{"id":"98a24033-a853-487e-89cf-49547ada997b","arxiv_id":"1908.09494","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalization of Kechris-Rosendal characterizes comeager diagonal conjugacy classes in arbitrary Polish permutation groups, and ball-preserving groups of ordered ultrametric spaces are shown to have a generic element but no generic pair.","lead":"This paper proves a general criterion for when a Polish permutation group of a countable structure admits a comeager diagonal conjugacy class, in terms of joint embedding and weak amalgamation of extendable partial maps. It also shows that ball-preserving bijection groups of ordered ultrametric spaces have a comeager conjugacy class but no comeager 2-diagonal one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.3's Rasiowa-Sikorski step uses uncountably many dense subsets when K_G is uncountable; the paper gives no argument that a countable subfamily suffices.","rationale":"The reader's CONDITIONAL verdict is appropriate. I agree that the Rasiowa-Sikorski step is the least secure point, but the precise obstruction is not that P = K_G must be countable; it is that the dense sets to be met are indexed by all S,T,U ∈ K_G and can form an uncountable family. This matters whenever the signature has function symbols, since finitely generated substructures can be infinite. The countability of finitely generated substructures is insufficient because infinitely many bijections can exist between a fixed pair of such substructures. The standard fix, passing to a relational expansion or proving that a countable cofinal family suffices, is mentioned neither in the theorem nor in the proof. The rest of the argument, including the game-theoretic equivalences and the ultrametric applications, appears coherent conditional on Theorem 3.3. Therefore I do not move the verdict.","tokens_in":18636,"tokens_out":20238,"duration_ms":211741,"concrete_test":"Take M = (Q,S) where S is a fixed Z-shift and G = Sym(Q). Compute K_G and verify that the family {D_{S,T,U} : S,T,U ∈ K_G} used in Theorem 3.3 is uncountable. Then attempt to construct a weakly K_G-injective Φ by meeting only the countable family of dense sets indexed by finite partial bijections, and check whether every infinite S ∈ K_G embeds into the resulting Φ. If some S fails to embed, the Rasiowa-Sikorski step in Theorem 3.3 has a genuine gap that the paper's present argument cannot close; if all embed, the concern is cosmetic and a clarifying remark would suffice.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 3.3, the proof sets P = K_G and applies Rasiowa-Sikorski to the family of all sets F_m, E_S, and D_{S,T,U} for m ∈ M and S,T,U ∈ K_G. Even on the charitable reading that Rasiowa-Sikorski does not require P itself to be countable, it does require the family of dense subsets to be countable. That family is not shown to be countable. The paper states the lemma only for countable P, and with function symbols the finitely generated substructures of a countable M can be infinite: for M = (N,S) and G = Sym(N), K_G contains continuum many bijections between N and N, so the sets E_S and D_{S,T,U} are indexed uncountably. Countability of the set of finitely generated substructures does not help, since a fixed pair of infinite finitely generated substructures can admit continuum many G-extendable bijections. No countable cofinal subfamily of K_G under inclusion is identified, and the proof of weak injectivity requires meeting D_{S,T,U} for every U extending T, not just a countable skeleton. Thus the main equivalence in Theorem 3.10 is not proved as written for arbitrary countable structures; a relational expansion or a countable-approximation argument is needed and is not supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework, based on Krawczyk-Kubis games, for characterizing when a Polish group of permutations of a countable structure has a comeager (or dense) n-diagonal conjugacy class. The main announced result, Theorem 3.10, states that for a countable structure M, a Polish group G <= Sym(M), and n >= 1, G has a comeager n-diagonal conjugacy class if and only if the family K_{G,n} of n-tuples of G-extendable bijections between finitely generated substructures has JEP and WAP. The paper also studies homogenizability of limits of weak Fraisse classes and applies the abstract results to ball-preserving bijections of ordered ultrametric spaces, recovering and extending results on ordered boron trees and Wazewski dendrites.","tokens_in":18951,"tokens_out":30814,"duration_ms":338832,"significance":"If the main theorem is correct, it provides a uniform and substantial generalization of the Kechris-Rosendal characterization to arbitrary countable structures and arbitrary Polish permutation groups, not only automorphism groups of Fraisse limits. The paper supplies detailed game-theoretic arguments, identifies a useful notion of weak K-injectivity, and gives new examples of groups with a comeager conjugacy class but no comeager 2-diagonal conjugacy class. The ultrametric applications are original and connect the abstract machinery to natural topological groups. However, the central equivalence currently rests on a Rasiowa-Sikorski step whose hypotheses are not verified, so the main theorem is not fully proved as stated.","major_comments":[{"comment":"The Rasiowa-Sikorski step is not justified. The proof sets P = K_G and applies the lemma to the family of all sets F_m, E_S, and D_{S,T,U}. The lemma as stated in the proof requires P to be countable, and in any case the family of dense subsets must be countable. The paper does not show that K_G is countable. For a countable structure with function symbols this can fail: if M = (N, S) and G = Sym(M), the substructure N itself is finitely generated, and every permutation of N is a G-extendable bijection from N to N, so K_G has cardinality continuum. No countable cofinal subfamily of K_G is identified, and the proof of weak injectivity requires meeting D_{S,T,U} for every U extending T, not just for a countable skeleton. Consequently the equivalence (1) iff (4) in Theorem 3.10 is not proved as written for arbitrary countable structures; a restriction to a relational expansion or an additional countability/finiteness argument is needed.","section":"Section 3, Theorem 3.3"},{"comment":"The final 'In particular' assertion is not supported by the preceding theorems. Theorems 5.8 and 5.9 are proved for the classes U^\\prec_N, i.e., for ordered rational N-ultrametric Urysohn spaces with uniformly bounded polygon size. The theorem, however, states that for every ordered ultrahomogeneous Polish ultrametric space X, BP(X) has a comeager conjugacy class and no comeager 2-diagonal conjugacy class. No reduction of an arbitrary such space to some U^\\prec_N is given, and the proofs of Theorems 5.8 and 5.9 do not obviously apply to spaces whose distance sets are not the rationals. This claim therefore needs either a proof or an explicit restriction of the statement.","section":"Section 5, Theorem 5.13"}],"minor_comments":[{"comment":"The statement begins 'LetG be a countable structure'; this should be 'Let M be a countable structure'.","section":"Section 3, Theorem 3.10"},{"comment":"The first sentence contains a typo: 'Let us can consider' should be 'Let us consider'.","section":"Section 1, Introduction"},{"comment":"The dense sets are introduced as D_{S,T,U}, but a later sentence refers to 'the sets D_S,f'; this notation is undefined and should be D_{S,T,U}.","section":"Section 3, proof of Theorem 3.3"},{"comment":"The transition from the countable Fraisse limits U^\\prec_N to 'Polish ultrametric spaces' is not explained: U^\\prec_N with its ultrametric is not complete. Please clarify whether X is meant to be a metric completion and why the family of finite subspaces of the completion is the same as that of U^\\prec_N.","section":"Section 5, Theorem 5.13"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper generalizes Kechris-Rosendal to arbitrary Polish permutation groups of countable structures, and the ultrametric applications are genuinely interesting. The central theorem may be true, but as written the proof has a real gap.\n\nThe new content is mostly in Theorem 3.10: for any Polish G ≤ Sym(M), a comeager n-diagonal conjugacy class is equivalent to JEP+WAP for K_{G,n}. That extends the earlier relational/finitely-generated setting, and the Krawczyk-Kubis game formalism is a natural fit. The homogenizability part (Theorem 4.3 and the example) is a clean application of Ahlman's criterion. The ultrametric Section 5 contains new results on BP(U_N^≺): comeager conjugacy class but no comeager 2-diagonal one. These recover older theorems from the author's own paper in a shorter way, and the ball-preserving perspective is worthwhile.\n\nThe soft spot is Theorem 3.3. The proof invokes Rasiowa-Sikorski with P = K_G. The version the paper states requires P countable; the more general standard lemma does not require P countable, but it still needs a countable family of dense subsets. Here the sets E_S and D_{S,T,U} are indexed by all S,T,U in K_G. When M has function symbols, finitely generated substructures can be infinite, and K_G can contain uncountably many mappings between two infinite substructures (e.g., Sym(N) acting on (N,S)). No countable subfamily of K_G is identified that would make the construction go through. This is not a cosmetic gap: the (1)⇒(2) direction of Theorem 3.10 is exactly what needs that construction. The rest of the paper seems to avoid the issue because Section 5 works with relational ultrametric structures, where finitely generated means finite and K_G is countable. I could not find an explicit countability assumption or a workaround in the paper, so the main theorem is not fully proved as stated.\n\nI would still send this to a serious referee. The generalization is significant enough, and the applications are solid enough, that the paper deserves engagement. A referee should ask the author to either state a countability/finiteness assumption or repair the argument. This is a revamp, not a reject.\n\nIf I were to bring it to a reading group, it would be to discuss the gap and whether the theorem has a hidden counterexample. I'd cite the ultrametric results cautiously.","headline":"A promising generalization of Kechris–Rosendal whose main proof has a genuine countability gap; the ultrametric applications are solid.","tokens_in":19417,"tokens_out":9236,"would_cite":true,"duration_ms":91808,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E15","54H11"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every countable structure $M$ and every Polish group $G$ of its permutations, a comeager $n$-diagonal conjugacy class exists in $G$ exactly when the family $K_{G,n}$ of $n$-tuples of $G$-extendable bijections between finitely…","keywords":["weak amalgamation","joint embedding property","diagonal conjugacy classes","ample generics","Polish permutation groups","non-archimedean groups","ultrametric spaces","homogenizable structures"],"falsifier":"Find a countable structure $M$ and a Polish group $G \\le \\mathrm{Sym}(M)$ for which $K_{G,1}$ satisfies JEP and WAP but $G$ has no comeager conjugacy class; such an example would refute the central equivalence as stated. A more immediate check is whether $K_{\\mathrm{Sym}(\\mathbb{N})}$ for the successor structure on $\\mathbb{N}$ is uncountable; if so, the generic-selection step of the proof needs an additional countability argument before Theorem 3.10 can cover that case.","tokens_in":18445,"feed_emoji":"🔄","tokens_out":13278,"duration_ms":118777,"temperature":0.7,"pith_summary":"This paper establishes a direct dictionary between large conjugacy classes in Polish permutation groups and the combinatorics of finite partial maps. For a countable structure $M$ and a Polish group $G$ of permutations of $M$, a comeager $n$-diagonal conjugacy class exists exactly when the family $K_{G,n}$ of $n$-tuples of $G$-extendable bijections between finitely generated substructures satisfies the joint embedding property and the weak amalgamation property. This widens a characterization that was previously available only for automorphism groups of classical limits of finite structures, and it also yields a dense-class analogue. The paper then applies the criterion to groups of ball-preserving bijections of ordered ultrametric spaces: these groups always have a comeager conjugacy class but never a comeager $2$-diagonal one. It also characterizes homogenizability of weak limits and gives an example of a weak limit that is not homogenizable.","feed_headline":"Comeager conjugacy classes equal JEP plus WAP in permutation groups","feed_subtitle":"Generic n-tuples in any Polish permutation group exist exactly when finite partial maps have JEP and WAP.","key_machinery":"The load-bearing object is $K_{G,n}$, the family of $n$-tuples of partial bijections between finitely generated substructures of $M$ that extend to elements of $G$, equipped with embeddings that are themselves $G$-extendable. Chains in this family are the finite approximations of group elements, and the paper's dictionary converts conjugacy in $G$ into isomorphism of chains. The main equivalence passes through three intermediate stations: JEP plus WAP for $K_{G,n}$; the existence of a weakly $K_{G,n}$-injective tuple in $G$, built with a Rasiowa-Sikorski-style generic chain; and a winning strategy for Odd in the associated infinite game, which is equivalent to the target conjugacy class being comeager. For the ultrametric results, partial ball-preserving bijections are coded as bijections between finite families of balls, and the proofs of the cofinal amalgamation property use encompassing and monotone orbits, while the failure of weak amalgamation for pairs is engineered with words in a free group.","core_discovery":"The paper's central claim is Theorem 3.10: if $M$ is countable, $G \\le \\mathrm{Sym}(M)$ is Polish, and $n \\ge 1$, then $G$ has a comeager $n$-diagonal conjugacy class if and only if $K_{G,n}$ has JEP and WAP. The proof treats $K_{G,n}$ as a category whose chains are finite approximations of elements of $G$; isomorphic chains correspond to conjugate group elements, and a weakly injective chain obtained by a generic construction supplies a comeager class. The dense version replaces WAP by JEP alone, and taking all $n$ together characterizes ample generics for $\\mathrm{Aut}(M)$. In the ultrametric part, the paper proves that partial ball-preserving bijections of the ordered rational $N$-ultrametric Urysohn spaces have the cofinal amalgamation property, while pairs of them fail weak amalgamation; consequently $\\mathrm{BP}(X)$ has a comeager conjugacy class but no comeager $2$-diagonal conjugacy class for every ordered ultrahomogeneous Polish ultrametric space $X$.","pith_inferences":["Editorial extension: the proof's countability step for $K_{G,n}$ is a genuine gap for structures with function symbols; passing to a relational expansion that preserves $G$ would make finitely generated substructures finite and appears to repair it, but the paper does not state this.","Editorial extension: the free-group-word obstruction to weak amalgamation for pairs of ball-preserving maps may be a template for other hierarchically ordered Fraïssé-like classes, suggesting that 'generic element but no generic pair' is common among tree-like structures.","Editorial extension: the dense version of the criterion means that JEP alone can certify a dense orbit in the space of $n$-tuples of group elements; this could be used to detect non-classifiable orbit equivalence relations in non-archimedean groups without computing full amalgamation."],"forward_implications":["A purely combinatorial condition on finite partial maps decides whether a Polish permutation group has a comeager $n$-diagonal conjugacy class, so no information outside $K_{G,n}$ is needed.","For automorphism groups of countable structures, ample generics is equivalent to $K_{\\mathrm{Aut}(M),n}$ having JEP and WAP for every $n$, making a single failed WAP for some $n$ a uniform obstruction to ample generics.","The ordered ultrametric examples give a family of non-archimedean groups with a generic element but no generic pair: ball-preserving bijection groups of ordered rational $N$-ultrametric Urysohn spaces and generalized ordered Ważewski dendrites.","The homogenizability criterion identifies when a weak Fraïssé limit can be expanded by finitely many definable relations into an ultrahomogeneous structure with the same automorphism group."],"supporting_citations":[{"why":"The original Fraïssé-limit characterization of comeager diagonal conjugacy classes that Theorem 3.10 generalizes.","marker":"[7]"},{"why":"Supplies weak Fraïssé limits, the generic-chain construction, and the infinite game used in Section 3.","marker":"[11]"},{"why":"Provides the SEAP criterion for homogenizable structures invoked in Theorem 4.3.","marker":"[1]"},{"why":"The construction of weak amalgamation classes adapted to produce the non-homogenizable example.","marker":"[12]"},{"why":"Ordered boron tree results that Corollary 5.11 recovers through the ultrametric theorems.","marker":"[9]"},{"why":"Rooted-tree ample generics result used to show the rooted version of the example has ample generics.","marker":"[10]"}],"fun_headline_variants":["JEP and WAP fully characterize comeager conjugacy classes","Weak amalgamation resolves generic tuples in Polish groups","Ultrametric spaces: one generic class, but not for pairs","Comeager n-diagonal classes hinge on amalgamation properties"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main equivalence assumes the family $K_{G,n}$ is countable when it applies a generic-selection lemma, and the paper does not verify that countability; for structures with function symbols, finitely generated substructures can be infinite, so $K_{G,n}$ can be uncountable.","fun_headline_variants_meta":{"raw":{"variants":["JEP and WAP fully characterize comeager conjugacy classes","Weak amalgamation resolves generic tuples in Polish groups","Ultrametric spaces: one generic class, but not for pairs","Comeager n-diagonal classes hinge on amalgamation properties"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1380,"prompt_tokens":934,"completion_tokens":446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":378}},"tokens_in":550,"tokens_out":446,"duration_ms":4704,"temperature":1.0,"reasoning_tokens":378,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:12:14.786815+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a countable structure $M$ and a Polish group $G \\le \\mathrm{Sym}(M)$ for which $K_{G,1}$ satisfies JEP and WAP but $G$ has no comeager conjugacy class; such an example would refute the central equivalence as stated. A more immediate check is whether $K_{\\mathrm{Sym}(\\mathbb{N})}$ for the successor structure on $\\mathbb{N}$ is uncountable; if so, the generic-selection step of the proof needs an additional countability argument before Theorem 3.10 can cover that case.","supporting_citations":[{"cited_title":"Rosendal, Turbulence, amalgamation, and generic automorphisms of ho mogeneous structures, Proc","cited_arxiv_id":null,"evidence_quote":"The original Fraïssé-limit characterization of comeager diagonal conjugacy classes that Theorem 3.10 generalizes."},{"cited_title":"Games with finitely generated structures","cited_arxiv_id":"1701.05756","evidence_quote":"Supplies weak Fraïssé limits, the generic-chain construction, and the infinite game used in Section 3."},{"cited_title":"Ahlman, Homogenizable structures and model completeness , Arch","cited_arxiv_id":null,"evidence_quote":"Provides the SEAP criterion for homogenizable structures invoked in Theorem 4.3."},{"cited_title":"Examples of weak amalgamation classes","cited_arxiv_id":"1907.09577","evidence_quote":"The construction of weak amalgamation classes adapted to produce the non-homogenizable example."},{"cited_title":"Ordered structures and large conjugacy classes","cited_arxiv_id":"1903.00936","evidence_quote":"Ordered boron tree results that Corollary 5.11 recovers through the ultrametric theorems."},{"cited_title":"Malicki, Rooted trees, strong cofinality and ample generics, M ath","cited_arxiv_id":null,"evidence_quote":"Rooted-tree ample generics result used to show the rooted version of the example has ample generics."}],"review_version":1}