{"id":"29b0be83-d36f-44b2-99f4-e330c23f9430","arxiv_id":"2409.11603","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces BSP surgeries on Legendrians, proves wall-crossing for disk potentials under Bohr-Sommerfeld profiles, and applies it to construct exotic monotone Lagrangian tori in P^n.","lead":"The paper defines a new surgery operation on Legendrians called BSP surgery by switching exact fillings and proves a wall-crossing formula for how disk potentials change under it with Bohr-Sommerfeld profiles. A smart generalist might read it to see new constructions of exotic monotone Lagrangian tori in projective space that add to known examples in symplectic topology.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Applicability of BSP surgery to Biran lifts depends on existence of exact filling pairs preserving wall-crossing hypotheses","rationale":"The reader's weakest_assumption directly identifies the same point: the existence and compatibility of the filling pairs needed for the Biran application. Without independent verification of that step, the construction of exotic tori remains conditional on an unconfirmed geometric input, leaving the overall claim unverified at present.","tokens_in":1633,"tokens_out":317,"duration_ms":21475,"concrete_test":"For the Biran lift in CP^2, explicitly exhibit the two exact fillings of the Legendrian, perform the BSP surgery, compute the disk potential before and after, and check whether the change matches the wall-crossing formula while monotonicity is retained.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim requires both a general wall-crossing formula for disk potentials under BSP surgery and its application to Biran circle-bundle lifts to produce exotic monotone tori in P^n. The latter rests on the existence of pairs of exact fillings of the relevant Legendrians that can be interchanged by the defined BSP surgery while keeping monotonicity and satisfying all hypotheses of the wall-crossing theorem (exactness of the fillings, appropriate bounding cochains, and preservation of the monotonicity condition). If no such pairs exist for the Biran lifts, or if the surgery alters the Legendrian in a way that invalidates the formula, the application does not follow even if the abstract wall-crossing statement is correct.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces Bohr-Sommerfeld profile (BSP) surgery, a new operation obtained by switching between two exact fillings of a Legendrian. It proves a wall-crossing formula describing the change in the disk potential under BSP surgery with Bohr-Sommerfeld profiles. As an application, the authors show that Biran circle-bundle lifts admit BSP surgery and apply the wall-crossing result to produce exotic monotone Lagrangian tori in projective space P^n.","tokens_in":1758,"tokens_out":487,"duration_ms":19773,"significance":"If the wall-crossing formula is correctly derived and the application to Biran lifts is valid, the work supplies a new constructive technique for modifying monotone Lagrangians while tracking their disk potentials. This could be useful for producing and distinguishing exotic examples in toric symplectic manifolds and for studying wall-crossing phenomena in Lagrangian Floer theory.","major_comments":[{"comment":"§4 (application to Biran lifts): the claim that Biran circle-bundle lifts admit BSP surgery requires explicit verification that there exist pairs of exact fillings of the relevant Legendrians that can be interchanged by the defined surgery while preserving exactness, the existence of suitable bounding cochains, and the monotonicity condition needed for the wall-crossing theorem to apply. The manuscript does not appear to supply a concrete check that these hypotheses remain satisfied after the surgery on the Biran lifts.","section":"§4"},{"comment":"Theorem on the wall-crossing formula (likely §3): the statement should make clear whether the formula holds for arbitrary exact fillings or only under additional restrictions on the profiles and the choice of fillings; if the latter, the restrictions must be stated explicitly so that the reader can confirm they are met in the Biran-lift application.","section":"§3"}],"minor_comments":[{"comment":"Notation for the disk potential and the surgery parameters should be introduced with a single consistent definition early in the paper rather than piecemeal.","section":"§2"},{"comment":"The introduction would benefit from a brief comparison table or diagram contrasting BSP surgery with existing Lagrangian surgery operations (e.g., Polterovich surgery).","section":"§1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments. We address the major comments point by point below.","responses":[{"response":"We agree that an explicit verification of the hypotheses is needed for the Biran-lift application. In the revised manuscript we will add a dedicated paragraph (or short subsection) in §4 that checks the existence of the required pairs of exact fillings, confirms preservation of exactness and monotonicity, and verifies the existence of suitable bounding cochains for the Legendrians arising from Biran circle-bundle lifts.","revision_made":"yes","referee_comment":"[§4] §4 (application to Biran lifts): the claim that Biran circle-bundle lifts admit BSP surgery requires explicit verification that there exist pairs of exact fillings of the relevant Legendrians that can be interchanged by the defined surgery while preserving exactness, the existence of suitable bounding cochains, and the monotonicity condition needed for the wall-crossing theorem to apply. The manuscript does not appear to supply a concrete check that these hypotheses remain satisfied after the surgery on the Biran lifts."},{"response":"We will revise the statement of the wall-crossing theorem in §3 to explicitly list the restrictions on the Bohr-Sommerfeld profiles and on the choice of exact fillings under which the formula is proved. The revised statement will also note that these restrictions are satisfied by the fillings used in the Biran-lift construction.","revision_made":"yes","referee_comment":"[§3] Theorem on the wall-crossing formula (likely §3): the statement should make clear whether the formula holds for arbitrary exact fillings or only under additional restrictions on the profiles and the choice of fillings; if the latter, the restrictions must be stated explicitly so that the reader can confirm they are met in the Biran-lift application."}],"tokens_in":1250,"tokens_out":403,"duration_ms":15217,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a new operation, BSP surgery, that switches between two exact fillings of a Legendrian, together with a wall-crossing formula that tracks the change in disk potential when the profile satisfies the Bohr-Sommerfeld condition. The authors then use this on Biran circle-bundle lifts to produce exotic monotone Lagrangian tori in projective space. Both the surgery and the specific formula appear new relative to the cited prior work, and the application is presented as a direct consequence rather than an ad-hoc construction. That gives a systematic route to more examples in an area where people already hunt for monotone tori with controlled potentials. The general formula could be reusable even outside this particular application. The soft spot sits in the application step. The wall-crossing theorem requires pairs of exact fillings that can be swapped while keeping monotonicity, exactness, and the bounding cochain conditions intact. The abstract does not exhibit such pairs for the Biran lifts, so it is not yet clear whether the surgery can actually be performed on those objects without violating the hypotheses. If no suitable pairs exist or if the surgery alters the Legendrian in a way that breaks the formula, the exotic tori do not follow. This is a concrete gap rather than a minor technicality. The paper is aimed at researchers already working on disk potentials, Legendrian fillings, and constructions of exotic tori in symplectic and contact geometry. Someone who knows Biran’s lifts and has computed potentials before will be able to check the details and see whether the new operation produces genuinely new examples. It deserves a serious referee because it supplies both a new operation and concrete new objects, even if the application needs verification in the full proofs.","headline":"The paper introduces BSP surgery on Legendrians with exact fillings and derives a wall-crossing formula for disk potentials, then applies it to Biran lifts for exotic tori in P^n.","tokens_in":2242,"tokens_out":419,"would_cite":false,"duration_ms":24969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Symplectic surgery, disk potentials, and combinatorial mutations in Floer theory show no structural overlap with RS cost functions or forcing chain","alignment":"orthogonal","rationale":"The paper develops BSP surgery on Lagrangians via exact fillings of Bohr-Sommerfeld Legendrians, proves a wall-crossing formula WBSP(x,z,...) = WL(x, z W_Λ(x), ...), and applies it to Biran lifts and Vianna tori via Newton-polytope mutations. None of its central objects (disk potentials, Reeb chords, Maslov indices, combinatorial mutations with factors W_Λ, monotonicity preservation under zero-area surgery) invoke or parallel the RS recognition cost J(x) = ½(x + x^{-1}) - 1, golden-ratio fixed points, 8-tick periodicity, or parameter-free derivation of constants. The constructions remain inside classical symplectic geometry and SFT neck-stretching; they neither echo nor contradict any theorem in the RS chain.","tokens_in":62621,"confidence":"high","tokens_out":214,"duration_ms":7693,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Wall-crossing formula describes how disk potentials change under Bohr-Sommerfeld profile surgery.","keywords":["Bohr-Sommerfeld profile surgery","disk potential","wall-crossing formula","monotone Lagrangian tori","Legendrian fillings","symplectic surgery","projective space"],"falsifier":"Explicit computation of the disk potential of a specific Biran circle-bundle lift before and after BSP surgery, checking whether the observed change matches the predicted wall-crossing formula.","tokens_in":2507,"feed_emoji":"","tokens_out":670,"duration_ms":21447,"temperature":0.7,"pith_summary":"The paper defines BSP surgery as an operation that switches between two exact fillings of a Legendrian while sometimes preserving monotonicity of the resulting Lagrangian. It proves a wall-crossing formula that determines the exact change in the disk potential when the surgery profile satisfies the Bohr-Sommerfeld condition. The formula is applied to Biran circle-bundle lifts, which are shown to admit this surgery, and the same relation is used to produce monotone Lagrangian tori in projective space that lie outside the standard Hamiltonian isotopy class. A reader would care because disk potentials serve as computable invariants that can distinguish Lagrangians, and controlled surgery operations supply a method to generate new examples from known ones.","feed_headline":"Wall-crossing formula tracks disk potentials under profile surgery","feed_subtitle":"Switching exact Legendrian fillings changes the potential in a controlled way and produces exotic tori in P^n","key_machinery":"BSP surgery, defined by switching between two exact fillings of Legendrians, together with the associated wall-crossing formula that tracks the resulting change in disk potential.","core_discovery":"We construct a new surgery type operation by switching between two exact fillings of Legendrians which we call a BSP surgery. In certain cases, this surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type formula for the change of the disk-potential under surgery with Bohr-Sommerfeld profiles. As an application, we show that Biran's circle-bundle lifts admit a Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about disk-potentials to construct exotic monotone Lagrangian tori in P^n.","pith_inferences":["The formula supplies a systematic way to relate disk potentials of Lagrangians obtained from different fillings of the same Legendrian.","Similar wall-crossing behavior may appear when the surgery is performed in other symplectic manifolds containing suitable Legendrians."],"forward_implications":["The disk potential of a Lagrangian after BSP surgery is related to the original potential by the wall-crossing formula.","Biran circle-bundle lifts admit Bohr-Sommerfeld type surgery.","Exotic monotone Lagrangian tori exist in projective space P^n.","The wall-crossing relation can be used to track invariants across sequences of such surgeries."],"fun_headline_variants":["BSP surgery alters disk potentials via wall-crossing","Wall-crossing formula derived from Legendrian filling switch","Biran lifts admit new Bohr-Sommerfeld type surgery","Monotone tori in P^n built using disk potential surgery"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"There exist pairs of exact fillings of the relevant Legendrians that can be switched via the defined BSP surgery while preserving the conditions needed for the wall-crossing formula to apply, particularly for the Biran circle-bundle lifts.","fun_headline_variants_meta":{"raw":{"variants":["BSP surgery alters disk potentials via wall-crossing","Wall-crossing formula derived from Legendrian filling switch","Biran lifts admit new Bohr-Sommerfeld type surgery","Monotone tori in P^n built using disk potential surgery"]},"model":"grok-4.3","cost_usd":0.005967,"raw_usage":{"total_tokens":2774,"prompt_tokens":559,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":59674500,"prompt_tokens_details":{"text_tokens":559,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2152,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":559,"tokens_out":63,"duration_ms":15881,"temperature":1.0,"reasoning_tokens":2152,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T20:11:56.048536+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Explicit computation of the disk potential of a specific Biran circle-bundle lift before and after BSP surgery, checking whether the observed change matches the predicted wall-crossing formula.","supporting_citations":[],"review_version":1}