{"id":"0f6aa58b-bd33-484e-a9f0-9620478143be","arxiv_id":"2601.07714","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Contractible positive loops define immaterial subsets whose Reeb-invariant complements are large in contact geometry via symplectic homology and contact quasi-measures.","lead":"This note defines immaterial subsets of contact manifolds as non-empty closed sets where contractible loops of contactomorphisms are positive somewhere. It claims the complement of any Reeb-invariant immaterial subset counts as large according to symplectic homology of fillings and contact quasi-measures.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Whether symplectic homology and contact quasi-measure results actually imply the complement is 'big' in the claimed contact-geometric sense","rationale":"The reader's weakest assumption correctly isolates the inference from the two technical results to the geometric bigness claim. Because the review was performed on the abstract, the full manuscript may contain the missing bridge; the concrete test above would confirm whether that bridge exists or whether the argument remains interpretive.","tokens_in":1577,"tokens_out":311,"duration_ms":25433,"concrete_test":"Locate the precise theorem or proposition that concludes the complement is big; check whether its proof invokes only the two cited results plus Reeb-invariance, or whether it inserts an unstated comparison (e.g., via a duality or capacity argument) between the subset and its complement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the complement of a Reeb-invariant immaterial subset is 'big'. This is asserted to follow from two results (symplectic homology of a filling and properties of contact quasi-measures). The load-bearing step is the inference that these algebraic or measure-theoretic conclusions translate into the geometric notion of bigness for the complement; if the paper only shows non-vanishing or positivity on the subset itself without a direct comparison or duality argument for the complement, the interpretation does not follow. Reeb-invariance is used to set up the results but may not bridge to the largeness statement without an additional step that is not automatic.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper considers contractible loops of contactomorphisms that are positive over a non-empty closed subset of a contact manifold, termed an immaterial subset. It argues that the complement of a Reeb-invariant immaterial subset can be viewed as big in contact-geometric terms. This is supported by two results: one on the symplectic homology of a filling and the other on recently introduced contact quasi-measures.","tokens_in":1713,"tokens_out":535,"duration_ms":48966,"significance":"If the supporting results establish the claimed largeness, the note offers a new way to detect big sets in contact manifolds via positive loops, algebraic invariants from symplectic homology, and quasi-measures. This could strengthen connections between dynamics of contactomorphisms and geometric notions of size, with the use of contact quasi-measures as a potentially useful tool if the implications are made precise.","major_comments":[{"comment":"Abstract and central argument: the claim that the complement of a Reeb-invariant immaterial subset is 'big' is asserted to follow from the two supporting results, but the manuscript must explicitly detail the inference step. It is not automatic that non-vanishing symplectic homology of the filling or positivity properties of contact quasi-measures on the subset imply a geometric bigness statement for the complement; a direct comparison, duality, or complement-specific argument appears needed and should be stated in the main text (e.g., after the statements of the two results).","section":"Abstract and introduction"},{"comment":"Symplectic homology result: clarify whether the result establishes a property that directly transfers to the complement being large (via Reeb-invariance or otherwise) or only concerns the filling associated to the immaterial subset itself. If the former, state the precise comparison or vanishing/non-vanishing statement used for the complement.","section":"Section on symplectic homology of the filling"}],"minor_comments":[{"comment":"Define 'immaterial subset' and 'Reeb-invariant' with precise notation at first use to ensure the reader can follow how invariance is used to set up the two results.","section":"Introduction"},{"comment":"Add a reference to the source introducing contact quasi-measures for context on the second supporting result.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a short note; the journal's scope for brief communications should be checked, but the topic fits math.SG. The citation pattern for quasi-measures appears appropriate if the connection to bigness is strengthened."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the thoughtful report and for identifying points where the logical connections in our arguments can be made more explicit. The comments focus on clarifying how our two supporting results imply that the complement of a Reeb-invariant immaterial subset is geometrically large. We agree these clarifications will improve the manuscript and will incorporate them in the revised version. Our responses to the major comments are as follows.","responses":[{"response":"We agree that the inference step linking the two results to the bigness of the complement should be stated explicitly rather than left implicit. The abstract and introduction assert support from the results but do not spell out the precise mechanism (e.g., via Reeb-invariance inducing a duality or vanishing on the complement). In the revised manuscript we will add a short paragraph immediately after the statements of the two main results. This paragraph will explain that Reeb-invariance of the immaterial subset allows the non-vanishing symplectic homology of its filling to imply, by standard properties of symplectic homology under decomposition, that the complement cannot admit a filling with vanishing homology in the relevant degree, thereby establishing geometric largeness; an analogous argument using the positivity of contact quasi-measures on the subset (and their vanishing on the complement under Reeb-invariance) will also be included.","revision_made":"yes","referee_comment":"[Abstract and introduction] Abstract and central argument: the claim that the complement of a Reeb-invariant immaterial subset is 'big' is asserted to follow from the two supporting results, but the manuscript must explicitly detail the inference step. It is not automatic that non-vanishing symplectic homology of the filling or positivity properties of contact quasi-measures on the subset imply a geometric bigness statement for the complement; a direct comparison, duality, or complement-specific argument appears needed and should be stated in the main text (e.g., after the statements of the two results)."},{"response":"The symplectic homology result is formulated for a filling of the immaterial subset and establishes non-vanishing in a specific degree under the assumption of a contractible positive loop. Because the subset is Reeb-invariant, this non-vanishing transfers directly to a statement about the complement: any filling of the complement would force vanishing of symplectic homology in the complementary degree by the long exact sequence or Mayer-Vietoris type argument for the decomposition of the filling, contradicting known non-vanishing results for contact manifolds with positive loops. We will revise the relevant section to state this precise transfer explicitly, including the non-vanishing statement for the complement.","revision_made":"yes","referee_comment":"[Section on symplectic homology of the filling] Symplectic homology result: clarify whether the result establishes a property that directly transfers to the complement being large (via Reeb-invariance or otherwise) or only concerns the filling associated to the immaterial subset itself. If the former, state the precise comparison or vanishing/non-vanishing statement used for the complement."}],"tokens_in":1218,"tokens_out":623,"duration_ms":54975,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Hi there, the main thing to know is that this short note introduces the term immaterial subset for a non-empty closed set in a contact manifold that admits a contractible loop of contactomorphisms positive over it. It then argues that when the subset is Reeb-invariant, the complement counts as big in contact-geometric terms, and this rests on two results: one about symplectic homology of the filling and one about recently introduced contact quasi-measures. The definition and the specific Reeb-invariance link look new within the cited literature, and the note does a reasonable job of being concise while pointing to how existing tools might give a fresh way to talk about largeness inside contact manifolds. The soft spot is exactly the central inference the stress-test flags. The abstract presents the homology and quasi-measure results as support for bigness of the complement, but it is not automatic that positivity or non-vanishing on the subset or filling directly translates into a geometric notion of bigness for the complement without an explicit comparison or duality step. Reeb-invariance sets up the application of those tools, yet the paper would be stronger if it spelled out why that step follows rather than assuming it. If the proofs contain a direct argument that closes this gap, the claim holds; otherwise it is a place that could use tightening. This is aimed at specialists in contact and symplectic geometry who already work with symplectic homology and quasi-measures. A reader looking for new descriptive language around size in contact manifolds could get some value from it. I would send it to peer review so the details of the two supporting results and the inference can be checked by people in the area.","headline":"This note defines immaterial subsets via positive contractible loops of contactomorphisms and claims Reeb-invariant ones make their complements big, but the inference from homology and quasi-measures needs checking.","tokens_in":2196,"tokens_out":410,"would_cite":false,"duration_ms":68920,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"Theorem 1.1. Let A ⊂ ∂W be the preimage of a closed subset under a smooth Reeb-invariant map. Assume A is immaterial and denote Ω := ∂W ∖ A. Then, the continuation map SH^Ω_* (W) → SH_*(W) is surjective."},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"Theorem 1.2 ... τ(B) ⩾ m_B / (m_B − m)"}],"headline":"Contact geometry note on immaterial subsets and selective symplectic homology has no overlap with RS forcing chain","alignment":"orthogonal","rationale":"The paper's central results (Theorem 1.1 on surjectivity of continuation maps SH^Ω_* (W) → SH_*(W) for Reeb-invariant immaterial A, and Theorem 1.2 on lower bounds for contact quasi-measures τ(B) via contractible loops) operate entirely within symplectic/contact geometry using Floer homology, spectral invariants, and Reeb flows. No J-cost, cosh identities, φ-ladder, 8-tick periodicity, or parameter-free constant derivations appear. RS modules such as Cost.FunctionalEquation (J-uniqueness), Foundation.DimensionForcing (D=3 via Alexander duality), and Foundation.RealityFromDistinction have no bearing on or contradiction with these statements.","tokens_in":44266,"confidence":"high","tokens_out":386,"duration_ms":14626,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The complement of a Reeb-invariant immaterial subset is big in contact geometric terms.","keywords":["contactomorphisms","immaterial subsets","Reeb-invariant sets","symplectic homology","contact quasi-measures","contact geometry","contractible loops"],"falsifier":"An explicit Reeb-invariant immaterial subset whose complement fails to be large when measured by either the symplectic homology of a filling or by contact quasi-measures.","tokens_in":2461,"feed_emoji":"","tokens_out":623,"duration_ms":37365,"temperature":0.7,"pith_summary":"This paper examines contractible loops of contactomorphisms that remain positive over certain non-empty closed subsets of a contact manifold; these subsets are called immaterial. It focuses on the case where the immaterial subset is invariant under the Reeb flow and claims that its complement then qualifies as large when measured by standard tools of contact geometry. A sympathetic reader would care because the result offers a concrete way to distinguish large and small subsets inside contact manifolds, with potential consequences for understanding periodic orbits and fillings. The argument rests on two independent supporting results, one from symplectic homology of a filling and one from contact quasi-measures.","feed_headline":"Reeb-invariant immaterial sets have large complements","feed_subtitle":"Symplectic homology and contact quasi-measures show the complement is big in contact geometric terms.","key_machinery":"Immaterial subset: a non-empty closed subset of the contact manifold over which there exists a contractible loop of contactomorphisms that is positive everywhere on the subset; Reeb-invariance of this subset is the extra condition that makes the complement large via the two cited invariants.","core_discovery":"We consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one regarding symplectic homology of the filling and the other regarding recently introduced contact quasi-measures.","pith_inferences":["The same largeness statement might hold for immaterial sets that are not Reeb-invariant if additional dynamical assumptions are imposed.","Similar arguments could be tested on standard contact manifolds such as the sphere to produce explicit examples.","The notion of bigness defined here may interact with other invariants already used to study contactomorphisms."],"forward_implications":["Symplectic homology of the filling detects the largeness of the complement.","Contact quasi-measures assign values consistent with the complement being large.","The dynamics generated by positive loops on immaterial sets respect this geometric size distinction.","Reeb-invariant immaterial sets can be treated as negligible in certain contact invariants."],"fun_headline_variants":["Large complements for Reeb-invariant immaterial sets","Immaterial sets with Reeb invariance have big complements","Big complements of immaterial sets via contact quasi-measures","Symplectic homology links to large complements for immaterial sets"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The results on symplectic homology of the filling and on contact quasi-measures actually establish the claimed largeness for the complement.","fun_headline_variants_meta":{"raw":{"variants":["Large complements for Reeb-invariant immaterial sets","Immaterial sets with Reeb invariance have big complements","Big complements of immaterial sets via contact quasi-measures","Symplectic homology links to large complements for immaterial sets"]},"model":"grok-4.3","cost_usd":0.016614,"raw_usage":{"total_tokens":6919,"prompt_tokens":493,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":166140500,"prompt_tokens_details":{"text_tokens":493,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":6366,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":493,"tokens_out":60,"duration_ms":84400,"temperature":1.0,"reasoning_tokens":6366,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-21T15:48:14.668170+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit Reeb-invariant immaterial subset whose complement fails to be large when measured by either the symplectic homology of a filling or by contact quasi-measures.","supporting_citations":[],"review_version":1}