{"id":"78a7d925-9024-4845-a572-901c3b3b369c","arxiv_id":"2502.10073","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under OCA and MA_ℵ1, homeomorphic Higson coronas of uniformly locally finite metric spaces force coarse equivalence, a statement independent of ZFC.","lead":"This mathematics paper proves that, under two common set-theoretic axioms (OCA and MA_ℵ1), any two uniformly locally finite metric spaces with homeomorphic Higson coronas, or identical boundaries at infinity, must be coarsely equivalent. It settles an open rigidity question in coarse geometry that was already known to fail under the continuum hypothesis.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"OCA uniformization in Section 4 uses a non-separable space of maps as a Polish space; this invalidates the stated application of OCA, blocking the construction of the global product-form lift.","rationale":"The reader's verdict of CONDITIONAL with 'medium' correctness risk is appropriate: the paper's strategy is plausible and the ZFC section is careful, but Section 4 contains multiple gaps. I focused on the Polish-space assertion because it is an internal false statement, not merely an unverified external input. The set of all arbitrary maps between two uncountable compact metric spaces with the sup metric is well known to be non-separable, so it cannot serve as the underlying space for OCA as stated. This directly affects the proof of Theorem 4.1, from which Theorems 1.4 and 1.5 follow. The other flagged issues (the off-by-one in Claim 4.6 and the sketchiness of Theorem 4.4) are also serious, but they appear more likely to be repairable by routine corrections, whereas the Polish-space problem may require a genuinely new technique or a different set-theoretic input. That said, the existence of Dow's ZFC example and the plausibility of the lifting theorems mean the central claim could still be true; hence the verdict should remain CONDITIONAL rather than moving to REJECT. I partially agree with the reader because they did list this exact Polish-space issue in their rationale, but their stated weakest assumption points instead to the imported lifting theorems, which is a somewhat different locus of risk.","tokens_in":25817,"tokens_out":12507,"duration_ms":115904,"concrete_test":"Check whether K^ε_0, defined on pairs (f,f')∈[N^N]^2 by ∃n∃g∈M_{f,n}∩M_{f',n} with ‖α_{f,n}(g)-α_{f',n}(g)‖>ε, is open in the product topology of N^N. If it is not open (or if the α_{f,n} cannot be chosen continuous), the OCA uniformization cannot be salvaged by moving to N^N, and the proof of Theorem 4.1 as written is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.1, after Claim 4.9, the author considers Maps(DX_i_n, DY_i_n), 'the set of all maps from DX_i_n to DY_i_n', and asserts 'This is a Polish space with the sup-distance'. For nonempty finite X_i_n, DX_i_n = D^{X_i_n} is an uncountable compact metric space (a finite-dimensional cube), and Maps(DX_i_n, DY_i_n) with the sup metric is not separable: the functions f_x(t)=δ_{x,t} (or characteristic functions of singletons) form an uncountable family pairwise at sup-distance ≥1. Hence it is not a Polish space. The paper then views the colouring K^ε_0 as a subset of [∏_n Maps(...) × N^N]^2 and applies OCA, which is stated for separably metrizable spaces. This is not justified. The failure is load-bearing: the OCA application (Claims 4.10 and 4.11) is exactly what produces the uniform family of maps α_n forming the product-form lift Λ in Claim 4.12. Without a separable metrizable domain for the colouring, the theorem's proof does not go through as written. A repair would require either a different OCA formulation (e.g., colouring [N^N]^2 directly, with a proof that K^ε_0 is open) or a proof that the lifts α_{f,n} can be chosen from a separable family, such as continuous maps.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the rigidity of Higson coronas for uniformly locally finite (u.l.f.) metric spaces. It introduces a notion of trivial *-homomorphisms between algebras of slowly oscillating functions modulo compact functions, and the coarse weak Extension Principle (cwEP). The main theorem (Theorem 1.4) asserts that under OCA and MA_ℵ1, if two u.l.f. metric spaces have homeomorphic Higson coronas, then they are coarsely equivalent. The proof architecture is: (i) in ZFC, show that a surjective *-homomorphism which is of product form on two sparse sequences is trivial (Section 3); (ii) under OCA+MA, show that every *-homomorphism between such algebras is of product form on the relevant sparse sequences (Theorem 4.1); (iii) combine to get the rigidity theorem and a companion theorem on injective maps (Theorem 1.5). The paper relies on two imported lifting results: Theorem 4.3 (De Bondt–Vignati, to appear) and Theorem 4.4, derived by a two-sentence sketch from [24] and [39].","tokens_in":25929,"tokens_out":12200,"duration_ms":109037,"significance":"If the main result is correct, it provides a positive answer to Question 1.3 under mild set-theoretic hypotheses, complementing the CH-based counterexamples of Protasov and Brian–Farah. The paper also introduces a useful framework (cwEP) and proves a substantial ZFC result (Theorem 3.12) showing that local triviality implies triviality for surjective maps. The architecture is clear, and the connection to known lifting theorems for reduced products is natural. However, the proof of the crucial local-existence step (Theorem 4.1) contains a serious gap in the application of OCA, and there is an off-by-one error in Claim 4.6. As written, the main theorem is not established.","major_comments":[{"comment":"The application of OCA is invalid. The paper states that Maps(DX_i_n, DY_i_n), the set of all maps from DX_i_n to DY_i_n, 'is a Polish space with the sup-distance'. For nonempty finite X_i_n, DX_i_n = D^{X_i_n} is an uncountable compact metric space (a finite-dimensional cube), and the set of all functions from this uncountable space to any space containing two distinct points is not separable in the sup metric (e.g., the characteristic functions of singletons form an uncountable discrete family). Hence Maps(DX_i_n, DY_i_n) is not Polish, and the product ∏_n Maps(DX_i_n, DY_i_n) × N^N is not separably metrizable. Consequently, OCA cannot be applied to the colouring K^ε_0 as stated. This is load-bearing: the OCA application produces the uniform family of maps α_n used to define the product-form lift Λ in Claim 4.12. A repair would require either applying OCA directly to [N^N]^2 with a proof that K^ε_0 is open in the usual topology, or proving that the lifts α_{f,n} can be chosen from a separable family (e.g., continuous maps).","section":"Section 4, after Claim 4.9"},{"comment":"There is an off-by-one error in the proof. The set T is defined as {k_n + 1 | n ∈ N}, but the geometry of the annuli in Definition 3.10 shows that X^e_{k_n} is contained in the union of X^o_{k_n} and X^o_{k_n - 1} (since X^e_n = [2^{6n+1}, 2^{6n+5}] and X^o_{n-1} = [2^{6n-2}, 2^{6n+2}]), not in X^o_{k_n} ∪ X^o_{k_n+1}. The argument as written therefore does not establish the claimed containment. The proof should use T = {k_n - 1} (after passing to a subsequence with k_n > 0). This affects the proof that χ_{\\tilde{U}} q_{N,i} =_{C0(Y)} q_{N,i}, which is used to show that U is clopen.","section":"Claim 4.6"},{"comment":"The paper's proof of Theorem 4.4 is only a two-sentence sketch, and Theorem 4.3 is cited as 'to appear' without a proof or preprint reference. Since these lifting results are the entire engine behind the local-existence step (Theorem 4.1), the paper is not self-contained. In particular, it is not verified that the hypotheses of Theorem 4.3 are satisfied by the spaces N_f and ∏ D^{Y_n} / ⊕ D^{Y_n}, especially regarding the quotient equivalence relation used in the definition of N_f. The author should either include complete proofs of Theorems 4.3 and 4.4, or state and prove the precise special cases needed here, rather than referring to unpublished work.","section":"Theorem 4.4"}],"minor_comments":[{"comment":"In the proof of part (2), the sets A and B are both defined as {y'_n}. This is clearly a typo; one should be {y_n} and the other {y'_n}. As written, the displayed equality of norms is nonsensical.","section":"Proposition 2.4, proof of (2)"},{"comment":"In the definition of the relation R_n, the text reads 'if f(k) ≥ m then g(k) ≤ f(k)', but m is not defined; it should be 'if f(k) ≥ n'.","section":"Claim 4.11"},{"comment":"There is a duplicated word: 'we say the the coarse weak Extension Principle holds'.","section":"Definition 2.7"},{"comment":"The construction of \\tilde{ρ} from the values on characteristic functions is not fully rigorous: it is stated that \\tilde{ρ} = ∑ \\tilde{ρ}_n is positive and order zero, but the extension from the specified values to all of ℓ∞ is only implicit. Since simple functions are dense in ℓ∞, this can be filled in, but the details should be supplied.","section":"Section 4, proof of Theorem 4.4"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an interesting and timely question, and the ZFC portion (Section 3) is a solid contribution. However, the main set-theoretic proof has a genuine gap (the OCA application) and an index error that need to be fixed. The heavy reliance on the author's own unpublished lifting theorems is also a concern for verification; the author should make those results available or prove the needed cases. I would recommend a major revision, with the expectation that the author either repairs the OCA argument or clearly states a corrected approach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper answers the Higson corona rigidity question under OCA+MA_aleph1, and shows the statement is independent of ZFC by combining with the CH-era counterexamples. That is a real result, and the ZFC portion (local triviality implies triviality) is careful and genuinely interesting. But the proof of the key set-theoretic step, Theorem 4.1, has a gap that blocks the argument as written: the space of all maps between finite-dimensional cubes is not separable in the sup metric, hence not Polish. The OCA application at the end of Section 4 needs a separable metrizable domain; the text uses a non-separable one. The stress-test note is right. This is not cosmetic—it is exactly the step that produces the uniform family of product-form lifts in Claim 4.12.\n\nWhat is new: the notion of cwEP, the sparse-sequence local triviality machinery, the ZFC theorem 3.12, and the main rigidity theorem 1.4. The paper states that no partial positive answer to Question 1.3 existed before, which checks out. The CH-based negative results are properly cited, and the overall architecture—prove local triviality in ZFC, then use OCA/MA to get local triviality—is credible.\n\nSoft spots beyond the Polish-space gap: the two-sentence proof of the lifting theorem 4.4 leans on the author's own unpublished results, including a preprint 'to appear'; that is heavy imported machinery. The off-by-one in Claim 4.6 (T = {k_n+1} vs {k_n-1}) looks like a typo but needs checking. The paper also does not supply a way to choose the lifts from a separable family, which is precisely what the Polish-space assertion was trying to provide.\n\nOverall: the result is important and likely true, but the proof has a repairable yet load-bearing gap. I would send this to a serious referee—the referee should focus on Section 4. If you work on corona rigidity, the paper is worth a careful read; the gap itself is instructive.\n\nRecommendation: engage with it, send it to peer review, and expect the author to fix the OCA application.","headline":"Significant result with a repairable but load-bearing gap: the OCA application in Section 4 uses a non-Polish space of maps.","tokens_in":26698,"tokens_out":2965,"would_cite":true,"duration_ms":27903,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E35","03E50","03E65","46L85","54D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under $\\mathsf{OCA}$ and $\\mathsf{MA}_{\\aleph_1}$, homeomorphic Higson coronas force coarse equivalence.","keywords":["Higson corona","coarse equivalence","rigidity","Open Colouring Axiom","Martin's Axiom","uniformly locally finite metric spaces","slowly oscillating functions","C*-algebras"],"falsifier":"Build, in a model of $\\mathsf{OCA}$ plus $\\mathsf{MA}_{\\aleph_1}$, two uniformly locally finite metric spaces with homeomorphic Higson coronas that are not coarsely equivalent; any such pair would refute Theorem 1.4 directly. Alternatively, exhibit a coordinate-fixing function between the reduced products $N_f$ of slowly oscillating functions defined in Section 4 that is not of product form; that would break the specific application of the metric lifting theorem on which local triviality depends.","tokens_in":25394,"feed_emoji":"🔭","tokens_out":10811,"duration_ms":93953,"temperature":0.7,"pith_summary":"The Higson corona of a discrete metric space is its boundary at infinity: the leftover part of a canonical compactification after the space itself is removed. This paper asks how much of the original geometry survives in that boundary, and answers that under the set-theoretic axioms $\\mathsf{OCA}$ and $\\mathsf{MA}_{\\aleph_1}$ all of it does: two uniformly locally finite metric spaces — bounded-geometry spaces in which balls of any fixed radius are uniformly finite — with homeomorphic Higson coronas are coarsely equivalent, meaning their large-scale geometries agree. That statement is not provable in $\\mathsf{ZFC}$ alone: under the Continuum Hypothesis there are many non-coarsely-equivalent spaces with isomorphic Higson coronas, so the axioms are doing genuine work. The proof shows that every isomorphism between the function algebras of Higson coronas is trivial in the paper's sense, namely induced by a coarse map of the underlying metric spaces.","feed_headline":"Homeomorphic Higson coronas force coarse equivalence","feed_subtitle":"Under two mild set-theoretic axioms, homeomorphic boundaries at infinity make discrete metric spaces coarsely equivalent.","key_machinery":"The central device is the coarse weak Extension Principle (cwEP), an analogue for Higson coronas of a weak Extension Principle for remainders of compactifications. A map satisfies cwEP when, off a nowhere-dense set, it is trivial, meaning it is induced by a coarse proper map of the underlying metric spaces. The proof of local triviality uses two specially chosen sparse sequences of annuli, the reduced products of finite-diameter metric spaces that they generate, and two imported lifting theorems: coordinate-fixing functions between such reduced products are of product form, and positive order-zero contractions from $\\ell^\\infty/c_0$ to itself lift on a nonmeager dense ideal. Once a homomorphism has product form on the two sparse sequences, a $\\mathsf{ZFC}$ argument assembles the local coordinate data, using marker functions, into a single coarse proper map that induces the homomorphism.","core_discovery":"Formally, the main result (Theorem 1.4) is that, assuming $\\mathsf{OCA}$ and $\\mathsf{MA}_{\\aleph_1}$, if $X$ and $Y$ are uniformly locally finite metric spaces and their Higson coronas $\\nu X$ and $\\nu Y$ are homeomorphic, then $X$ and $Y$ are coarsely equivalent. The Higson corona $\\nu X$ is the spectrum of the quotient $C^*$-algebra $C_\\nu(X)=\\mathrm{Ch}(X)/C_0(X)$, where $\\mathrm{Ch}(X)$ is the algebra of bounded slowly oscillating functions. Along the way the paper proves a coarse weak Extension Principle for surjective maps, showing that any unital surjective $*$-homomorphism between such algebras is trivial off a nowhere-dense set, where a trivial homomorphism is one induced by a coarse proper map. It also proves an embedding version: a continuous injection between coronas restricts to a coarse embedding on a clopen subspace. Since a trivial isomorphism is induced by a coarse equivalence, Theorem 1.4 follows.","pith_inferences":["A natural next step would be to test whether the same rigidity survives for non-uniformly locally finite or non-discrete bounded-geometry spaces, where the reduced-product machinery does not directly apply.","If the metric lifting theorem used here is later extended to weaker continuity assumptions, the hypothesis could perhaps be weakened from $\\mathsf{OCA}$ plus $\\mathsf{MA}_{\\aleph_1}$ to $\\mathsf{OCA}$ alone; the paper's comments suggest this is open.","The cwEP formulation may transfer to other coarse-geometry quotients, such as stable Higson coronas, where analogous rigidity questions remain open.","One could look for a $\\mathsf{ZFC}$ example of non-coarsely-equivalent spaces with homeomorphic Higson coronas; finding one would show the axioms are necessary rather than merely sufficient."],"forward_implications":["Under $\\mathsf{OCA}$ and $\\mathsf{MA}_{\\aleph_1}$, homeomorphic Higson coronas imply coarse equivalence for uniformly locally finite metric spaces, resolving the open rigidity question in that setting.","The rigidity statement is independent of $\\mathsf{ZFC}$: it holds under $\\mathsf{OCA}$ and $\\mathsf{MA}_{\\aleph_1}$, and known constructions under the Continuum Hypothesis produce non-coarsely-equivalent spaces with isomorphic Higson coronas.","Every surjective $*$-homomorphism between Higson-corona algebras is trivial in the paper's sense, so its dual map comes from a coarse proper map off a nowhere-dense set.","A continuous injection between Higson coronas restricts to a coarse embedding from a clopen subspace, with the complement mapped to a nowhere-dense set.","The coarse weak Extension Principle for surjective maps follows from the same axioms, giving a structural description of all such homomorphisms."],"supporting_citations":[{"why":"Supplies the metric lifting theorem: under OCA and MA_ℵ1 every coordinate-fixing function between reduced products of separable uniformly bounded metric spaces is of product form; this turns local coordinate data into a global lift.","marker":"[11]"},{"why":"Supplies the noncommutative OCA lifting theorem used in Theorem 4.4 to lift positive order-zero contractions on ℓ∞/c0 on a nonmeager dense ideal.","marker":"[24]"},{"why":"Used together with [24] in the sketch deriving the order-zero lifting result that Theorem 4.4 needs.","marker":"[39]"},{"why":"Provides the CH-consistent construction of a continuum of non-coarsely-equivalent asymptotic dimension one spaces with isomorphic Higson coronas, showing the positive theorem rests on extra set-theoretic assumptions.","marker":"[9]"},{"why":"Provides the CH result that all asymptotic dimension zero spaces have isomorphic Higson coronas, giving the model-theoretic obstruction behind the negative side under CH.","marker":"[28]"},{"why":"Introduces the weak Extension Principle and the analytic-quotient lifting toolbox that the paper's cwEP adapts to coarse geometry.","marker":"[16]"},{"why":"Gives the ZFC example of a nontrivial copy of βN\\N inside itself, which forces the clopen nowhere-dense refinement in the definition of cwEP.","marker":"[12]"}],"fun_headline_variants":["Higson corona homeomorphy forces coarse equivalence","Homeomorphic Higson coronas yield coarse equivalence","Coarse equivalence from Higson corona homeomorphy","Homeomorphic Higson coronas ensure coarse equivalence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that two imported lifting results — one stated from the author's own unreviewed preprint — really apply to the reduced products of slowly oscillating functions used in Section 4; if either theorem fails, or covers fewer maps than claimed, the local triviality step that powers the rigidity theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Higson corona homeomorphy forces coarse equivalence","Homeomorphic Higson coronas yield coarse equivalence","Coarse equivalence from Higson corona homeomorphy","Homeomorphic Higson coronas ensure coarse equivalence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.004456,"raw_usage":{"total_tokens":16524,"prompt_tokens":845,"completion_tokens":15679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":15619}},"tokens_in":461,"tokens_out":15679,"duration_ms":124578,"temperature":1.0,"reasoning_tokens":15619,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T19:33:22.280006+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build, in a model of $\\mathsf{OCA}$ plus $\\mathsf{MA}_{\\aleph_1}$, two uniformly locally finite metric spaces with homeomorphic Higson coronas that are not coarsely equivalent; any such pair would refute Theorem 1.4 directly. Alternatively, exhibit a coordinate-fixing function between the reduced products $N_f$ of slowly oscillating functions defined in Section 4 that is not of product form; that would break the specific application of the metric lifting theorem on which local triviality depends.","supporting_citations":[{"cited_title":"A metric lifting theorem","cited_arxiv_id":"2411.11127","evidence_quote":"Supplies the metric lifting theorem: under OCA and MA_ℵ1 every coordinate-fixing function between reduced products of separable uniformly bounded metric spaces is of product form; this turns local coordinate data into a global lift."},{"cited_title":"McKenney and A","cited_arxiv_id":null,"evidence_quote":"Supplies the noncommutative OCA lifting theorem used in Theorem 4.4 to lift positive order-zero contractions on ℓ∞/c0 on a nonmeager dense ideal."},{"cited_title":"Vignati, Rigidity conjectures for continuous quotients , Ann","cited_arxiv_id":null,"evidence_quote":"Used together with [24] in the sketch deriving the order-zero lifting result that Theorem 4.4 needs."},{"cited_title":"Conjugating trivial automorphisms of $\\mathcal P(\\mathbb N)/\\mathrm{Fin}$","cited_arxiv_id":"2410.08789","evidence_quote":"Provides the CH-consistent construction of a continuum of non-coarsely-equivalent asymptotic dimension one spaces with isomorphic Higson coronas, showing the positive theorem rests on extra set-theoretic assumptions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the CH result that all asymptotic dimension zero spaces have isomorphic Higson coronas, giving the model-theoretic obstruction behind the negative side under CH."},{"cited_title":"Farah, Analytic quotients: theory of liftings for quotients over a nalytic ideals on the inte- gers, Mem","cited_arxiv_id":null,"evidence_quote":"Introduces the weak Extension Principle and the analytic-quotient lifting toolbox that the paper's cwEP adapts to coarse geometry."},{"cited_title":"Dow, A non-trivial copy of βN \\ N , Proc","cited_arxiv_id":null,"evidence_quote":"Gives the ZFC example of a nontrivial copy of βN\\N inside itself, which forces the clopen nowhere-dense refinement in the definition of cwEP."}],"review_version":1}