{"id":"4514f713-c21a-4685-b067-7ec1808ff141","arxiv_id":"2507.13234","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Contact spectral invariants are constructed from contact Hamiltonian Floer homology and used to prove rigidity theorems for contact manifolds.","lead":"This paper builds a quantitative theory of contact Hamiltonian dynamics using contact Hamiltonian Floer homology, producing numerical spectral invariants and a new persistence-module structure. It uses these invariants to prove results on contact non-displaceability, orderability, and translated points.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Triangle inequality rests on a pair-of-pants product whose full weakly+ monotone/Novikov-field case is deferred; if the 'analogous' construction fails, the advertised generality of Theorem 2.11 and the big-fiber application collapses.","rationale":"The reader's weakest_assumption was the pair-of-pants product and the conjectural chain model. I agree that the first of these is truly load-bearing; the second is less central because the spectral invariants c(h, θ) are defined directly in Definition 2.9 and Section 6 proves their properties without the gapped-module identification in Section 10.2. What makes Theorem 2.12 load-bearing is that the triangle inequality — item (5) — is not a standalone result: it is used in Lemma 2.5 to control eternal classes, in Theorem 2.17 to produce the partial contact quasi-state (through superadditivity and the vanishing argument), and therefore in Theorem 1.4. The paper's own Section 11.3 explicitly limits the detailed construction to Liouville domains over Z2; Theorem 11.2's maximum principle is more general, but a complete product also requires Novikov-coefficient transversality and continuation compatibility that are delegated as 'analogous'. There is no sign of a mechanical error in the parts that are written out; the issue is a deferred proof at the exact point where the paper claims more than it supplies. A conditional acceptance is therefore the right verdict: either fill in the general construction or restrict the statement of Theorem 2.11(5) and the affected applications to the Liouville/Z2 case. The proposed test is a concrete minimal instance of the missing generality.","tokens_in":42387,"tokens_out":6194,"duration_ms":70679,"concrete_test":"Carry out Theorem 2.12 for a non-Liouville weakly+ monotone filling with Novikov field coefficients, e.g., the disk bundle of O(-1) over CP^{n-1} whose boundary is the standard contact sphere, verifying: (a) the moduli space counts with Novikov weights are finite for generic product data, using the compactness estimates of §11.4; (b) the chain-level product respects the continuation maps used in §6, so the diagram in Theorem 2.12 commutes with the canonical morphisms; and (c) the induced limit product equals the usual pair-of-pants product on SH(W). If any of (a)–(c) fails or requires additional monotonicity assumptions, Theorem 2.11(5) must be restated with a Liouville/Z2 hypothesis and the applications in §1.2 adjusted accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is Theorem 2.11, and its most consequential innovation is the triangle inequality (item 5), which drives Lemma 2.5, the quasi-state/quasi-measure construction (Theorems 2.17 and 2.20), and hence Theorem 1.4. The proof of the triangle inequality in Section 6 invokes Theorem 2.12, the pair-of-pants product HF(h) ⊗ HF(g) → HF(h • g). Section 11.3 defines this product explicitly only for Liouville domains with Z2 coefficients, and says the general case 'involves Novikov rings and is analogous to the closed case'. The maximum-principle compactness in Section 11.4 is stated for arbitrary weakly+ monotone fillings, so the main analytic bound may be in place; what is not supplied is the rest of a rigorous construction in that generality: Novikov-field counts with weak+ monotone bubbling control, orientations, the compatibility of the product with continuation maps and zig-zag isomorphisms, and the identification of the induced map on direct limits with the usual symplectic-homology product. Without these, Theorem 2.12 — and therefore Theorem 2.11(5) — is proven only in the Liouville/Z2 case. Since Theorem 1.4 and Theorem 2.17 both rely on the triangle inequality for arbitrary weakly+ monotone fillings, the advertised scope of the paper's headline applications is not yet justified. The paper itself acknowledges related gaps, e.g., Section 10.2 calls the weakly+ 'chain model' conjectural, although that specific gap is not needed for Theorem 2.11; the load-bearing issue is the completeness of Theorem 2.12's proof in the non-Liouville, Novikov-coefficient setting.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops quantitative invariants for contact Hamiltonian dynamics on the boundary of a weakly+ monotone strong filling W. It defines contact spectral invariants c(W,h,θ) via images of the contact Hamiltonian Floer groups HF∗(η#h) in symplectic homology, and proves for them spectrality, shift, monotonicity, stability, descent, and a triangle inequality (Theorem 2.11). The triangle inequality is derived from a pair-of-pants product in contact Hamiltonian Floer homology (Theorem 2.12). The framework is then applied to prove the contact big fiber theorem (Theorem 1.4), orderability (Theorem 1.6), and existence results for translated points (Theorems 1.7 and 1.8), and to construct partial contact quasi-states and quasi-measures (Theorems 2.17 and 2.20). The paper also introduces gapped persistence modules parametrized by a partially ordered set and gives a computational example for unit cotangent bundles of spheres.","tokens_in":42677,"tokens_out":7443,"duration_ms":87271,"significance":"If the results hold in the stated generality, this is a substantial contribution to contact rigidity: it provides a unified quantitative framework that goes beyond Liouville fillings and recovers or strengthens several known results in contact non-squeezing, orderability, and translated points. The paper is careful to distinguish its approach from concurrent work, and the spectral invariants are defined intrinsically from Floer-theoretic data with no fitted parameters. The maximum-principle argument in Section 11.4 is detailed and appears to be the key new analytic input. However, the significance is conditional on the pair-of-pants product being available in the full weakly+ monotone/Novikov-field setting, since Theorem 2.11(5), and hence the quasi-state, quasi-measure, and big-fiber applications, rely on it.","major_comments":[{"comment":"The pair-of-pants product that underlies the triangle inequality is constructed in detail only for Liouville domains with Z2 coefficients. Section 11.3 states that the general weakly+ monotone case involving Novikov rings is 'analogous' to the closed case. Theorem 2.12, however, is stated for arbitrary weakly+ monotone strong fillings. The missing ingredients are load-bearing: Novikov-field counts with control of weakly+ monotone bubbling, orientations, compatibility of the product with continuation maps and zig-zag isomorphisms, and the identification of the induced map on direct limits with the usual symplectic homology product. Since Theorem 2.11(5) is invoked in the weakly+ setting to prove Theorem 2.17, Theorem 2.20, and Theorem 1.4, the advertised scope of the main applications is not justified as written. This is the main gap the authors need to close, either by supplying the full construction or by restricting the theorems to the case actually proven.","section":"§11.3, Theorem 2.12"},{"comment":"The proof of Lemma 2.5 applies the product of Theorem 2.12 to constant contact Hamiltonians −2ε and a+ε, concluding that HF∗(−2ε) ⊗ HF∗(a+ε) → HF∗(a−ε) exists. Theorem 2.12 is stated under the hypothesis that the contact Hamiltonians vanish near t=0 and t=1, which constants do not satisfy. A reparametrization argument of the type used in the proof of Theorem 2.11 would likely supply the missing hypothesis, but that step is not written. Since Lemma 2.5 is used in the proof of Theorem 1.8, this appeal should be made explicit and justified.","section":"§2.1.4, Lemma 2.5"}],"minor_comments":[{"comment":"The text explicitly concedes that a filtered chain model for HF∗(η#h) over a Novikov field for general weakly+ monotone fillings is conjectural. This concession should be stated as a restriction on the gapped-module interpretation: the equality (10.4) between the algebraic spectral invariant and c(W,h,θ) is only established in the Liouville setting, and Example 10.17 should not be read as covering weakly+ monotone fillings.","section":"§10.2, Example 10.17"},{"comment":"There are several typos: 'Riemannian surface' should be 'Riemann surface' in §11.1, 'Reimannian' appears in the introduction to Section 11, 'Rloer homology' in §10.2 should be 'Rabinowitz Floer homology', 'a priory' in §8.4 should be 'a priori', and 'elemenot' in Definition 2.16(6) should be 'element'.","section":"§11.1 and §10.2"},{"comment":"Given the status of Theorem 2.12 in Section 11.3, the abstract and introduction should explicitly indicate the precise setting in which the triangle inequality is proven, or else the reader is led to expect more generality than the manuscript currently supplies.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The central gap is local and the authors appear aware of it: Section 11.3 explicitly limits the detailed construction to Liouville domains with Z2 coefficients, and Section 10.2 admits a related conjectural chain model. If the product construction is completed in the weakly+ monotone/Novikov-field setting, or if the main statements are restricted to the proven case, the paper would be suitable for publication. The concurrent-work discussion with Cant is handled transparently and does not raise novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is substantial, not a repackaging. The genuinely new content is a spectral invariant c(h,theta) for admissible contact Hamiltonians on the boundary of a weakly+ monotone strong filling, with spectrality, shift, monotonicity, stability, descent, and a triangle inequality obtained from a pair-of-pants product. The gapped module formalism is also new, and the applications—big fiber theorem, orderability, translated points—follow formally from the invariants. The paper is also honest in a useful way: it says where the technical details stop.\n\nWhat is solid: the formal development is coherent. The proofs of spectrality, shift, monotonicity, stability, and descent are in place. The pair-of-pants compactness argument in Section 11 is proved for weakly+ monotone fillings via the Alexandrov maximum principle, and the computation for cosphere bundles in Section 3 is a real check. The acknowledgements record and fix earlier sign inconsistencies in the triangle inequality, which is a good sign. There are no fitted parameters; the invariants are defined from Floer theory, not tuned to the applications. The self-citations point to prior published or earlier work and are not a problem.\n\nThe soft spot is scope. Section 11.3 defines the pair-of-pants product only for Liouville domains with Z2 coefficients and says the general weakly+ monotone Novikov field case is analogous to the closed case. The stress test is right that this matters: Theorem 2.11(5) depends on that product, and the big fiber theorem and quasi-state construction in their advertised generality depend on the triangle inequality. Section 10.2 likewise marks the filtered chain model for the gapped-module interpretation of c as conjectural for general weakly+ monotone fillings. That conjectural piece is not needed for the main direct-limit definition of c, so it is less central. The missing analytic work in Section 11 is standard in outline—Novikov counts, orientations, compatibility with continuation maps—but it is genuinely absent. The paper's own text limits the detailed proof to the Liouville/Z2 case.\n\nWho is it for: symplectic and contact topologists working on quantitative rigidity, spectral invariants, or persistence modules. The gapped module part may outlive the specific applications. It deserves a serious referee; I would send it out with a request to either supply the general case or restrict the main theorems to the case where all details are actually written. That is a conditional path, not a desk reject.","headline":"A genuinely new contact spectral invariant framework with real applications, but the advertised weakly+ monotone generality is propped up by a pair-of-pants product proven in detail only for Liouville domains over Z2.","tokens_in":43260,"tokens_out":2422,"would_cite":true,"duration_ms":29392,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","53D10","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Contact spectral invariants obey a triangle inequality","keywords":["contact Hamiltonian Floer homology","spectral invariants","contact rigidity","orderability","translated points","gapped modules","pair-of-pants product","symplectic homology"],"falsifier":"Run the Section 3 computation on the unit cosphere bundle D*S^n: the paper predicts c(0,u^k)=−2π⌊k/2⌋ and c(0,au^k)=−2π⌊k/2⌋ for all k≥0. A mismatch there, or a weakly+ monotone filling where the pair-of-pants product fails to be compact, would refute Theorem 2.11.","tokens_in":42158,"feed_emoji":"📐","tokens_out":7340,"duration_ms":76899,"temperature":0.7,"pith_summary":"Contact Hamiltonian Floer homology assigns groups to an admissible contact Hamiltonian on the boundary of a symplectic filling that satisfies a mild monotonicity condition. This paper extracts real-valued spectral invariants c(W,h,θ) from those groups and proves they satisfy spectrality, shift, monotonicity, stability, descent, and a triangle inequality. With those properties, it gives alternative proofs of the contact big fiber theorem, orderability of boundaries whose symplectic-homology unit is not eternal, and existence of translated points for contactomorphisms of small oscillation energy. The relevance is that these are quantitative rigidity results for arbitrary contact Hamiltonian flows, going beyond the Reeb dynamics that ordinary symplectic homology sees.","feed_headline":"Contact spectral invariants obey a triangle inequality","feed_subtitle":"Real-valued invariants from contact Hamiltonian Floer theory drive big-fiber, orderability, and translated-point results.","key_machinery":"The central object is the contact spectral invariant c(W,h,θ):=−inf{η | θ∈im(HF∗(W,η#h)→SH∗(W))}, where η#h is the contact Hamiltonian obtained by twisting h with the η-shifted Reeb flow. The filtration parameter η is a time-shift of a translated point, so the parameter set R∖S_h is not linearly ordered; the paper packages this into an oscRh-gapped module, a persistence module indexed by a poset. The load-bearing identity is the pair-of-pants product HF∗(h)⊗HF∗(g)→HF∗(h•g), built from a maximum principle on a pair-of-pants with a slit; that product makes the triangle inequality possible and also upgrades the spectral invariants into partial contact quasi-states and quasi-measures.","core_discovery":"The paper's central claim is that for any closed contact manifold M strongly filled by a weakly+ monotone symplectic manifold W, every admissible contact Hamiltonian h on M produces a persistence module P(W,h)={HF∗(W,η#h)}_η over the non-totally-ordered parameter set R∖S_h, and the resulting spectral invariants c(h,θ) are finite exactly for non-eternal classes θ∈SH∗(W). Theorem 2.11 asserts that c(h,θ) has spectrality, shift, monotonicity, stability, descent, and the triangle inequality c(h,θ1)+c(g,θ2)≤c(h#g,θ1∗θ2)+2max{oscRh,oscRg}. From these properties the paper obtains the contact big fiber theorem for weakly+ monotone fillings, orderability when the unit is not eternal, and translated-point existence below the minimal Reeb period; along the way it introduces 'gapped modules', persistence modules parametrized by a partially ordered set, as the algebraic structure that houses this filtration.","pith_inferences":["The gapped-module formalism suggests that contact rigidity is governed by the geometry of the shift set S_h rather than by a linearly ordered action spectrum, so one could try to define barcode-type invariants for contactomorphisms without a total order.","If the conjectural filtered chain model for general weakly+ monotone fillings is constructed, the same invariants should extend to non-exact fillings such as negative line bundles, where Liouville-domain arguments do not apply.","The explicit correction term 2max{oscRh,oscRg} in the triangle inequality hints that a sharper, correction-free inequality should hold for Reeb-invariant Hamiltonians; systematic examples could show whether the correction is optimal.","Testing the contact quasi-measure axioms on prequantization spaces, where Reeb-invariant functions and displaceable sets are explicit, could connect these invariants to known contact rigidity thresholds."],"forward_implications":["Every boundary of a weakly+ monotone strong filling whose symplectic-homology unit is not eternal is orderable: it admits no contractible positive loop of contactomorphisms.","Every contact involutive map on such a boundary has a non-displaceable fibre; for Liouville fillings with nonzero symplectic homology this is the contact big fiber theorem.","Every contactomorphism with oscillation energy below the minimal Reeb period has a translated point, and conversely a contactomorphism with no translated points forces the unit in symplectic homology to be eternal.","The invariants satisfy |c(φ,θ)−c([id],θ)|≤|φ|_S, linking them to the natural norm on the universal cover of the contactomorphism group.","For Reeb-invariant, Poisson-commuting Hamiltonians the invariants define a partial contact quasi-state and quasi-measure, giving contact analogues of the rigidity functionals used in symplectic topology."],"supporting_citations":[{"why":"Supplies the maximum principle that makes contact Hamiltonian Floer homology well-defined on the symplectization.","marker":"[35]"},{"why":"Provides the symplectic homology framework over which the spectral invariants are defined as a direct limit.","marker":"[68]"},{"why":"Supplies the zig-zag isomorphisms and selective symplectic homology used for spectrality and descent.","marker":"[63]"},{"why":"Contains the contact big fiber theorem that this paper generalizes to weakly+ monotone fillings.","marker":"[61]"},{"why":"Introduces the norm and translated-point conjecture proved in Theorem 1.7.","marker":"[60]"},{"why":"Establishes Rabinowitz Floer homology and the symplectic (co)homology duality that underpins the eternal/non-eternal dichotomy.","marker":"[19]"},{"why":"Provides the filtered chain model and algebraic spectral invariants that the gapped-module theory adapts.","marker":"[67]"},{"why":"Gives the pair-of-pants-with-a-slit model used to define the product.","marker":"[4]"}],"fun_headline_variants":["Triangle inequality proven for contact spectral invariants","Contact big fiber theorem from quantitative Floer theory","Orderability and translated points via spectral invariants","Gapped modules: persistence parametrized by posets","Contact Hamiltonian Floer yields new persistence structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes the pair-of-pants product and the gapped-module chain model work for arbitrary weakly+ monotone strong fillings with Novikov coefficients, while the detailed proof in the paper covers Liouville domains with Z2 coefficients and calls the general case analogous.","fun_headline_variants_meta":{"raw":{"variants":["Triangle inequality proven for contact spectral invariants","Contact big fiber theorem from quantitative Floer theory","Orderability and translated points via spectral invariants","Gapped modules: persistence parametrized by posets","Contact Hamiltonian Floer yields new persistence structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1512,"prompt_tokens":938,"completion_tokens":574,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":501}},"tokens_in":554,"tokens_out":574,"duration_ms":5896,"temperature":1.0,"reasoning_tokens":501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:27:35.333904+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Section 3 computation on the unit cosphere bundle D*S^n: the paper predicts c(0,u^k)=−2π⌊k/2⌋ and c(0,au^k)=−2π⌊k/2⌋ for all k≥0. A mismatch there, or a weakly+ monotone filling where the pair-of-pants product fails to be compact, would refute Theorem 2.11.","supporting_citations":[{"cited_title":"Merry and Igor Uljarević,Maximum principles in symplectic homology, Israel J","cited_arxiv_id":null,"evidence_quote":"Supplies the maximum principle that makes contact Hamiltonian Floer homology well-defined on the symplectization."},{"cited_title":"Ann.292 (1992), 685–710","cited_arxiv_id":null,"evidence_quote":"Provides the symplectic homology framework over which the spectral invariants are defined as a direct limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the zig-zag isomorphisms and selective symplectic homology used for spectrality and descent."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the contact big fiber theorem that this paper generalizes to weakly+ monotone fillings."},{"cited_title":"Symplectic Geom.15 (2017), 1173– 1208","cited_arxiv_id":null,"evidence_quote":"Introduces the norm and translated-point conjecture proved in Theorem 1.7."},{"cited_title":"43, 2010, pp","cited_arxiv_id":null,"evidence_quote":"Establishes Rabinowitz Floer homology and the symplectic (co)homology duality that underpins the eternal/non-eternal dichotomy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the filtered chain model and algebraic spectral invariants that the gapped-module theory adapts."},{"cited_title":"Topol.14 (2010), 1569–1722","cited_arxiv_id":null,"evidence_quote":"Gives the pair-of-pants-with-a-slit model used to define the product."}],"review_version":1}