{"id":"03b93f0f-5333-4fc8-8312-0973412687a7","arxiv_id":"2607.01896","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of topologically free minimal F_∞-actions on the Cantor set without dynamical comparison (with or without invariant measures), plus separation of strict comparison in the crossed product from dynamical comparison, via monoid construction and embedding into a refinement monoid realized as a","lead":"The paper shows there exist topologically free minimal actions of the infinite free group on the Cantor set that lack dynamical comparison, whether or not they preserve a measure. It also shows that strict comparison in the reduced crossed-product C*-algebra does not force dynamical comparison.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Realization step may introduce extra relations making the monoid almost unperforated despite the abstract construction","rationale":"The reader's weakest assumption matches the load-bearing step exactly. No other internal inconsistency is visible from the given material, and the monoid-to-dynamics transfer is the only place where the algebraic counterexample could fail to produce a genuine dynamical one.","tokens_in":1606,"tokens_out":338,"duration_ms":12456,"concrete_test":"In the section constructing the action from the monoid (likely the final technical section), extract the explicit generators and relations used to define the type semigroup; recompute whether the image of the original non-almost-unperforated element remains non-almost-unperforated after all dynamical equivalences are imposed. If the perforation witness survives, the claim holds; if it is killed, the counterexample collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The argument proceeds by building a monoid M that is not almost unperforated, embedding M into a countable refinement monoid N, and then producing a topologically free minimal action of F_∞ on the Cantor set whose type semigroup is isomorphic to N. For the counterexample to hold, this isomorphism must be exact: no additional dynamical relations can be forced by the concrete action that would make N almost unperforated. The abstract only sketches the technique; if the realization (via, e.g., a suitable subshift or inverse-limit construction) adds perforation-detecting relations or collapses elements in a way that restores almost unperforation, the non-comparison property fails to transfer.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs a topologically free minimal action of the free group F_∞ on the Cantor set whose type semigroup is a countable refinement monoid that is not almost unperforated, thereby providing a counterexample to dynamical comparison. The same phenomenon is realized both in the presence and absence of invariant measures. It is further shown that strict comparison of the reduced crossed product does not imply dynamical comparison. The argument proceeds by exhibiting an explicit non-almost-unperforated monoid M, embedding M into a countable refinement monoid N, and realizing N as the type semigroup of a concrete dynamical system.","tokens_in":1760,"tokens_out":530,"duration_ms":23797,"significance":"If the realization step preserves the exact monoid structure, the result separates dynamical comparison from both the existence of invariant measures and from strict comparison of the crossed product, supplying concrete counterexamples in the theory of minimal actions on zero-dimensional spaces. The explicit algebraic construction of M and its embedding into a refinement monoid is a clear technical strength that makes the counterexamples potentially verifiable and usable for further work.","major_comments":[{"comment":"§4 (Realization of the monoid): the claim that the constructed action realizes N exactly as its type semigroup (without additional relations forced by minimality or topological freeness) is load-bearing for the non-almost-unperforation property. The sketch does not contain an explicit verification that the generators corresponding to the perforation-witnessing elements of N remain distinct and satisfy no extra inequalities in the dynamical type semigroup.","section":"§4"},{"comment":"§3.2 (Embedding M ↪ N): while the embedding is stated to be order-preserving, it is not shown that the image of the non-almost-unperforated pair in M remains non-almost-unperforated inside the refinement monoid N; refinement could in principle introduce new relations that restore almost unperforation before the dynamical realization step.","section":"§3.2"}],"minor_comments":[{"comment":"The notation for the type semigroup is introduced without a displayed definition; a displayed equation would clarify the precise monoid operation used throughout.","section":"§2"},{"comment":"Several references to prior work on almost unperforated monoids are given only by author names; full citations should be added in the bibliography.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the thorough report and the recommendation for major revision. The two major comments identify places where additional explicit verification is needed to make the arguments fully rigorous. We agree that these points require expansion and will revise the manuscript accordingly. Our point-by-point responses follow.","responses":[{"response":"We agree that the current presentation in §4 provides only a sketch and lacks an explicit verification that the type semigroup of the realized action coincides exactly with N. In the revised manuscript we will add a detailed argument in §4 showing that the clopen sets corresponding to the generators of N can be chosen so that the only relations enforced by the minimal topologically free action are those already present in N. The construction proceeds by first realizing the free refinement monoid on the generators and then using the freeness of F_∞ to ensure no unintended dynamical relations are introduced among the perforation-witnessing elements; this will be verified by direct computation of the type semigroup using the explicit partition of the Cantor set.","revision_made":"yes","referee_comment":"[§4] §4 (Realization of the monoid): the claim that the constructed action realizes N exactly as its type semigroup (without additional relations forced by minimality or topological freeness) is load-bearing for the non-almost-unperforation property. The sketch does not contain an explicit verification that the generators corresponding to the perforation-witnessing elements of N remain distinct and satisfy no extra inequalities in the dynamical type semigroup."},{"response":"We acknowledge that the current text in §3.2 asserts the embedding is order-preserving but does not explicitly prove preservation of non-almost-unperforation. In the revision we will insert a short lemma immediately after the construction of N showing that the specific pair witnessing non-almost-unperforation in M remains non-almost-unperforated in N. The argument relies on the fact that the refinement relations added to obtain N are generated by elements outside the submonoid generated by the image of M, so no new inequalities are forced between multiples of the original perforation-witnessing elements.","revision_made":"yes","referee_comment":"[§3.2] §3.2 (Embedding M ↪ N): while the embedding is stated to be order-preserving, it is not shown that the image of the non-almost-unperforated pair in M remains non-almost-unperforated inside the refinement monoid N; refinement could in principle introduce new relations that restore almost unperforation before the dynamical realization step."}],"tokens_in":1314,"tokens_out":554,"duration_ms":15188,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is an explicit construction of topologically free minimal actions of the free group on countably many generators on the Cantor set that lack dynamical comparison. The same holds both when invariant measures exist and when they do not, and the crossed-product C*-algebra can still have strict comparison. This is obtained by starting with a monoid that is not almost unperforated, embedding it into a countable refinement monoid, and realizing the latter as the type semigroup of the action.\n\nThe algebraic side uses standard monoid techniques and appears to be carried out cleanly. The separation is new in this setting; earlier work in the area often treated dynamical comparison as following from the other properties or bundled them together in classification arguments.\n\nThe soft spot is the passage from the abstract monoid to the concrete dynamical system. The stress-test note flags that the realization (via subshift or inverse-limit methods) could introduce extra relations that make the type semigroup almost unperforated after all. Without the full details of how the embedding and realization are done, it is not possible to rule that out. If that step preserves the perforation exactly, the counterexample works; if not, the claim collapses.\n\nThe paper is aimed at people working on C*-algebras from minimal actions and on the comparison properties that appear in classification programs. It is the sort of targeted counterexample that deserves a serious referee, mainly to verify the realization step and to confirm that no hidden assumptions crept into the monoid embedding.","headline":"The paper builds monoid examples to separate dynamical comparison from topological freeness and minimality in actions of F_infty, but the dynamical realization step is the part that needs the closest check.","tokens_in":2232,"tokens_out":388,"would_cite":false,"duration_ms":16934,"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":"There exist topologically free minimal actions of the infinite free group on the Cantor space without dynamical comparison.","keywords":["dynamical comparison","topologically free actions","minimal actions","Cantor space","free group","type semigroup","crossed products","invariant measures"],"falsifier":"Finding a specific topologically free minimal action of F_∞ on the Cantor set and verifying directly whether its type semigroup is almost unperforated.","tokens_in":2504,"feed_emoji":"","tokens_out":611,"duration_ms":22637,"temperature":0.7,"pith_summary":"This paper shows that dynamical comparison need not hold for topologically free minimal actions of F_infinity on the Cantor set. Examples are constructed both when invariant measures exist and when they do not. It is also shown that strict comparison of the reduced crossed product C*-algebra fails to imply dynamical comparison. The method uses an algebraic construction of a suitable monoid embedded into a refinement monoid that is then realized dynamically.","feed_headline":"Free group minimal actions on Cantor sets can lack dynamical comparison","feed_subtitle":"Such actions exist with or without invariant measures, and crossed product comparison does not imply the dynamical version.","key_machinery":"A non-almost-unperforated monoid embedded into a countable refinement monoid and realized as the type semigroup of the action.","core_discovery":"The authors prove the existence of a topologically free minimal action of F_∞ on the Cantor space that lacks dynamical comparison. This occurs in both the measure-preserving and non-measure-preserving cases. They additionally demonstrate that strict comparison in the associated reduced crossed product does not entail dynamical comparison. Their approach involves constructing a monoid which is not almost unperforated, embedding it into a countable refinement monoid, and realizing it as the type semigroup of the action.","pith_inferences":["Actions of other amenable groups might admit similar counterexamples if their type semigroups can be engineered similarly.","Classification programs for crossed products by minimal actions may need to account for this separation between comparison properties.","This construction technique could be adapted to produce examples on other compact spaces beyond the Cantor set."],"forward_implications":["Dynamical comparison can fail even for topologically free minimal actions on zero-dimensional compact spaces.","The failure is possible regardless of the existence of invariant probability measures.","Algebraic strict comparison in crossed products is strictly weaker than dynamical comparison for the action.","The type semigroup can encode the absence of dynamical comparison while preserving minimality and topological freeness."],"fun_headline_variants":["Minimal F_∞ actions on Cantor sets lack dynamical comparison","Free group actions on Cantor space without dynamical comparison","Dynamical comparison fails for some minimal Cantor actions","C* strict comparison does not imply dynamical comparison","Topologically free actions lack dynamical comparison"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A monoid that fails to be almost unperforated can be embedded into a countable refinement monoid and realized as the type semigroup for a topologically free minimal action on the Cantor set.","fun_headline_variants_meta":{"raw":{"variants":["Minimal F_∞ actions on Cantor sets lack dynamical comparison","Free group actions on Cantor space without dynamical comparison","Dynamical comparison fails for some minimal Cantor actions","C* strict comparison does not imply dynamical comparison","Topologically free actions lack dynamical comparison"]},"model":"grok-4.3","cost_usd":0.007151,"raw_usage":{"total_tokens":3240,"prompt_tokens":544,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":71512000,"prompt_tokens_details":{"text_tokens":544,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2626,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":544,"tokens_out":70,"duration_ms":24038,"temperature":1.0,"reasoning_tokens":2626,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T04:27:40.789177+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a specific topologically free minimal action of F_∞ on the Cantor set and verifying directly whether its type semigroup is almost unperforated.","supporting_citations":[],"review_version":1}