ideal with b(J)=cf(d) under d=c, used in Theorem 5.3(1)."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Provides maximal ideals with prescribed b(J) in the Cohen model, used in Theorem 5.6."},{"cited_title":"MR 2632174","cited_arxiv_id":null,"evidence_quote":"Provides maximal ideals with prescribed b(J) in the Cohen model, used in Theorem 5.6."},{"cited_title":"67 (1980), no","cited_arxiv_id":null,"evidence_quote":"Supplies the Baire-property criterion (if infinitely many intervals [m_k,m_{k+1}) are contained in A then A∉I) used in Lemma 6.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the definition of p and the consistency of add(N)<p used in Theorem 7.1's 'in particular' conclusion."}],"review_version":1}