{"id":"60135379-aed8-4c44-ba9d-7c53e310246d","arxiv_id":"2509.01029","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For every n≥2, the statement 'there is an n-entangled set of reals but no (n+1)-entangled set' is consistent with PID plus Martin's Axiom.","lead":"This paper proves a consistency result in set theory: for any number n, it is consistent that a set of real numbers can be entangled in all n-way patterns but fail an (n+1)-way pattern. It answers an open question from a 2025 paper and also shows limitations of construction-scheme axioms.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem rests on Theorem 2.6, imported from [13] with no proof and no stated large-cardinal hypothesis; if the Neeman-iteration consistency proof needs a supercompact, the abstract's unqualified 'is consistent' is incomplete.","rationale":"The reader's weakest_assumption is exactly the reliance on Theorem 2.6 from [13]. This is the most load-bearing link in the chain: without a correct consistency proof of PFA_{n-ent}(E), Theorem 0.2 collapses, since Section 3 only shows that PFA_{n-ent}(E) destroys higher entangled sets while preserving E. The concern is not merely about style: PFA itself has the consistency strength of a supercompact, and despite the restricted 'preserves E' class, the cited Theorem 97 may require such a cardinal. The paper gives no proof, so the strength is not verifiable from the manuscript. I do not treat disagreement with consensus as a flaw, and I do not accuse the authors of dishonesty; this is a standard completeness/attribution issue. The internal Lemma 3.7/3.8 case in Theorem 3.10 is a secondary but real written gap; it is likely repairable, so it does not change the conditional verdict. Therefore the appropriate judgment remains CONDITIONAL, matching the reader, so verdict_should_be is UNCHANGED. The proposed test—reading the cited proof and checking its large-cardinal hypotheses—would settle whether the main concern lands: if no large cardinal is needed, the unqualified consistency claim is fine; if a supercompact is required, the central claim must be rephrased as relative consistency.","tokens_in":24728,"tokens_out":16758,"duration_ms":203925,"concrete_test":"Obtain the proof of Theorem 97 in [13] (and the Neeman iteration background in [22]) and check the exact hypotheses. Specifically: does the construction of a model of PFA_{n-ent}(E) use a supercompact cardinal, a Laver diamond, or only CH/inaccessibility? If it uses a supercompact, restate Theorem 0.2 as a relative consistency result with that hypothesis and update the abstract accordingly. Separately, re-verify the recursive step of Theorem 3.10 in the case where a_k = a and a is of the second kind: Lemma 3.7 does not apply as stated; check whether Lemma 3.8 supplies the needed (y,z) and whether hypotheses (IV) and (V) are still obtained.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 0.2 is obtained by combining Theorem 3.10 with the parametrized axiom PFA_{n-ent}(E). The consistency of PFA_{n-ent}(E) is Theorem 2.6, declared 'Essentially Theorem 97, [13]' and given no proof here. The paper also imports the Neeman preservation theorem (Theorem 2.5) and the derivation of PID (Theorem 2.8) from [13]. Thus the central claim's lower bound is not established inside this paper. If the argument in [13] relies on a supercompact cardinal or another large-cardinal assumption, then Theorem 0.2 and the abstract's 'is consistent' are only relative-consistency statements; as written they assert consistency in the unqualified ZFC sense. No large-cardinal hypothesis appears in Theorem 0.2 or the abstract, and Theorem 2.6 is not accompanied by a proof, so the consistency strength of the main theorem is left unverified. An additional, subordinate gap is that the proof of Theorem 3.10 invokes Lemma 3.7 in the case 'a_k = a', but Lemma 3.7 requires a to be of the first kind over the relevant chain; a may be of the second kind in that case, and the proof does not switch to Lemma 3.8, which covers second-kind sets. This does not replace the imported-theorem concern but shows the preservation proof also needs a careful repair.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper answers Question 3.5 of Carroy-Levin-Notaro [6] in the positive: for every n >= 2, the statement 'there is an n-entangled set but no n+1-entangled set' is shown consistent together with PID and MA (Theorem 0.2). The proof passes through a parametrized proper forcing axiom PFA_{n-ent}(E), argued to be consistent in [13] (Theorem 2.6) and to imply MA and PID (Theorems 2.7-2.8). The new technical core is a forcing P(X,E) built from finite chains of 'good pairs' (Section 3), which is proper, preserves the n-entangledness of a fixed E, and makes an arbitrary uncountable X non-(n+1)-entangled. The second half of the paper gives construction-scheme results: FCA^Delta produces n-entangled sets that are not n+1-entangled for each n (Theorem 0.3), and 2-capturing schemes are shown insufficient for constructing entangled sets (Theorems 0.4 and 0.5).","tokens_in":25133,"tokens_out":33217,"duration_ms":340085,"significance":"If Theorem 0.2 is established, it is a substantial result: it answers the open question from [6] and shows the finite n-entangledness hierarchy can be terminated at any prescribed level in models of two strong forcing axioms. The forcing P(X,E) and the preservation argument are original and constitute the main technical contribution; the construction-scheme results are clean and useful, and Theorem 0.3 unifies prior CH-based constructions. The paper is proof-based; no machine-checked proofs or code are provided. The significance is tempered by the fact that the lower-bound consistency theorem is imported from an unpublished source and the preservation proof currently has a gap; both are fixable but are load-bearing.","major_comments":[{"comment":"The consistency of PFA_{n-ent}(E) is the lower bound for the main theorem, but it is quoted ('Essentially Theorem 97, [13]') without proof, without an explicit large-cardinal hypothesis, and deferred to an upcoming joint paper. The abstract and Theorem 0.2 state 'is consistent' unqualified; if the [13] proof uses a supercompact cardinal, as is typical for PFA-type axioms, the consistency claim must be qualified. Please state the exact hypothesis, give the proof or a detailed reference with the argument, and adjust the abstract accordingly.","section":"§2, Theorem 2.6"},{"comment":"In the recursive construction, the case 'k ∈ P and a_k = a' is handled 'using Lemma 3.7'. Lemma 3.7 requires a to be of the first kind over the chain F_{n_r+k}; the proof has not shown this, and it can fail (a may have several elements in one F(i+1)∖F(i) block). In that situation Lemma 3.7 is inapplicable; Lemma 3.8 does apply because a ∩ F_{n_r+k}(0)=∅, and since σ_k=τ when a_k=a, its conclusion type(a,b)=τ is exactly what the recursion needs. The case split on first/second kind is therefore necessary.","section":"§3, proof of Theorem 3.10"},{"comment":"The genericity criterion for P(X,E) — hence its properness — is stated and its proof omitted ('we leave it to the reader'). Properness is required to apply PFA_{n-ent}(E), and the analogous preservation proof (Theorem 3.10) contains the gap described above, so this omission cannot be regarded as routine. A proof or at least a detailed sketch should be included. The paper should also explicitly describe the ω1-many dense sets that turn the generic-filter object Q_G of Corollary 3.5 into a witness of non-(n+1)-entangledness under PFA.","section":"§3, Theorem 3.4"}],"minor_comments":[{"comment":"The displayed definition of PFA_{n-ent}(E) ends with 'then m(P)' with no conclusion; presumably 'm(P)=ω1'. As printed the axiom is incomplete.","section":"§2, PFA definition"},{"comment":"The definition of a_i^F uses F(i+1) for i ≤ n; the convention that the top model F(|F|) means the universe (stated earlier for chains) should be repeated here to avoid ambiguity.","section":"§3, Definition 3.6"},{"comment":"The recursion bound 'Assume k ≤ n' should be over the number n_r^p of new good pairs, not the fixed integer n; as written it is confusing.","section":"§3, proof of Theorem 3.10"},{"comment":"The homeomorphism between E_F, E_e, and E_o is asserted solely from the equality of the Δ-values. Since the definitions of e_α and o_α allow sign flips (e.g., e_α(l) = -Ξ_α(l)), the order at the first difference can be reversed, so the claimed homeomorphism needs a proof.","section":"§5, after Prop. 5.5"},{"comment":"The inequality 'n = |C_odd| < |C_{l-1}^j| - 1 = n - 1' is garbled; the contradiction should be n ≤ n - 1 (or |C_odd| ≤ |C_{l-1}^j| - 1). Please fix.","section":"§5, proof of Prop. 5.4, Case 2"},{"comment":"'It is easy to see that P is a ccc-forcing provided that P is an entangled (thus, increasing) set' should say 'provided that E is an entangled set' — P is the forcing notion, not the entangled set.","section":"§6, proof of Theorem 6.1"},{"comment":"The proof uses 'Claim 1 of Lemma 8.5 in [1]' in a generalized n-entangled form. Provide a reference or a proof of this generalization, since [1] states it for entangled sets only.","section":"§2, proof of Theorem 2.7"},{"comment":"[13] is cited without a venue. If it is a preprint, please state its status; given that Theorem 2.6 is essential, a published reference or an appendix with the argument would be preferable.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main theorem's lower bound rests on Theorem 2.6, attributed to [13], which is cited without a publication venue and whose full proof is promised in an upcoming joint paper with the author as a co-author. The editor may want to confirm the status of [13] and the large-cardinal hypothesis. The paper also relies on the author's own prior work for FCA^Δ and construction schemes; those are separate papers and appear appropriate, but their availability should be checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem answers Question 3.5 of [6] positively: for every n≥2, you can consistently have an n-entangled set and no (n+1)-entangled sets, together with PID and MA. That is a real result. The forcing P(X,E) that destroys (n+1)-entangledness while preserving n-entangledness is a serious construction, and the preservation proof in Theorem 3.10 is dense but looks like a genuine argument. The FCA^Δ results and the m_F > ω1 results also do real work: the first gives a uniform construction of the whole hierarchy from one axiom, the second shows 2-capturing schemes alone are insufficient to produce entangled sets. There is substance here.\n\nThe main soft spot is the foundation of Theorem 0.2. The consistency of PFA_{n-ent}(E) is imported as Theorem 2.6, credited as \"Essentially Theorem 97, [13]\" with no proof and no stated large-cardinal hypothesis. That is load-bearing. If the proof in [13] needs a supercompact, the abstract's unqualified \"is consistent\" overstates the result—it is at best a relative consistency claim. The same applies to the imported Neeman preservation theorem and the derivation of PID. The author should either prove Theorem 2.6 in this paper or give a precise citation with explicit hypotheses and a source that is actually available.\n\nThere is also a smaller gap in the proof of Theorem 3.10. In the case a_k = a, the proof invokes Lemma 3.7, but that lemma requires a to be of the first kind over the relevant chain. The text does not verify that, and if a is of the second kind the proof should switch to Lemma 3.8. This looks repairable but needs to be written out.\n\nThe scheme section has a couple of rough edges: the piecewise definition of e has a line that looks like a sign typo, and Proposition 5.4's Case 2 inequality is compressed. On a careful read the contradiction can be recovered, but the section needs editing.\n\nBottom line: this paper is for set theorists working on entangled sets, MA/PID, or construction schemes. It deserves a serious referee—conditional accept, not desk reject. The referee should demand (a) a complete statement and proof, or an explicit citation with hypotheses, for PFA_{n-ent}(E); (b) a patch to the Lemma 3.7/3.8 case split; (c) cleanup of the scheme section. I would cite the main theorem once those are in place.","headline":"Answers a real open question, but the main theorem leans on an unproven imported consistency proof; worth refereeing with conditions.","tokens_in":25608,"tokens_out":8561,"would_cite":true,"duration_ms":89846,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E65","03E75","03E50","03E05","03E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every n ≥ 2, there is a consistent model with an n-entangled set but no (n+1)-entangled set.","keywords":["entangled sets","forcing axioms","Martin's axiom","P-ideal dichotomy","proper forcing","construction schemes","FCA^Delta","consistency"],"falsifier":"Read Theorem 97 of [13] (cited here as Theorem 2.6) and check whether its consistency proof for $\\mathrm{PFA}_{n\\text{-ent}}(E)$ uses large cardinals beyond ZFC; if it does, the unqualified consistency claim in Theorem 0.2 is false as stated. Alternatively, attempt to construct an $(n+1)$-entangled set inside a model of $\\mathrm{PFA}_{n\\text{-ent}}(E)$; any such set would refute the paper's main implication.","tokens_in":24647,"feed_emoji":"♾️","tokens_out":9145,"duration_ms":97818,"temperature":0.7,"texified_at":"2026-08-05T20:19:42.272633+00:00","pith_summary":"This paper proves that the hierarchy of entangledness among sets of reals can stop at any prescribed finite level: for every $n \\ge 2$ there is a model of set theory in which an $n$-entangled set exists but no $(n+1)$-entangled set exists, and the model can also satisfy the P-ideal dichotomy (PID) and Martin's axiom (MA). The proof imports a parametrized forcing axiom, $\\mathrm{PFA}_{n\\text{-ent}}(E)$, from prior work by Guzmán and Todorcevic, and shows it forces away all $(n+1)$-entangled sets while preserving the $n$-entangledness of a chosen exemplar E. A second thread shows that the axiom $\\mathrm{FCA}^\\Delta$ yields pairwise homeomorphic sets at every exact level of entangledness, and two limitative theorems say that $2$-capturing construction schemes are too weak to produce entangled sets. Together the results answer an open question of Carroy, Levine, and Notaro and sharpen the known behaviour of Abraham–Shelah entangled sets under strong forcing axioms.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5345,"prompt_tokens":899,"completion_tokens":4446,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":899,"completion_tokens_details":{"reasoning_tokens":3561}},"feed_headline":"Consistent: n-entangled sets exist with no (n+1)-entangled sets","feed_subtitle":"A forcing construction preserves an n-entangled set while killing all (n+1)-entangled ones, even under PID and Martin's axiom.","key_machinery":"The key mechanism is the forcing $\\mathbb{P}(X,E)$, whose conditions are finite $\\in$-chains of countable elementary submodels — 'good pairs' $(s, x)$ with $x(i) \\in s(i+1) \\setminus s(i)$ — together with the constraint that the types of any two distinct $(n+1)$-tuples $x, y$ are non-constant. The chain of submodels makes the forcing proper and supports two duplication lemmas (Lemmas 3.7 and 3.8) that split the work according to whether a small piece of E sits in one or many submodel layers; these lemmas preserve the $n$-entangledness of E while the generic filter adds an uncountable family of $(n+1)$-tuples from X with no constant type, so X is no longer $(n+1)$-entangled. The parametrized axiom $\\mathrm{PFA}_{n\\text{-ent}}(E)$ then supplies the","core_discovery":"The paper's central claim is that for each fixed integer $n \\ge 2$, the statement 'there is an $n$-entangled set of reals, but there is no $(n+1)$-entangled set of reals' is consistent with ZFC plus the P-ideal dichotomy and Martin's axiom. The proof works through a parametrized forcing axiom, $\\mathrm{PFA}_{n\\text{-ent}}(E)$, which asserts that a fixed set E is $n$-entangled and that every proper forcing preserving E's $n$-entangledness has a filter meeting any family of $\\aleph_1$ dense sets. This axiom, imported from Guzmán and Todorcevic, is consistent and implies both MA and PID. The genuinely new piece is a forcing $\\mathbb{P}(X,E)$ that, for any uncountable X, is proper, preserves E's $n$-entangledness, and forces X to stop being $(n+1$","pith_inferences":["The good-pairs forcing ℙ(X,E) is a template: any property of X that can be killed by an uncountable family of finite tuples avoiding a definable set of types may be destructible this way while preserving a hereditary property of E.","If the imported consistency of PFA_{n-ent}(E) turns out to require large cardinals, the main theorem can likely be repaired by re-running the argument inside a model of PFA for the right class of forcings, but the unqualified ZFC-consistency claim would need revision.","The FCA^Δ result suggests that the Cohen model, where FCA^Δ holds, contains a complete ladder of exact entangledness levels; checking this would give a CH-based witness without any forcing axioms.","Problem 0.6 — whether every 2-entangled set can be made n-entangled — might be approachable by iterating ℙ(X,E) along the uncountable targets X, destroying their higher entangledness one by one."],"forward_implications":["The hierarchy of entangledness is not well-ordered by consistency strength: each level n can be the top level in a model of PID+MA.","Question 3.5 of [6] has a positive answer for every n ≥ 2.","Under FCA^Δ, the full spectrum of exact entangledness levels occurs simultaneously among pairwise homeomorphic subsets of ℝ.","Neither the existence of a 2-capturing scheme nor a 2-Δ-capturing scheme implies the existence of an entangled set, and a 2-Δ-capturing scheme can coexist with MA.","The boundary between finite and infinite entangledness is not merely a matter of taking all n; each finite level behaves like an independent phenomenon."],"supporting_citations":[{"why":"Supplies the preservation theorem for Neeman iterations and the consistency of the parametrized forcing axiom PFA_{n-ent}(E) on which the main theorem rests.","marker":"[13]"},{"why":"Defines the Neeman iteration with two-type side conditions that Theorem 2.5 uses in the preservation result.","marker":"[22]"},{"why":"Provides the duplication lemmas and the claim that a ccc forcing failing to preserve n-entangledness can be corrected by another ccc forcing, used to show PFA_{n-ent}(E) implies MA.","marker":"[1]"},{"why":"Introduces n-entangled sets and proves the base consistency of having n-entangled sets without entangled sets under MA.","marker":"[2]"},{"why":"Poses Question 3.5, the open problem that Theorem 0.2 answers positively.","marker":"[6]"},{"why":"Introduces the axiom FCA^Δ and the Δ-capturing framework used in Theorem 0.3 to build the exact-level entangled sets.","marker":"[8]"}],"fun_headline_variants":["For every n≥2, consistent: n-entangled, no (n+1)-entangled","n-entangled set exists without any (n+1)-entangled, consistently","Consistency: n-entangled but no (n+1)-entangled for any n≥2","No (n+1)-entangled sets even when n-entangled ones exist (consistent)","Consistent with MA and PID: n-entangled but no (n+1)-entangled"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The paper's main theorem rests on the imported consistency of the parametrized forcing axiom $\\mathrm{PFA}_{n\\text{-ent}}(E)$ from [13], a theorem stated with no proof here and no explicit large-cardinal hypothesis, together with the omitted proof that $\\mathbb{P}(X,E)$ is proper; if either fails, the consistency claim does not go through.","fun_headline_variants_meta":{"raw":{"variants":["For every n≥2, consistent: n-entangled, no (n+1)-entangled","n-entangled set exists without any (n+1)-entangled, consistently","Consistency: n-entangled but no (n+1)-entangled for any n≥2","No (n+1)-entangled sets even when n-entangled ones exist (consistent)","Consistent with MA and PID: n-entangled but no (n+1)-entangled"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000676,"raw_usage":{"total_tokens":2905,"prompt_tokens":731,"completion_tokens":2174,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":2055}},"tokens_in":475,"tokens_out":2174,"duration_ms":15979,"temperature":1.0,"reasoning_tokens":2055,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:59:41.692038+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Read Theorem 97 of [13] (cited here as Theorem 2.6) and check whether its consistency proof for $\\mathrm{PFA}_{n\\text{-ent}}(E)$ uses large cardinals beyond ZFC; if it does, the unqualified consistency claim in Theorem 0.2 is false as stated. Alternatively, attempt to construct an $(n+1)$-entangled set inside a model of $\\mathrm{PFA}_{n\\text{-ent}}(E)$; any such set would refute the paper's main implication.","supporting_citations":[{"cited_title":"The p-ideal dichotomy, martin’s axiom and entangled sets","cited_arxiv_id":null,"evidence_quote":"Supplies the preservation theorem for Neeman iterations and the consistency of the parametrized forcing axiom PFA_{n-ent}(E) on which the main theorem rests."},{"cited_title":"Forcing with sequences of models of two types.Notre Dame J","cited_arxiv_id":null,"evidence_quote":"Defines the Neeman iteration with two-type side conditions that Theorem 2.5 uses in the preservation result."},{"cited_title":"On the consistency of some partition theorems for continuouscolorings,andthestructureof ℵ1-denserealordertypes","cited_arxiv_id":null,"evidence_quote":"Provides the duplication lemmas and the claim that a ccc forcing failing to preserve n-entangledness can be corrected by another ccc forcing, used to show PFA_{n-ent}(E) implies MA."},{"cited_title":"Martin’s axiom does not imply that every twoℵ1-dense sets of reals are isomorphic","cited_arxiv_id":null,"evidence_quote":"Introduces n-entangled sets and proves the base consistency of having n-entangled sets without entangled sets under MA."},{"cited_title":"Some questions on entangled linear orders, 2025","cited_arxiv_id":null,"evidence_quote":"Poses Question 3.5, the open problem that Theorem 0.2 answers positively."},{"cited_title":"Multiple gaps and some finitizations of club and ch, 2025","cited_arxiv_id":null,"evidence_quote":"Introduces the axiom FCA^Δ and the Δ-capturing framework used in Theorem 0.3 to build the exact-level entangled sets."}],"review_version":1}