{"id":"3c774d83-2a48-485f-8e1f-de099234553b","arxiv_id":"2607.27400","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For minimal actions of countably infinite discrete amenable groups, r-point mean equicontinuity (Weyl or along a Følner sequence) holds iff the conditional topomorphic degree is at most r−1; an exact degree formula and a full realization theorem follow.","lead":"Proves that multivariate mean equicontinuity of minimal amenable group actions is controlled exactly by a fiber multiplicity — the conditional topomorphic degree — resolving a recent conjecture. Also gives an exact formula for this degree in terms of measure sequence entropy and realizes every finite multiplicity profile with a zero-entropy minimal system.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central hard direction depends on imported Lemma 2.6 (I_μ ⊆ product of compact σ-algebras); if that inclusion fails, Proposition 3.3 and Theorem 1.1's (ii)⇒(iii) collapse.","rationale":"I read the paper in good faith. The main theorem is Theorem 1.1. The proof of (i)⇒(ii) is trivial; (iii)⇒(i) is Theorem 4.1, which I checked line-by-line and found correct: the finite-to-one branch argument, the uniform continuity, and the averaging via Lemma 2.2 all work. The hard direction (ii)⇒(iii) is Theorem 4.4. Its proof uses Lemma 3.7 and Lemma 3.8 to bound the number of ergodic measures and the residual multiplicities, and Corollary 3.6 to convert positive off-diagonal phase-joining mass into mean-sensitivity. The passage from positive phase-joining mass to a mean-sensitive tuple is Theorem 3.5, which relies on Proposition 3.3 for the existence of an ergodic lifted joining. Proposition 3.3 depends on Lemma 2.6, the inclusion of the invariant σ-algebra of the product in the product of the maximal compact σ-algebras. This lemma is imported without proof from [41, Lemma 3.5]. I could not find an internal proof or a detailed derivation; the text merely cites the external source. This is the weakest link in the chain: if the inclusion is false or has hidden hypotheses, Prop 3.3 does not hold, and the hard direction collapses. I therefore agree with the reader's weakest_assumption. The other flagged items (Prop 2.11, Lemma 6.2(ii)) are less central: Prop 2.11 affects the exact exponential formula in Theorem 1.2, but Theorem 1.1 only uses the support-cardinality b_μ, not its identification with exp(h*); Lemma 6.2(ii) affects only the realization theorem 1.5. The proofs of the new technical results (Lemmas 3.1, 3.4, Theorem 3.5, Corollary 3.6, Theorem 4.1, Theorem 4.3, Theorem 4.4) appear internally coherent and I found no contradiction. Thus the appropriate verdict is CONDITIONAL, conditional on confirmation of Lemma 2.6 (and the cited Host–Kra theorem). Since the reader already issued CONDITIONAL, my read does not change the verdict. Hence UNCHANGED.","tokens_in":29641,"tokens_out":18015,"duration_ms":166092,"concrete_test":"Independently re-derive Lemma 2.6 for arbitrary countably infinite discrete amenable groups: verify that for any N ergodic measures μ_i on a minimal G-system, every invariant set of the product measure μ lies in the product of the maximal compact σ-algebras. Specifically, check the proof in [41, Lemma 3.5] and confirm that the Host–Kra theorem [25, Thm 16] used there is stated for all such G, not only abelian G; if the proof requires commutativity or another unstated assumption, the hard direction of Theorem 1.1 lacks support. A minimal sanity check: for G=Z, take μ_1, μ_2 two ergodic measures with the same non-trivial Kronecker factor and confirm that the invariant σ-algebra of the product is exactly the product of the Kronecker σ-algebras.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The hard direction (ii)⇒(iii) of Theorem 1.1 flows through Theorem 4.4 → Corollary 3.6 → Theorem 3.5 → Proposition 3.3, whose proof hinges entirely on Lemma 2.6: I_μ ⊆ K_{μ_1}⊗…⊗K_{μ_N} (mod μ) for the product measure μ=μ_1⊗…⊗μ_N of ergodic measures. This lemma is not proved in the text; it is imported from the author's earlier paper [41, Lemma 3.5] (with Host–Kra [25, Thm 16] as corroboration). If the inclusion were false or had an extra hypothesis (e.g., commutativity of G or finite-dimensionality of the compact factors), the ergodicity of a.e. phase joining Λ_t would fail; the local realization Theorem 3.5 could not be applied, and the proof of tdeg_G(X)≤r−1 would break. The text gives no independent verification for countably infinite discrete amenable groups, and a referee must rely on the external source. This is not a claim of internal inconsistency; it is a correctness-risk dependency. The reader's weakest_assumption identifies Prop 3.3 as the engine and Lemma 2.6 as its external support; I concur. A secondary under-derivation is Lemma 6.2(ii), which affects only Theorem 1.5, not the central equivalence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies minimal actions of a countably infinite discrete amenable group G on a compact metric space X. It introduces the conditional topomorphic degree tdeg_G(X) (Eq. (1)) as the supremum, over invariant measures, of the essential supremum of conditional fiber support cardinalities over the maximal equicontinuous factor. The central result (Theorem 1.1) is a three-way equivalence, for every r≥2, between Weyl mean r-equicontinuity, F-mean r-equicontinuity for some Følner sequence, and tdeg_G(X)≤r−1. This resolves the Breitenbücher–Haupt–Jäger conjecture for minimal Z-systems and extends it to all countably infinite discrete amenable groups. The proof of the hard direction (ii)⇒(iii) proceeds through mixed compact-phase joinings: Lemma 3.1, Proposition 3.3, Lemma 3.4, the local realization Theorem 3.5, and Corollary 3.6. Theorem 1.2 gives the exact degree decomposition tdeg_G(X)=Σ_{μ∈M^e_G(X)} ι_μ exp(h*_μ(G)); Theorem 1.3 derives essential IT N-tuples and the lower bound h*_top(X,G)≥log d; Theorem 1.5 realizes every finite multiplicity profile by a zero-entropy minimal almost one-to-one extension of an irrational circle rotation.","tokens_in":29846,"tokens_out":15609,"duration_ms":167517,"significance":"If correct, this is a substantial contribution to the structure theory of minimal amenable group actions. It resolves a previously open conjecture, introduces a clean numerical invariant separating compact and residual multiplicities, and sharpens known sequence-entropy lower bounds. The main line of proof is detailed and mostly self-contained; the two significant external imports (Lemma 2.6 and Proposition 2.11 from [41]) are explicitly disclosed and are used in a form matching their statements. The paper does not provide machine-checked artifacts, but the central chain of implications is checkable by hand and the arguments are well organized.","major_comments":[],"minor_comments":[{"comment":"The proof of Proposition 3.3, and hence the hard implication (ii)⇒(iii) of Theorem 1.1, depends entirely on the inclusion I_μ⊆K_{μ_1}⊗⋯⊗K_{μ_N} (mod μ) stated as Lemma 2.6. This lemma is imported from the author's earlier paper [41, Lemma 3.5] with a corroborating reference to Host–Kra, but it is not proved in the present text. I do not regard this as circular or as an internal inconsistency, since the citation is explicit and the source is published. However, because this is the single most load-bearing external input, the authors should add either a proof of Lemma 2.6 in the present notation or a precise restatement of [41, Lemma 3.5] listing all hypotheses, and a sentence explaining why those hypotheses hold for arbitrary countably infinite discrete amenable G and arbitrary ergodic μ_1,…,μ_N.","section":"§2.4.2, Lemma 2.6; §3.2, Proposition 3.3"},{"comment":"The assertion that \\hat κ_{ι,b}: \\hat X_{ι,b}→Z_ι \\represents the measure-theoretic maximal compact factor] is dispatched in one sentence. Since Theorem 1.5 relies on this identification, the proof should be expanded: one should explain how the finite cyclic coordinate C_ι interacts with the totally strictly ergodic block B_b, why no additional measure-theoretic eigenvalues appear, and how strict ergodicity of S_b^ι yields the stated maximal compact factor. This is likely standard, but as written it is under-derived.","section":"§6.1, Lemma 6.2(ii)"},{"comment":"In the paragraph following Eq. (23), the positivity of ∫_{\\widetilde W_1×⋯×\\widetilde W_N} Λ_t(U_1×⋯×U_N) dm_P(t) is justified by continuity of each J_q at the identity and Tonelli's theorem. For completeness, the authors should explicitly state that the set of ℓ∈L for which all J_q(ℓ_q)>0 has positive Haar measure; this uses the denseness of α(G) in L and continuity, and it is the step that makes the displayed integral strictly positive.","section":"§3.3, Theorem 3.5"},{"comment":"There is a small typesetting issue: the notation for the finite multiset P appears as \\{P=\\{\\{(ι_1,b_1),\\ldots\\}} \\}] with extra braces in both the abstract and Theorem 1.5. It should simply read \\{P=\\{(ι_1,b_1),\\ldots,(ι_ℓ,b_ℓ)\\}]. The symbol \\widertilde W] is also rendered inconsistently as \\fW] in a few places.","section":"Abstract and §6.1"}],"recommendation":"minor_revision","confidential_remarks":"The main theorem depends heavily on two results from the author's own earlier work [41]: Lemma 2.6 and Proposition 2.11. Both are explicitly cited and the reader's take confirms they are used in matching form, so I do not view this as circular. An editor may nevertheless wish to ensure that Lemma 2.6 is proved for the full class of countably infinite discrete amenable groups in [41], since the hard direction of the central equivalence rests on it. The remaining concern, Lemma 6.2(ii), affects Theorem 1.5 rather than the central equivalence and can be fixed by a short argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this one for real: it proves the converse of the Breitenbucher–Haupt–Jager conjecture for all countably infinite discrete amenable groups, not just Z. The equivalence between Weyl mean r-equicontinuity, F-mean r-equicontinuity along some Følner sequence, and conditional topomorphic degree ≤ r−1 is the headline, and it holds up on close reading. The exact degree formula tdeg = Σ ι_μ exp(h*_μ) is also new and strictly sharpens the 2025 Liu–Wang–Xu bound by capturing the compact multiplicities. The realization theorem (every finite multiplicity profile occurs in a zero-entropy minimal almost one-to-one extension of an irrational rotation) is a nice bonus.\n\nThe genuinely new technique is the mixed compact-phase joinings construction. The idea of transporting compact phase data rather than the original fiber tuple is what breaks the obstruction identified in [5, Remark 5.2], and it is coherent. I walked through Theorem 4.1, Lemma 3.4, Proposition 3.3, Theorem 3.5, Lemma 3.7/3.8, and Theorem 4.4 carefully and found no internal contradiction. The proof of Theorem 4.4, where the full multiplicity packet is reassembled and an M-sensitive point is produced, is the core argument and it works. The almost-everywhere ergodicity of the lifted phase joining (Prop 3.3) is the engine, and its proof is sound provided Lemma 2.6 is true.\n\nThe soft spots are exactly the two the stress-test flags. Lemma 2.6 (I_μ ⊆ K_{μ_1}⊗···⊗K_{μ_N}) is imported from the author's own [41, Lemma 3.5], with Host–Kra cited as corroboration. It is not re-derived here, and the entire hard direction (ii)⇒(iii) rests on it. That is not circular, but it is a correctness-risk dependency: if the lemma needs an extra hypothesis for general amenable groups, the proof collapses at Prop 3.3. A referee should reconfirm it. The second soft spot is Lemma 6.2(ii), where the maximal compact factor of the cyclic product B_b × C_ι is identified in one sentence; it affects Theorem 1.5, not the main theorem. I agree these are the only two real worries.\n\nThis is a serious paper. The main theorem is important, the technique is new, and the proof is mostly careful. The imports are disclosed and published, not hidden. I would send it to a referee. The referee should focus on Lemma 2.6 and on expanding or sourcing 6.2(ii). I would cite this if the imports check out.","headline":"A strong paper that likely resolves the BHJ conjecture for all countable amenable groups, with the main risk being reliance on two imported lemmas from the author's prior work.","tokens_in":30497,"tokens_out":745,"would_cite":true,"duration_ms":9645,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","37B40","37A35","37A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For minimal amenable group actions, multivariate mean r-equicontinuity is equivalent to the conditional topomorphic degree being at most r−1.","keywords":["amenable group actions","multivariate mean equicontinuity","conditional topomorphic degree","maximal equicontinuous factor","IT-tuples","maximal pattern entropy","sequence entropy","almost one-to-one extensions"],"falsifier":"Construct (or exhibit) the Haupt–Jäger degree-2 extension of an irrational circle rotation—the canonical two-to-one topomorphic extension that is not almost automorphic—and test it for Weyl mean 2-equicontinuity and for maximal pattern entropy. The theorem predicts that this system has tdeg = 2, is not Weyl mean 2-equicontinuous, and satisfies h*_top ≥ log 2; a computation showing it is Weyl mean 2-equicontinuous, or showing h*_top < log 2, would refute the main equivalence and the entropy bound directly. As a more surgical check, one can test the imported Lemma 2.6 on a skew-product example o","tokens_in":29354,"feed_emoji":"📐","tokens_out":7613,"duration_ms":68859,"temperature":0.7,"pith_summary":"This paper establishes, for every r≥2, a three-way equivalence for minimal actions of any countably infinite discrete amenable group on a compact metric space: Weyl mean r-equicontinuity, mean r-equicontinuity along some Følner sequence, and conditional topomorphic degree at most r−1. The conditional topomorphic degree counts, in the worst case over invariant measures, how many points of a fiber of the maximal equicontinuous factor map are needed to support the conditional measure. This resolves a 2026 conjecture for Z-actions and extends it to all countably infinite discrete amenable groups. The paper also proves an exact decomposition of that degree into compact multiplicities and residual multiplicities, each tied to maximal measure sequence entropy, and shows that every finite multiset of such multiplicity pairs is realized by a zero-entropy minimal almost one-to-one extension of an irrational circle rotation.","feed_headline":"One integer degree decides mean r-equicontinuity","feed_subtitle":"The equivalence settles a conjecture for Z-actions and extends to all countably infinite discrete amenable groups.","key_machinery":"The engine is a family of mixed compact-phase joinings Λ_t on X^N. For ergodic measures μ_1,...,μ_N, one takes their measure-theoretic maximal compact factors Z_i=K_i/H_i, forms the joint phase group L = closure of the diagonal image of G in the product of the K_i, and integrates orbit measures λ_t over L to get ergodic phase measures on the product of compact factors; lifting these through the disintegrations of the μ_i over the Z_i yields Λ_t. The key properties are that almost every Λ_t is ergodic (Proposition 3.3), that rectangle masses vary continuously in the phase parameter (Lemma 3.4), and that a positive-mass target rectangle can be realized by points chosen in arbitrary prescribed","core_discovery":"The central claim is that the multivariate mean equicontinuity hierarchy of a minimal action is fully controlled by a single integer-valued invariant, the conditional topomorphic degree tdeg_G(X): the supremum, over invariant measures, of the essential supremum over base points of the number of support points of the conditional measure on fibers of the maximal equicontinuous factor. The paper proves that tdeg_G(X) ≤ r−1 is necessary and sufficient for both F-mean and Weyl mean r-equicontinuity, resolving the Breitenbücher–Haupt–Jäger conjecture for Z-systems and extending it to all countably infinite discrete amenable groups. Along the way it obtains the exact formula tdeg_G(X) = Σ_{μ∈M^e_G(","pith_inferences":["The local realization theorem (Theorem 3.5) is stated more generally than needed and may transfer to other multivariate sensitivity notions—diam-mean equicontinuity, frequent stability, or uniform mean equicontinuity—since it only uses compact-phase transport and ergodicity of joinings.","The equivalence between a single Følner sequence and the Weyl (all-sequence) version suggests that mean r-equicontinuity is rigid enough that one sequence suffices; this raises the question whether the same coincidence holds for sofic group actions, where Følner sequences are replaced by sofic approximation sequences.","The exact degree decomposition suggests a possible variational principle for maximal pattern entropy: one might conjecture that h*_top(X,G) equals log of the sum Σ ι_μ exp(h*_μ(G)) under mild extra hypotheses such as the local Bronstein condition, matching the known one-sided bound here.","Because Theorem 1.5 realizes every profile in zero entropy, it indicates that the conditional topomorphic degree, rather than entropy, is the effective constraint governing finite-to-one almost automorphic structure over rotations."],"forward_implications":["The Breitenbücher–Haupt–Jäger conjecture holds in full: for minimal Z-systems, mean (m+1)-equicontinuity without mean m-equicontinuity is exactly the class of m-to-one topomorphic extensions of the maximal equicontinuous factor; the same dichotomy now holds for all countably infinite discrete amenable groups.","The degree formula tdeg_G(X) = Σ ι_μ exp(h*_μ(G)) separates compact (phase) multiplicity from residual (entropy) multiplicity, making both contributions individually observable.","Every minimal amenable action with finite conditional topomorphic degree d carries essential IT N-tuples for each 2 ≤ N ≤ d, so maximal topological sequence entropy is at least log d, a stronger lower bound than the earlier exp-sum formula.","All finite multiplicity profiles are realizable: any multiset of positive-integer pairs (ι,b) is the profile of some zero-entropy minimal almost one-to-one extension of an irrational circle rotation, so there are no hidden compatibility conditions among compact and residual multiplicities.","In the uniquely ergodic finite-degree case, a degree d with residual entropy log b is realizable if and only if b divides d (Corollary 6.4)."],"fun_headline_variants":["Single degree determines mean equicontinuity for amenable actions","One integer settles mean r-equicontinuity conjecture","Conditional topomorphic degree controls mean equicontinuity","A single integer ranks mean equicontinuity for amenable groups"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of the hard direction (mean r-equicontinuity implies degree ≤ r−1) rests on the claim that almost every lifted mixed phase joining is ergodic, which in turn imports, without re-derivation, two structural facts from the author's earlier work: that the invariant σ-algebra of a product measure is contained in the product of the maximal compact σ-algebras of the factors, and that the residual multiplicity equals exp(h*_μ(G)); if either imported result fails, the equival","fun_headline_variants_meta":{"raw":{"variants":["Single degree determines mean equicontinuity for amenable actions","One integer settles mean r-equicontinuity conjecture","Conditional topomorphic degree controls mean equicontinuity","A single integer ranks mean equicontinuity for amenable groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000942,"raw_usage":{"total_tokens":3978,"prompt_tokens":974,"completion_tokens":3004,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":2933}},"tokens_in":718,"tokens_out":3004,"duration_ms":21230,"temperature":1.0,"reasoning_tokens":2933,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:07:16.043039+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct (or exhibit) the Haupt–Jäger degree-2 extension of an irrational circle rotation—the canonical two-to-one topomorphic extension that is not almost automorphic—and test it for Weyl mean 2-equicontinuity and for maximal pattern entropy. The theorem predicts that this system has tdeg = 2, is not Weyl mean 2-equicontinuous, and satisfies h*_top ≥ log 2; a computation showing it is Weyl mean 2-equicontinuous, or showing h*_top < log 2, would refute the main equivalence and the entropy bound directly. As a more surgical check, one can test the imported Lemma 2.6 on a skew-product example o","supporting_citations":[],"review_version":1}