{"id":"418ee35f-2dfd-47f5-9036-5e2ad0951d1c","arxiv_id":"2606.27568","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves new theorems on the two-disjoint-copies property for compact spaces, including its absence from all scattered spaces and equivalence to uncountability for metric compact spaces.","lead":"The paper defines the two-disjoint-copies property (2DCP) for Tychonoff spaces and proves that no scattered compact space has it, that compact perfect spaces without 2DCP exist in ZFC, and that for compact metric spaces 2DCP is equivalent to the space being uncountable. A smart generalist might read it to see concrete progress on connections between compact space properties and open questions about quotients of function spaces C_p(X) tied to Efimov's problem.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption concerns a background implication used only to situate the problem; the load-bearing steps for the new theorems (scattered non-existence, ZFC counterexamples, and metric equivalence) are independent of that citation and rely instead on explicit constructions and the standard embedding of 2^ω. The unverdicted status is therefore driven solely by lack of manuscript access rather than by any identified gap in the argument.","tokens_in":1855,"tokens_out":378,"duration_ms":32937,"concrete_test":"Apply the Cantor-Bendixson derivative to an arbitrary uncountable compact metric space X; verify that if the process does not terminate then the perfect kernel is non-empty and contains a homeomorphic copy of 2^ω (standard metric argument via nested clopen sets of diameter <1/n). If this embedding holds, the claimed equivalence for metric spaces follows immediately from the paper's other statements.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims rest on two new results: (1) no scattered compact has 2DCP, and (2) explicit ZFC examples of perfect compacta without 2DCP (e.g., Isbell-Mrówka compacta). These are combined with the already-stated fact that every space containing a copy of 2^ω has 2DCP. For compact metric spaces the equivalence then follows from the standard Cantor-Bendixson theorem: countable compact metric spaces are scattered, while uncountable ones contain a perfect subset and hence a copy of 2^ω. No hidden assumption in this chain is visible from the abstract or the described argument structure; the background citation to Banakh-Kąkol-Śliwa is used only for motivation and is not invoked in the new theorems.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines the two-disjoint-copies property (2DCP) for Tychonoff spaces: a space X has 2DCP if there is a sequence (K_n) of nonempty compact subsets such that each K_n contains two disjoint subsets homeomorphic to K_{n+1}. It proves that no scattered compact space has 2DCP, constructs explicit ZFC examples of perfect compacta without 2DCP (including Isbell-Mrówka compacta), shows that for compact metric spaces 2DCP is equivalent to uncountability, identifies classes that satisfy 2DCP (locally homogeneous spaces, spaces containing βω or 2^ω, Brech spaces, Sobota-Zdomskyy Efimov-type spaces) and classes that do not, and discusses the link to infinite-dimensional metrizable quotients of C_p(X) and Efimov's problem.","tokens_in":2007,"tokens_out":518,"duration_ms":30987,"significance":"The results supply ZFC counterexamples to 2DCP among perfect compacta and a clean characterization for the metric case via the Cantor-Bendixson theorem. The explicit constructions (Isbell-Mrówka spaces) and positive results for specific zero-dimensional classes are concrete contributions that separate 2DCP from the mere presence of 2^ω copies and advance the motivating question about C_p(X) quotients.","major_comments":[],"minor_comments":[{"comment":"§1 (or wherever the definition of 2DCP is formalized): the recursive clause 'each K_n contains two disjoint subsets homeomorphic to K_{n+1}' should be stated with an explicit quantifier over the two subsets to avoid any ambiguity about whether the homeomorphisms are required to be onto the whole K_{n+1}.","section":"Definition section"},{"comment":"The statement that 'compact h-homogeneous spaces with 2DCP without such copies exist in ZFC' is asserted without a reference or section pointer in the abstract; ensure the construction is cross-referenced to the relevant theorem number in the body.","section":"Abstract / introduction"},{"comment":"Table or list of examples (if present): the Isbell-Mrówka compacta are cited as failing 2DCP; confirm that the argument uses only the standard properties of these spaces and does not rely on additional set-theoretic assumptions beyond ZFC.","section":"Examples section"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive evaluation of the manuscript, the clear summary of its contributions, and the recommendation to accept. We are pleased that the results on the two-disjoint-copies property, the ZFC examples, the metric characterization, and the connections to C_p-theory and Efimov's problem were viewed as concrete advances.","responses":[],"tokens_in":1429,"tokens_out":86,"duration_ms":10852,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main news is that no scattered compact space has the two-disjoint-copies property and that there are ZFC examples of perfect compacta without it, such as certain Isbell-Mrówka spaces. This immediately yields the equivalence of 2DCP with uncountability for compact metric spaces. The argument chain looks direct: every space with a copy of 2^ω has 2DCP, countable compact metric spaces are scattered, and uncountable ones contain a perfect set. The stress-test note confirms no hidden assumption breaks this.\n\nWhat the paper does well is supply concrete positive and negative classes among zero-dimensional compacta. The Brech spaces and the Sobota-Zdomskyy Efimov-type spaces are shown to have 2DCP, while the Isbell-Mrówka examples fail it. It also records that locally homogeneous compacta and spaces containing βω or 2^ω have the property, and notes that compact h-homogeneous spaces with 2DCP but without those copies exist in ZFC. These distinctions organize material around the open question of infinite-dimensional metrizable quotients of C_p(X) for infinite compact X.\n\nThe soft spot is that the link from 2DCP to the C_p quotient still rests on the earlier Banakh-Kąkol-Śliwa result rather than being re-derived here. That is not a flaw for the new theorems, which appear independent, but it means the paper's motivational framing depends on prior work. The abstract mentions open questions, which is appropriate.\n\nThis is for readers working in set-theoretic topology or C_p-theory who want ZFC separations rather than consistency results. The results are grounded enough and the examples explicit enough that the paper deserves a serious referee rather than a desk reject.","headline":"The paper cleanly separates 2DCP from scattered compacta in ZFC and gives explicit perfect counterexamples, which settles the metric case via Cantor-Bendixson.","tokens_in":2576,"tokens_out":440,"would_cite":false,"duration_ms":16246,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"No scattered compact space has the two-disjoint-copies property and ZFC yields perfect counterexamples.","keywords":["two-disjoint-copies property","compact spaces","scattered spaces","perfect spaces","C_p-theory","Efimov problem","homogeneous spaces","zero-dimensional spaces"],"falsifier":"An explicit construction of a scattered compact space that admits a sequence of nonempty compact subsets each containing two disjoint homeomorphic copies of the next one would falsify the claim that no scattered compact space has 2DCP.","tokens_in":2752,"feed_emoji":"","tokens_out":825,"duration_ms":48311,"temperature":0.7,"pith_summary":"The paper examines the two-disjoint-copies property on Tychonoff spaces, defined via a sequence of nonempty compact sets where each splits into two disjoint homeomorphic copies of the next. It proves that scattered compact spaces never satisfy this property. It also gives ZFC constructions of compact perfect spaces that fail the property. These facts together show that the property holds for a compact metric space exactly when the space is uncountable. The work further identifies sufficient conditions such as local homogeneity and supplies positive examples among certain zero-dimensional Efimov-type spaces while connecting the property to questions about metrizable quotients of C_p(X).","feed_headline":"Scattered compact spaces fail the two-disjoint-copies property","feed_subtitle":"ZFC examples show some perfect compacts also lack it, making the property equivalent to uncountability among metric cases.","key_machinery":"The two-disjoint-copies property (2DCP), the existence of a sequence (K_n) of nonempty compact subsets of X such that each K_n contains two disjoint subsets homeomorphic to K_{n+1}.","core_discovery":"No scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces X the 2DCP is equivalent to uncountability of X. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mrówka compacta. Positive classes among zero-dimensional compact spaces include the Brech as well as the Sobota-Zdomskyy compact spaces of Efimov type, which have 2DCP. Locally homogeneous compact spaces and spaces containing a copy of βω or 2^ω also have 2DCP, though compact h-homogeneous spaces with 2DCP that contain neither such copy exist in ZFC.","pith_inferences":["Because 2DCP is sufficient but not necessary for all infinite compact spaces to produce the desired C_p(X) quotient, the open question whether every infinite compact X yields an infinite-dimensional metrizable quotient of C_p(X) remains unresolved by these examples.","The Isbell-Mrówka compacta supply concrete separable uncountable test spaces that can be checked directly for the presence or absence of infinite-dimensional metrizable quotients of their C_p spaces.","The existence of h-homogeneous compact spaces with 2DCP but without copies of βω or 2^ω shows that standard copies are sufficient but not required for the property."],"forward_implications":["Scattered compact spaces fail to have 2DCP.","Compact perfect spaces without 2DCP exist in ZFC.","For compact metric spaces, 2DCP holds precisely when the space is uncountable.","Locally homogeneous compact spaces have 2DCP.","Compact spaces containing a copy of βω or 2^ω have 2DCP."],"fun_headline_variants":["Scattered compacts fail two-disjoint-copies property","No scattered compact has two-disjoint-copies property","Compact metrics have two-disjoint-copies property iff uncountable","Perfect compacts without two-disjoint-copies property in ZFC","Isbell-Mrówka compacta fail two-disjoint-copies property"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The earlier result that 2DCP produces an infinite-dimensional metrizable quotient of C_p(X) continues to hold for the spaces considered here.","fun_headline_variants_meta":{"raw":{"variants":["Scattered compacts fail two-disjoint-copies property","No scattered compact has two-disjoint-copies property","Compact metrics have two-disjoint-copies property iff uncountable","Perfect compacts without two-disjoint-copies property in ZFC","Isbell-Mrówka compacta fail two-disjoint-copies property"]},"model":"grok-4.3","cost_usd":0.007054,"raw_usage":{"total_tokens":3354,"prompt_tokens":850,"num_sources_used":0,"completion_tokens":85,"cost_in_usd_ticks":70537000,"prompt_tokens_details":{"text_tokens":850,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2419,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":850,"tokens_out":85,"duration_ms":31356,"temperature":1.0,"reasoning_tokens":2419,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T01:23:04.591900+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit construction of a scattered compact space that admits a sequence of nonempty compact subsets each containing two disjoint homeomorphic copies of the next one would falsify the claim that no scattered compact space has 2DCP.","supporting_citations":[],"review_version":1}